Progress in Optics

PROGRESS IN OPTICS VOLUME 49 EDITORIAL ADVISORY BOARD G.S. Agarwal Stillwater, USA G. Agrawal Rochester, USA T. ...

Author: Emil Wolf

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PROGRESS IN OPTICS VOLUME 49

EDITORIAL ADVISORY BOARD

G.S. Agarwal

Stillwater, USA

G. Agrawal

Rochester, USA

T. Asakura

Sapporo, Japan

A. Aspect

Orsay, France

M.V. Berry

Bristol, England

A.T. Friberg

Stockholm, Sweden

V.L. Ginzburg

Moscow, Russia

F. Gori

Rome, Italy

P. Knight

London, England

L.M. Narducci

Philadelphia, USA

J. Peˇrina

Olomouc, Czech Republic

H. Walther

Garching, Germany

PROGRESS IN OPTICS VOLUME 49

EDITED BY

E. Wolf University of Rochester, N.Y., U.S.A.

Contributors H. Benisty, C. Brosseau, N.J. Cerf, A. Dogariu, M. Dušek, J. Fiurášek, M. Hendrych, A. Joshi, N. Lütkenhaus, V.N. Mahajan, C. Weisbuch, M. Xiao

Amsterdam · Boston · Heidelberg · London · New York · Oxford · Paris San Diego · San Francisco · Singapore · Sydney · Tokyo

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK

First edition 2006 Copyright © 2006 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; e-mail: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/ permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Catalog Card number: 61-19297 ISBN-13: 978-0-444-52732-5 ISBN-10: 0-444-52732-X ISSN: 0079-6638 For information on all Elsevier publications visit our web site at books.elsevier.com 06 07 08 09 10 10 9 8 7 6 5 4 3 2 1 Printed and bound in The Netherlands

Preface This volume of Progress in Optics contains six review articles on a wide range of topics. The first article by V. Mahajan deals with Gaussian apodization and beam propagation. The point-spread functions and the optical transfer functions of optical systems with Gaussian pupils are discussed and are compared with those for a uniformly illuminated pupil. The results are also applicable to the propagation of Gaussian beams, such as are encountered in laser transmitters. The analytical results are illustrated by numerical examples. The next article by A. Joshi and M. Xio reviews recent investigations regarding the use of electromagnetically-induced transparency to manipulate and to control linear and nonlinear optical properties of atomic systems near resonance. Emphasis is given to enhanced four-wave mixing in three- and four-level atomic systems and to controlling nonlinear optical processes with three-level atoms inside an optical cavity. The third article by H. Benisty and C. Weisbuch is concerned with photonic crystals and covers a broad range of topics, from physical properties of such crystals to some of their uses, for example in integrated optics. In the article which follows, C. Brosseu and A. Dogariu discuss some basic mathematical aspects of three-dimensional electromagnetic fields, especially with regards to their polarization properties. The traditional theory of polarization is restricted to planar wavefields, but recent developments in optics, particularly in near-field optics, require broader theory. This article discusses the mathematical basis for such a generalization. The fifth article by M. Dušek, N. Lütkenhaus and M. Hendrych deals with the relatively new field of quantum cryptography. This is a technique for secure communications based on quantum mechanics. The article explains the underlying principle of quantum cryptography, discusses the security of realistic systems and presents reviews of different experimental methods for practical implementation of this new technique. The concluding article by N. Cerf and J. Fiurášek gives an account of researches on optical quantum cloning. After a brief introduction of the so-called “no-cloning v

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theorem” and its relationship with the linearity and causality of quantum mechanics, the concept of quantum cloning machine is explained. In particular stateindependent and state-dependent cloning machines are discussed. It is clear that this volume covers a broad range of subjects, some rather practical, other somewhat abstract. It seems, therefore, likely that the reader will find in this volume some reviews which will be of special interest to him. Emil Wolf Department of Physics and Astronomy and The Institute of Optics University of Rochester Rochester, NY 14627, USA April 2006

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1. Gaussian apodization and beam propagation, Virendra N. Mahajan (El Segundo, CA and Tucson, AZ, USA) . . . . . . . . . . . .

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§ 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Pupil function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Point-spread function . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Optical transfer function . . . . . . . . . . . . . . . . . . . . . . . . § 3. Aberration-free Gaussian pupil . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Pupil function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Point-spread function . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Optical transfer function . . . . . . . . . . . . . . . . . . . . . . . . § 4. Defocused Gaussian pupil . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Pupil function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Point-spread function . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Axial irradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Optical transfer function . . . . . . . . . . . . . . . . . . . . . . . . § 5. Strehl ratio, aberration tolerance, and Zernike–Gauss polynomials . . . . 5.1. Strehl ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Depth of focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Balanced aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Zernike–Gauss polynomials . . . . . . . . . . . . . . . . . . . . . . 5.5. Strehl ratio for primary aberrations . . . . . . . . . . . . . . . . . . § 6. Balancing of defocus aberration with spherical aberration or astigmatism 6.1. Focused beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Collimated beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 7. Aberrated Gaussian pupil . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Spherical aberration . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Symmetry properties . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Line of sight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 8. Weakly-truncated Gaussian pupils and beams . . . . . . . . . . . . . . . . 8.1. Pupil function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Point-spread function . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Radius of curvature of the propagating wavefront . . . . . . . . . . 8.4. Collimated beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. Beam focusing and waist imaging by a lens . . . . . . . . . . . . . 8.6. Optical transfer function . . . . . . . . . . . . . . . . . . . . . . . . 8.7. Strehl ratio, aberration balancing, and orthogonal polynomials . . . vii

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8.8. Beam characterization and measurement 8.9. Nonparaxial Gaussian beams . . . . . . § 9. Conclusions . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 2. Controlling nonlinear optical processes in multi-level atomic systems, Amitabh Joshi and Min Xiao (Fayetteville, AR, USA) . .

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§ 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. Modified linear and nonlinear optical properties in multi-level atomic systems . . . . . . 2.1. Absorption reduction and dispersion enhancement . . . . . . . . . . . . . . . . . 2.2. Enhanced nonlinearity in EIT systems . . . . . . . . . . . . . . . . . . . . . . . . § 3. Enhanced four-wave mixing processes with induced atomic coherence . . . . . . . . . . 3.1. Three-level atomic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Four-level atomic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 4. Controlled optical bistability and optical multistability with three-level atoms inside an optical cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Theoretical calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Controllable optical bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Controllable optical multistability . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Controllable direction of hysteresis cycle in optical bistability/optical multistability 4.5. Dynamic hysteresis in optical bistability . . . . . . . . . . . . . . . . . . . . . . . § 5. Controlled optical switching in three-level atomic systems . . . . . . . . . . . . . . . . . 5.1. Controlled switching between bistable states . . . . . . . . . . . . . . . . . . . . . 5.2. All-optical switching controlled by coupling laser beam . . . . . . . . . . . . . . 5.3. Controlled optical switching in four-level N-type atomic system . . . . . . . . . . § 6. Controlled optical instability with three-level atoms inside an optical cavity . . . . . . . 6.1. Theoretical calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Experimental measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 7. Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3. Photonic crystals, Henri Benisty (Orsay, France) and Claude Weisbuch (Palaiseau, France and Santa Barbara, CA, USA) . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . Main variables and notations . . . . . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . § 2. Basics of periodic dielectric media and theoretical tools 2.1. 1D Bragg structure . . . . . . . . . . . . . . . . . 2.2. Density of states, local DOS . . . . . . . . . . . . 2.3. Generalizing to 2D and 3D periodic crystals . . . 2.4. Basics of band calculation . . . . . . . . . . . . . 2.5. The discovery of PBG . . . . . . . . . . . . . . . 2.6. Bloch waves . . . . . . . . . . . . . . . . . . . . 2.7. Theoretical tools: a brief overview . . . . . . . .

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Contents § 3. Three-dimensional photonic crystals . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Fabrication of 3D photonic crystals . . . . . . . . . . . . . . . . . . . . . . 3.3. Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Metallodielectric systems and microwave regime . . . . . . . . . . . . . . § 4. Two-dimensional photonic crystals . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Theoretical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. 2D photonic crystal with vertical guidance . . . . . . . . . . . . . . . . . . 4.3. 2D realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Applications and characterization methods . . . . . . . . . . . . . . . . . . § 5. Photonic crystal fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Issues of fibre world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The playing field of photonic crystal fibers . . . . . . . . . . . . . . . . . . Single-mode fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . True PC guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-Delta fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special applications: nonlinear optics, supercontinuum generation, etc. . . Brief overview of PCF modeling tools . . . . . . . . . . . . . . . . . . . . § 6. Conclusion and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and special issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special . issues of regular journals on photonic crystals, in chronological order References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 4. Symmetry properties and polarization descriptors for an arbitrary electromagnetic waveﬁeld, Christian Brosseau (Brest, France) and Aristide Dogariu (Orlando, FL, USA) . . . . . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. A brief survey of the interplay between polarization concepts and geometry . . . . . . . 2.1. From Bartholinus (1669) to Stokes (1852): polarization is a geometric property of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. From Stokes (1852) to Poincaré (1892): polarization is an electromagnetic property of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. From Poincaré (1892) to Wolf (1954): polarization is a statistical property of light 2.4. And now: geometric algebra is back . . . . . . . . . . . . . . . . . . . . . . . . . § 3. Density matrix and the convexity property of the states of polarization . . . . . . . . . . 3.1. Polarization and the density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Scalar invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The convex set of polarization states . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Summary of geometric algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 4. Polarization of a plane wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Density matrix and degree of polarization . . . . . . . . . . . . . . . . . . . . . . 4.2. Some important consequences and concrete applications . . . . . . . . . . . . . . § 5. Polarization of an arbitrary wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Density matrix and polarization descriptors . . . . . . . . . . . . . . . . . . . . . 5.2. Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 6. Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 7. Summary and prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Lie groups in polarization optics . . . . . . . . . . . . . . . . . . . . . . . . . A.1. SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B: Madison convention for the density matrix of massive spin-1 particles . . . . Appendix C: Degree of polarization of a field consisting of a superposition of an ensemble of evanescent waves of random amplitude and of black-body radiation . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 5. Quantum cryptography, Miloslav Dušek (Olomouc, Czech Republic), Norbert Lütkenhaus (Erlangen, Germany) and Martin Hendrych (Castelldefels, Barcelona, Spain) . . . . . . . . . . . . . . . . . . . . . . . § 1. Ciphering . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Introduction, cryptographic tasks . . . . . . . . . 1.2. Asymmetrical ciphers (public-key cryptography) 1.3. Symmetrical ciphers (secret-key cryptography) . 1.4. Vernam cipher, key distribution problem . . . . . § 2. Quantum key distribution . . . . . . . . . . . . . . . . . 2.1. The principle, eavesdropping can be detected . . 2.2. Quantum measurement . . . . . . . . . . . . . . . 2.3. Quantum states cannot be cloned . . . . . . . . . 2.4. Protocol BB84 . . . . . . . . . . . . . . . . . . . 2.5. Eavesdropping, intercept–resend attack . . . . . . § 3. Some other discrete protocols for QKD . . . . . . . . . 3.1. Two-state protocol, B92 . . . . . . . . . . . . . . 3.2. B92 protocol with a strong reference pulse . . . . 3.3. Six-state protocol . . . . . . . . . . . . . . . . . . 3.4. SARG protocol . . . . . . . . . . . . . . . . . . . 3.5. Decoy-state protocols . . . . . . . . . . . . . . . 3.6. Entanglement-based protocols . . . . . . . . . . . § 4. Experiments . . . . . . . . . . . . . . . . . . . . . . . . 4.1. QKD with weak laser pulses . . . . . . . . . . . . 4.2. Entanglement-based protocols . . . . . . . . . . . § 5. Technology . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Light sources . . . . . . . . . . . . . . . . . . . . 5.2. Detectors . . . . . . . . . . . . . . . . . . . . . . 5.3. Quantum channels . . . . . . . . . . . . . . . . . § 6. Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Transmission rate . . . . . . . . . . . . . . . . . . 6.2. Limit on the distance . . . . . . . . . . . . . . . . 6.3. Quantum repeaters . . . . . . . . . . . . . . . . . § 7. Supporting procedures . . . . . . . . . . . . . . . . . . 7.1. Estimation of leaked information . . . . . . . . . 7.2. Error correction for classical bit strings . . . . . . 7.3. Privacy amplification for classical bit strings . . . 7.4. Advantage distillation for classical bit strings . . 7.5. Authentication of public discussion . . . . . . . .

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xi

Contents § 8. Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Attacks on ideal protocols . . . . . . . . . . . . . . . . 8.2. Secure key rates from classical three-party correlations 8.3. Bounds on quantum key distribution . . . . . . . . . . 8.4. Security proofs . . . . . . . . . . . . . . . . . . . . . . 8.5. Specific attacks . . . . . . . . . . . . . . . . . . . . . . 8.6. Results . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7. Side channels and other imperfections . . . . . . . . . § 9. Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 6. Optical quantum cloning, Nicolas J. Cerf (Bruxelles, Belgium) and Jaromír Fiurášek (Olomouc, Czech Republic) . . . . . . . . . . § 1. Introduction and history . . . . . . . . . . . . . . . . . . 1.1. The no-cloning theorem . . . . . . . . . . . . . . 1.2. Beyond the no-cloning theorem . . . . . . . . . . 1.3. Quantum cloning without signaling . . . . . . . . 1.4. Content of this review . . . . . . . . . . . . . . . § 2. Overview of quantum cloning machines . . . . . . . . . 2.1. Universal cloning machines . . . . . . . . . . . . 2.2. Pauli and Heisenberg cloning machines . . . . . 2.3. Phase- and Fourier-covariant cloning machines . 2.4. Group-covariant cloning machines . . . . . . . . 2.5. High-d state-dependent cloning machines . . . . 2.6. Cloning a pair of orthogonal qubits . . . . . . . . 2.7. Entanglement cloning machines . . . . . . . . . . 2.8. Real cloning machines . . . . . . . . . . . . . . . 2.9. Highly-asymmetric cloning machines . . . . . . . 2.10. Continuous-variable cloning machines . . . . . . 2.11. Probabilistic cloning machines . . . . . . . . . . 2.12. Economical cloning machines . . . . . . . . . . . § 3. One-to-two quantum cloning as a CP map . . . . . . . . 3.1. Isomorphism between CP maps and operators . . 3.2. Covariance condition . . . . . . . . . . . . . . . . 3.3. Cloning as a semidefinite programming problem 3.4. Double-Bell ansatz . . . . . . . . . . . . . . . . . 3.5. Heisenberg cloning machines . . . . . . . . . . . 3.6. Three special cases of Heisenberg cloners . . . . § 4. N -to-M universal quantum cloning . . . . . . . . . . . 4.1. Optimal cloning transformation . . . . . . . . . . 4.2. Optimality proof for 1 → M cloning of qubits . . 4.3. Universal asymmetric quantum cloning . . . . . . 4.4. Universal-NOT gate . . . . . . . . . . . . . . . . § 5. Universal cloning of photons . . . . . . . . . . . . . . . 5.1. Amplification of light . . . . . . . . . . . . . . . 5.2. Symmetrization . . . . . . . . . . . . . . . . . . . 5.3. Universal asymmetric cloning of photons . . . . 5.4. Cloning of orthogonally polarized photons . . . .

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xii

Contents

§ 6. Phase-covariant cloning of photons . . . . . . . . . 6.1. Phase-covariant cloning of qubits . . . . . . . 6.2. Phase-covariant cloning of qudits . . . . . . . 6.3. Optical phase-covariant cloning . . . . . . . . 6.4. Experimental 1-to-3 phase-covariant cloning . § 7. Cloning of optical continuous variables . . . . . . . 7.1. Cloning of coherent states . . . . . . . . . . . 7.2. Cloning by phase-insensitive amplification . . 7.3. Experimental cloning of coherent states . . . 7.4. Gaussian distribution with finite width . . . . 7.5. Cloning of conjugate coherent states . . . . . § 8. Conclusions . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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Author index for Volume 49 . . . . Subject index for Volume 49 . . . . Contents of previous volumes . . . Cumulative index – Volumes 1–49

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E. Wolf, Progress in Optics 49 © 2006 Elsevier B.V. All rights reserved

Chapter 1

Gaussian apodization and beam propagation by

Virendra N. Mahajan The Aerospace Corporation, 2350 E. El Sugundo Blvd., El Segundo, CA 90245, USA

and College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA e-mail: [email protected]

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(06)49001-6 1

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

§ 2. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

§ 3. Aberration-free Gaussian pupil . . . . . . . . . . . . . . . . . . . . .

7

§ 4. Defocused Gaussian pupil . . . . . . . . . . . . . . . . . . . . . . . .

16

§ 5. Strehl ratio, aberration tolerance, and Zernike–Gauss polynomials . .

34

§ 6. Balancing of defocus aberration with spherical aberration or astigmatism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

§ 7. Aberrated Gaussian pupil . . . . . . . . . . . . . . . . . . . . . . . .

66

§ 8. Weakly-truncated Gaussian pupils and beams . . . . . . . . . . . . .

71

§ 9. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2

§ 1. Introduction We consider optical systems with Gaussian apodization or Gaussian pupils, i.e., those with a Gaussian amplitude across the wavefront at their exit pupils. The mathematical treatment is applicable equally to an imaging system with Gaussian transmission (obtained, for example, by placing a Gaussian filter at its exit pupil) as well as a laser transmitter in which the laser beam has a Gaussian distribution across its exit pupil. In Section 2 we outline the general theory for obtaining the point-spread and optical transfer functions of a system from its pupil function. A Gaussian illumination, which in principle extends to infinity, is truncated by the finite size of the pupil. An aberration-free system with a Gaussian pupil is considered in Section 3. We show that it yields a point-spread function (PSF) with a broader central bright spot but lower secondary maxima compared to the Airy pattern obtained for a uniform pupil (Jacquinot and Roizen-Dossier [1964], Buck [1967], Campbell and DeShazer [1969], Olaofe [1970], Dickson [1970], Schell and Tyras [1971], Williams [1973], Mahajan [1986]). Its central irradiance is smaller than that for a uniform pupil of the same total power (Mahajan [1980, 1986]). The corresponding optical transfer function (OTF) is higher for low spatial frequencies and lower for high spatial frequencies (Chung and Hopkins [1989], Mahajan [2004]). In Section 4 we discuss a defocused system and show that the principal maximum of axial irradiance of a focused beam with a small Fresnel number lies at a point that is closer to the pupil and not at the geometrical focus (Li and Wolf [1982], Carter [1982], Sucha and Carter [1984], Dementev and Domarkene [1984]). However, as in the case of a uniform pupil, the maximum central irradiance on a target at a fixed distance is obtained when the beam is focused on it (Mahajan [1986, 2004]). The effect of aberrations on the central irradiance is considered in Section 5. It is shown that the Strehl ratio for a given amount of primary aberration is higher for a Gaussian pupil than that for a corresponding uniform pupil, or the aberration tolerance for a given Strehl ratio is higher for a Gaussian pupil (Lowenthal [1974], Mahajan [1995, 2003, 2005a]). Aberration balancing to reduce aberration variance and thus improve Strehl ratio for small aberrations is explained, and Zernike–Gauss polynomials that represent balanced aberrations are discussed. For 3

4

Gaussian apodization and beam propagation

[1, § 1

systems with small Fresnel numbers, it is shown in Section 6 that the axial irradiance closer to the pupil increases when the defocus aberration is balanced with spherical aberration (Yoshida and Asakura [1996], Jiang and Stamnes [1997], Mahajan [2005b]) or astigmatism (Mahajan [2005b]). Both focused and collimated beams are discussed. The effect of an aberration on the PSF is considered in Section 7. It is shown that while apodization broadens the central bright spot but reduces the secondary maxima, balanced spherical aberration does not change the size of the bright spot but increases the secondary maxima, thus obliterating the positive attribute of apodization (Sklar [1975], Lowenthal [1975]). The line of sight of an aberrated system, defined as the centroid of its PSF, is shown to be affected by coma-type aberrations only, and lies farther from its center for a Gaussian pupil than for a uniform pupil (Mahajan [1985]). Narrow or weakly-truncated Gaussian illumination is considered in Section 8. It is shown that a Gaussian beam exiting from the pupil remains Gaussian as it propagates (Siegman [1971], Gaskill [1978], Belland and Crenn [1982], Mahajan [2004]). The rings of the diffraction pattern disappear as the truncation decreases. Since the beam remains Gaussian as it propagates, the OTF of such a Gaussian pupil is also a Gaussian (Mahajan [2004]). It is shown that a focused beam has the smallest radius, called its waist, in a plane that is closer to the pupil than the focal plane. Considering the waist of a beam incident on a lens as an object, an imaging equation is developed in which the waist of the transmitted beam acts as the image (Williams [1973], Self [1983], Mahajan [1986]). It is shown that when the waist of the incident beam lies in the front focal plane, the waist of the transmitted beam lies in its back focal plane, and not at infinity as in conventional imaging. Similarly, whereas in conventional imaging of a real object by a positive lens forming a real image there is a minimum separation between an object and its image, there is no minimum separation between a real object waist and a real image waist. Whereas for small truncations the approximate expression exp(−σΦ2 ) in terms of the phase aberration variance σΦ2 estimates the true value of the Strehl ratio quite well for wave aberration standard deviation σw λ/4, it significantly underestimates it for large truncations, unless the Strehl ratio is greater than or equal to 0.9 or σw λ/20 (Mahajan [2005a]). A beam quality factor of M 2 is defined that accounts for the difference in the divergence of a practical beam and that of an ideal Gaussian beam. Although most of our discussion is based on paraxial beams, i.e., for apertures or beam radii much larger than the optical wavelength, nonparaxial beams for which the beam radius is comparable to the wavelength are also discussed briefly.

1, § 2]

Theory

5

§ 2. Theory 2.1. Pupil function If A(rp ) is the amplitude and Φ(rp ) is the phase aberration defined with respect to a reference sphere of radius of curvature R at a point rp in the plane of the exit pupil of the system, its pupil function can be written P (rp ) = A(rp ) exp iΦ(rp ) . (2.1) The total power in the exit pupil and, therefore, in the image is given by Pex = A2 (rp ) drp ,

(2.2)

where A2 (rp ) is the irradiance at a pupil point.

2.2. Point-spread function The point-spread function (PSF) of the system in a plane at a distance R from the plane of the pupil is given by (Goodman [1996], Born and Wolf [1999], Mahajan [2004]) 2 2πi 1 PSF(ri ) = (2.3) P (rp ) exp − rp · ri drp , 2 2 λR Pex λ R

where ri is the position vector of a point in the image plane and λ is the wavelength of object radiation. The corresponding irradiance distribution of the image of a point object is given by I (ri ) = Pex PSF(ri ).

(2.4)

2.3. Optical transfer function The optical transfer function (OTF) τ ( vi ) of a system is, by definition, the Fourier transform of its PSF, vi · ri ) dri , τ ( vi ) = PSF(ri ) exp(2πi (2.5)

where vi is a spatial frequency vector. Substituting eq. (2.3) into eq. (2.5), the OTF can also be written as the autocorrelation of its pupil function P (rp )P ∗ (rp − λR vi ) drp τ ( vi ) = (2.6) , |P (rp )|2 drp

6

Gaussian apodization and beam propagation

[1, § 2

Fig. 1. Geometry for evaluating the OTF of a system with a circular pupil. The centers of the two pupils are located at (0, 0) and λR(ξ, η) in the (xp , yp ) coordinate system and ∓(λR/2)(vi , 0) in the (p, q) coordinate system, where vi = (ξ 2 + η2 )1/2 and φ = tan−1 (η/ξ ). The shaded area is the overlap area of the two pupils. When normalized by the pupil radius a, the centers of the two pupils of unity radius lie at ∓v along the p axis.

where the integration is across the overlap area of two intersecting pupils centered at rp = 0 and rp = λR vi , shown shaded in fig.1 for a circular pupil. It is convenient to consider a (p, q) coordinate system whose origin lies at the midpoint of the line joining the centers of the two pupils, but whose axes are rotated by the polar angle φ of the spatial frequency vector with respect to those of the (xp , yp ) coordinate system, as illustrated in fig. 1 for a system with a circular pupil of radius a or diameter D = 2a. The centers of the two pupils in this coordinate system are located at ∓(λR/2)(vi , 0). The corresponding pupil function P (p, q) may be obtained from the pupil function P (xp , yp ) by replacing xp with p cos φ − q sin φ and yp with p sin φ + q cos φ. For a radially symmetric pupil function, it is obtained simply by replacing xp2 + yp2 by p 2 + q 2 . It is evident from fig. 1 that the overlap of the two pupils reduces to zero as λRvi → D. Accordingly, the OTF is zero for vi vc , where vc = 1/(λF ) is the cutoff spatial frequency of the system. Here F = R/D is the f -number of the image-forming light cone. Using normalized quantities ρ = rp /a and v = vi /vc , the OTF in polar coordinates (v, φ) may be written a2 P (p + v, q)P ∗ (p − v, q) dp dq, 0 v 1. (2.7) τ (v, φ) = Pex

1, § 3]

Aberration-free Gaussian pupil

7

When p and q are normalized by the pupil radius a, as in eq. (2.7), then the two unit circles are centered√at ∓v along the p axis, and the limits of p and q are ∓( 1 − q 2 − v) and ∓ 1 − v 2 , respectively. Their overlap region, which forms the region of integration in the above integral, is symmetric in p and q. § 3. Aberration-free Gaussian pupil 3.1. Pupil function A pupil is referred to as a Gaussian pupil if the amplitude variation across it with rp has the form of a Gaussian. It can be obtained in two different ways. In imaging applications, the wave incident on the exit pupil has a uniform amplitude, but its transmission varies as a Gaussian. In applications of Gaussian beam propagation, the incident wave has a Gaussian amplitude, but its transmission is uniform. In either case, the transmitted amplitude immediately to the right of the pupil can be written

A(ρ) = A0 exp −γρ 2 , 0 ρ 1, (3.1)

where A0 is a constant (with A20 having dimensions of W/m2 ) and 2 a γ = ω

(3.2)

is the truncation parameter of the pupil with ω as the Gaussian radius representing the radial distance at which the amplitude decreases to 1/e of its value at the center. The case of a uniform pupil is obtained by letting γ → 0. A large value of γ or ω ≪ a represents a narrow Gaussian or a weakly-truncated pupil. Substituting eq. (3.1) into eq. (2.2), the total power transmitted by the pupil is given by 1

exp −2γρ 2 ρ dρ Pex = 2Sex A20 (3.3a) 0

where

Sex 2 = A0 1 − exp(−2γ ) , 2γ

Sex = πa 2

(3.3b)

(3.4)

is the area of the exit pupil. The pupil irradiance I (ρ) = A2 (ρ) in units of Pex /Sex is given by I (ρ) =

2γ exp(−2γρ 2 ) . 1 − exp(−2γ )

(3.5)

8

Gaussian apodization and beam propagation

[1, § 3

(a)

(b) √ Fig. 2. Amplitude and irradiance distributions at the exit pupil for γ = a/ω = 0, 1, 2 and 3: (a) amplitude; (b) irradiance. The amplitude A(ρ) is normalized to unity at the center, but the irradiance I (ρ) is in units of Pex /Sex .

Figure 2 shows how the amplitude and irradiance vary across the pupil for several values of γ . The amplitude A(ρ) has a value of 1/e (or approximately 0.368) √ at ρ = 1, 1/2 and 1/3 for γ = 1, 2 and 3, respectively, compared to a value of unity at the center. The case of uniform amplitude corresponds to γ = 0. The corresponding value of irradiance at the center of the pupil is 2.31, 8.00 and 18.00, compared to a value of unity for a uniform pupil. The corresponding irradiance at the edge of the pupil relative to its value at the center is given by 1/e2 = 0.1353, 1/e4 = 0.0183 and 1/e6 = 2.479 × 10−3 .

1, § 3]

Aberration-free Gaussian pupil

9

If we let the upper limit of the integral in eq. (3.3a) approach infinity, we obtain the total power incident on the pupil plane ∞

Pinc = 2Sex A20 exp −2γρ 2 ρ dρ 0

Sex A20 = . 2γ

(3.6)

Thus, the irradiance at the center of the pupil may be written A20 =

2Pinc . πω2

(3.7)

Hence, from eqs. (3.3b) and (3.6), the fraction of the power transmitted by the pupil is given by Ptrans =

Pex = 1 − exp(−2γ ). Pinc

(3.8)

√ The fractional power transmitted by the pupil as a function of γ or a/ω is illustrated in fig. 3 labeled as beam. Its value is 86.47% when γ = 1, 98.89% √ √ √ when γ = 1.5, 99.97% when γ = 2, and practically 100% when γ = 3. Thus, the fractional power exp(−2γ ) lying outside the pupil is negligible when √ γ 2.

√ Fig. 3. Fractional power Ptrans transmitted by the exit pupil as a function of γ = a/ω. The curve labeled “Beam” is for a Gaussian beam incident on a pupil which has uniform transmission, and the curve labeled “Apodized pupil” is for uniform illumination of a pupil with a Gaussian transmission.

10

Gaussian apodization and beam propagation

[1, § 3

For a uniformly illuminated pupil that transmits in the form of a Gaussian, e.g., by placing a Gaussian amplitude filter, the power incident on it is given by Pinc = A20 Sex .

(3.9)

The power transmitted by the pupil is again given by eq. (3.3a). Hence, the fractional transmitted power is given by Pex 1 − exp(−2γ ) (3.10) = . Pinc 2γ It decreases as γ increases, as illustrated by the curve labeled “Apodized pupil” in √ fig. 3. Its value for γ = 1, 2 and 3 is 43.23%, 12.50% and 5.56%, respectively. The pupil irradiance in units of Pex /Sex is again given by eq. (3.5). Ptrans =

3.2. Point-spread function It should be evident that the problem of determining the irradiance distribution of the image of a point object is identical to the problem of determining the focal-plane irradiance distribution of a focused beam. In either case, substituting eq. (3.1) into eq. (2.3) and utilizing eq. (2.4), we may write the irradiance distribution in the case of an unaberrated pupil 2 1 2π −2 I (r, θi ; γ ) = π I (ρ) exp −πiρr cos(θp − θi ) ρ dρ dθp , 0

0

(3.11) where (ri , θi ) are the polar coordinates of a point in the observation plane, r = |ri |/(λF ), (ρ, θp ) are the polar coordinates of a point in the pupil plane, and the irradiance is in units of the focal-point irradiance Pex Sex /(λ2 R 2 ) for a uniform pupil. Carrying out the angle integration, we obtain the radially symmetric irradiance distribution 1

2 I (r; γ ) = 4 (3.12) I (ρ)J0 (πrρ)ρ dρ . 0

It is shown in the Appendix that the integral in eq. (3.12) can be written as the sum of a Gaussian term and other terms that vanish for large values of γ . Thus, for large values of γ , the focal-plane distribution is also Gaussian. Indeed, a Gaussian beam remains Gaussian as it propagates (Siegman [1971], Gaskill [1978], Belland and Crenn [1982], Nourrit, de Bougrenet de la Tocnaye and Chanclou [2001], Mahajan [2004]). Letting r = 0 in eq. (3.12), we obtain the focal-point irradiance I (0; γ ) =

tanh(γ /2) . γ /2

(3.13)

1, § 3]

Aberration-free Gaussian pupil

11

The fractional power contained in a circle of radius rc in units of λF is given by π2 rc P (rc ; γ ) = (3.14) I (r; γ )r dr. 2 0

For a uniformly illuminated pupil, γ = 0, I (ρ) → 1, and eqs. (3.12) and (3.14) reduce to

2J1 (πr) 2 I (r) = (3.15) πr and P (rc ) = 1 − J02 (πrc ) − J12 (πrc ),

(3.16)

respectively. Equation (3.15) represents the well-known Airy pattern (Airy [1835]), and eq. (3.16) was first given by Rayleigh [1879]. Figure 4 shows how the focal-point irradiance I (0; γ ) varies with γ . It has a maximum value of unity for γ = 0, i.e., for uniform illumination, and decreases monotonically as γ increases. It shows that, for a fixed total power Pex in the pupil, any amplitude variations across it (but without any aberrations) reduce the focal-point irradiance. Now the increase in γ can be due to an increase in a or a decrease in ω. If a is kept fixed and ω is decreased, the value of the focal√ point irradiance for γ = 1, 2 and 3 is 0.924, 0.482 and 0.222, respectively, compared to its value of unity for a uniform pupil. However, if ω is kept fixed and a is increased, then the focal-point irradiance for a Gaussian pupil increases

√ Fig. 4. Focal-point irradiance I (0; γ ) as a function of γ normalized by Pex Sex /(λ2 R 2 ) when a is 2 2 fixed but ω varies, and by 2πPex ω /(λ R 2 ) when a varies but ω is fixed.

12

Gaussian apodization and beam propagation

[1, § 3

√ Fig. 5. Variation of I (0; γ ) normalized by Pinc Sex /(λ2 R 2 ) as a function of γ , showing that its value √ is maximum when γ = 1.120 or ω = 0.893a.

as tanh(γ /2), as may be seen by multiplying the right-hand side of eq. (3.13) by the normalizing factor Pex Sex /(λ2 R 2 ). This is also illustrated in fig. 4, showing √ that the increase is less than 4% for γ 2. The normalizing factor in this case is 2πPex ω2 /(λ2 R 2 ). For a uniform pupil, however, the focal-point irradiance increases quadratically with a. For a given total beam power Pinc incident on a pupil of fixed radius a, the transmitted power Pex increases as ω decreases, but the corresponding focal-point irradiance decreases. Hence, for a given total power Pinc , there is an optimum value of ω that yields the maximum focal-point irradiance (Buck [1967]). To determine this value, we write the focal-point irradiance given by eq. (3.13) in units of Pinc Sex /(λ2 R 2 ): tanh(γ /2) 2 2 I (0; γ ) = 1 − exp(−2γ ) = 1 − exp(−γ ) . γ /2 γ √ Figure 5 shows how I (0; γ ) varies with γ . Letting ∂I (0; γ ) = 0, ∂γ

(3.17)

(3.18)

we find that I (0; γ ) has a maximum equal to 0.8145 when γ = 1.255 or ω = 0.893a. The corresponding irradiance at the edge of the pupil is 8.1% and the transmitted power Ptrans is 91.87%. Figure 6 shows the image-plane irradiance and encircled-power distributions √ for γ = 0, 1, 2 and 3. For clarity, the irradiance distributions are also plotted on

1, § 3]

Aberration-free Gaussian pupil

13

(a)

(b) √ Fig. 6. (a) PSF and encircled power for a Gaussian pupil with γ = 0, 1, 2 and 3. The irradiance is in 2 2 units of Pex Sex /(λ R ) and the encircled power is in units of Pex . r and rc are in units of λF . (b) The irradiance distribution normalized to unity at the center is also shown on a log scale to highlight the differences between the secondary maxima of uniform and Gaussian beams.

14

Gaussian apodization and beam propagation

[1, § 3

Table 1 Maxima and minima of image-plane irradiance distribution and corresponding encircled powers for a Gaussian pupil with γ = 1 compared with those for a uniform (γ = 0) pupil, which are given in parentheses Max/Min

r, rc

I (r)

P (rc )

Max

0 (0) 1.43 (1.22) 1.79 (1.64) 2.33 (2.23) 2.76 (2.68) 3.30 (3.24)

0.924 (1) 0 (0) 0.0044 (0.0175) 0 (0) 0.0012 (0.0042) 0 (0)

0 (0) 0.955 (0.838) 0.962 (0.867) 0.973 (0.910) 0.976 (0.922) 0.981 (0.938)

Max

3.76 (3.70)

0.0005 (0.0016)

0.983 (0.944)

Min

4.29 (4.24)

0 (0)

0.985 (0.952)

Max

4.75 (4.71)

0.0002 (0.0008)

0.986 (0.957)

Min Max Min Max Min

a logarithmic scale. At and near the focal point, a uniform beam gives a higher irradiance than a Gaussian beam. Similarly, the encircled power is greater for a uniform beam for small values of rc , but smaller for large values. Of course, P (rc ) → 1 as rc → ∞ regardless of the value of γ . The positions of maxima and minima and the corresponding irradiance and encircled-power values are given in Table 1 for γ = 1. The corresponding results for a uniform beam are given in parentheses. It is evident that the corresponding maxima and minima for a Gaussian beam are located at higher values of r than those for a uniform beam. Thus, the diffraction rings for a Gaussian pupil are larger. For example, the radius of the central bright spot is 1.43 for the Gaussian beam compared to 1.22 for a uniform beam. Moreover, whereas the principal maximum for a Gaussian beam is only slightly lower (0.924 compared with 1), the secondary maxima are lower by a factor greater than 3 compared with the corresponding maxima for a uniform beam. Lower secondary maxima is indeed the objective of apodization. The Gaussian illumination broadens the central disc but reduces the power in the secondary rings. Thus, as γ increases the central disc broadens and the secondary maxima disappear. It is shown in Section 8.2

1, § 3]

Aberration-free Gaussian pupil

15

that the aberration-free PSF can be approximated by a Gaussian with negligible √ error when γ 2. A detailed examination of how the Airy pattern evolves into a Gaussian distribution shows, for example, that its first two dark rings merge into one at the position of the second dark ring as γ → 2.63, or its first two secondary maxima merge into one at the position of the second maximum as γ → 5.60 (Li [1987]). Moreover, the radius of the central bright disc increases discontinuously with γ with steps at the aforementioned γ values and γ = 8.64 and 11.71, its value being greater than 2γ for γ ≫ 1. 3.3. Optical transfer function Substituting eqs. (3.1) and (3.3b) into eq. (2.7), we obtain the aberration-free OTF (Mahajan [2004]) a2 A(p + v, q)A(p − v, q) dp dq τ (v; γ ) = Pex √1−v 2

8γ exp(−2γ v 2 ) = exp −2γ q 2 dq π[1 − exp(−2γ )] 0 √1−q 2 −v

exp −2γp 2 dp, 0 v 1. × (3.19) 0

Letting γ = 0 in eq. (3.19) yields the OTF for a uniform pupil, which is simply equal to the fractional overlap area in fig. 1 (Goodman [1996], Born and Wolf [1999], Mahajan [2004])

1/2 2 −1 , 0 v 1. cos v − v 1 − v 2 (3.20) π Figure 7 shows how the OTF varies with v for several values of γ . We note that, compared to a uniform pupil (i.e., for γ = 0), the OTF of a Gaussian pupil is higher for low spatial frequencies, and lower for high spatial frequencies. Moreover, as γ increases, the bandwidth of low frequencies for which the OTF is higher decreases and the OTF at high frequencies becomes increasingly smaller. This is due to the fact that the Gaussian weighting across the overlap region of two pupils whose centers are separated by small values of v is higher than that for large values of v. If we consider an apodization such that the amplitude increases from the center toward the edge of the pupil, then the OTF is lower for low frequencies and higher for high frequencies (Chung and Hopkins [1989]). Thus, unlike aberrations, which reduce the modulation transfer function (MTF) of a system at all τ (v; 0) =

16

Gaussian apodization and beam propagation

[1, § 4

Fig. 7. The OTF of a Gaussian pupil. The spatial frequency v is normalized by the cutoff frequency √ 1/(λF ). A uniform pupil corresponds to γ = 0, and a large value of γ represents a weakly-truncated pupil.

frequencies within its passband, the amplitude variations can increase or decrease √ the MTF at any of those frequencies. It is shown in Section 8.6 that for γ 2, the OTF can be approximated by a Gaussian.

§ 4. Defocused Gaussian pupil 4.1. Pupil function When the image is observed in a defocused image plane at a distance z from the pupil, then the pupil function may be written P (ρ) = A0 exp −(γ − iBd ) ρ 2 , (4.1) where

R −1 Bd (z) = πN z

(4.2)

is the coefficient of defocus phase aberration representing its peak value. Here N = a 2 /(λR) is the Fresnel number representing the number of Fresnel zones in the pupil as observed from the focus. The defocus aberration is generally not symmetric about the Gaussian image plane. For small values of N (e.g., less than

1, § 4]

Defocused Gaussian pupil

17

or equal to 10), z can be much different from R for Bd to achieve a significant value. Hence, the defocus aberration for two image planes located symmetrically about the Gaussian image plane has different magnitude as well as different sign. However, when the Fresnel number is very large (≫10), Bd becomes large even for very small differences between z and R. Accordingly, the small defocus aberration tolerance dictates that z be approximately equal to R. Hence, eq. (4.2) may be written z , Bd (z) = πN 1 − (4.3) R which for a longitudinal defocus Δ = z − R yields πNΔ . (4.4) R Hence, the defocus aberrations for two image planes located symmetrically about the Gaussian image plane have the same magnitude but opposite signs. The longitudinal defocus may be written in terms of the focal ratio F = R/(2a) of the image-forming light cone as Bd (R + Δ) = −

Δ = −8Bd F 2 ,

(4.5)

where Bd is now the coefficient of wave aberration [obtained by multiplying the phase aberration coefficient Bd in eq. (4.3) by λ/(2π)]. An observation made in a plane beyond the focal plane so that z > R introduces a negative defocus aberration, and one closer to the pupil so that z < R introduces a positive aberration, as may be seen from eq. (4.3).

4.2. Point-spread function Substituting eq. (4.1) into eq. (2.3) and replacing R by z, we obtain the PSF in a defocused image plane. The irradiance distribution in a defocused image plane at a distance z from the plane of the exit pupil may be written 2 2 1

2R 2 I (ρ) exp iBd ρ J0 (πρr)ρ dρ , I (r; z; γ ) = (4.6) z 0

where the irradiance is in units of Pex Sex /(λ2 R 2 ), as in eq. (3.11), but r is in units of λz/D. Equation (4.6) represents the modified form of the defocus-free eq. (3.12) due to an observation in a defocused plane. The modification consists of the inverse-square law dependence on the distance z and the defocus aberration Bd ρ 2 . The encircled power is obtained by substituting eq. (4.6) into eq. (3.14).

18

Gaussian apodization and beam propagation

[1, § 4

From eq. (4.6) we note that the irradiance distribution is independent of the sign of Bd . However, it is asymmetric about the Gaussian image plane; i.e., the irradiance distributions in two observation planes located at z = R ± Δ are not identical. There are three reasons for this asymmetry (Mahajan [1983b, 1994, 2004]). First, the inverse square law dependence on z increases I (z) for z < R and decreases it for z > R. Second, Bd is asymmetric. Third, the argument of the Bessel function J0 (·) depends on z. For small Fresnel numbers, all three factors contribute to the asymmetry. The defocus aberration reduces the central irradiance, but the inverse-square dependence on z increases it for z < R. Since the smaller the Fresnel number the larger the depth of focus, the irradiance on and near the axis can be significantly higher for z < R than the focal-point irradiance. This is illustrated by considering the axial irradiance of a beam with different Fresnel numbers. However, when the Fresnel number is very large, the depth of focus is very small and the effect of inverse-square dependence on z is negligible. Thus, the (R/z)2 factor in eq. (4.6) can be replaced by unity. Moreover, since Bd only changes sign in going from one symmetric plane to the other, the irradiance distribution, which is independent of the sign of Bd , is symmetric about the Gaussian image plane. For large Fresnel numbers, fig. 8 shows the defocused irradiance and encircled power distributions for a defocus of 0.5, 1, 1.5 and 2 waves. The corresponding longitudinal defocus is obtained from eq. (4.5) by multiplying the wave aberration by −8F 2 . For clarity, the irradiance values have been multiplied by 5 in figs. 8(b–d). For small values of defocus, the central irradiance is higher for the uniform beam, but the encircled power is higher for the Gaussian (except for very small circles). However, its value in the case of a uniform beam is zero for an integral number of waves of defocus. The cancellation of contributions of the Fresnel zones is complete. This is not true for a Gaussian pupil, since the amplitudes of the √ various zones are different. For one wave of defocus, the PSFs for γ = 2 and 3 are practically the same for r 0.8. It is shown in Section 4.4 that the corresponding OTFs are practically equal to each other for any spatial frequency. Defocused PSFs in the form of isophotes (i.e., contours of constant irradiance) have been given by Li and Yu for γ = 1, 2, 3 and ∞ (Li and Yu [1989]). Since a given value of z corresponds to a certain value of Bd according to eq. (4.3), the PSFs given in fig. 8 can be applied to systems with small Fresnel numbers provided they are multiplied by (R/z)2 . Of course, the units of r also change from λR/D to λz/D. Results for such systems have been observed experimentally for N ≃ 0.5 and 1, γ ≃ 0.5 and 1, and various values of z (Horng and Li [1988]), and for γ = 0 and N ≃ 0.5, 1 and 5 (Li and Platzer [1983]).

1, § 4]

Defocused Gaussian pupil

19

(a)

(b) Fig. 8. Defocused irradiance and encircled-power distributions. The Fresnel number N is assumed to be large so that defocus has negligible impact on the irradiance due to the inverse-square law dependence on z. The units of r, rc , I (r) and P (rc ) are the same as in fig. 6. For clarity, the irradiance values have been multiplied by 5 in figures (b)–(d).

20

Gaussian apodization and beam propagation

(c)

(d) Fig. 8. Continued.

[1, § 4

1, § 4]

Defocused Gaussian pupil

21

4.3. Axial irradiance 4.3.1. Focused beam If we let r = 0 in eq. (4.6), we obtain the axial irradiance of the beam (Mahajan [1994, 2004, 2005b]): 2 R 2γ 1 (cosh γ − cos Bd ). I (0; z; γ ) = (4.7) z Bd2 + γ 2 sinh γ It goes through a series of maxima and minima as a function of z because of the cos Bd term. By equating its derivative with respect to z to zero, we obtain the positions of its maxima and minima as the solutions of λz Bd 2 (4.8) − 2 (cosh γ − cos Bd ) = − sin Bd . Sex Bd + γ 2 They occur at approximately those z values at which the pupil subtends an odd or an even number of Fresnel zones, respectively. Since cosh γ > 1 and cos Bd 1, the minima are not equal to zero [unless γ = 0, as shown by eq. (4.9)]. The corresponding equations for a uniform beam are I (0; z) =

2

R sin(Bd /2) 2 z Bd /2

(4.9)

and tan

Bd R Bd = , 2 z 2

z = R.

(4.10)

Multiplying the right-hand sides of eqs. (4.7) and (4.9) by the normalizing factor Pex Sex /(λ2 R 2 ), we find that, as will be discussed in Section 5.2, the factor multiplying (R/z)2 represents the Strehl ratio of the beam due to defocus aberration, i.e., the ratio of the central irradiance at a distance z when the beam is focused at a distance R to that when it is focused at a distance z. It should be evident that defocus decreases the central irradiance, but the inverse-square law increases it for z < R and decreases it for z > R. Figure 9 shows how the axial irradiance of a focused Gaussian beam with γ = 1 differs from that of a corresponding uniform beam for Fresnel numbers N = 1, 10 and 100 (Mahajan [2005b]). The numbers 0.924, 0.482 and 0.222 on the curves in fig. 9(a) for N = 1 represent the irradiance value at the focal point relative to a value of unity for a uniform pupil. We note that the principal maximum is higher for the uniform beam compared with that for the Gaussian beam.

22

Gaussian apodization and beam propagation

[1, § 4

(a)

(b) Fig. 9. Axial irradiance of a beam focused at a fixed distance R with a Fresnel number N = a 2 /(λR) = 1, 10 and 100. The irradiance is in units of the focal-point irradiance Pex Sex /(λ2 R 2 ) for a uniform beam.

1, § 4]

Defocused Gaussian pupil

23

(c) Fig. 9. Continued.

It lies closer and closer to the pupil as γ increases. However, the secondary maxima are higher for the Gaussian beam. Moreover, the axial minima for the uniform beam have a value of zero at z values given by Bd = 2πn or z/R = (1+2n/N )−1 , where n is a nonzero positive or a negative integer. For example, when N = 1, the minima are located at z/R = 1/3, 1/5, 1/7, . . . , and there is no minimum for z < R. Qualitatively, as stated in Section 4.2, the even number of Fresnel zones that the pupil subtends at these locations cancel out each other to yield a zero irradiance. Since these zones have different weights in the case of a Gaussian pupil, the cancellation is not complete and the irradiance is a minimum but not zero. We note from figs. 9(b,c) that the curves become symmetric about the focal point z = R as N increases, regardless of the value of γ , as expected from the discussion in Section 4.2. Just as the Airy pattern evolves into a Gaussian distribution as γ increases, similarly the axial irradiance of a uniform beam with its minima and secondary maxima evolves into a smooth distribution (Li [1988a]). For example, when N ≫ 1, the first pair of minimum and maximum merge into a much less pronounced maximum as γ → 2.13. Similarly, the first two pairs of minima and maxima merge into a single but low maximum as γ → 2.68.

24

Gaussian apodization and beam propagation

[1, § 4

4.3.2. Diffraction focus It should be noted that even though the principal maximum of axial irradiance does not lie at the focus, the maximum central irradiance on a target at a given distance from the pupil is obtained when the beam is focused on it. This can be seen by equating to zero the derivative of the axial irradiance given by eq. (4.7) with respect to R and determining the value of R that yields the maximum irradiance. When doing so, the normalization factor Pex Sex /(λ2 R 2 ) should be substituted in this equation with the consequence that the R 2 factor in front of its right-hand side disappears. We find that R must equal z. Since the observation distance z is fixed, the inverse-square law dependence of irradiance on the distance is also fixed. Hence, focusing the beam at any distance other than z introduces defocus aberration and reduces the central irradiance. Figure 10 illustrates this point, where the central irradiance Iz (0; R) on a target at a fixed distance z is shown as a function of the focal distance R. The irradiance in units of Pex Sex /(λ2 z2 ) and Nz = a 2 /(λz) represents the Fresnel number of the pupil as observed from the target. As in fig. 6, the maximum irradiance values for uniform and Gaussian (γ = 1) beams are 1 and 0.924, respectively. Moreover, as in fig. 9, the curves become symmetric about the axial point R = z as Nz increases. Thus, regardless of the value of the Fresnel number, the focal point is indeed the diffraction focus, even though a higher central irradiance is obtained for small Fresnel numbers at axial points closer to the pupil. This is different from, for example, the case of a beam aberrated by a small amount of spherical aberration, in which case an appropriate amount of defocus improves the central irradiance (see Section 5.3). In other words, defocussing the beam in such a case yields a higher central irradi-

Fig. 10. Central irradiance Iu of a uniform beam and Ig of a Gaussian beam with γ = 1 on a target at a fixed distance z from the pupil plane when the beam is focused at various distances R. The irradiance is in units of Pex Sex /(λ2 z2 ) and Nz = a 2 /(λz) represents the Fresnel number of the pupil as observed from the target.

1, § 4]

Defocused Gaussian pupil

25

ance, or the corresponding defocused image is better. Hence, the defocused point (whose location depends on the amount of spherical aberration and the type of pupil illumination, e.g., uniform or Gaussian) is the diffraction focus. Although we cannot improve the central irradiance on a target by defocussing the beam, the fact that the principal maximum of axial irradiance lies at a point other than the focal point implies a large depth of focus. For example, for a uniform beam with N = 1, as in fig. 9(a), the axial irradiance is larger than the focal-point irradiance for 0.453 < z/R < 1. As will be discussed in Section 5.2, defocus tolerance or depth of focus for a Gaussian beam is even higher. From a practical standpoint, the quantity of interest in a certain observation plane is the encircled power, and not just the central irradiance. Unless the radius of the circle is very small, the plane of the principal maximum of encircled power does not coincide with the plane of the principal maximum of axial irradiance. This is illustrated in fig. 11 for a uniform beam, where it is seen that the plane of maximum or principal maximum (when there is more than one maximum) of encircled power depends on the value of the radius rc of the circle (Mahajan [1983a, 1983b, 1983c, 2004]). Predictions of this kind have been verified experimentally (Li [1983], Tanaka and Kanzaki [1987], Horng and Li [1988]). For a focused Gaussian beam, the locations of the planes of minimum field spread (i.e., the spot radius at which the irradiance is 1/e2 of its value at the center)

Fig. 11. Encircled power in a circle of radius rc in units of λF for a focused beam with N = 1 as a function of the axial distance z from the pupil.

26

Gaussian apodization and beam propagation

[1, § 4

(Tanaka, Saga and Hauchi [1985], Tanaka, Saga and Mizokami [1985]) and of minimum root mean square spot radius (Nicola, Anderson and Lisak [1998]) have also been investigated. For small Fresnel numbers, these locations are also closer to the pupil and not at the focal plane. It is also interesting to note that in the case of a uniform beam with N > 12, the principal maximum of encircled power for any radius rc lies at the focal plane, i.e., the Airy pattern yields the highest encircled power. For N > 1.31, the Airy disc contains the highest encircled power, i.e., the power in a circle of radius 1.22 (in units of λF ) as a function of z has the highest value in the focal plane. Similarly, for N > 2.68, the principal maximum of encircled power in the second ring, i.e., for rc = 2.23, also lies in the focal plane (Li [1984]). 4.3.3. Collimated beam The results for a collimated beam can be obtained from those for a focused beam by letting R → ∞. Thus, eq. (4.6) for the irradiance distribution in a plane at a distance z from the pupil reduces to 2 1

2 2 I (ρ) exp iBd ρ J0 (πrρ)ρ dρ , I (r; z) = 4Bd (4.11) 0

where

Sex (4.12) λz represents the phase aberration of a plane wavefront with respect to a reference sphere centered at a distance z and passing through the center of the pupil. In eq. (4.11), the irradiance in both the pupil and the observation planes is in units of the pupil irradiance Pex /Sex for a uniform circular beam. As in eq. (4.6), r is in units of λz/D. In the far field, i.e., for z D 2 /λ, the phase aberration is Bd π/4 (corresponding to a wave aberration of less than or equal to λ/8) and may be neglected. Hence the irradiance distribution and, correspondingly, the encircled power distribution in a far-field plane is similar to the focal-plane distribution discussed earlier. The only difference is in scaling of the diffraction pattern. Similarly, in the near field, i.e., for z < D 2 /λ, the irradiance and encircled-power distributions correspond to defocused distributions discussed earlier. The only significant difference is in the definition of Bd . If z is in units of the far-field distance D 2 /λ and we let r = 0 in eq. (4.11), we obtain the axial irradiance (in units of Pex /Sex ) 2γ π . I (0; z) = (4.13) cosh γ − cos 4z [1 + (4γ z/π)2 ] sinh γ Bd =

1, § 4]

Defocused Gaussian pupil

27

Its maxima and minima are located at z values given by π 2(4z/π)3 γ 2 π = − sin . cosh γ − cos (4.14) 4z 4z 1 + (4γ z/π)2 The results for a uniform beam can be obtained by letting γ → 0. Thus, eq. (4.13) reduces to π I (0; z) = 4 sin2 . (4.15) 8z The maxima of axial irradiance have a value of 4 at z values given by 1 , n = 0, 1, 2, . . . . 4(2n + 1) Its minima have a value of zero at z values given by z=

(4.16)

1 , n = 0, 1, 2, . . . . (4.17) 8(n + 1) These z values for the location of maxima and minima correspond to those axial positions at which the circular exit pupil subtends an odd or an even number of Fresnel’s half-wave zones, respectively. For z > a 2 /λ, the axial irradiance decreases monotonically to zero. For z D 2 /λ, called the far-field distance, it decreases approximately as z−2 . Figure 12 illustrates how the axial irradiance of collimated uniform and Gaussian beams varies with distance z from the pupil. Compared to fig. 9, fig. 12 corresponds to N = 0. Unlike the principal maximum of a focused beam in fig. 9(a), the maximum farthest from the pupil has a lower value than those closer to it in the case of a Gaussian beam. Moreover, whereas the maxima of axial irradiance of a collimated uniform beam have the same value of 4 and minima have a value of zero, the maxima in the case of a Gaussian beam are higher and the minima are nonzero. Because of their different amplitudes, the Fresnel zones do not cancel completely at the location of the minima. Now we consider a beam focused at the far-field distance and compare its axial irradiance with that of a corresponding collimated beam. In particular, we show that beam focusing yields a negligible improvement over a collimated beam, which further decreases as γ increases. From eq. (4.7), the axial irradiance of a beam focused at the far-field distance D 2 /λ may be written 2 2γ π 1 (cosh γ − cos Bd ), I (0; z; γ ) = (4.18) 2 2 4z sinh γ Bd + γ z=

where

Bd =

π 1 −1 4 z

(4.19)

28

Gaussian apodization and beam propagation

[1, § 4

Fig. 12. Axial irradiance of a collimated Gaussian beam in units of the pupil irradiance Pex /Sex at a uniform pupil. The distance z is units of the far-field distance D 2 /λ.

and the irradiance is in units of Pex /Sex representing the illumination of a uniform pupil. For a uniform beam, eq. (4.18) reduces to I (0; z) =

π 4z

2

sin Bd /2 Bd /2

2

.

(4.20)

Figure 13 shows a comparison of the axial irradiance of a focused beam with that of a collimated beam. We note that while they differ in the near field, they yield practically the same irradiance on a target lying in the far field. In other words, beam focusing does not significantly increase the power concentration on the target. In the case of a uniform beam, a collimated beam gives a value of 0.5858 at the far-field distance D 2 /λ compared to a value of 0.6168 for a focused beam. Thus a collimated beam gives an axial irradiance at a distance z in the far field that is 0.95 times the irradiance at this point when the beam is focused at it. Similarly, a collimated beam gives an irradiance of 0.5428, 0.2895 and 0.1360 compared to a value of 0.5700, 0.2973 and 0.1370 for a focused beam √ when γ = 1, 2 and 3, respectively. Thus, as γ increases, the axial irradiance of a collimated beam at the far-field distance D 2 /λ approaches the corresponding value for a beam focused at this point. The results for γ = 0 and 1 are quite

1, § 4]

Defocused Gaussian pupil

29

(a)

(b) Fig. 13. Axial irradiance of a collimated beam compared with a corresponding beam focused at the far-field distance D 2 /λ. The axial distance z is in units of the far-field distance, and the irradiance is in units of the pupil irradiance Pex /Sex for a uniform pupil.

30

Gaussian apodization and beam propagation

(c)

(d) Fig. 13. Continued.

[1, § 4

1, § 4]

Defocused Gaussian pupil

31

similar, the only significant difference being that the axial minima are zero when γ = 0 and nonzero when γ = 1. 4.4. Optical transfer function Substituting eqs. (4.1) and (3.3b) into eq. (2.7), the defocused OTF for Gaussian pupil can be written (Mahajan [2004]) √1−v 2

4γ exp(−2γ v 2 ) dq exp −2γ q 2 τ (v; γ ; Bd ) = π[1 − exp(−2γ )] 0 √1−q 2 −v

× (4.21) exp −2γp 2 + 4iBd vp dp. √ −(

1−q 2 −v)

Letting exp(ix) = cos x + i sin x, the imaginary part of the integral vanishes upon integration being an odd function of p with symmetric limits of integration, and eq. (4.21) reduces to √1−v 2

8γ exp(−2γ v 2 ) dq exp −2γ q 2 τ (v; γ ; Bd ) = π[1 − exp(−2γ )] 0 √1−q 2 −v

× (4.22) exp −2γp 2 cos(4Bd vp) dp, 0

yielding a real OTF, as expected for a symmetric pupil function. The defocused OTF given by eq. (4.22) differs from the in-focus OTF given by eq. (3.19) only by the cos(4Bd vp) factor in the p-integral. Letting γ = 0, the corresponding OTF for a uniform pupil is given by (Mahajan [2004]) √1−v 2

1 τ (v; 0; Bd ) = (4.23) dq sin 4Bd v 1 − q 2 − v . πBd v 0

Both eqs. (4.22) and (4.23) show that the defocused OTF is independent of the sign of Bd . Hence it is asymmetric about the Gaussian image plane for small Fresnel numbers and symmetric for large ones, just like the PSF. For small Fresnel numbers, the cutoff frequency is vc = D/(λz) instead of D/(λR), as may be seen from eqs. (2.3) and (2.6) by replacing R with z. Figure 14 shows the defocused OTF corresponding to the PSFs considered in fig. 8. As the amount of defocus increases, the OTF becomes negative in the low-frequency region, representing reversal of image contrast, and nearly zero in the high-frequency region. As γ increases, the contrast reversal disappears. The OTF is positive for all frequencies √ when γ 2. It is a coincidence that for one wave of defocus, the OTFs for

32

Gaussian apodization and beam propagation

[1, § 4

(a)

(b) Fig. 14. OTF of a defocused pupil. For large values of γ , the OTF can be approximated by a simple expression given by eq. (8.46).

1, § 4]

Defocused Gaussian pupil

(c)

(d) Fig. 14. Continued.

33

34

Gaussian apodization and beam propagation

[1, § 5

√ γ = 2 and 3 are practically identical. From the approximate expression given later [see eq. (8.46)], the OTFs are given by exp(−27.74v 2 ) and exp(−26.77v 2 ), respectively.

§ 5. Strehl ratio, aberration tolerance, and Zernike–Gauss polynomials 5.1. Strehl ratio For an aberrated system, the irradiance distribution in a defocused image plane at a distance z is given by 2 1 2π R I (ρ) exp iΦ(ρ, θp ) I (r; θi ; z; γ ) = πz 0 0 2 × exp −πiρr cos(θp − θi )ρ dρ dθp ,

(5.1) where Φ(ρ; θp ) is the phase aberration of the system at a point (ρ, θp ) in the plane of its exit pupil, including the defocus aberration whose peak value is given by eq. (4.2). As in eq. (4.6), the irradiance is normalized by the aberration-free focalpoint irradiance Pex Sex /(λ2 R 2 ) for a uniform beam and r is in units of λz/D. In fact, eq. (5.1) reduces to eq. (4.6) if we let Φ(ρ, θp ) = Bd ρ 2 . By definition, the Strehl ratio of the system is the ratio of the central irradiance at a distance z when an aberrated beam is focused at a distance R to that when it is focused at a distance z in the absence of any aberration. The aberration-free focal-point irradiance is given by eq. (3.13). Hence, the Strehl ratio is given by 1 2π | 0 0 A(ρ) exp[iΦ(ρ, θ)]ρ dρ dθ|2 S= 1 2π [ 0 0 A(ρ)ρ dρ dθ ]2 2 2 1 2π

γ 2 exp −γρ exp iΦ(ρ, θ) ρ dρ dθ , = π[1 − exp(−γ )] 0 0 (5.2) where we have dropped the subscript p on θp . The value of S is less than one. For small aberrations, approximate expressions for the Strehl ratio may be written (Lowenthal [1974], Szapiel [1982], Mahajan [1986, 2004, 2005b]):

σΦ2 2 ≃ 1 − σΦ2 ≃ exp −σΦ2 , S ≃ 1− (5.3) 2

1, § 5]

35

Strehl ratio, aberration tolerance, and Zernike–Gauss polynomials

Table 2 Primary aberrations and their standard deviations for optical systems with Gaussian pupils. For comparison, the results for a uniform pupil, i.e., γ = 0, are also given √ √ Seidel aberration σΦ (γ = 0) σΦ (γ = 1) σΦ ( γ = 2) σΦ ( γ 3) Spherical, As ρ 4 Coma, Ac ρ 3 cos θ Astigmatism, Aa ρ 2 cos2 θ Defocus, Bd ρ 2 Tilt, Bt ρ cos θ

2A √s = As 3.35 3 5 A √c = Ac 2.83 2 2 Aa 4 Bd B √ = d 3.46 2 3 Bt 2

As 3.67 Ac 3.33 Aa 4.40 Bd 3.55 Bt 2.19

As 6.20 Ac 6.08 Aa 6.59 Bd 4.79 Bt 2.94

√ 2 5As 2 √γ 3Ac γ 3/2 √Aa 2γ Bd γ √Bt 2γ

where σΦ2 = Φ 2 − Φ 2

(5.4)

is the variance of the phase aberration across the Gaussian-amplitude weighted pupil. The mean and the mean square values of the aberration are obtained from the expression 1 2π n A(ρ)[Φ(ρ, θ)]n ρ dρ dθ Φ = 0 0 1 2π 0 0 A(ρ)ρ dρ dθ 1 2π n

γ = (5.5) exp −γρ 2 Φ(ρ, θ) ρ dρ dθ π[1 − exp(−γ )] 0 0 with n = 1 and 2, respectively. Letting γ = 0 yields the results for a uniform pupil (Nijboer [1947], Maréchal [1947], Born and Wolf [1999], Mahajan [1982, 2004]). Table 2 lists the primary aberrations and their standard deviations for increasing values of γ . It is evident that the standard deviation of an aberration decreases as γ increases. This is due to the fact that while an aberration increases as ρ increases, the amplitude decreases more and more rapidly as γ increases, thus reducing its effect more and more compared to that for a uniform pupil. Accordingly, for a given small amount of aberration Ai the Strehl ratio for a Gaussian pupil is higher than that for a uniform pupil. Similarly, the aberration tolerance for a given Strehl ratio is higher for a Gaussian pupil.

5.2. Depth of focus The approximate aberration tolerance for a certain value of the Strehl ratio can be determined from eq. (5.3) if its variance is known. For example, for S 0.8, the

36

Gaussian apodization and beam propagation

[1, § 5

standard deviation σw of the wave aberration must be less than or equal to λ/14, where σw = (λ/2π)σΦ . For convenience, the axial irradiance shown in fig. 9 has been normalized to unity at the focal point in fig. 15. A cursory look at the horizontal scale in this figure shows that the depth of focus increases as N decreases, regardless of the value of γ . For small values of N, the depth of focus is determined by the defocus aberration Bd as well as the inverse-square law dependence on z. For large values of N, the depth of focus is given by ±8Bd F 2 , where Bd is the peak defocus wave aberration. Within the depth of focus, the inverse-square law dependence on z has a negligible effect on the axial irradiance. Since Bd is smaller for a larger value of γ , the depth of focus is correspondingly larger. Letting Φ(ρ, θ) = Bd ρ 2 in eq. (5.2), we obtain the Strehl ratio of a defocused but otherwise aberration-free beam

2 1 2 γ exp −(γ − iBd )x dx S= 1 − exp(−γ ) 0

2 γ 1 = 1 − exp(−2γ ) − 2 exp(−γ ) cos Bd . 2 2 1 − exp(−γ ) γ + Bd

(a) Fig. 15. Axial irradiance of a focused beam normalized to unity at the focal point.

(5.6)

1, § 5]

Strehl ratio, aberration tolerance, and Zernike–Gauss polynomials

(b)

(c) Fig. 15. Continued.

37

38

Gaussian apodization and beam propagation

[1, § 5

It is evident that S is independent of the sign of Bd . For a uniform or a weaklytruncated Gaussian beam, eq. (5.6) reduces to sin Bd /2 2 S= (5.7) Bd /2

or

S=

1 , 1 + (Bd /γ )2

(5.8)

respectively. Figure 16 shows how the Strehl ratio varies with the defocus aberration Bd in units of wavelength λ for various values of γ . It is evident that for a given value of S, the defocus tolerance increases as γ increases. Unlike for a Gaussian pupil, the Strehl ratio for a uniform pupil is zero when Bd is an integral number of wavelengths. For a certain Strehl ratio, the value of Bd can be obtained from this figure, and the corresponding depth of focus can be obtained from eq. (4.5). For example, a Strehl ratio of 0.8 is obtained for Bd = λ/4 when γ 1. Hence, the depth of focus is approximately ±2λF 2 . However, its value is ±5λF 2 when √ γ = 3.

Fig. 16. Strehl ratio of a Gaussian beam as a function of the defocus wave aberration coefficient Bd in units of wavelength λ, showing how it increases as γ increases. For a certain value of the Strehl ratio, the value of Bd is obtained from this figure which, in turn, is used to obtain the corresponding value of the depth of focus from eq. (4.5).

1, § 5]

39

Strehl ratio, aberration tolerance, and Zernike–Gauss polynomials

5.3. Balanced aberrations Since the Strehl ratio depends on the aberration variance, we balance a given aberration with lower-order aberrations to minimize its variance. Thus, we balance spherical aberration and astigmatism with defocus aberration, and coma with tilt aberration. We calculate the variance of the balanced aberration and equate its derivative with respect to the balancing aberration to zero to determine its optimum value to minimize the variance. The balanced primary aberrations thus obtained are listed in Table 3. The corresponding standard deviations are given in Table 4. The factor by which the standard deviation of a primary aberration is reduced by balancing it with another is listed in Table 5. The diffraction focus representing the point of maximum irradiance for a small aberration is listed in Table 6. We note that, although aberration balancing in the case of a uniform pupil reduces the standard deviation of spherical aberration and coma by factors of 4 and 3, respectively, the reduction in the case of astigmatism is only a factor of 1.22. For a Gaussian pupil, the trend is similar but the reduction factors are smaller for spherical aberration and coma, and larger for astigmatism. For a Gaussian beam with γ = 1, they are 3.74, 2.64 and 1.27, corresponding to spherical aberration, coma and astigmatism, respectively.

Table 3 Balanced primary aberrations Balanced aberration

Φ(ρ, θ ; γ = 0)

Φ(ρ, θ ; γ = 1)

Φ(ρ, θ ;

√

γ = 2)

Φ(ρ, θ ;

√

γ 3)

As (ρ 4 − ρ 2 ) As (ρ 4 − 0.933ρ 2 ) As (ρ 4 − 0.728ρ 2 ) As ρ 4 − γ4 ρ 2

3 2 Coma Ac ρ − 3 ρ cos θ Ac (ρ 3 − 0.608ρ) cos θ Ac (ρ 3 − 0.419ρ) cos θ Ac ρ 3 − γ2 ρ cos θ

Aa ρ 2 cos2 θ − 21 Aa ρ 2 cos2 θ − 12 Astigmatism Aa ρ 2 cos2 θ − 21 Aa ρ 2 cos2 θ − 21 Spherical

Balanced aberration Spherical Coma Astigmatism

Table 4 Standard deviation of balanced primary aberrations √ σΦ (γ = 0) σΦ (γ = 1) σΦ ( γ = 2) A √s = As 13.42 6 5 A √c = Ac 8.49 6 2 A √a = Aa 4.90 2 6

As 13.71 Ac 8.80 Aa 5.61

As 18.29 Ac 12.21 Aa 9.08

√ σΦ ( γ 3) 2As γ2 Ac γ 3/2 Aa 2γ

40

[1, § 5

Gaussian apodization and beam propagation

Table 5 Factor by which the standard deviation of a Seidel aberration across a pupil is reduced when it is optimally balanced with other aberrations Balanced aberration

Reduction factor Uniform (γ = 0)

Gaussian (γ = 1)

Gaussian √ ( γ = 2)

Spherical Coma Astigmatism

4 3 1.22

3.74 2.64 1.27

2.95 2.01 1.38

Weakly-truncated √ Gaussian ( γ 3) √ √5 = 2.24 √3 = 1.73 2 = 1.41

Table 6 Diffraction focus Balanced aberration

Diffraction focus

Spherical

(0, 0, 8F 2 As )

4 3 FAc , 0, 0 (0, 0, 4F 2 Aa )

Coma Astigmatism

Uniform (γ = 0)

Gaussian (γ = 1)

Gaussian √ ( γ = 2)

(1.22FAc , 0, 0) (0, 0, 4F 2 Aa )

(0.84FAc , 0, 0) (0, 0, 4F 2 Aa )

(0, 0, 7.46F 2 As )

(0, 0, 5.82F 2 As )

Weakly-truncated √ Gaussian ( γ 3)

2 0, 0, 32 γ F As

4 γ FAc , 0, 0 (0, 0, 4F 2 Aa )

5.4. Zernike–Gauss polynomials When a classical aberration of a certain order is combined with aberrations of lower order to minimize its variance across the pupil, the balanced aberration thus obtained can be identified with a corresponding Zernike–Gauss polynomial. The phase aberration function can be expanded in terms of a complete set of Zernike– Gauss circle polynomials Rnm (ρ; γ ) cos mθ in the form (Mahajan [1986, 1995, 2003, 2004]) Φ(ρ, θ ; γ ) =

n ∞

n=0 m=0

cnm

2(n + 1) (1 + δm0 )

1/2

Rnm (ρ; γ ) cos mθ,

(5.9)

where cnm are the expansion coefficients, n and m are positive integers including zero, n − m 0 and even, δij is a Kronecker delta. The radial polynomial Rnm (ρ; γ ) is a polynomial of degree n in ρ containing terms in ρ n , ρ n−2 , . . . , and ρ m , whose coefficients depend on the Gaussian amplitude through γ ; i.e., it has the form Rnm (ρ; γ ) = anm ρ n + bnm ρ n−2 + · · · + dnm ρ m ,

(5.10)

1, § 5]

Strehl ratio, aberration tolerance, and Zernike–Gauss polynomials

41

where the coefficients anm , etc., depend on γ . The polynomials obey the orthogonality relation 1 m m 1 0 Rn (ρ; γ )Rn′ (ρ; γ )A(ρ)ρ dρ = (5.11) δmn′ . 1 n+1 A(ρ)ρ dρ 0

Consider a typical Zernike–Gauss aberration term in eq. (5.9) Φnm (ρ, θ ; γ ) = cnm

2(n + 1) 1 + δm0

1/2

Rnm (ρ; γ ) cos mθ.

(5.12)

Unless n = m = 0, its mean value is zero. For m = 0, this may be seen with the help of eq. (5.11) and the fact that R00 (ρ; γ ) = 1 is a member of the polynomial set. The orthogonality eq. (5.11) yields the result that the mean value of Rn0 (ρ; γ ) is zero. When m = 0, the average value of cos mθ is zero. Similarly, the mean square value of the aberration is given by 1 2π m m 2 [Φ (ρ, θ ; γ )]2 A(ρ)ρ dρ dθ Φn (ρ, θ ; γ ) = 0 0 1n 2π 0 0 A(ρ)ρ dρ dθ 2 . = cnm

(5.13)

Hence, its variance is given by 2 2 2 σnm = Φnm − Φnm 2 , = cnm

n = 0, m = 0.

(5.14)

Thus, each expansion coefficient, with the exception of c00 , represents the standard deviation of the corresponding aberration term. The variance of the aberration function is accordingly given by n ∞ 2 2 σΦ2 = Φ 2 (ρ, θ ; γ ) − Φ(ρ, θ ; γ ) = . cnm

(5.15)

n=1 m=0

The radial polynomials corresponding to balanced primary aberrations are listed in Table 7. Thus, the balancing defocus for spherical aberration given by Bd = (b40 /a40 )As and the balancing tilt for coma given by Bt = (b31 /a31 )Ac are in agreement with the corresponding values given in Table 3. From the form of the Zernike–Gauss polynomial R22 (ρ) cos 2θ , it is evident that the balancing defocus in the case of astigmatism is independent of γ . Similarly, comparing the form of a primary aberration with the corresponding Zernike polynomial, we √ can immediately write its standard deviation. Thus, the standard deviations As / 5a40 ,

42

Gaussian*

Aberration

Radial polynomial

Piston

R00

1

Distortion (tilt)

R11

a11 ρ

Field curvature (defocus) Astigmatism

R20

a20 ρ 2 + b20

2ρ 2 − 1

2.04989ρ 2 − 0.85690

R22

a22 ρ 2

ρ2

1.14541ρ 2

Coma

R31

a31 ρ 3 + b31 ρ

3ρ 3 − 2ρ

3.11213ρ 3 − 1.89152ρ

Spherical aberration

R40

6ρ 4 − 6ρ 2 + 1

6.12902ρ 4 − 5.71948ρ 2 + 0.83368

a40 ρ 4 + b40 ρ 2 + c40

Uniform (γ = 0)

Gaussian (γ = 1)

Weakly-truncated Gaussian

1

1

ρ

1.09367ρ

1

γ 2ρ γρ√2 −1 3

γ 2 √ ρ 6

γ γ 3 2 2ρ −ρ 2 +2 γ 2 ρ 4 −4γρ √ 2 5

Gaussian apodization and beam propagation

Table 7 Zernike–Gauss radial polynomials representing balanced primary aberrations for Gaussian beams. Polynomials for special cases of γ = 0 (corresponding to a uniform beam), γ = 1, and weakly-truncated Gaussian beams are also given

* a 1 = (2p )−1/2 , a 0 = [3(p − p 2 )]−1/2 , b0 = −p a 0 , a 2 = (3p )−1/2 , a 1 = 1 (p − p 2 /p )−1/2 , b1 = −(p /p )a 1 , a 0 = {5[p − 2K p + 4 2 3 4 8 1 6 2 4 2 2 2 4 4 2 3 1 2 3 2 2 2 6 (K12 + 2K2 )p4 − 2K1 K2 p2 + K22 ]}−1/2 , b40 = −K1 a40 , c40 = K2 a40 , ps = ρ s = (1 − exp γ )−1 + (s/2γ )ps−2 , s is an even integer, p0 = 1, K1 = (p6 − p2 p4 )/(p4 − p22 ), K2 = (p2 p6 − p42 )/(p4 − p22 ).

[1, § 5

1, § 5]

Strehl ratio, aberration tolerance, and Zernike–Gauss polynomials

43

Table 8 Standard deviation factor for primary aberrations for a Gaussian circular beam with various values of γ * √ γ

Balanced spherical

Balanced coma

Balanced astigmatism

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

13.42 13.69 13.71 14.90 18.29 26.33 43.52 75.78 128.09

8.49 8.53 8.80 9.74 12.21 17.62 27.57 42.96 64.01

4.90 5.06 5.61 6.81 9.08 12.82 18.06 24.51 32.00

* The numbers given in this table represent the factor by which the

peak aberration coefficient Ai must be divided by in order to obtain the standard deviation.

√ √ Ac /2 2a31 and Aa /2 6a22 , of balanced spherical aberration, coma and astigmatism, respectively, are in agreement with their values given in Table 4. From Tables 2 and 4 we note that, for a weakly-truncated beam, the √ √ balancing 5, 3 and of a primary aberration reduces its standard deviation by a factor of √ 2 in the case of spherical aberration, coma and astigmatism, respectively. These reduction factors are listed in Table 5 for the uniform (γ = 0), e−2 truncated (γ = 1) and weakly-truncated (large γ ) Gaussian beams. The reduction factors decrease in the case of spherical aberration and coma, but increase in the case of astigmatism. Table 8 gives the reduction factors that relate the peak value Ai of a primary aberration at the edge of a circular aperture and the standard deviation of its corresponding balanced aberration for various of √of γ1 . In the √ case √ values 0 2 balanced aberrations, these numbers are given by 5a4 , 2 2a3 and 2 6a2 for spherical aberration, coma and astigmatism, respectively.

5.5. Strehl ratio for primary aberrations For spherical aberration As ρ 4 balanced with defocus Bd ρ 2 , coma Ac ρ 3 cos θ balanced with tilt Bt ρ cos θ, and astigmatism Aa ρ 2 cos2 θ balanced with defocus Bd ρ 2 , eq. (5.2) reduces to (Mahajan [2005a])

γ S= 1 − exp(−γ )

2

1 0

2 exp(−γ x)f (x) dx ,

(5.16)

44

Gaussian apodization and beam propagation

[1, § 5

where ⎧

2 spherical + defocus, ⎪ ⎨ exp i As x + Bd x 3/2 1/2 f (x) = J0 Ac x + Bt x (5.17) coma + tilt, ⎪ ⎩ exp i(0.5Aa + Bd )x J0 (0.5Aa x) astigmatism + defocus.

The coefficient of a balancing aberration for different values of γ is given in Table 2. For example, the defocus aberration that balances spherical aberration is √ given by Bd /As = −1, −0.933 and −4/γ when γ = 0, 1 and greater than or equal to 3, respectively. Similarly, the tilt aberration that balances coma for these values of γ is given by Bt /Ac = −2/3, −0.608 and −2/γ , respectively. The defocus coefficient given by Bd = −Aa /2 to balance astigmatism is independent of the value of γ . Figure 17 shows how the Strehl ratio varies with spherical aberration As with and without balancing with defocus. We note from figs. 1(a,b) that the balanced aberration yields a much higher Strehl ratio for small values of γ but As 3λ. Thus, when spherical aberration As ρ 4 is present in a system, the central irradiance in a defocused observation plane yielding minimum aberration variance is higher than that at the Gaussian image point or the beam focus. For a weakly-truncated Gaussian beam, however, the improvement in Strehl ratio due to balancing is not √ as significant, as illustrated in Figure 17(c) for γ = 3. Moreover, in this case,

(a) Fig. 17. Strehl ratio of a beam aberrated by spherical aberration As in units of wavelength λ. (a) uniform beam (γ = 0); (b) Gaussian beam with γ = 1; (c) weakly-truncated Gaussian beam with √ γ = 3.

1, § 5]

Strehl ratio, aberration tolerance, and Zernike–Gauss polynomials

45

(b)

(c) Fig. 17. Continued.

the balancing of spherical aberration reduces the Strehl ratio when As 8.3λ. Of course, the value of Bd defining the defocused plane for minimum aberration variance depends on the value of γ . Although for moderate values of As , the Strehl ratio for balanced spherical aberration is greater than its corresponding value without balancing with defocus, a larger value is obtained in a plane that is different from the one that yields minimum variance when the aberration is large. Figure 18 shows how the Strehl ratio of a uniform beam for various values of balanced spherical aberration varies

46

Gaussian apodization and beam propagation

[1, § 5

(a)

(b) Fig. 18. Strehl ratio for balanced spherical aberration as a function of defocus aberration δ, which represents the deviation of defocus aberration from its value for optimum balancing of spherical aberration for minimum variance.

1, § 5]

Strehl ratio, aberration tolerance, and Zernike–Gauss polynomials

47

(c) Fig. 18. Continued.

as a function of the defocus aberration δ. It should be evident that δ represents the deviation of defocus aberration from its value required for minimum variance of spherical aberration. We note that the Strehl ratio is symmetrical about δ = 0, as expected (Mahajan [1994]). However, when As 2.3λ, minimum variance does not yield maximum Strehl ratio. For example, when As = 3λ, the maximum central irradiance is obtained in a plane corresponding to δ = ±1λ, or Bd = −2λ or −4λ. In fact, a minimum of irradiance is obtained in the plane δ = 0, or Bd = −3λ, corresponding to minimum variance. A similar result is obtained for a Gaussian beam with γ = 1, as illustrated in fig. 18(b), except that the Strehl ratio is not symmetric about δ = 0. For a weakly-truncated Gaussian beam with √ γ = 3, maximum Strehl ratio is obtained for δ > 0 even for As = 2λ, as shown in fig. 18(c). The peak shifts to larger values of δ as As increases. For a uniform pupil, balanced coma yields a higher Strehl ratio than the Seidel coma only for Ac 2.3λ, as illustrated in fig. 19. For larger aberrations, Seidel coma yields a higher Strehl ratio than balanced coma, i.e., the irradiance at the Gaussian image point is higher than at the point with respect to which the aberration variance is minimum. Similar results are obtained for a Gaussian pupil, although the cross-over point occurs for larger values of Ac as γ increases. For √ γ = 3, the cross-over occurs at Ac = 7λ. For larger aberrations, the differ-

48

Gaussian apodization and beam propagation

[1, § 5

(a)

(b) Fig. 19. Strehl ratio of a beam aberrated by coma Ac in units of wavelength λ: (a) uniform beam √ (γ = 0); (b) Gaussian beam with γ = 1; (c) weakly-truncated Gaussian beam with γ = 3. Bt represents the coefficient of the balancing tilt aberration.

ence between the Strehl ratios for balanced coma and Seidel coma increases as γ increases. Figure 20 shows the Strehl ratio for astigmatism. The increase in Strehl ratio by balancing astigmatism with defocus is not as significant as in the case of balanced spherical aberration or balanced coma. As in the case of coma, balanced astigmatism yields a higher Strehl ratio than Seidel astigmatism for Aa < 1λ for small

1, § 5]

Strehl ratio, aberration tolerance, and Zernike–Gauss polynomials

49

[t]

(c) Fig. 19. Continued.

(a) Fig. 20. Strehl ratio of a beam aberrated by astigmatism Aa in units of wavelength λ: (a) uniform √ beam (γ = 0); (b) Gaussian beam with γ = 1; (c) weakly-truncated Gaussian beam with γ = 3. Bd represents the coefficient of the balancing defocus aberration.

√ values of γ . For a weakly-truncated Gaussian beam with γ = 3, the cross-over point is Aa ≃ 4λ. Since for small aberrations the Strehl ratio depends on the aberration variance regardless of the type of the aberration, it is useful to determine the range of va-

50

Gaussian apodization and beam propagation

[1, § 5

(b)

(c) Fig. 20. Continued.

lidity of the approximate eq. (5.3). It has been shown that for a uniform pupil the approximate expression exp(−σΦ2 ) for the Strehl ratio yields a value that is within 10% of its true value as long as it is greater than or equal to 0.3 (Mahajan [1983c]). For small values of γ , the approximate expression estimates the true value of Strehl ratio quite well for σw λ/4, as may be seen from figs. 21(a–d). However, figs. 22 (a) and (b) show that unless the Strehl ratio is greater than or equal to 0.9 or σw λ/20, exp(−σΦ2 ) significantly underestimates the true value of the Strehl ratio (Mahajan [2005a]). Moreover, the Strehl ratios for a given value

1, § 5]

Strehl ratio, aberration tolerance, and Zernike–Gauss polynomials

51

(a)

(b) Fig. 21. Strehl ratio S of a primary aberration with and without balancing as a function of its standard deviation σw in units of wavelength λ: (a) uniform beam (γ = 0) with a primary aberration; (b) uniform beam with a balanced primary aberration; (c) Gaussian beam with γ = 1 and a primary aberration; (d) Gaussian beam with γ = 1 and a balanced primary aberration.

52

Gaussian apodization and beam propagation

(c)

(d) Fig. 21. Continued.

[1, § 5

1, § 5]

Strehl ratio, aberration tolerance, and Zernike–Gauss polynomials

53

(a)

(b) Fig. 22. Strehl ratio of a Gaussian beam aberrated by a primary aberration compared with its value for a corresponding balanced primary aberration as a function of standard deviation σw in units of √ √ wavelength λ: (a) γ = 2; (b) weakly-truncated beam with γ = 3. S: spherical, BS: balanced spherical; C: coma, BC: balanced coma; A: astigmatism, BA: balanced astigmatism.

54

Gaussian apodization and beam propagation

[1, § 6

of σw of the various aberrations are quite different from each other. In particular, the Strehl ratios for balanced spherical aberration or balanced astigmatism are significantly smaller than the corresponding values without balancing. For a uniform pupil, however, the Strehl ratio for a given value of σw is the same for both spherical and balanced spherical aberrations. Of course, the aberration coefficient As for a given value of σw is quite different for the aberration with and without balancing.

§ 6. Balancing of defocus aberration with spherical aberration or astigmatism 6.1. Focused beam We have seen that the depth of focus is quite large when the Fresnel number is small. Hence, the axial irradiance at a certain distance z < R in the vicinity of focus can be increased if an appropriate amount of spherical aberration or astigmatism is introduced to balance the defocus aberration. This is similar to the fact that the Strehl ratio degraded by a small amount of spherical aberration or astigmatism improves when an appropriate amount of defocus aberration is added by observing in a slightly defocused image plane. Thus we consider the axial irradiance of a beam aberrated by spherical aberration (Yoshida and Asakura [1996], Mahajan [2005b]) or astigmatism (Mahajan [2005b]). Both focused and collimated beams are considered. Table 9 lists the standard deviation of defocus aberration with and without balancing with spherical aberration or astigmatism, showing that it decreases as γ increases.√Spherical aberration reduces the standard deviation by a factor of 4, √ 3.74 and 5 when γ = 0, 1 and 3, respectively. The corresponding reduction √ √ factors for astigmatism are 3/2, 1.27 and 2. The reduction factor is the same whether spherical aberration is balanced with defocus aberration or defocus aberration is balanced with spherical aberration, but the balanced aberrations have different forms in the two cases. For example, for a uniform beam, spherical aberration As ρ 4 balanced with defocus aberration is As (ρ 4 − ρ 2 ), but defocus aberration Bd ρ 2 balanced with spherical aberration is Bd [ρ 2 − (15/16)ρ 4 ]. Similarly, astigmatism Aa ρ 2 cos2 θ balanced with defocus aberration is Aa (ρ 2 cos2 θ − ρ 2 ), but defocus aberration balanced with astigmatism is Bd [ρ 2 −(2/3)ρ 2 cos2 θ]. The difference comes from the fact that optimal balancing is done with respect to defocus aberration in one case and spherical aberration or astigmatism in the other.

1, § 6]

Balancing of defocus aberration with spherical aberration or astigmatism

55

Table 9 Standard deviation of defocus aberration, and defocus aberration balanced with spherical aberration or astigmatism for minimum variance. The amount of the balancing aberration is listed for each case in parentheses Aberration Bd ρ 2 + Φ(ρ, θ)

Standard deviation

Defocus, Bd ρ 2

Bd B √ = d 3.46 2 3 Bd Bd √ = 13.86

8 3 As = − 15 16 Bd

γ =0

Defocus aberration balanced with spherical aberration, Bd ρ 2 + As ρ 4 Defocus aberration balanced with astigmatism, Bd ρ 2 + Aa ρ 2 cos2 θ

Bd B √ = d 4.24

3 2 Aa = − 23 Bd

γ =1 Bd 3.55

Bd 13.27

(As = −Bd ) Bd 4.53

(Aa = −0.77Bd )

√ γ 3 Bd γ B √d

5γ As = − γ5 Bd B √d 2γ

(Aa = −Bd )

If the defocus aberration is balanced with an aberration Φ(ρ, θ), then eq. (4.6) is modified and yields the axial irradiance according to 2 R 2γ I (0; z; γ ) = 1 − exp(−2γ ) πz 2 1 2π

2 2 exp −γρ exp i Φ(ρ, θ) + Bd ρ ρ dρ dθ . × 0

0

(6.1)

For a beam aberrated by spherical aberration As ρ 4 , eq. (6.1) reduces to 2 R 2γ I (0; z; γ ) = 1 − exp(−2γ ) z 1

2 2 exp(−γ x) exp i As x + Bd x dx . ×

(6.2)

0

Similarly, for a beam aberrated by astigmatism Aa ρ 2 cos2 θ , it reduces to 2 R 2γ I (z; γ ) = 1 − exp(−2γ ) z 2 1 exp(−γ x) exp i(0.5Aa + Bd )x J0 (0.5Aa x) dx , × 0

(6.3) where we have used the fact that 2π

2π 2 2 2 exp 0.5iAa ρ 2 cos 2θ dθ exp iAa ρ cos θ dθ = exp 0.5iAa ρ 0 0

= 2π exp 0.5iAa ρ 2 J0 0.5Aa ρ 2 . (6.4)

56

Gaussian apodization and beam propagation

[1, § 6

When N = 1 and γ = 0, the axial irradiance is maximum at z = 0.6R corresponding to a defocus aberration of λ/3. Figure 23 shows how spherical aberration modifies the axial irradiance of a beam. The variation of |Bd | with z is also shown in this figure. The value of Bd is positive for z/R < 1 and negative for z/R > 1. The solid curve in fig. 23(a) shows the axial irradiance of a uniform beam aberrated by spherical aberration As = −5π/8 or −(5/16)λ that minimizes the variance of defocus aberration of λ/3 corresponding to the z value where the principal maximum lies in the absence of spherical aberration. Compared to when As = 0, the irradiance is smaller in the vicinity of the focal point and larger in the vicinity of the principal maximum. The peak moves closer to the pupil, the secondary maxima are higher and the minima are no longer zero. If the amount of spherical aberration varies as Bd varies with z/R so that As = −(15/16)Bd , then the aberration variance is minimum and the axial irradiance is higher for any value of z (although the difference in the vicinity of the focal point is negligible), as illustrated by the dashed curve. The peak value now lies even closer to the pupil, which is more than ten times the focal-point irradiance. Thus, the peak value with

(a) Fig. 23. Axial irradiance of a focused beam aberrated by spherical aberration As . The defocus aberration |Bd | in units of wavelength is also shown: (a) uniform beam (γ = 0); (b) Gaussian beam with √ γ = 1; (c) weakly-truncated Gaussian beam with γ = 3.

1, § 6]

Balancing of defocus aberration with spherical aberration or astigmatism

(b)

(c) Fig. 23. Continued.

57

58

Gaussian apodization and beam propagation

[1, § 6

spherical aberration is more than five times the peak value without it. Figure 23(b) shows similar results for a Gaussian beam with γ = 1. Minimum variance of the defocus aberration when balanced with spherical aberration is obtained in this case when As = −Bd (a more precise relationship is As = −0.995Bd ). The maxima for a Gaussian beam are lower than those for a uniform beam. √ As illustrated in fig. 23(c) for a weakly-truncated Gaussian beam with γ = 3, the peak moves toward the focus and its value more than doubles when spherical aberration As = −729/5π or −7.39λ is introduced, minimizing the variance of the defocus aberration of 4.10λ corresponding to the peak location. Aberration balancing reduces the standard deviation of the defocus aberration by a factor of √ √ 5 from a value of Bd /γ to Bd / 5γ . If the amount of spherical aberration varies as Bd varies with z/R so that As = −(9/5)Bd , then the peak location does not change but its value increases significantly. A peak with a four-field increase that is slightly closer to the pupil is obtained when As = −9.23λ. Figure 24 shows how the axial irradiance is modified when defocus aberration is balanced with astigmatism. The increase is not as dramatic as in the case of

(a) Fig. 24. Axial irradiance of a focused beam aberrated by astigmatism Aa . The defocus aberration |Bd | in units of wavelength is also shown: (a) uniform beam (γ = 0); (b) Gaussian beam with γ = 1; √ (c) weakly-truncated Gaussian beam with γ = 3.

1, § 6]

Balancing of defocus aberration with spherical aberration or astigmatism

(b)

(c) Fig. 24. Continued.

59

60

Gaussian apodization and beam propagation

[1, § 6

spherical aberration when γ = 0 or 1. This is consistent with the fact that aberration balancing also in this case does not reduce the standard deviation of the defocus aberration significantly. However, the axial irradiance of a weakly-truncated √ beam with γ = 3 increases significantly and monotonically as z decreases. Figure 25 shows the Strehl ratio for a given value of defocus aberration Bd as a function of spherical aberration As . Both Bd and As are in units of wavelength λ. It is evident that, as the amount of defocus aberration increases, the maximum Strehl ratio is obtained for an As value that is different from the one yielding minimum variance. For example, when Bd = 3λ, the maximum Strehl ratio is obtained when As = −2.2λ instead of −2.8λ when γ = 0, and As = −2λ √ instead of 3λ when γ = 1. When γ = 3, the Strehl ratio peaks to the right of the expected value of As = −(9/5)Bd .

(a) Fig. 25. Strehl ratio of a beam for a given value of defocus aberration Bd as a function of spherical aberration As . Both Bd and As are in units of wavelength λ: (a) Uniform beam (γ = 0); the Strehl ratio in this case is zero when Bd is an integral number of wavelengths, as may be seen from eq. (5.7); √ (b) Gaussian beam with γ = 1; (c) weakly-truncated Gaussian beam with γ = 3.

1, § 6]

Balancing of defocus aberration with spherical aberration or astigmatism

(b)

(c) Fig. 25. Continued.

61

62

Gaussian apodization and beam propagation

[1, § 6

6.2. Collimated beam A collimated beam is equivalent to a beam focused at infinity, or one with a Fresnel number of zero. Thus, letting R → ∞ in eqs. (4.2) and (6.1), we obtain I (z; γ ) =

2γ (Bd /π)2 1 − exp(−2γ ) 1 2π 2

2 2 × exp −γρ exp i Φ(ρ, θ) + Bd ρ ρ dρ dθ , 0

0

(6.5)

where Bd =

π . 4z

(6.6)

Now the irradiance is in units of the pupil irradiance P /S for a uniform pupil and z is in units of the far-field distance D 2 /λ, where D = 2a is the pupil diameter. The defocus aberration can be balanced with spherical aberration or astigmatism in exactly the same manner as for a focused beam. Thus, the results of Table 9 apply equally well for a collimated beam. The only significant difference is in the definition of the defocus coefficient Bd . Hence, the axial irradiance of a beam optimally balanced with spherical aberration or astigmatism (whose amount varies with z) is given by 1

2 2γ Bd2 2 I (z; γ ) = (6.7) exp(−γ x) exp i As x + Bd x dx 1 − exp(−2γ ) 0 and

I (z; γ ) =

2γ Bd2 1 − exp(−2γ ) 1 2 × exp(−γ x) exp i(0.5Aa + Bd ) J0 (0.5Aa x) dx .

(6.8)

0

Figure 26 illustrates the effect of spherical aberration on the axial irradiance of a collimated beam. The aberration-free axial irradiance is included in this figure for comparison. The solid curves show that the axial irradiance increases dramatically near the pupil, especially for small values of γ , as a varying amount of spherical aberration minimizing the variance of the defocus aberration is introduced. Similar results are obtained when astigmatism is introduced, as illustrated in fig. 27, though the increase at axial points near the pupil is not as dramatic.

1, § 6]

Balancing of defocus aberration with spherical aberration or astigmatism

63

(a)

(b) Fig. 26. Axial irradiance of a collimated beam, i.e., one with a Fresnel number N = 0, aberrated by spherical aberration As . The axial irradiance when As = 0 is shown for comparison. The defocus aberration Bd in units of wavelength is also shown: (a) uniform beam (γ = 0); (b) Gaussian beam √ with γ = 1; (c) weakly-truncated Gaussian beam with γ = 3. In figures (a) and (b), the right-hand scale is for As = 0 and Bd .

64

Gaussian apodization and beam propagation

[1, § 6

(c) Fig. 26. Continued.

(a) Fig. 27. Axial irradiance of a collimated beam, i.e., one with a Fresnel number N = 0, aberrated by astigmatism Aa . The axial irradiance when Aa = 0 is shown for comparison. The defocus aberration Bd in units of wavelength is also shown: (a) uniform beam (γ = 0); (b) Gaussian beam with γ = 1; √ (c) weakly-truncated Gaussian beam with γ = 3.

1, § 6]

Balancing of defocus aberration with spherical aberration or astigmatism

65

(b)

(c) Fig. 27. Continued.

Of course, the irradiance at a certain distance in the absence of aberration is higher if the beam is focused at that distance. For example, the irradiance of a uniform

66

Gaussian apodization and beam propagation

[1, § 7

beam focused at a distance z is (π/4z)2 . Its value at z = 0.1, corresponding to Bd = 1.25λ, is 62 compared to a value of 2 or 44 for a collimated beam with As = 0 or As = −(15/16)Bd , respectively. § 7. Aberrated Gaussian pupil 7.1. Spherical aberration We have seen from fig. 6 that the secondary maxima of an aberration-free PSF decrease as γ increases. We now consider how they are affected by aberrations. We illustrate this by considering PSFs aberrated by spherical aberration. For systems with large Fresnel numbers, the aberrated PSF is given by 1 2

I (r; γ ) = 4 (7.1) I (ρ) exp i As ρ 4 + Bd ρ 2 J0 (πrρ)ρ dρ . 0

Figure 28 shows the aberrated PSFs for As = 1, 2, 3 and 4 waves with and without balancing by defocus. The PSFs are normalized to unity at the center. As in the case of uniform pupils (Mahajan [1983a], Szapiel [1986]), spherical aberration broadens the PSF, makes the diffraction minima nonzero, and increases the value of the secondary maxima (Sklar [1975], Lowenthal [1975]). For balanced aberrations, however, the radius of the central bright spot and positions of the minima and maxima do not change, but power flows from the central spot to the diffraction rings. Thus, whereas the Gaussian amplitude at the pupil increases the size of the central bright spot but decreases the secondary maxima, the balanced aberration does not change the size of the central bright spot but increases the secondary maxima. Hence, the advantage of the Gaussian apodization in reducing the secondary maxima is lost if spherical aberration is present. The amount of spherical aberration must be less than or equal to λ/4 to take advantage of the positive attribute of apodization. For a weakly-truncated pupil, the effect of spherical aberration on the spot size is similar, but there are no diffraction rings.

7.2. Symmetry properties For large Fresnel numbers, eq. (4.6) shows that the aberration-free PSF is symmetric about the Gaussian image plane. Accordingly, the axial irradiance is also symmetric. However, a PSF aberrated by spherical aberration is not symmetric (Mahajan [1994]). In the case of a uniform pupil though the axial irradiance is symmetric about the point with respect to which the aberration variance is mini-

1, § 7]

Aberrated Gaussian pupil

67

(a)

(b) Fig. 28. PSFs aberrated by spherical and balanced spherical aberrations: (a,b) uniform pupil (γ = 0); √ (c,d) Gaussian pupil with γ = 1; (e,f) weakly-truncated Gaussian pupil with γ = 3. Aberration-free PSFs are included for comparison.

68

Gaussian apodization and beam propagation

(c)

(d) Fig. 28. Continued.

[1, § 7

1, § 7]

Aberrated Gaussian pupil

(e)

(f) Fig. 28. Continued.

69

70

Gaussian apodization and beam propagation

[1, § 7

mum, as illustrated in fig. 18(a). The axial distance in this figure is represented by the deviation of defocus aberration δ from its value for minimizing the aberration variance. For a Gaussian beam, however, even the axial irradiance is asymmetric, √ as illustrated in figs. 18(b) and (c) for γ = 1 and 3, respectively. The axial irradiance of a PSF aberrated by astigmatism Aa ρ 2 cos2 θ is symmetric about the point Bd = −Aa /2 for both uniform and Gaussian pupils. 7.3. Line of sight By definition, the coordinates of the centroid of a PSF Ii (xi , yi ) are given by −1 (xi , yi )Ii (xi , yi ) dxi dyi . xi , yi = Pex (7.2)

It can be shown that the centroid is affected only by those aberrations that vary as cos θ (Mahajan [1985, 2004]). For a Gaussian pupil, the centroid of a PSF aberrated by coma Ac ρ 3 cos θ is given by

1 1 . + xi = 4Ac F (7.3) 2γ 1 − exp(2γ ) The PSF is symmetric about the xi axis and, therefore, yi = 0. From the form of the radial polynomial R31 (ρ; γ ; ε), the point xm in the image plane with respect to which the aberration variance is minimum is given by Mahajan [1985, 2004]

2 γ xm = 2Ac F (7.4) + . γ 1 + γ − exp(γ )

Letting γ = 0 in eqs. (7.3) and (7.4), we obtain the corresponding results for a uniform pupil: xi = 2Ac F

(7.5)

and 4 (7.6) Ac F. 3 For small values of Ac , the peak value Ip of the aberrated PSF occurs at a point xp that is (approximately) equal to xm . The values of xm , xp and xi in units of λF and the corresponding irradiances Im , Ip and Ic for typical values of Ac are given in Table 10 for a Gaussian pupil with γ = 1. The values of I (0) are also included in this table. The corresponding results for a uniform pupil are included in parentheses to illustrate the effect of Gaussian apodization. The irradiances given are normalized by the aberration-free central irradiance for the pupil under consideration. xm =

1, § 8]

71

Weakly-truncated Gaussian pupils and beams

Table 10 Typical values of the minimum variance point xm , the peak value point xp and the centroid xi in units of λF , and the corresponding irradiances Im , Ip and Ic in units of the aberration-free central irradiance for a Gaussian pupil with γ = 1 aberrated by primary coma. The numbers in parentheses are for a corresponding uniform pupil with the irradiance normalized to unity at the center for an aberration-free pupil Ac

xm

xp

xi

Im

Ip

Ic

I (0)

0

0 (0) 0.61 (0.67) 1.22 (1.33) 1.82 (2.00) 2.43 (2.67)

0 (0) 0.60 (0.67) 1.15 (1.30) 1.40 (1.80) 1.46 (1.57)

0 (0) 0.69 (1) 1.37 (2) 2.06 (3.00) 2.75 (4.00)

1 (1) 0.8805 (0.8712) 0.6013 (0.5708) 0.3205 (0.2715) 0.1305 (0.0864)

1 (1) 0.8806 (0.8712) 0.6062 (0.5717) 0.3672 (0.2844) 0.2947 (0.1978)

1 (1) 0.8670 (0.6535) 0.5590 (0.1445) 0.2479 (0.0004) 0.0624 (0.0061)

1 (1) 0.4567 (0.3175) 0.1708 (0.0791) 0.1199 (0.0618) 0.0733 (0.0341)

0.50 1.00 1.50 2.00

§ 8. Weakly-truncated Gaussian pupils and beams When the Gaussian illumination is very narrow compared to the radius of the pupil, i.e., when ω ≪ a or γ is very large, it is said to be weakly truncated. Although in earlier sections we have already given numerical results for large values of γ using the general equations, we now show that simple closed-form equations can be obtained for such cases. We write these equations in two equivalent forms. Equations (a) are written in a normalized form so that they can be investigated parametrically. Equations (b) are written without any normalization, and these are more suitable for evaluating results when the specific parameters involved are known. Moreover, by letting z → 0, these equations reduce to the corresponding equations in the pupil plane.

8.1. Pupil function For a weakly-truncated beam, the pupil may be assumed to be infinitely wide with negligible error. Thus, we let ρ → ∞ in the radial integrations. Moreover, the fractional power exp(−2γ ) blocked by the pupil may be neglected compared to unity and the aberration-free pupil function may be written

A(ρ) = 2γ exp −γρ 2 . (8.1) Similarly, eq. (3.5) for the pupil irradiance distribution may be written

I (ρ) = 2γ exp −2γρ 2 ,

(8.2a)

72

Gaussian apodization and beam propagation

[1, § 8

or I (rp ) =

2 rp 2Pex exp −2 . 2 ω πω

(8.2b)

In eq. (8.2a), I (ρ) is in units of Pex /Sex , as in eq. (3.8). In eq. (8.2b), no such normalization is used.

8.2. Point-spread function Replacing the upper limit of integration in eq. (4.6) by infinity, we obtain 2 2 ∞ R 2 . exp −(γ − iB )ρ J (πρr)ρ dρ I (r; z; γ ) = 8γ d 0 z 0

Hence, if we let β = πr and α = γ − iBd and follow ∞

1 β2 −αρ 2 J0 (βρ)ρ dρ = exp − , Re α > 0, 2α 4α 0

eq. (8.3) reduces to

2 γ π2 r 2 2γ R exp − I (r; z) = z Bd2 + γ 2 2(Bd2 + γ 2 ) 2Pex −2r 2 = exp , πωz2 ωz2

(8.3)

(8.4)

(8.5a) (8.5b)

where ωz2

=

λz πω

2

+ω

2

z 1− R

2

(8.5c)

is the beam radius at a distance z. In eq. (8.5a), the irradiance is in units of Pex Sex /(λ2 R 2 ) and r is in units of λz/D, as was the case in eq. (4.6). In eq. (8.5b) these quantities are not normalized to illustrate that the diffracted irradiance is independent of the pupil radius a. Comparing eqs. (8.2a) and (8.5a), we note that, when the truncation of the beam by the pupil is negligible, the diffraction rings disappear and a Gaussian beam remains Gaussian as it propagates (Siegman [1971], Gaskill [1978], Mahajan [2004]). The encircled-power distribution in an observation plane is given by

−γ π2 rc2 P (rc ; z) = 1 − exp (8.6a) 2(Bd2 + γ 2 ) 2rc2 = Pex 1 − exp − 2 (8.6b) , ωz

1, § 8]

Weakly-truncated Gaussian pupils and beams

73

where rc is in units of λz/D. Equation (8.6a) is obtained by substituting eq. (8.5a) into eq. (3.14), and eq. (8.6b) is obtained by substituting eq. (8.5b) into rc I (r; z)r dr. P (rc ; z) = 2π (8.7) 0

If we let r = 0 in eqs. (8.5a) and (8.5b), we obtain the axial irradiance 2 2γ R I (0; r) = z Bd2 + γ 2 2Pex . = πωz2

(8.8a) (8.8b)

Of course, for large values of γ eq. (4.7) also reduces to eq. (8.8a), as expected. If we let z = R in eqs. (8.5a, 8.5b), we obtain the focal-plane irradiance distribution 2 π2 r 2 I (r; R) = exp − (8.9a) γ 2γ 2r 2 2Pex exp − , = (8.9b) 2 2 πωR ωR where λR ω (8.9c) π is the beam radius in the focal plane. The focal-point irradiance is given by ωR =

I (0; R) =

2 γ

(8.10a)

2πPex ω2 (8.10b) , λ2 R 2 a result that may also be obtained from eq. (3.13) for large values of γ . If we equate to zero the derivative of eq. (8.5a) with respect to z, we obtain the z value zp at which the axial irradiance is maximum: zp γ 2 −1 = 1+ (8.11a) R πN ωR 2 −1 . = 1+ (8.11b) ω =

It is evident that zp < R, i.e., the peak value lies at a point that is closer to the pupil compared to the focal point. Substituting eqs. (8.11) into eqs. (8.8), we

74

Gaussian apodization and beam propagation

[1, § 8

obtain the peak value of the axial irradiance 2 2γ + 2 2 γ π N 2Pex = , 2 πωzp

I (0; zp ) =

(8.12a) (8.12b)

where 2 ωzp

2

=ω 1+

ω ωR

2 −1

.

Equation (8.12c) may also be written ωR 2 −1 2 2 ωzp = ωR , 1+ ω

(8.12c)

(8.12d)

showing that ωzp < ωR . Thus, the radius of the beam waist is smaller than the beam radius in the focal plane. Comparing eqs. (8.10a) and (8.12a), we note that the peak axial irradiance is higher than the corresponding focal-point irradiance by 2γ /(π2 N 2 ) (in units of √ Pex Sex /(λ2 R 2 )) or by 2Pex /(πω2 ). As illustrated in fig. 9(a) for γ = 3 and N = 1, the value at z = 0 is given by 2γ /π2 or 1.824. The peak is located at z = 0.1086R with a value of 2.046, which is 9.2 times the corresponding focalpoint irradiance of 0.222 and 1.12 times the central pupil irradiance of 1.824. Equation (8.8a) shows that the defocus aberration of Bd = 81/π or 4.10λ for this point reduces the irradiance by a factor of 0.1086, but the inverse-square law increases it by a factor of 84.79. Equations (8.11a) and (8.12c) can also be written in the form −1 zp = 1 + (πNg )−2 (8.13a) R and −1 2 ωzp (8.13b) = ω2 1 + (πNg )2 ,

respectively, where Ng = ω2 /(λR) is the Gaussian Fresnel number. It represents the number of Fresnel zones in the pupil plane within the Gaussian-beam radius, as observed from the focus, just as N represents the number of zones within the full pupil. Since the axial irradiance is maximum at zp , the beam radius is minimum at this position. This may also be seen by equating to zero the derivative of eq. (8.5c) with respect to z. The minimum beam radius ωzp is referred to as the beam waist. Even though the peak of axial irradiance does not lie at the focal point, maximum central irradiance on a target at a fixed distance z is obtained when the beam

1, § 8]

Weakly-truncated Gaussian pupils and beams

75

is focused on it, as discussed in Section 4.3.1. Similarly, although the beam waist does not lie in the focal plane, the radius of the beam in the plane of the target is minimum when the beam is focused on it. This may be seen by considering R as a variable and letting ∂ωz /∂R = 0 from eq. (8.5c). We find that the beam radius ωz on the target is minimum with a value of λz/(πω) when R = z. Correspondingly, the axial irradiance I (0; z) on it is maximum, as may be seen from eq. (8.5b). If we choose R such that the beam waist lies on the target, its value is given by eq. (8.11b) with zp = z ωR 2 R = 1+ > 1, (8.14) z ω

where ωR = λR/(πω) is the beam radius in the focal plane. As expected, the beam is focused beyond the target. The beam radius on the target is the radius of the beam waist, which according to eq. (8.12d) is given by ωR 2 −1 2 2 ω z = ωR 1 + ω 2 λz ωR 2 = (8.15) 1+ , πω ω showing, as expected, that it is larger than the beam radius λz/(πω) when the beam is focused on the target. Accordingly, the axial irradiance on the target is smaller when the waist of the beam lies on it compared to when the beam is focused on it. Again, as explained in Section 4.3.1, the reason for this is that, while the effect of the inverse-square law dependence on the target distance is fixed, the defocus aberration reduces the axial irradiance when the beam is focused anywhere except on the target. Figure 29 shows the aberration-free focal-plane irradiance and encircled-power √ distributions for γ = 2. The solid curves have been obtained by using eqs. (3.12) and (3.14), and the dashed curves represent their corresponding approximations given by eqs. (8.5a) and (8.6a), respectively, with Bd = 0. We note that the approximate results agree well with the true results. The maximum difference, which occurs at the focus, is less than 4%. For larger γ , the agreement is found to be even better. However, it will be shown in Section 8.7 that when the beam is √ aberrated, a larger value of γ , namely γ = 3, is required for the validity of the weakly-truncated approximation. Figure 30 shows how the axial irradiance of a focused Gaussian beam varies √ when γ = 2 and N = 1, 10 and 100. Once again, the solid curves in this figure have been obtained by using eq. (4.6); the dashed curves represent their corresponding approximations given by eq. (8.8a). It is evident that eq. (8.8a)

76

Gaussian apodization and beam propagation

[1, § 8

√ Fig. 29. Focal-plane irradiance and encircled-power distributions for a Gaussian beam with γ = 2. 2 2 As in fig. 6, the irradiance and encircled power are in units of Pex Sex /(λ R ) and Pex , respectively. The radial distance r or rc in the focal plane is in units of λF . The focal point is at r = 0. Solid curves: exact results; dashed curves: corresponding approximations neglecting the beam truncation.

√ Fig. 30. Axial irradiance of a Gaussian beam with γ = 2 focused at a distance R with a Fresnel number N = 1, 10 and 100. The irradiance is in units of Pex Sex /(λ2 R 2 ). Solid curves: exact results; dashed curves: corresponding approximations neglecting the beam truncation.

approximates the true axial irradiance quite well. The only significant difference occurs when N = 1, in that the true results show secondary maxima and minima, but the approximate result shows only the principal maximum. For larger values of √ γ , e.g., γ = 2.5, the secondary maxima and minima disappear and the true and

1, § 8]

Weakly-truncated Gaussian pupils and beams

77

approximate results overlap each other at the scale of fig. 30. Hence, we conclude that the truncation of an aberration-free Gaussian beam by a pupil has a negligible √ effect on the irradiance distribution as the beam propagates when γ 2.

8.3. Radius of curvature of the propagating wavefront The complex amplitude of an apertured converging spherical wave of radius of curvature R in the quadratic (or Fresnel) approximation varies as exp(−iπr 2 /(λR)), where r is the radial distance of a point in the aperture plane from its axis; i.e., its radius of curvature is given by the inverse of the coefficient of −iπr 2 /λ in the exponent of its complex amplitude representation. The complex amplitude of the diffracted Gaussian spherical wave consists of two factors that have iri2 dependence in the exponent. One of these, exp(iπri2 )/(λz), comes from free-space propagation through a distance z. The other comes about when the diffraction integral in eq. (8.3) is evaluated. The diffracted amplitude is given by R 1 ri2 U (ri ; z) = −2i 2γ exp ik z + z 2 z ∞ πρri D 2 exp −(γ − iBd )ρ J0 ρ dρ × λz 0 R i = − 2γ z γ − iBd 1 r2 −[πri D/(λz)]2 exp . × exp ik z + i 2 z 4(γ − iBd ) (8.16) Hence, the radius of curvature Rz of a diffracted Gaussian spherical wave at a distance z from the plane of the exit pupil is given by

Sex Bd −1 z/Rz = λz(Bd2 + γ 2 ) =

1 − z/R − 1. (1 − z/R)2 + [λz/(πω2 )]2

(8.17)

At the waist position zp , Rzp = ∞, implying a plane wave. Moreover, at the focal plane, Rz = −R. A negative value of Rz indicates a diverging spherical wave. For z > R, z/Rz < −1 and the beam continues to expand as it propagates. Substituting for γ and Bd , and noting that the units of diffracted irradiance are

78

Gaussian apodization and beam propagation

[1, § 8

Pex Sex /(λ2 R 2 ), the complex amplitude given by eq. (8.16) may also be written ri2 ω0 1 ri2 exp i kz + φ(z) exp iπ exp − , U (ri ; z) = −iA0 ωz λRz 2 ωz2 (8.18) where πω02 (8.19) λz is a phase factor. It is seen that the diffracted amplitude is independent of the pupil radius a, as expected for a weakly-truncated pupil, equivalent to free-space propagation. tan φ(z) =

8.4. Collimated beam The results for a weakly-truncated collimated Gaussian beam (equivalent to a plane wave incident on a Gaussian pupil) can be obtained from those for a focused beam by letting R → ∞. Thus, for example, eqs. (8.5a) and (8.5b) reduce to −8γ z2 r 2 2γ exp I (r; z) = (8.20a) 1 + (4γ z/π)2 1 + (4γ z/π)2 −2r 2 2Pex exp , = (8.20b) πωz2 ωz2 where λz 2 ωz2 = ω2 1 + . πω2

Similarly, eq. (8.17) reduces to 2 2 πω . Rz = −z 1 + λz

(8.20c)

(8.20d)

In eq. (8.20a) the irradiance is in units of Pex /Sex , r is in units of λz/D (z is not normalized here), and z is in units of D 2 /λ. As illustrated in fig. 11 for √ γ = 3, the axial irradiance has a peak value of 2γ or 18 at the pupil and decreases monotonically as z increases. If we consider a beam with a waist of radius ω0 located at z = 0 with a central irradiance I0 = 2P /(πω02 ), then its central irradiance Iz and beam radius ωz at a distance z may be written Iz =

I0 1 + (z/zR )2

(8.21)

1, § 8]

Weakly-truncated Gaussian pupils and beams

79

and 2 z , ωz2 = ω02 1 + zR

(8.22)

where zR =

πω02 λ

(8.23)

is the distance, called the Rayleigh range, from the plane of the beam waist to a plane in which the central irradiance decreases by a factor of 2 and the beam √ radius increases by a factor of 2. The beam radius ωz increases monotonically as z increases, and at large distances is approximately given by ω z = ω0

λz z = . zR πω0

(8.24)

As illustrated in fig. 31, the beam expands as a hyperbola with asymptotes, indicated by the dashed lines, inclined to its axis at an angle θ=

ωz λ , = z πω0

(8.25)

called the far-field divergence or diffraction angle of the beam. The backward propagation of the beam is also shown in the figure. For large values of z, the beam radius increases linearly with it. Equation (8.18) for the radius of curvature of the beam wavefront at a distance z may also be written

Rz = −z 1 +

zR z

2 .

(8.26)

It has a value of infinity at z = 0, and a value of −2zR at the Rayleigh range zR . At large distances z ≫ zR , Rz → z, as for a spherical wave.

Fig. 31. Beam radius and divergence angle. √ The radius of the beam is ω0 and its wavefront is planar at z = 0. Its radius increases by a factor of 2 and its wavefront becomes spherical with a radius of curvature ∓zR at z = ±zR .

80

Gaussian apodization and beam propagation

[1, § 8

8.5. Beam focusing and waist imaging by a lens Consider a lens of focal length f illuminated by a Gaussian beam which has a waist of radius ω0 at a (numerically negative) distance z0 from it, as illustrated in fig. 32. The beam radius ωl and the radius of curvature Rl− of the wavefront in the plane of the lens may be written from eqs. (8.22) and (8.26) by letting ω = ω0 and z = −z0 . Thus, 2 z0 2 2 ω l = ω0 1 + (8.27) zR and 2 zR Rl− = −z0 1 + , z0

(8.28)

where πω02 (8.29) λ is the Rayleigh range of the incident or object beam. We assume that the lens is large enough that its radius is 2ωl so that the weak-truncation approximation is satisfied. The radius of curvature Rl+ of the wavefront exiting from the lens is given by zR =

1 1 1 + = − + f. Rl Rl

(8.30)

The beam transmitted by the lens first converges and then expands as it propagates to the right of the lens. Its radius ωz′ at a distance z′ from the lens is given by

Fig. 32. Imaging of a Gaussian beam by a lens of focal length f . The incident (object) beam has a waist of radius ω0 at a (numerically negative) distance z0 from the lens, and the transmitted (image) beam has a waist of radius ω0′ at a distance z0′ .

1, § 8]

Weakly-truncated Gaussian pupils and beams

eq. (8.5c) with appropriate substitutions: ′ 2 λz z′ 2 ωz′2 = + ωl2 1 − + πωl Rl 2 2 2 ′ 2

1 (ω0 z ) z0 ′ 2 z0 + zR 1 − + ω z . = 2 − 0 2 2 2 z′ f z0 + zR zR z02 + zR

81

(8.31)

Letting z′ = f in eq. (8.31), we obtain f λf = , πω0 zR

ωf′ =

(8.32)

showing that the beam radius in the back focal plane of the lens is independent of the location z0 of the incident waist. The radius of the waist of the transmitted beam according to eq. (8.12c) is given by ωl 2 −1 ω0′2 = ωl2 1 + ωR + l πωl2 2 −1 2 = ωl 1 + λRl+ =

ω02 f 2

2 (z0 + f )2 + zR

(8.33)

.

It lies at a distance z0′ from the lens which, according to eq. (8.11b), is given by + 2 −1 z0′ λRl = 1 + + Rl πωl2 or z0′ =

2] f [z0 (z0 + f ) + zR 2 (z0 + f )2 + zR

.

(8.34)

Considering the incident and transmitted waists as an object and its corresponding image formed by the lens, eq. (8.34) relating their distances from the lens may be written 1 1 1 = , − 2 /(z + f ) z0′ f z0 + zR 0

(8.35)

which reduces to a conventional imaging equation in the limit zR /f → 0. If we plot z0′ /f as a function of z0 /f using zR /f as a parameter, as illustrated in fig. 33(a), the curves pass through an inflection point (−1, 1) with maxima and

82

Gaussian apodization and beam propagation

[1, § 8

(a)

(b) Fig. 33. Incident and transmitted waist relationships: (a) object and corresponding image distances; (b) transverse magnification.

1, § 8]

Weakly-truncated Gaussian pupils and beams

83

minima lying at (−1−zR /f, 1+f/(2zR )) and (−1+zR /f, 1−f/(2zR )), respectively, as may be seen by differentiating eq. (8.32) with respect to z0 and equating the result to zero. Thus, when the waist of the incident beam lies in the front focal plane of the lens, the waist of the transmitted beam lies in its back focal plane, and not at infinity as in conventional imaging. Accordingly, the transmitted waist lies in the back focal plane only if the incident waist lies in the front focal plane. Otherwise, the transmitted waist lies beyond the back focal plane if z0 /f < −1, or inside the back focal plane if z0 /f > −1. In conventional imaging of a real object by a positive lens forming a real image, there is a minimum separation of 4f between an object and its image, which corresponds to −z0 = z0′ = 2f and unity magnification. However, there is no minimum separation between a real object waist and a real image waist (Self [1983], Mahajan [2004]). For example, if the incident waist lies at the lens, i.e., if z0 = 0, then eq. (8.31) reduces to 1 1 2 1/2 ′ ′ 1 ′ . ωz = ω0 z 2 + ′ − (8.36) z f zR The radius and position of the corresponding transmitted waist are given by eqs. (8.33) and (8.34) according to ω0′ =

ω0 [1 + (zR /f )2 ]1/2

(8.37)

z0′ =

f . 1 + (f/zR )2

(8.38)

ω0′ 1 = , 2 ω0 {[1 + (z0 /f )] + (zR /f )2 }1/2

(8.39)

and

From eq. (8.33), the waist magnification is given by M=

which reduces to conventional image magnification as zR /f → 0, provided the negative sign associated with an inverted image is ignored. A magnification of unity is obtained when 2 zR z0 = −1 ± 1 − (8.40) , f f and correspondingly z0′ z0 =− , f f

(8.41)

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Gaussian apodization and beam propagation

[1, § 8

thus yielding the inflection point in fig. 33(a) when z0 /f = −1. The radii of curvature of the wavefronts incident on and exiting from the lens are given by Rl− = −2f = −Rl+ . As illustrated in fig. 33(b), the maximum value of magnification is f/zR , which occurs when the object waist lies in the front focal plane, i.e., when z0 = −f . Hence, unity magnification in this case occurs only if f = zR . Moreover, M 1 for zR /f 1. The Rayleigh range of the image beam is given by πω0′2 (8.42) = M 2 zR . λ Differentiating eq. (8.32), the longitudinal magnification of the waist is given by ′ zR =

Ml =

∂z0′ z2 − (z0 + f )2 = −M 2 R , 2 + (z + f )2 ∂z0 zR 0

(8.43)

which reduces to the expression for conventional imaging as zR → 0. 8.6. Optical transfer function If we write the p-integral in eq. (4.21) in the form 2

√1−q 2 −v 2 2 2 i exp − Bd v exp −2γ p − v B dp, d √ γ γ −( 1−q 2 −v) we see that for large values of γ , the contribution to the integral is negligible unless v = 0. Therefore, in such cases, eq. (4.21) may be written

8γ exp{−[2γ + (2/γ )Bd2 ]v 2 } 1 τ (v; γ ; Bd ) = dq exp −2γ q 2 π[1 − exp(−2γ )] 0 √1−q 2

× (8.44) exp −2γp 2 dp. 0

The 2D integral represents the Gaussian-weighted area of a quadrant of the pupil, which may also be written

π 1 π (8.45) exp −2γρ 2 ρ dρ = 1 − exp(−2γ ) . 2 0 8γ

Hence, for large values of γ , eq. (4.21) reduces to

2 2 2 τ (v; γ ; Bd ) = exp − 2γ + Bd v , 0 v 1, γ

(8.46)

1, § 8]

Weakly-truncated Gaussian pupils and beams

85

where the spatial frequency v is now normalized by the cutoff frequency D/(λz). It is evident that τ (v; γ ; Bd ) is independent of the sign of Bd . Since the PSF for large values of γ is Gaussian, the corresponding OTF is also Gaussian. Equation (8.46) may also be obtained by Fourier transforming the corresponding PSF (which represents the irradiance distribution in an image plane for unity total power). The Fourier transform of the radially symmetric PSF is equal to its zeroth-order Hankel transform, which yields the OTF τ (v) = 2π PSF(r)J0 (2πvr)r dr. (8.47) Substituting for the PSF from eq. (8.5b) into eq. (8.47), we obtain π2 ωz2 vi2 τ (vi ) = exp − , 2

(8.48)

which is the same as eq. (8.46). If we let Bd = 0 in eq. (8.46), we obtain the defocus-free OTF,

τ (v) = exp −2γ v 2 , 0 v 1, (8.49) which is a zeroth-order Hankle transform of the defocus-free PSF given by eq. (8.9a), as expected.

8.7. Strehl ratio, aberration balancing, and orthogonal polynomials Letting the upper limit on the radial integration approach infinity and neglecting exp(−γ ) compared to unity, eq. (5.2) for the Strehl ratio and eq. (5.5) for the mean and mean square values of the aberration may be written 2 2 ∞ 2π

γ 2 S= (8.50) exp −γρ exp iΦ(ρ, θ) ρ dρ dθ π 0 0

and

n γ Φ = π

0

∞ 2π 0

n exp −γρ 2 Φ(ρ, θ) ρ dρ dθ,

(8.51)

respectively. Letting Φ(ρ, θ) = Bd ρ 2 in eq. (8.50), where Bd is the coefficient of the defocus phase aberration, we obtain the Strehl ratio of a defocused but otherwise aberration free beam ∞ 2 1 2 exp −(γ − iBd )x dx = , S=γ (8.52) 1 + (Bd /γ )2 0

86

[1, § 8

Gaussian apodization and beam propagation

a result that was already obtained in eq. (5.8). Closed-form expressions for spherical aberration and for astigmatism can also be obtained. They are listed in Table 11 along with the integral expressions for coma and balanced spherical aberration. For comparison, the corresponding expressions for a uniform beam are also listed in the table. The standard deviation of a primary aberration for a large value of γ can be obtained by calculating its mean and mean square values according to eq. (8.51). The results thus obtained are given in Table 2. The corresponding balanced aberrations and their standard deviations are given in Tables 3 and 4, respectively. √ √ The 5, 3 balancing of an aberration reduces the standard deviation by a factor of √ and 2 in the case of spherical aberration, coma and astigmatism, respectively, as noted in Table 5. The diffraction focus for these aberrations is listed in Table 6. The amount of balancing aberration decreases as γ increases in the case of spherical aberration and coma, but does not change in the case of astigmatism. For example, in the case of spherical aberration, the amount of balancing defocus for a weakly-truncated Gaussian beam is 4/γ times the corresponding amount for a uniform beam. Similarly, in the case of coma, the balancing tilt for a weaklytruncated Gaussian beam is 3/γ times the corresponding amount for a uniform beam. The location of the diffraction focus is independent of the value of γ in

Aberration Spherical, As ρ 4

Balanced spherical, As ρ 4 + Bd ρ 2 Coma, Ac ρ 3 cos θ Balanced coma, (Ac ρ 3 + Bt ρ) cos θ

Table 11 Strehl ratio for primary aberrations √ S (γ = 0) S ( γ 3) √ √ 2 1 π 1 − S √1 2 2 2 * b {C ( b ) + S ( b )} 2 ′ b′

b 2 + 12 −C √1 ′ b √ √ 1 2 2 Equation (5.16) with b {C ( b ) + S ( b )} √ 3/2 ) dx 2 0 J0 (2 2σc x

1

2

Bd = − γ4 As and As = γ √σs 2 5 Equation (5.16) with 3/2

Bt = 0 and Ac = γ √ σc 3

3/2 2 1/2 2 √ dx Equation 6σ x − x (5.16) with J 2 bc 0 0 3 Bt = − γ2 Ac and Ac = γ 3/2 σbc

2 −1/2 2 2 1 + Aγa = [1 + 2σa2 ]−1/2 J0 (2σa ) + J1 (2σa ) 1

Astigmatism, Aa ρ 2 cos2 θ √ 2 2 ∞ J Balanced astigmatism, 2 k=0 2k+1 ( 6σba ) 3σ ba Aa ρ 2 (cos2 θ − 1/2) √

sin 3σd 2 2 √ Defocus, Bd ρ 3σd

a 2 −1 1+ A = 2γ

B 2 −1 = 1 + γd

1 2 1+σba 1 1+σd2

√ √ * b = 3 5σ /π, b′ = πσ / 5, C(b) = b cos(πx 2 /2) dx, S(b) = b sin(πx 2 /2) dx. s s 0 0

1, § 8]

Weakly-truncated Gaussian pupils and beams

87

the case of astigmatism, since the balancing defocus is the same regardless of the value of γ . Compared to the peak value of an aberration, its standard deviation is smaller by a factor of γ 2 /2, γ 3/2 and 2γ in the case of spherical aberration, coma and astigmatism, respectively. When a Gaussian beam is weakly truncated, i.e. when γ is large, the quantity ps in Table 7 reduces to (Mahajan [1986, 1995, 2003, 2004]) s s ps−2 = !γ −s/2 . ps = ρ s = (8.53) 2γ 2 As a result, we obtain simple expressions for the radial polynomials, which are listed in Table 7. They are similar to Laguerre polynomials (Szapiel [1982]). If we normalize rp by ω (instead of by a), then γ disappears from these expressions. The standard deviation of an aberration can be obtained by comparing its form with the corresponding orthonormal aberration of eq. (5.12). Since the power in a weakly-truncated Gaussian beam is concentrated in a small region near the center of the pupil, the effect of the aberration in its outer region is negligible. Accordingly, the aberration tolerances in terms of the peak value of the aberration at the edge of the pupil (ρ = 1) may not be very meaningful. They may instead be defined in terms of their value at the Gaussian radius (Herloski [1985], Mahajan [1986, 1995, 2003, 2004]). Comparing the standard deviation reduction factors given in Table 8 with those for a weakly-truncated Gaussian beam given in Table 4, we find that they agree √ with each other with negligible difference for γ 3. This provides a convenient definition of a weakly-truncated Gaussian beam, namely that a 3ω. Some √ authors have assumed that γ 2 provides a sufficient condition for the validity of the aberration analysis of a weakly-truncated Gaussian beam given here. √ When γ = 2, the standard deviation of balanced spherical aberration according to the weakly-truncated beam assumption is given by As /8, whereas the true √ value is given by As /18.29, which is significantly different. When γ = 3, the corresponding standard deviations are given by As /40.50 and As /43.52, which are nearly equal. The difference between the true and approximate results is even √ less for γ > 3. Similarly, the approximate and exact values of the standard √ deviation of balanced coma are Ac /8 and Ac /12.21 for γ = 2, and Ac /27 √ and Ac /27.57 for γ = 3. In the case of balanced astigmatism, however, even √ when γ = 2, the approximate value of Aa /8 is nearly equal to the true value √ of Aa /9.08. Of course, for γ = 3, the approximate and exact values of Aa /18 √ and Aa /18.06, respectively, are practically equal to each other. When γ = 2, the standard deviation of defocus aberration Bd ρ 2 is Bd /4.80 compared to a true √ value of Bd /4. When γ = 3, the true and approximate values are Bd /9.05 and Bd /9 respectively, which are practically equal.

88

Gaussian apodization and beam propagation

[1, § 8

√ When γ = 2, even though the true focal-plane distribution obtained from eq. (3.2) agrees quite well with the corresponding approximate result obtained from eq. (8.9a), the true and approximate standard deviations of primary aberrations are significantly different, as pointed out above. The reason for the discrepancy in the case of an aberrated beam is simple. Even though the irradiance in the region of the pupil ω/a ρ 1 is quite small compared with that at or near its center, the amplitude in this region is not as small. Moreover, the aberration in this region can be quite large and can thus have a significant effect on the standard deviation. In the case of spherical aberration, it increases as ρ 4 . In the case of coma and astigmatism, it increases as ρ 3 and ρ 2 , respectively. Hence, we √ require a larger value of γ , namely γ 3, for the aberrated-beam analysis of this section to be valid. This is also true of defocus, which varies as ρ 2 .

8.8. Beam characterization and measurement We have seen that a diffraction-limited weakly-truncated beam expands according to eq. (8.20b) as it propagates beyond its waist location. It is characterized by two parameters: the size and location of its waist. The beam radius in any plane can be determined from eq. (3.8) by placing in the plane a circular aperture concentric with the beam and measuring the fractional power passing through it. The difficulty of positioning an aperture in two dimensions can be avoided by using a slit or a wire and measuring the transmitted flux, as was done by Stijns [1980]. The size of the waist of a beam can be determined by focusing it with a lens of known focal length, measuring the spot size in the focal plane, and utilizing eq. (8.32). Its position can be calculated from eq. (8.25) by measuring its divergence angle. Since this involves large distances, it can be determined more conveniently by an interferometric method (Herman, Pardo and Wiggins [1985]). In practice, laser beams are not diffraction-limited Gaussian beams, i.e., the fundamental TEM00 mode, due to limitations of the optical cavity or nonuniformity of the lasing material. They are multimode beams which expand more than that predicted by eq. (8.22) for a Gaussian beam, although a spot radius can be defined for each mode (Nemoto and Makimoto [1979]) and its propagation can be investigated (Lenz [1996]). The amplitude of a mode of order p in the plane of the beam waist at z = 0 is given by (Kogelnik and Li [1966], Siegman [1986]) 2 r 1 r2 U (ri ; 0) = Ap Lp 2 i2 exp − i2 , 2 ω0 ω0

(8.54)

1, § 8]

Weakly-truncated Gaussian pupils and beams

89

where Lp (·) is the Laguerre polynomial of order p. The corresponding diffracted amplitude at a distance z is given by 2 r ω0 U (ri ; z) = −i(−1)p Ap Lp 2 i2 ωz ωz ri2 1 ri2 × exp i kz + φ(z) exp iπ (8.55) , exp − λRz 2 ωz2 where

πω02 (8.56) λz and ωz and Rz are given by the same eqs. (8.22) and (8.26) as for the fundamental Gaussian mode. For simplicity, a parameter called the M 2 factor is used to characterize multimode beams, where M (not to be confused with the magnification of a beam waist by a lens considered in Section 8.5) is the factor by which the beam crosssection in any plane, including the plane of the beam waist, is larger compared to its value if the beam were a diffraction-limited Gaussian beam (called the embedded Gaussian beam) (Sasnett [1989]). Such a beam is often referred to as the “M times the diffraction-limited beam”. Thus, the radius and the far-field divergence of a multimode beam are larger by a factor of M, but its Rayleigh range is the same as that of the Gaussian beam. If ω0 is the radius of the waist of the Gaussian beam, then the radius W0 of the multimode beam is simply Mω0 . Similarly, if θ = λ/(πω0 ) is the divergence of the Gaussian beam, then the divergence of the multimode beam is Θ = Mθ . However, when compared with a Gaussian beam (called the normalizing Gaussian beam) of the same width as the multimode beam, the divergence is larger and the Rayleigh range is smaller by a factor of M 2 according to φp (z) = (2p + 1) tan−1

Θ = Mθ = M

λ λ = M2 πω0 πW0

(8.57)

and πω02 1 πW02 W0 (8.58) = 2 = . λ λ Θ M Of course, the beam radius at a distance z from the beam waist, following eq. (8.22), is given by 2 2 λz z 2 M = W02 1 + . Wz2 = W02 1 + (8.59) zR πW02 zR =

90

Gaussian apodization and beam propagation

[1, § 8

Similarly, from eqs. (8.21) and (8.26), its central irradiance in terms of the central irradiance I0 at the beam waist and the radius of curvature of its wavefront may be written I0 Iz = (8.60) I + [λzM 2 /(πW02 )]2 and πW02 2 Rz = −z 1 + . λzM 2

(8.61)

Thus, M 2 is a measure of the quality of a beam, and the higher its value the poorer its quality. A value lying between 1 and 1.2 is considered very good. For a Gaussian distribution with a standard deviation of σ , its 1/e2 radius is equal to 2σ . Hence, if the standard deviation σ of a beam in a certain plane is calculated from its measured data, then its beam radius is simply 2σ . If σx and σy are the standard deviations along the x and y axes, then the corresponding 1/e2 beam widths are 2σx and 2σy (Johnston [1998]).

8.9. Nonparaxial Gaussian beams Our discussion so far has been limited to paraxial Gaussian beams, i.e., those with a beam radius much larger than the optical wavelength. These beams remain Gaussian as they propagate. They represent solutions of the spherical Helmholtz wave equation in the paraxial approximation, i.e., when the change in their amplitude over a propagation distance of a wavelength is small enough that the second derivative of the amplitude with respect to z is negligible. In other words, they are solutions of the parabolic wave equation. However, when the waist size of a Gaussian beam is comparable to the wavelength, as in the case of semiconductor lasers used in communication systems, it does not propagate as a Gaussian beam. Various authors have suggested corrections to the Gaussian amplitude solution using different approaches: perturbation method (Lax, Louisell and McKnight [1975]), angular spectrum approach (Agrawal and Pattanayak [1979]), method of stationary phase (Friberg, Jaakkola and Tuovinen [1992]), transition operators (Wünsche [1992]), frequency-domain and short-pulse time-domain beam methods (Heyman and Felson [2001]). Correction to the propagation of higherorder modes has also been discussed (Takenaka, Yokota and Fukumitsu [1985], Tuovinen [1992], Friberg, Jaakkola and Tuovinen [1992]. Agrawal and Pattanayak gave a series solution of the Helmholtz scalar wave equation for free-space propagation in powers of a parameter 1/(kω0 ). They

1, § 9]

Conclusions

91

showed that when ω0 2.3λ the zeroth-order term representing the paraxial solution gives a very good estimate. (Note that their ω0 represents the beam radius at which the√irradiance is 1/e of its value at the center and is, therefore, smaller by a factor of 2 than our ω0 , which is the beam radius at which the irradiance is 1/e2 of its value at the center.) For smaller values of ω0 , the paraxial solution overestimates the amplitude in the central region of the beam. The difference is relatively small in the outer region. The opposite is true of the phase. However, the firstorder correction gives a reasonable estimate of the exact amplitude and phase. Further analysis by Nemoto [1990] concluded that a lower bound for the paraxial solution is ω0 λ, first-order correction is effective when λ ω0 0.5λ and is partially effective when 0.5λ ω0 0.25λ, and the paraxial solution fails when ω0 0.25λ. Nonparaxial analysis of the far-field patterns of diode lasers with active-layer thickness much smaller than the wavelength has been discussed by Li and Katz [1996]. Using the Huygens–Fresnel principle they obtained numerical results substantially in agreement with observations on commercially available visible diode lasers. Any disagreement in the plane normal to the laser junction was attributed to the effect of the active layer, which was neglected in the calculations. Similarly, any disagreement in the plane parallel to the junction was attributed to the initial output beam not being purely Gaussian.

§ 9. Conclusions A truncated Gaussian illumination yields a PSF with a broader central disc and lower secondary maxima compared to the Airy pattern obtained for a uniform pupil. For a given total power, the central irradiance for a Gaussian pupil is smaller than that for a uniform pupil. The corresponding optical transfer function (OTF) is higher for low spatial frequencies and lower for high spatial frequencies. As in the case of uniformly illuminated pupils, the principal maximum of axial irradiance of a focused beam with a small Fresnel number lies at a point that is closer to the pupil and not at the geometrical focus. However, the maximum central irradiance on a target at a fixed distance is obtained when the beam is focused on it. In such cases, the axial irradiance closer to the pupil increases when the defocus aberration thus introduced is balanced with spherical aberration or astigmatism. A Gaussian beam with a weak truncation remains Gaussian as it propagates. Its diffraction pattern does not consist of any rings that are associated with truncated beams. Such a beam also yields maximum irradiance on a target when it is focused on the target, although its waist and therefore the principal maximum of axial

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Gaussian apodization and beam propagation

[1, § 9

irradiance lie in a plane that is much closer to the pupil. Since the beam remains Gaussian as it propagates, the OTF of a weakly-truncated Gaussian pupil is also a Gaussian. The Strehl ratio for a Gaussian pupil for a given amount of a primary aberration is higher than that for a uniform pupil, or the aberration tolerance for a given Strehl ratio is higher for a Gaussian pupil. This is especially true for weaklytruncated beams. In particular, the depth of focus increases rapidly as the pupil radius increases beyond twice the beam radius. However, whereas an aberrationfree beam may be considered weakly truncated when the beam radius is half of that of the pupil, an aberrated beam must be smaller than one third of the pupil radius. When the waist of a beam incident on a lens lies in its front focal plane, the waist of the transmitted beam lies in its back focal plane, and not at infinity as in conventional imaging. Similarly, whereas in conventional imaging of a real object by a positive lens forming a real image, there is a minimum separation between an object and its image, there is no minimum separation between a real object waist and a real image waist. A beam quality factor of M 2 is defined to account for the difference in the divergence of a practical beam and that of an ideal Gaussian beam. We point out that the paraxial beam analysis is valid unless the waist radius is smaller than the wavelength. We have considered systems or beams with circular pupils. The treatment given here can be extended to annular pupils by changing the lower limit on the radial integration from zero to the obscuration ratio (Holmes, Korka and Avizonis [1972], Nayyar and Verma [1978], Mahajan [1981, 2004] and Kathuria [1989]). The treatment can also be extended to elliptical beams produced by junction lasers (Li [1988b]).

Acknowledgements The author gratefully acknowledges computer plotting help from Yunsong Huang, figure editing by Victor Onouye, and helpful discussions with Drs. Bill Swantner and Yajun Li.

Appendix We show here that the diffracted beam consists of a Gaussian and other terms. The Gaussian term is exactly the same as that obtained for a weakly-truncated beam,

1]

References

93

and the other terms are negligible for such a beam (Kuttner [1986], Campbell √ [1987]). Substituting for I (ρ) from eq. (3.5), the integral in eq. (4.6) is proportional to 1

(A.1) exp −bρ 2 J0 (πrρ)ρ dρ, 0

where b = γ − iBd . Integrating by parts and noting that lim

ρ→0

and

Jn (πrρ) 1 = n ρn 2 n!

(A.2)

d Jn (πrρ) Jn+1 (πrρ) , = πr dρ ρn ρn

the above integral can be written ∞ π2 r 2 πr n 1 Jn (πr) . − exp(−b) exp − − 2b 4b 2b

(A.3)

(A.4)

n=0

Thus, the diffracted amplitude consists of the sum of a Gaussian term and other terms, which vanish in the limit of large γ . Hence, for a weakly-truncated beam, eq. (4.6) reduces to eq. (8.5a), as expected.

References Agrawal, G.P., Pattanayak, D.N., 1979, Gaussian beam propagation beyond the paraxial approximation, J. Opt. Soc. Am. 69, 575–578. Airy, G.B., 1835, On the diffraction of an object-glass with circular aperture, Trans. Camb. Phil. Soc. 5, 283–291. Belland, P., Crenn, J.P., 1982, Changes in the characteristics of a Gaussian beam weakly diffracted by a circular aperture, Appl. Opt. 21, 522–527. Born, M., Wolf, E., 1999, Principles of Optics, Oxford University Press, New York. Buck, A.L., 1967, The radiation pattern of a truncated Gaussian aperture distribution, Proc. IEEE 55, 448–450. Campbell, C., 1987, Fresnel diffraction of Gaussian laser beams by circular apertures, Opt. Eng. 26, 270–275. Campbell, J.P., DeShazer, L.G., 1969, Near fields of truncated-Gaussian apertures, J. Opt. Soc. Am. 59, 1427–1429. Carter, W.H., 1982, Focal shift and concept of effective Fresnel number for a Gaussian laser beam, Appl. Opt. 21, 1989–1994. Chung, C.S., Hopkins, H.H., 1989, Influence of nonuniform amplitude on the optical transfer function, Appl. Opt. 28, 1244–1250. Unlike our eq. (33), the corresponding OTF equation in this paper is not properly normalized to unity at the center, although the numerical results are. Dementev, A.S., Domarkene, D.P., 1984, Diffraction of converging spherical waves by a circular aperture, Opt. Spectrosc. 56, 532–534.

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Gaussian apodization and beam propagation

[1

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E. Wolf, Progress in Optics 49 © 2006 Elsevier B.V. All rights reserved

Chapter 2

Controlling nonlinear optical processes in multi-level atomic systems by

Amitabh Joshi, Min Xiao Department of Physics, University of Arkansas, Fayetteville, AR 72701, USA e-mail: [email protected]; [email protected]

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(06)49002-8 97

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 2. Modified linear and nonlinear optical properties in multi-level atomic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 3. Enhanced four-wave mixing processes with induced atomic coherence 126 § 4. Controlled optical bistability and optical multistability with three-level atoms inside an optical cavity . . . . . . . . . . . . . . . . . . . . . .

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§ 5. Controlled optical switching in three-level atomic systems . . . . . .

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§ 6. Controlled optical instability with three-level atoms inside an optical cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 7. Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . .

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Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 1. Introduction A medium absorbing a beam of electromagnetic field can be made transparent to this beam with the help of another beam of electromagnetic field in a threelevel atomic system. This phenomenon is better known as electromagnetically induced transparency (EIT) in the atomic physics and optics community. The concept of EIT started with a related topic of lasing without inversion (LWI) proposed more than fifteen years ago by Kocharovskaya and Khanin [1988], Harris [1989] and Scully, Zhu and Gavrielides [1989], and was used for enhancing nonlinear optical processes by Harris, Field and Imamoglu [1990], Hakuta, Marmet and Stoicheff [1991] and Thompson, Stoicheff, Zhang and Hakuta [1994]. Such absorption reduction is due to light-induced atomic coherence between the atomic energy levels, and in the dressed-state picture can be considered as destructive quantum interference in transition probability amplitudes. EIT was first experimentally demonstrated in a three-level -type configuration in strontium atomic vapor by Boller, Imamoglu and Harris [1991] using pulsed lasers. In that experiment the transmittance of the pulse probe field coupling between a lower-lying state and an autoionization state was increased from exp(−20) without a coupling field to exp(−1) in the presence of a coupling field (interacting with the other transition of the three-level atomic system). The absorption was basically suppressed with the use of a high-intensity pulsed coupling laser beam. The next experiment on EIT was carried out in lead vapor in a ladder-type configuration also with pulsed lasers (Field, Hahn and Harris [1991]). The basic requirement for observing EIT in these experiments is to have the coupling-beam Rabi frequency (proportional to the field strength) larger than the Doppler linewidth of the hot atomic vapor to overcome the large Doppler effect, which demands a very high intensity for the coupling laser beam (15 MW/cm2 ). Later, EIT effects were demonstrated in three-level ladder-type (Gea-Banacloche, Li, Jin and Xiao [1995]) and -type (Li and Xiao [1995a]) rubidium atomic systems with weak cw diode lasers, which were based on two-photon Doppler-free (TPDF) (counterpropagating for the ladder-type and co-propagating for the -type) configurations for the coupling and probe laser beams. With TPDF schemes, EIT could be observed with the coupling-field Rabi frequency much smaller than the Doppler 99

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linewidth of the atomic vapor (Field, Hahn and Harris [1991], Gea-Banacloche, Li, Jin and Xiao [1995]), and even with the coupling-field Rabi frequency smaller than the atomic natural linewidth, as a direct demonstration of quantum interference in such EIT system (Li and Xiao [1995b]). Such TPDF technique with weak cw diode lasers provides a solid base for advances in studying interesting EIT-related effects in atomic vapor cells. Soon after, EIT effects were also studied in cold atomic samples (Hopkins, Usadi, Chen and Durrant [1997], Yan, Rickey and Zhu [2001a]). For early EIT and related works, several good review articles provide the reader with early references (Arimondo [1996], Harris [1997] and Marangos [1998]). With the narrow EIT dip in the absorption profile as the probe frequency scans through the atomic resonance in the three-level atomic system, the dispersion properties of the probe transition are also greatly modified. A sharp change in dispersion, corresponding to the narrow EIT dip, was experimentally measured in a three-level atomic system in 1995, and a group velocity reduction of c/13 was inferred from the sharp dispersion slope measurement (Xiao, Li, Jin and Gea-Banacloche [1995]). In 1999, several groups demonstrated great group velocity reduction (to tens of meters per second) in EIT systems with a cold atomic sample (Hau, Harris, Dutton and Behroozi [1999]) or atomic vapor cells (Kash, Sautenkov, Zibrov, Hollberg, Welch, Lukin, Rostovtsev, Fry and Scully [1999], Budker, Kimball, Rochester and Yashchuk [1999]) with frequency-locked diode lasers. By using a double-peaked Raman gain in a four-level system to generate absorptionless anomalous dispersion, superluminal pulse propagation in an atomic vapor cell of cesium was demonstrated (Wang, Kuzmich and Dogariu [2000]). Recently, the field of “slow light” and “fast light” has become a very active research topic, and such phenomena have also been observed in solid and semiconductor materials. Relevant references can be found in a recent review by Boyd and Gauthier [2002]. The storage and memory of photons in such EIT media have also attracted great attention in recent years, as this could be useful in quantum information processing (Lukin and Imamoglu [2001]). When such a highly dispersive EIT medium is placed inside an optical cavity, the cavity transmission properties will be greatly modified. For example, owing to the slowing down of photons inside the cavity due to the sharp change in dispersion slope, the effective photon lifetime is substantially increased, which leads to narrower cavity output linewidth. Such a reduction of the cavity output linewidth was theoretically predicted (Lukin, Fleischhauer, Scully and Velichansky [1998]) and experimentally demonstrated (Wang, Goorskey, Burkett and Xiao [2000]) a few years ago. Also, such slowing down in photon speed inside an optical cavity will allow the cavity

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ringdown effect (CRE) to be observable with a much shorter optical cavity and a much slower cavity scanning speed (Yang, Joshi and Xiao [2004a]), which will boost the practical applications of this CRE in precision spectroscopy. However, since these topics are based solely on the linear absorption and dispersion properties, which are beyond the scope of the current review, we will not discuss them further in the following. One of the early hopes for making use of EIT and light-induced atomic coherence was to achieve LWI in multi-level atomic systems. Although gain without inversion has been experimentally demonstrated in several atomic systems (Zibrov, Lukin, Nikonov, Hollberg, Scully, Velichansky and Robinson [1995], Mompart and Corbalan [2000]), such inversionless gain seems not to be able to provide more efficient laser operation, and there are questions as to whether inversions actually exist in other representations (such as in dressed states: Alsing, Cardimona and Carmichael [1992]). By making use of the absorption and dispersion properties of the EIT medium, an electromagnetically induced grating can be formed which can diffract a weak probe beam coupled to one atomic transition by a strong standing wave coupled to another atomic transition in a three-level atomic system (Ling, Li and Xiao [1998]). This phenomenon was later demonstrated in cold three-level atomic samples (Cardoso and Tabosa [2002], Mitsunaga and Imoto [1999]) and even in an atomic vapor cell (Brown and Xiao [2005]). There are many other interesting effects in multi-level atomic systems related to EIT, such as electromagnetically induced absorption (EIA) (Lezama, Barreiro and Akulshin [1999]), dark-state polaritons (Lukin and Imamoglu [2001], Fleischhauer and Lukin [2002]), correlated photon pairs (van der Wal, Eisaman, Andre, Walsworth, Phillips, Zibrov and Lukin [2003], Kuzmich, Bowen, Boozer, Boca, Chou, Duan and Kimble [2003]), etc. The most promising potential applications of EIT are in enhancing nonlinear optical processes in multi-level systems by making use of light-induced atomic coherence and greatly reduced light speed to increase the effective interaction length. Typically nonlinear effects are stronger near resonance, however linear absorption near resonance is always too large to make such resonantlyenhanced nonlinear effects useful. On the other hand, when linear absorption is small off-resonance, the nonlinear coefficient is also very small. EIT is ideal in this regard, because it suppresses linear absorption at resonance and at the same time enhances nonlinear coefficients due to light-induced atomic coherence. Since the early demonstrations of enhanced harmonic generation (Hakuta, Marmet and Stoicheff [1991]) and wave-mixing processes (Thompson, Stoicheff, Zhang and Hakuta [1994]) in multi-level atomic hydrogen systems, many experimental demonstrations were reported in enhancing nonlinear optical processes,

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such as enhanced harmonic generation with optimal atomic coherence (Jain, Xia, Yin, Merriam and Harris [1996]), four-wave mixing in three-level -type (Li and Xiao [1996] and Hemmer, Katz, Donoghue, Cronin-Golomb, Shahriar and Kumar [1999]) and four-level double -type (Lu, Burkett and Xiao [1998] and Lukin, Matsko, Fleischhauer and Scully [1999]) atomic systems, electromagnetically induced focusing due to Kerr-index change (Moseley, Shepherd, Fulton, Sinclair and Dunn [1995]), suppressed linear absorption with enhanced thirdorder nonlinearity in a four-level N-type atomic system (Kang and Zhu [2003] and Braje, Balic, Yin and Harris [2003]), and even six-wave mixing in a fourlevel N-type atomic system (Kang, Hernandez and Zhu [2004a]). In a recent study using atomic sodium vapor, influence of coherent Raman scattering on coherent population trapping (CPT) has been demonstrated (Wong, Bennink, Marino, Boyd and Stroud Jr. [2004]). By using an optical cavity to remove contributions from the linear absorptions, the third-order nonlinear susceptibility or Kerr-nonlinear index of refraction (self-phase modulation) in a three-level atomic system was measured directly and was shown to be greatly enhanced relative to its two-level counterpart (Wang, Goorskey and Xiao [2001, 2002a]). The dependencies of the Kerr-nonlinear coefficient on the coupling beam frequency detuning and intensity were also carefully measured (Wang, Goorskey and Xiao [2002b]), as was the Kerr-nonlinear index due to cross-phase modulation in a similar system was also measured (Chang, Du, Yao, Xie and Wang [2004]). When multi-level atoms are placed inside an optical cavity, many interesting nonlinear optical phenomena appear, such as optical bistability, optical multistability, and optical dynamic instability. One of the biggest advantages of studying nonlinear optical processes in three- or four-level atomic systems, compared to two-level atomic system, is the ability to manipulate the absorption (Gea-Banacloche, Li, Jin and Xiao [1995] and Li and Xiao [1995a]), dispersion (Xiao, Li, Jin and Gea-Banacloche [1995]), and nonlinear (Wang, Goorskey and Xiao [2001, 2002a, 2002b]) optical properties of the probe beam by the additional coupling laser beam or beams (both frequency and intensity). With such controlled linear and nonlinear optical properties, the steady-state and dynamical behaviors of the composite systems with multi-level atoms inside an optical cavity can be well controlled experimentally. By adjusting the controlling beam frequency or intensity the threshold values and the shape of the steadystate bistable curve can be controlled (Joshi, Brown, Wang and Xiao [2003]), and transitions from bistable to multistable hysteresis curves have been observed (Joshi and Xiao [2003]). With increased frequency detuning of the coupling beam the rotation of the hysteresis loop could be made to change from forward direc-

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tion (counter-clockwise) to backward direction (clockwise) (Joshi, Yang and Xiao [2004]), which is very unusual in such systems. The hysteresis cycles occurred in many other systems (magnetic, electrical, biological, etc.) is in forward direction due to energy considerations. This system also exhibits a very interesting dynamic hysteresis cycle (Joshi, Yang and Xiao [2005]). In a certain parameter range, the cavity output field of this composite system could become unstable, e.g. dynamic instability due to the competition between the optical pumping in the coupling transition and the saturation in the probe transition (Wang, Goorskey and Xiao [2002c] and Yang, Joshi and Xiao [2004b]), again controllable by adjusting the coupling-beam frequency detuning or intensity. Such controllable nonlinear optical processes in multi-level EIT systems, especially inside an optical cavity, will be the main focus of this review. We neither intend to review the entire field of EIT-related phenomena, nor do we try to include all the experiments on nonlinear optical processes done in multi-level atomic systems. This review limits its scope to covering mainly the controlled nonlinear optical processes of three-level atoms inside an optical cavity, with excursions to a few closely related experiments such as four-wave-mixing in three- and four-level atomic systems. Section 2 reviews controllable linear absorption and dispersion properties in three-level atomic systems, which are relevant to the later discussions as well as the measurement of the Kerr-nonlinear index of refraction with dependence on controlling (coupling) beam intensity and frequency detuning. Section 3 discusses a few experiments of four-wave mixing in three- and fourlevel atomic systems, emphasizing the enhanced nonlinear optical processes due to induced atomic coherence in such EIT systems. In Section 4 we present steady-state optical bistability and optical multistability with three-level atoms inside an optical cavity, with both shape and rotation direction of the hysteresis cycles controlled by the coupling-beam frequency detuning and other experimental parameters. Section 5 presents one special application of the controlled nonlinearity in EIT systems, i.e. all-optical switching. With measured nonlinearity, simple all-optical switches were demonstrated in the composite system of three-level atoms inside an optical cavity, in which the cavity (probe) output field intensity can be controlled by coupling-field intensity or frequency detuning. Section 6 describes theoretical and experimental investigations of controlled optical dynamic instability in the cavity output field with threelevel atoms inside an optical cavity. In Section 7 we give summary and outlook.

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§ 2. Modiﬁed linear and nonlinear optical properties in multi-level atomic systems In a two-level system interacting with a coherent radiation field, the Rabi oscillation is a consequence of coherent excitation of atoms by a monochromatic (or near-monochromatic) light source, resonant with an atomic transition, on a time scale much less than the natural lifetime of the excited state. The decay of the excited state is an incoherent process. The populations of the two levels oscillate sinusoidally with a constant frequency, called the Rabi frequency, if the excitation field is strong. The Rabi frequency is derived using a quantum-mechanical treatment of the atom, based on the probability of a transition taking place at a given time rather than the more phenomenological approach used in rate equations. The Rabi frequency has a ubiquitous presence in quantum optics and is used as a measure for interaction strength, frequency of population oscillation, nutation frequency, and optical Larmor frequency among others. For a two-level atomic system one can define the Rabi frequency as ΩRabi = με/h¯ + Δ2 , where μ is the transition dipole moment, ε is the amplitude of the coherent field, and Δ is the detuning of the coherent field frequency from the interacting atomic transition frequency (Allen and Eberly [1987]). The strong laser field causes dynamic Stark splitting of the atomic levels (Allen and Eberly [1987] and Shore [1990]). The ac-Stark splitting or dynamic Stark splitting of the levels is directly related to Rabi frequency. For a probe field exactly on resonance with the atomic transition the splitting due to the field equals the Rabi frequency for a non-Doppler-broadened system and it is symmetric about the resonance condition. When the field is off-resonance with the transition the splitting becomes asymmetric. EIT (which will be discussed in subsequent paragraphs for three-level systems) will enhance the depth of the hole that is produced by Autler–Townes splitting (with a weak field) as quantum interference occurs between the Autler–Townes components, deepening the hole. The joint effect thus makes the medium transparent. EIT can be distinguished from Autler–Townes splitting by the level of dephasing occurring in the system. For large dephasing on the unlinked transition one observes Autler–Townes splitting but not EIT. An experiment carried out by Fano [1961] laid the foundation for the phenomena of CPT/EIT. In Fano’s experiment excitation was taking place between a lower state and a continuum ionizing state. Another path of excitation was between the lower state and an autoionized state. If the atom goes to the autoionizing state it relaxes to the ionizing state continuum. Thus, the final state can be reached by two different routes and interference between these two routes gives rise to asymmetric peaks in the excitation spectrum. In this experiment the tran-

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sition probability vanishes on one side of the resonance and the interference produces zero absorption in the medium. The condition of zero absorption leads to population trapping in particular energy levels. Fano’s early experiment was an inspiration for the formulation of CPT (Gray, Whitley and Stroud [1978]). Such CPT phenomena exist in three-level systems coupled by two laser fields. CPT was first observed in an atomic fluorescence experiment with three-level -type sodium atoms in a vapor cell, where fluorescence disappears due to the fact that the population has been trapped in the superposition of the two ground states due to Fano interference in transition probability amplitudes, unable to be excited into the upper state. More explicitly, let the atom initially be in the superposition of the two ground states and let a field be applied to this system so that the probability amplitude for being in the upper state is zero and hence the population remains trapped in the two lower ground states. CPT arises due to the destructive interference between the two routes coupling two ground states |g and |g ′ to the upper state |e . Hence CPT can be explained by considering two of the eigenstates of the Hamiltonian of the atom–field system. These eigenstates are coherent superpositions of the two lower levels only, and are known as bright and dark states, respectively. The former is coupled to the upper state by electric dipole interaction but the latter is uncoupled. Setting the field strengths (Rabi frequencies) of the two laser fields to an appropriate ratio will result in vanishing of the dipole moment from the dark state to the upper state |e ( dark|μ′ |e = 0). The conditions for CPT have been well established both theoretically and experimentally (Arimondo [1996] and Gray, Whitley and Stroud [1978]). EIT can be considered as a ‘subset’ of CPT phenomena. In original CPT experiments the strengths of the two fields interacting with the atoms are of the same order of magnitude and hence the interference effects arise from both fields. In EIT one of the fields is much weaker (known as the probe field) than the other (known as the coupling field). For -type systems, CPT in general has the levels |g and |g ′ as either Zeeman or hyperfine levels within the ground states of the atom and has both levels populated. In EIT, however, one of the levels will have no population at any time during the process (due to strong optical pumping of the coupling field). A dressed-state analysis also leads to the EIT result. A dressed state is defined as “an eigenstate of the time-independent form of the total Hamiltonian, including interactions” (Cohen-Tannoudji [1977]). Here the atom–field interaction is considered as a whole so that the Hamiltonian for the system is made up of components including both the bare-state atom and the atom–field interaction. Under the influence of the coupling field a dressed-state analysis leads to the formation of a coherent superposition of states by the two energy levels connected by the coupling beam. It is the destructive interference between two probe transi-

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tion amplitudes to these two upper dressed states (from the lower probe state) that results in EIT. EIT can be understood without invoking the dressed-state picture by using the concept of ‘coherence’ related to the phenomenon of interference. Semiclassically, coherence is associated with oscillating electric dipoles driven by a coupling field applied between a pair of quantum states. If there are several ways to excite the oscillating dipoles then interference arise between various contributions to this dipole and a summation results in a total electric dipole. This is analogous to the Fano interference in autoionization. The coherences are related to the offdiagonal elements of the density operator of the system, which will be discussed in more detail subsequently.

2.1. Absorption reduction and dispersion enhancement As discussed above, EIT and CPT are closely linked phenomena. Another closely associated phenomenon is that of coherent adiabatic population transfer, discussed extensively in the literature (Bergmann, Theuer and Shore [1998]). The common element in these processes is the presence of a three-level atomic system. There are three different configurations for three-level atomic systems, known as ladder- (or cascade)-type configuration, -type configuration, and V-type configuration, as shown in fig. 1. In all these configurations, two pairs of levels (|1 ←→ |2 ) and (|2 ←→ |3 ) are dipole-coupled while there is no dipoleallowed transition in the third pair of levels (|1 ←→ |3 ). The phenomenon of EIT is mostly investigated in either the ladder- or the -type scheme since there is no need for population transfer in these configurations. In order to understand the phenomenon of EIT in these three-level schemes we follow the density-matrix approach as we can naturally incorporate damping processes into the equations for the decays of populations and coherence of the system. The probability-amplitude calculation technique gives identical results (Harris [1989], Harris, Field and Imamoglu [1990] and Li and Xiao [1995b]). We consider a three-level system in ladder configuration [E3 > E2 > E1 ; Ei , i = 1, 2, 3, represents the level energy] as depicted in fig. 1(a). The probe laser (frequency ωP ) interacts with the atomic transition |1 → |2 (frequency ω21 ) with frequency detuning ΔP = ω21 − ωP . The coupling laser (frequency ωC ) interacts with the other atomic transition |2 → |3 (frequency ω23 ) with frequency detuning ΔC = ω23 − ωC . In the semiclassical approach the Liouville equation for the density operator in dipole and rotating-wave approximations is given by

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Fig. 1. Three-level Rb atomic system in (a) ladder configuration; (b) -configuration; (c) V-configuration.

Gea-Banacloche, Li, Jin and Xiao [1995]: ρ˙11 = γ2 ρ22 + ig21 EP ρ12 − ig21 EP∗ ρ21 , ρ˙22 = γ3 ρ33 − γ2 ρ22 + ig32 EC ρ23

− ig32 EC∗ ρ32 − ig21 EP ρ12 + ig21 EP∗ ρ21 ,

ρ˙33 = −γ3 ρ33 − ig32 EC ρ23 + ig32 EC∗ ρ32 ,

ρ˙32 = −(γ32 + iΔC )ρ32 + ig32 EC (ρ33 − ρ22 ) + ig21 EP∗ ρ31 ,

ρ˙21 = −(γ21 + iΔP )ρ21 + ig21 EP (ρ22 − ρ11 ) − ig32 EC∗ ρ31 ,

ρ˙31 = − γ31 + i(ΔC + ΔP ) ρ31 − ig32 EC ρ21 + ig21 EP ρ32 .

(2.1)

The populations of this closed system satisfy the relation ρ11 + ρ22 + ρ33 = 1. The dipole moment matrix elements for the two transitions are 2h¯ g21 and 2h¯ g23 , respectively, and for simplicity they are considered to be real. EP (EC ) is the amplitude of the probe (coupling) field. The decay rates for diagonal elements are γi (i = 1, 2, 3 such that γ1 = 0, i.e., ground state) and for off-diagonal elements γij = (γi + γj )/2. The EIT arises from the existence of coherence terms ρ23 and ρ13 coupled to ρ12 . We will see that it is the presence of the coupling field EC that is responsible for the creation of ρ31 in steady state.

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The above density-matrix equations can be solved numerically if there are time-dependent laser pulses interacting with the atomic system. Such results provide transient effects related to EIT (Li and Xiao [1995c]). Sometimes Maxwell’s equations are also needed along with the density-matrix equation in order to compute propagations of the matched pulses (Harris [1993]), propagation losses and pulse shape modification (Harris and Luo [1995]). Using cw lasers for exciting the EIT medium, the steady-state limit is good enough; all the time derivatives of the density-matrix elements vanish, and the solution of the coupled density-matrix equations gets simplified considerably. For example, in steady state one can write (Gea-Banacloche, Li, Jin and Xiao [1995]) ig32 EC ig21 EP ρ21 + ρ32 γ31 + i(ΔP + ΔC ) γ31 + i(ΔP + ΔC ) ig32 EC ≃− ρ21 . γ31 + i(ΔP + ΔC )

ρ31 = −

(2.2)

The term proportional to ρ32 is neglected because it is multiplied by the probe field EP considered to be weak for the EIT system, and also because the population in level |2 (as well as in level |3 ) is negligibly small and hence should be the value of ρ32 . Further substitution and simplification in steady state gives (Gea-Banacloche, Li, Jin and Xiao [1995]) ρ21 = −

ig21 EP γ21 + iΔP +

ΩC2 /4 γ31 +i(ΔP +ΔC )

,

(2.3)

where the Rabi frequency of the coupling field is defined as ΩC = 2g32 EC (assuming the amplitude of the coupling field to be real). We have also assumed that in steady state to first order in EP /EC , ρ11 ∼ = 0. The complex sus= ρ33 ∼ = 1, ρ22 ∼ ceptibility at the probe field frequency is obtained from the medium polarization as (in which c.c. stands for complex conjugate) P = such that χ=

1 ε0 EP χ(ωP )e−iωP t + c.c. = −2h¯ g21 Nρ21 e−iωP t + c.c., 2 2 N/ε 4ih¯ g21 0

γ21 + iΔP +

ΩC2 /4 γ31 +i(ΔP +ΔC )

,

(2.4)

(2.5)

where N is the density of atoms. Here χ = χ ′ + iχ ′′ , in which the real (χ ′ ) and imaginary (χ ′′ ) parts are responsible for dispersion and absorption characteristics of the medium, respectively. The intensity absorption coefficient is defined as α = ωP n0 χ ′′ /c, and the dispersion coefficient as β = ωP n0 χ ′ /2c, where n0 is the

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background index of refraction. In the above expression for χ one can easily incorporate the effect of Doppler broadening in an atomic vapor sample. In a laddertype EIT system the probe and coupling beams are arranged to counter-propagate and their frequencies are assumed to be very close. An atom moving towards the probe (coupling) beam with velocity v is affected by the probe (coupling) frequency detuning with an upshift (downshift) to ΔP + ωP v/c (ΔP − ωP v/c). The √ atomic density with velocity v is √ N (v) dv = (N0 /u π ) exp(−v 2 /u2 ) dv in the Maxwellian distribution, with u/ 2 the root mean squared velocity (depending on temperature). For a purely Doppler medium, the full width √ at half maximum (FWHM) of the absorption profile is ωD = 2ωP (u/c) ln( 2 ). In the case of ωP ∼ ωC , one can integrate the expression for χ over all velocities to obtain (Gea-Banacloche, Li, Jin and Xiao [1995]) 2 N √π 4ih¯ cg12 2 0 ez (1 − erf z), χ= (2.6) ε0 uωP where erf z is the error function with the complex argument

ΩC2 /4 c . γ21 + iΔP + z= uωP γ31 + i(ΔP + ΔC )

(2.7)

Due to the counterpropagation of the coupling and probe beams in such a ladder system, the first-order Doppler effect is canceled in the quantity ΔP + ΔC , which ensures the appearance of the EIT effect even with relatively low cw coupling laser beams (Gea-Banacloche, Li, Jin and Xiao [1995]). The first experiment with cw diode lasers was performed with rubidium atoms (in a three-level ladder-type system using the D2 line of 85 Rb) in a vapor cell at room temperature (Gea-Banacloche, Li, Jin and Xiao [1995]). The coupling laser (wavelength 775.8 nm) couples the upper transition from the state 5P3/2 , F ′ = 4 (state |2 ) to the state 5D5/2 , F ′ = 5 (state |3 ) and the probe laser (wavelength 780 nm) couples the lower transition 5S1/2 , F = 3 (state |1 ) to 5P3/2 , F ′ = 4 (state |2 ). This three-level system is exactly the one used above in the theoretical model with γ2 = 6.0 MHz, γ3 = 0.97 MHz, and a Doppler width of about 540 MHz at room temperature. In that experiment the probe and the coupling beams are orthogonally polarized and propagating in opposite directions through a rubidium vapor cell. The two diode lasers used for the probe and coupling fields are both temperature and current stabilized with a free-running linewidth of about 5 MHz. When the coupling beam was blocked, a typical absorption curve was obtained for the transition from the state 5S1/2 , F = 3 to the state 5P3/2 , F ′ = 4, as shown in fig. 2(a). The maximum absorption at the center of the Doppler-broadened line

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Fig. 2. Absorption coefficient α as a function of probe detuning Δ1 (= ΔP ). Thick black curve, experiment; gray curve, theory. (a) No coupling laser. (b) Coupling laser on resonance ΔC = 0. Theoretical parameters: γ21 = 3 MHz, γ31 = 0.5 MHz, ΩC = 92 MHz, ωD = 540 MHz, δωD = −2.97 MHz, laser (half) linewidth γC = γP = 2.5 MHz. (After Gea-Banacloche, Li, Jin and Xiao [1995].)

was measured to be α = 8.2 × 10−2 cm−1 at T = 21◦ C. When the coupling field was turned on and tuned to the resonance frequency ΔC = 0, a narrow dip at the center of the absorption profile appeared as shown in fig. 2(b). The new absorption coefficient was measured to be α = 2.9 × 10−2 cm−1 at the center frequency giving an absorption reduction of 64.4% as a result of EIT or the atomic coherence induced by the coupling field. The experimental results match well with the theoretical curves in figs. 2(a,b) with only the coupling-field Rabi frequency as a fitting parameter. The value of 92 MHz for ΩC in fig. 2(b) is much smaller than the Doppler width in this system, which shows the advantage of using two-photon Doppler-free configuration in such EIT system. The main limit in absorption re-

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duction (EIT) in this ladder-type system is due to the large value of the dephasing rate γ31 and the relatively broad free-running linewidths of the diode lasers. If the coupling beam is off resonance, a dispersive-like structure appears at the side of the Doppler-broadened absorption curve as a result of contributions from absorption reduction due to atomic coherence and enhancement due to twophoton absorption. The absorption peak at far detuning (ΔC outside Doppler profile) is due to two-photon absorption from state |1 to state |3 . These observed phenomena can all be well explained by the theoretical results of eq. (2.6) (Gea-Banacloche, Li, Jin and Xiao [1995]). One can control probe-beam absorption (Gea-Banacloche, Li, Jin and Xiao [1995] and Li, Jin and Xiao [1995]) by adjusting the coupling-beam intensity at ΔC = 0 [fig. 3(a)] or ΔC = 0 [fig. 3(b)]. The absorption coefficient of the probe transition is reduced at the center fre-

Fig. 3. Absorption coefficient α as a function of coupling (pumping) beam power: (a) for ΔC = ΔP = 0; (b) for ΔC = −550 MHz and ΔP ∼ 550 MHz (the two-photon absorption peak is slightly shifted from ΔP = −ΔC ). Solid line, theoretical prediction. (After Gea-Banacloche, Li, Jin and Xiao [1995].)

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quency of the probe beam as the coupling power increases. This absorption reduction shows the typical EIT behavior and is due to the atomic coherence induced by the coupling beam [fig. 3(a)]. However, with off-resonance ΔC ∼ = −550 MHz for the coupling beam, the two-photon absorption coefficient increases with increasing power of the coupling beam [fig. 3(b)]. One can also control the dispersion properties of the probe beam in such EIT system. In order to measure the dispersive properties of the atomic medium a Mach–Zehnder interferometer is used for the probe beam (Xiao, Li, Jin and Gea-Banacloche [1995]). The small phase shift β(ω)L due to the atoms in the atomic cell is detected using a homodyne arrangement. The differential signal of the balanced homodyne detectors (D1 and D2 ) is given by Id (ω) ∝ 2|ELO ||ES |e−α(ω)L/2 cos φLO + β(ω)L ,

(2.8)

where ES is the signal field passing through the cell and ELO is the local oscillator field passing through the other arm (reference beam). The signal beam is far below the saturation intensity of the rubidium D2 transition and the condition |ES | ≪ |ELO | is satisfied. α(ω) is the absorption coefficient, β(ω) is the dispersion coefficient, L is the length of the rubidium cell, and φLO is the reference phase of the interferometer, which is reset to π/2 by a piezoelectric transducer for each frequency scan, so that Id ∝ e−α(ω)L/2 β(ω)L for |β(ω)|L ≪ 1. Each frequency scan of the probe laser takes 50 ms during which time the interferometer has negligible drift. When the coupling beam is blocked, a typical dispersion curve of probe transition for a two-level atomic system is recorded [fig. 4(a)]. However, when the coupling field is applied and tuned to the resonance frequency (ΔC = 0) the dispersion curve shows a sharp variation near ΔP = 0 [fig. 4(b)]. Using the dispersion curve, the change in dispersion near −9 which yields the group ∼ center frequency is found to be d(βL) dν |ν0 = 19.4 × 10 velocity vg = c/13.2. This expected slowing down of the group velocity in the probe beam is the result of the rapid change in refractive index (enhanced normal dispersion) due to the atomic coherence induced by the coupling beam in the EIT system. With improved laser sources (narrow linewidth and locked frequency between the coupling and probe lasers), better choice of atomic level system, and inclusion of buffer gases in the atomic vapor cell, the dispersion change can be greatly enhanced. For example, in a -type atomic system (hot rubidium gas at 360 K) the group velocity of light was controlled by coupling laser power and measured to slow down to 90 m/s (Kash, Sautenkov, Zibrov, Hollberg, Welch, Lukin, Rostovtsev, Fry and Scully [1999]). With a cold atomic sample, the group velocity

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Fig. 4. Measured dispersion βL of the rubidium vapor vs probe detuning ΔP : (a) dispersion without coupling laser; (b) dispersion with coupling laser on resonance ΔC = 0. (After Xiao, Li, Jin and Gea-Banacloche [1995].)

of a probe pulse was reduced down to 17 m/s (Hau, Harris, Dutton and Behroozi [1999]). For a closed three-level -type system [shown in fig. 1(b)] where the probe (coupling) laser with frequency ωP (ωC ) near the ω21 (ω23 ) resonance couples level |2 and level |1 (|3 ), the Liouville equations for density-matrix elements in the dipole and rotating-wave approximations take the form (Gea-Banacloche, Li, Jin and Xiao [1995]) i i ρ˙11 = γ31 (ρ33 − ρ11 ) + γ21 ρ22 − ΩP∗ ρ21 + ΩP ρ12 , 2 2 i i i i ρ˙22 = −(γ23 + γ21 )ρ22 + ΩP∗ ρ21 − ΩP ρ12 + ΩC∗ ρ23 − ΩC ρ32 , 2 2 2 2 i i ρ˙33 = γ31 (ρ11 − ρ33 ) + γ23 ρ22 − ΩC∗ ρ23 + ΩC ρ32 , 2 2 i i γ21 + γ32 + γ31 ρ˙23 = − + iΔC ρ23 + ΩC (ρ22 − ρ33 ) − ΩP ρ13 , 2 2 2 γ21 + γ32 + γ31 i i ρ˙21 = − + iΔP ρ21 + ΩP (ρ22 − ρ11 ) − ΩC ρ31 , 2 2 2

i i ∗ ρ˙31 = − γ31 + i(ΔP − ΔC ) ρ31 − ΩC ρ21 + ΩP ρ32 , (2.9) 2 2 where ΩP (μ21 ) and ΩC (μ23 ) are complex Rabi frequencies (dipole moments) of the probe and coupling fields (transitions), respectively. The radiative decay

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[2, § 2

rate from level |2 to level |1 (|3 ) is γ21 (γ23 ), and the nonradiative decay rate between levels |3 and |1 is γ31 . Under the EIT condition – strong coupling field, weak probe field, i.e., ΩC ≫ ΩP – in the steady state, almost all atoms are in state |1 so that ρ11 ∼ = 1, ρ22 ∼ = 0 (to first order in ΩP /ΩC ). The Doppler-free configuration for = ρ33 ∼ coherence effects in such -type systems is that the probe beam and the coupling beams propagate collinearly. In this arrangement the first-order Doppler shifts of the probe and coupling lasers for the same group of atoms with velocity v can be canceled (Gea-Banacloche, Li, Jin and Xiao [1995]). As shown earlier for the ladder-type EIT system, the effect of Doppler broadening in this -type EIT system can also easily be taken into account under the assumption of ωP ∼ = ωC . Hence in the steady state (Gea-Banacloche, Li, Jin and Xiao [1995]) √ ich¯ μ221 N0 π z2 e (1 − erf z) χ= (2.10) ε0 uωP with

ΩC2 /4 c , z= (2.11) γ + iΔP + uωP γ31 + i(ΔP − ΔC )

where γ = (γ21 + γ23 + γ31 )/2. The experimental arrangement for observing EIT in -type systems is slightly different from that for ladder-type systems. Here, both the probe and coupling beams propagate collinearly to overcome the firstorder Doppler effect. The probe and coupling lasers are orthogonally polarized. These two beams are combined by a polarization cube beamsplitter before entering the rubidium vapor cell, and are separated by another polarization cube beamsplitter after passing through the rubidium cell. A three-level -type system as shown in fig. 1(b) can be found in 87 Rb by considering the D1 line as follows. The hyperfine level F ′ = 2 of 5P1/2 serves as the excited state |2 . The hyperfine levels F = 1 and F = 2 of the ground state 5S1/2 serve as the two ground states |1 and |3 of the -type system, respectively. Another hyperfine level F ′ = 1 of the excited state 5P1/2 is 812 MHz away (outside the Doppler width of the transition line) and can be neglected. The probe laser ωP interacts with the 5S1/2 , F = 1 and 5P1/2 , F ′ = 2 states with Rabi frequency ΩP , while the coupling laser interacts with the 5S1/2 , F = 2 and 5P1/2 , F ′ = 2 states with Rabi frequency ΩC . In fig. 5 the absorption peak of 5S1/2 , F = 1 → 5P1/2 , F ′ = 2 transition is shown. The lower solid curve without a dip is when the coupling field is blocked. When the coupling field is turned on, it first pumps the population in level |3 to level |1 , so the absorption of the probe beam increases but with a big dip at the line center due to the induced atomic coherence, as shown in the upper solid

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Fig. 5. Absorption coefficient for the probe beam versus probe frequency for the 5S1/2 , F = 1 → 5P1/2 , F ′ = 2 transition of 87 Rb. The lower solid curve is without coupling field. The upper solid curve is for the coupling field tuned at the 5S1/2 , F = 2 → 5P1/2 , F ′ = 2 transition with intensity IC = 19.6 W/cm2 at the cell center. The dotted curve is the corresponding theoretical result with γ21 + γ23 = 6.0 MHz, γ31 = 0.1 MHz, ωD = 530 MHz and ΩC = 105 MHz. (After Li and Xiao [1995a].)

curve. The absorption coefficient at the peak without (with) coupling field is α = 0.92 × 10−2 cm−1 (α = 0.56 × 10−1 cm−1 ). When measuring relative to the new absorption peak after taking into account the absorption increase due to optical pumping (α = 1.92 × 10−2 cm−1 ), the absorption reduction is measured to be 70.8% (Li and Xiao [1995a]). The fact that the dip reaches below the unpumped level (lower solid curve in fig. 5) clearly indicates the EIT effect due to atomic coherence generated in this system. The theoretically fitted and experimentally measured values agree within uncertainty. When the coupling laser is tuned to the 5S1/2 , F = 2 → 5P1/2 , F ′ = 1 transition, the EIT effect is also observed in the 5S1/2 , F = 1 → 5P1/2 , F ′ = 1 transition. When the pumping intensity is 560 mW/cm2 , the absorption reduction is measured to be about 85% (Li and Xiao [1995a]). After the initial demonstrations of EIT with low-power cw diode lasers in twophoton Doppler-free configurations (Gea-Banacloche, Li, Jin and Xiao [1995]

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[2, § 2

and Li and Xiao [1995a]), absorption reductions of more than 90% in atomic vapor cells and cold atomic samples were reported. Experimental observation (Yan, Rickey and Zhu [2001a]) of doubly dressed states was made in 87 Rb atoms cooled and confined in a magneto-optical trap. The doubly dressed states are produced by a strong-coupling laser and a moderate pump laser in an effective three-level atomic configuration. The absorption spectrum of a weak probe laser reveals a three-peaked spectral profile that can be interpreted by a dressed-state picture which agrees well with the calculation based on the density-matrix equations. These experiments have shown that the absorption and dispersion properties of the probe beam can be modified and controlled by the coupling beam in threelevel atomic systems due to induced atomic coherence. Such controllability can be very useful in modifying nonlinear optical processes in these systems, as we will show later.

2.2. Enhanced nonlinearity in EIT systems As we have seen in Section 2.1, the atomic coherence induced by the coupling and probe beams can reduce the resonant absorption of the probe beam through EIT, and the steep dispersion slope in such systems can significantly reduce the group velocity of probe pulses, which will greatly increase the effective interaction length of a probe pulse with the atomic medium. The induced atomic coherence can also enhance nonlinearity in the multi-level systems, which makes it possible to perform nonlinear optical processes with very low light intensities. Normally in nonlinear optical processes if the nonlinear susceptibility is resonantly enhanced, the medium will have a large absorption and get opaque. Generated atomic coherence in multi-level systems provides an opportunity to create a nonlinear medium with resonantly enhanced nonlinear susceptibility and, at the same time, EIT to have reduced absorption for the laser beams involved. One such example was given by Harris, Field and Imamoglu [1990], with the energy-level diagram shown in fig. 6. A strong coupling beam of frequency ωc is applied between a metastable state |2 and a state |3 . When the transition between levels |1 and |2 (dipole-forbidden) is pumped with a two-photon absorption (ωa + ωb ), the sum frequency ωd = ωa + ωb + ωc is generated. The transition |1 → |3 is a resonance transition, and in the absence of ωc , radiation at ωd is strongly absorbed. When the Rabi frequency of the coupling field exceeds the Doppler width of the |1 → |3 transition, the medium becomes transparent at the line center. Such transparency occurs because of destructive interference of the split (Autler–Townes) components of the |1 → |3 transition induced by the coupling

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Fig. 6. Schematics of energy levels for the sum-frequency generation process ωd = ωa + ωb + ωc . The state |3 has spontaneous decay rate Γ3 . When a strong field at frequency ωc is tuned to the line center of the |2 → |3 transition, the medium becomes transparent on the |1 → |3 transition for the generated field. (After Harris, Field and Imamoglu [1990].)

field at ωc . Such interference surprisingly does not negate the nonlinearity that causes the generation of ωd , due to a change of sign in the dressed eigenvector. For the generated (up-converted) frequency lying between the Autler–Townes components, there is a constructive rather than destructive interference in the nonlinear susceptibility (Harris, Field and Imamoglu [1990]). This system gives a very good example of how induced atomic coherence can reduce linear absorption and enhance nonlinearity at the same time in a multi-level atomic system. 2.2.1. Measured Kerr-nonlinear index in a three-level -type atomic system In the past decade, several experimental demonstrations of enhancing nonlinear optical processes were reported in various multi-level atomic systems. Although experimental demonstrations of enhanced nonlinear optical processes are not so difficult to realize, direct measurements of nonlinear coefficients associated with such nonlinear processes have been a difficult task because of the existence of

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residual linear absorption and dispersion effects. Standard techniques for measuring nonlinearity, such as Z-scan, do not work well in the system of atoms in a long atomic vapor cell. The enhanced Kerr-nonlinear effect in multi-level systems can give rise to interesting applications such as self-phase modulation for optical shutters, four-wave mixing process for frequency conversion, and entangled state for quantum information processing. Direct measurement of the Kerr-nonlinear coefficient is important for understanding and optimizing nonlinear optical processes with controllable parameters and will allow one to control nonlinear optical processes with desired precision. A suitable technique for measuring the Kerr-nonlinear index of refraction in such an atomic vapor cell is to place it inside an optical cavity and measure the effect of the nonlinear phase shift on the cavity output profile, which will not be affected by linear absorption (Wang, Goorskey and Xiao [2001]). It is easy to calculate the nonlinear susceptibility (for the probe beam) of a three-level atomic medium using the density-matrix equations. For this purpose one can use the same procedure as for calculating the linear susceptibility (Section 2.1), however, one needs to keep the probe intensity to higher (third) order; one gets the following expression for the total susceptibility between states |1 and |2 in a three-level -type system (Wang, Goorskey and Xiao [2001, 2002a]):

iN|μ21 |2 1 |ΩP |2 F + F ∗ 2γ31 ∼ χ= (2.12) − 1− , F 2γ + γ21 2γ + γ21 |F |2 h¯ with F ≡ γ + iΔP + (|ΩC |2 /4)/[γ31 + i(ΔP − ΔC )]. The first term in eq. (2.12) is the linear susceptibility, the second term is the contribution to the linear susceptibility from the higher-order density-matrix element, and the last term is the third-order (or Kerr-like) nonlinearity due to the finite probe intensity χ ∼ = χ (1) + 3χ (3) |EP |2 which is modified by the atomic coherence. The Kerrnonlinear index of refraction n2 (here n0 is the linear index) is given by (Wang, Goorskey and Xiao [2001] and Boyd [2003]) 12π2 Re χ (3) n20 c

1 4iNπ2 |μ21 |4 F + F∗ = Re − . 2γ + γ21 |F |2 F h¯ 3 cn20

n2 =

(2.13)

For atoms in a vapor cell, one can take into account the higher-order Doppler effect by integrating eq. (2.13) over a Maxwellian velocity distribution as described in Section 2.1 for the linear susceptibility. The first-order Doppler effect is eliminated by co-propagating the coupling and probe beams through the atomic cell inside the optical ring cavity. By neglecting the Doppler effect and

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∼ ±ΩC /2 for ΔC = 0. Also, γ31 in eq. (2.13), the maxima of n2 appear at ΔP = n2 > 0 for ΔP = −ΩC /2 and n2 < 0 for ΔP = ΩC /2. On the other hand, if ΔP = 0 and ΔC is changed, n2 behaves in the opposite way since ΔC acts similar to ΔP but with the opposite sign (Wang, Goorskey and Xiao [2001]). The experimental arrangement for measuring n2 in a three-level EIT system is shown in fig. 7. The rubidium atomic vapor is contained in a 5 cm-long cell having Brewster windows, which is wrapped in a μ-metal sheet to shield magnetic field, and heated to about 67.5◦ C. The cell is placed in an optical ring cavity comprised of three mirrors. The flat mirror M1 has a reflectivity of 99% while the concave mirrors M2 and M3 (both with R = 10 cm) have reflectivities of 97% and 99.5%, respectively. Mirror M3 is mounted on a piezoelectric transducer (PZT). The cavity finesse (F ) with the Rb atomic vapor cell is measured to be about 55 (far from any resonant absorption line) and the free spectral range is 822 MHz (for total cavity length ∼37 cm). The probe laser beam enters through mirror M2 and cir-

Fig. 7. Experimental setup. LD1 and LD2 – coupling and probe diode lasers, respectively; PBS1–PBS4 – polarizing cubic beam splitters; λ/2 – half-wave plates; FR – Faraday rotators; FP1–FP3 – Fabry–Perot cavities; D1–D4 – detectors; SAS – unit for saturation absorption spectroscopy; APD – avalanche photodiode detector. (After Wang, Goorskey and Xiao [2001].)

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culates in the cavity in one direction. The coupling beam is introduced through a polarizing beam splitter with an orthogonal polarization with respect to the probe beam and does not circulate in the cavity. The measurement of frequency detuning ΔC or ΔP can easily be carried out by another Fabry–Perot cavity in conjunction with a saturation absorption spectroscopy (SAS) setup. First the coupling beam was tuned and locked to the transition 5S1/2 , F = 2 → 5P1/2 , F ′ = 2, e.g., ΔC = 0 of the 87 Rb atom as shown in fig. 1(b). Then the probe beam was tuned and locked to the transition 5S1/2 , F = 1 → 5P1/2 , F ′ = 2 slightly below or above resonance. The average Rabi frequencies corresponding to the coupling and probe laser fields were ΩC = 2π × 72 MHz and ΩC = 2π × 11 MHz, respectively. The length of the cavity was scanned by the mirror mounted on the PZT, and the cavity transmission was monitored by an APD. In the absence of the coupling field (ΩC = 0, no EIT) the cavity transmission profile is basically symmetric. With the coupling beam turned on (with EIT), the cavity transmission profile becomes asymmetric. The degree of asymmetry in the cavity transmission profile is a direct measure of nonlinear phase shift which is proportional to n2 and given by δ=

2π[(n0 − 1)L + LC ] 2πL + n2 IP + Φ0 − 2πm, λ λ

(2.14)

where L is the length of the medium, LC is the cavity length, m is an integer, λ is the wavelength of the probe light, φ0 is the phase offset of the cavity, and IP is the intracavity intensity of the probe beam. The direction of the asymmetry gives the sign of the nonlinear coefficient n2 . As the cavity length is scanned from shorter to longer, the intracavity intensity will take longer to reach the peak power if n2 is negative because the first and second terms in eq. (2.14) have opposite signs. When n2 is positive, the asymmetry in cavity transmission will behave in the opposite way. The Kerr-nonlinear index of refraction n2 can be obtained directly by n2 =

[(t1 − tr ) − (tr − t2 )] , 2τ (Ir − Iδ )L

(2.15)

where Ir is the peak intensity of the cavity transmission at time tr ; t1 , t2 are the values of scan time when intensity falls to Ir /2 (∼ = Iδ ). In fig. 8 the measured Kerr-nonlinear index of refraction n2 is plotted for varying probe-beam frequency detuning while keeping the coupling beam on resonance (solid squares) or blocking it off (open circles). In the experimental measurements the same intracavity peak intensity at different ΔP was maintained

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Fig. 8. Measured Kerr-nonlinear coefficient n2 versus frequency detuning of probe beam with ΔC = 0 and ΩC = 2π × 72 MHz. Solid squares are with coupling beam, open circles are without coupling beam. (After Wang, Goorskey and Xiao [2001].)

by altering the probe power entering the cavity. Clearly, the Kerr-nonlinear index of refraction n2 is greatly modified near the probe resonance frequency at the EIT condition and varies sharply there. Both shape and magnitude of this curve match well with the theoretical model with Doppler broadening included (Wang, Goorskey and Xiao [2002a]). The enhancement in nonlinearity near resonance (ΔP ∼ = ±7 MHz for given ΩC ) is about 2 orders of magnitude compared to the two-level system. The positions of the maximal nonlinear coefficient n2 can be tuned away from near resonant frequency by increasing ΩC , which is consistent with the theoretical predictions (Wang, Goorskey and Xiao [2002a, 2002b]). Another interesting study of n2 is with respect to the variation of coupling field detuning ΔC with ΔP = 0 as depicted in fig. 9. This behavior is also predicted by eq. (2.13) as the roles of ΔP and ΔC differ only by sign. The importance of this result is that the sign of n2 can easily be changed by simply tuning ΔC (frequency of the controlling field) within a very small range. The change of n2 was measured as a function of coupling beam power PC (Wang, Goorskey and Xiao [2002b]) by keeping ΔC = 0, ΩP = 2π × 11 MHz and a certain value of ΔP . The measured data for ΔP = 40 MHz are plotted in fig. 10(a). The Kerr nonlinearity increases quickly as the coupling power increases, reaches a maximum, and then decreases slowly. However, near reso-

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Fig. 9. Measured Kerr-nonlinear coefficient n2 versus frequency detuning of coupling beam with ΔP = 0 and ΩC = 2π × 72 MHz. (After Wang, Goorskey and Xiao [2001].)

nance, n2 behaves differently. For example, at ΔP = 7 MHz, n2 keeps increasing as the coupling power increases, as shown in fig. 10(b), and is limited only by the available coupling power in that experiment. Notice that n2 is positive for ΔP = 40 MHz but becomes negative for ΔP = 7 MHz, as indicated in fig. 8. It should be noted that the measured n2 (about 7 × 10−6 cm2 /W) is quite large compared to the value achievable in a regular two-level atomic medium. This enhancement of nonlinearity in probe transition is due to the atomic coherence effect induced by the coupling beam (or constructive interference). Thus, the measured curves of n2 as functions of ΔP , ΔC , PC , etc. show that one can easily manipulate and control the Kerr-nonlinear coefficient of the probe transition by adjusting experimental parameters (frequency detuning and power of the coupling beam). The n2 measured in this experiment is related to n2 IP (which arises from the nonlinear susceptibility χ (3) and is responsible for self-phase modulation). The earlier reported work on n2 is related to n′2 IC [determined from the slope of the linear dispersion, and responsible for cross-phase modulation (XPM)] where reduced group velocity was observed (Kash, Sautenkov, Zibrov, Hollberg, Welch, Lukin, Rostovtsev, Fry and Scully [1999] and Budker, Kimball, Rochester and Yashchuk [1999]). Also, control of self-focusing and self-defocussing were reported earlier (Moseley, Shepherd, Fulton, Sinclair and Dunn [1995]). Recently,

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Fig. 10. Measured Kerr-nonlinear index of refraction n2 as a function of coupling-beam power for (a) ΔP = 40 MHz and (b) ΔP = 7 MHz. Other parameters: ΩP = 2π × 11 MHz; ΔC = 0. (After Wang, Goorskey and Xiao [2002b].)

direct measurement of cross-phase nonlinear refractive index n′2 was reported in a three-level -type 87 Rb atomic system using the same methodology as described above (Chang, Du, Yao, Xie and Wang [2004]). 2.2.2. Enhancement of nonlinearity in four-level atomic systems Schmidt and Imamoglu [1996] predicted giant Kerr nonlinearity mediated by EIT in a four-level N-type system. They analyzed a XPM scheme in such four-level system that exhibits a giant, resonantly enhanced nonlinearity along with a vanishing linear susceptibility. This system, shown in fig. 11(a), uses an EIT condi-

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Fig. 11. Schematics of (a) a four-level EIT system showing enhanced Kerr nonlinearity, and (b) a three-level system showing XPM. (After Kang and Zhu [2003].)

tion and is limited by two-photon absorption giving rise to resonant XPM without dissipation, which leads to a large conditional phase shift at extremely low pump power. The N-type four-level system contains a scheme consisting of levels |1 , |2 and |3 . Levels |2 and |3 are coherently coupled by a laser of frequency ωc with Rabi frequency ΩC . This results in EIT if the probe beam at ωa and the coupling beam at ωc are at two-photon resonance with the |1 ←→ |2 transition. In addition a signal field ωb is also present with a frequency detuning Δ from the |2 → |4 transition. Level |2 is assumed to be metastable so that there is no decay from this level, which can be achieved by considering the two hyperfine split levels of a ground state. The coupling field ωc introduces dressed states, and XPM occurs when the probe field is tuned between these two dressed states. These dressed states are separated by hΩ ¯ C with ωc on resonance which could be much smaller than the original atomic linewidth. In this four-level scheme the EIT on the |1 ←→ |3 transition has several important consequences. The most important one is the elimination of one-photon loss, which in a three-level scheme [fig. 11(b)] needs an undesirably large detuning from the intermediate level. Also, there is no one-photon absorption on the |2 ←→ |4 transition so this scheme is limited only by the small two-photon absorption given by Im[χ (3) ] allowing for the significantly longer interaction length required for efficient XPM. Secondly, no self-phase modulation is present in the system as all higher-order nonlinearities in the |1 ←→ |3 transition get canceled. So, any phase shift in ωa is caused purely by ωb and thus useful in quantum nondemolition measurements.

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The real part of χ (3) in this four-level scheme is given by (Schmidt and Imamoglu [1996]) N|μ13 |2 |μ24 |2 Γ4 , Re χ (3) FOUR = ΩC2 Δ 2ε0 h¯ 3

(2.16)

while the corresponding expression for the three-level scheme with two-photon absorption is given by (Schmidt and Imamoglu [1996]) N|μgi |2 |μiu |2 1 . Re χ (3) THREE = Δ2P Δ 8ε0 h¯ 3

(2.17)

By comparing these two expressions, one can see that in the four-level EIT scheme, detuning (ΔP ) from the intermediate level is replaced by the Rabi frequency (ΩC ) of the coupling beam. This Rabi frequency can be chosen to be much smaller than the one-photon detuning limited by the linewidth. This leads to a dramatic enhancement of χ (3) . Since the lower limit of ΩC is given by ΩC2 > Γ2 Γ3 for the EIT condition (Gea-Banacloche, Li, Jin and Xiao [1995]), ΩC can be chosen to be much smaller than Γ3 provided Γ2 ∼ = 0, which is easily achievable in rubidium or sodium system (Schmidt and Imamoglu [1996]). It was found that XPM nonlinearity is several orders of magnitude larger in this four-level system than in the usual three-level scheme involving two-photon transition without EIT, and the required laser intensity in the former scheme is several orders of magnitude smaller (Schmidt and Imamoglu [1996]). Also, in this scheme it is possible to make a conditional phase shift of π with just one photon and thus it is a promising candidate for quantum nondemolition measurements of weak signals and for quantum logic gate operations (Schmidt and Imamoglu [1996]). The experimental observation of the enhanced Kerr nonlinearity in terms of XPM in the four-level N-type EIT scheme was reported by Kang and Zhu [2003]. The experiment was done with cold 87 Rb atoms confined in a magneto-optical trap (MOT). The trapped 87 Rb atom cloud is ∼3 mm in diameter and contains ∼109 atoms. The schematic level configuration is shown in fig. 11(a). Here the probe, signal, and coupling transitions are: D1 line, 5S1/2 , F = 1 → 5P1/2 , F ′ = 1, D2 line, 5S1/2 , F = 2 → 5P3/2 , F ′ = 3, and D1 line, 5S1/2 , F = 2 → 5P1/2 , F ′ = 1, respectively, and these transitions are driven by three separate extended-cavity diode lasers, which overlap in the cold 87 Rb atomic cloud. In order to simultaneously measure the phase shift and the amplitude attenuation experienced by the probe laser, the frequency modulation (FM) spectroscopic technique was used. The transmitted probe beam was collected by a fast photodiode and processed with a lock-in amplifier and, simultaneously, two DC output signals, one in phase and the quadrature recorded with a digital oscilloscope, which give the nonlinear absorption coefficient and the XPM phase shift.

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Since the EIT suppresses the linear susceptibility, the measurements on quadrature and in-phase signals of the probe laser give the nonlinear susceptibilities. The peak XPM phase shift from measured quadrature signals was found to be 67% of the phase shift from linear dispersion experienced by a weak probe laser in a two-level system. Thus the resonantly enhanced Kerr nonlinearity derived from the measurement is several orders of magnitude greater than that of the conventional three-level XPM scheme in confirmation of the prediction of Schmidt and Imamoglu [1996]. Two-photon nonlinearity, which is the absorptive analogue of the Kerr nonlinearity of Schmidt and Imamoglu [1996] in a four-level N-type system, has been described by Harris and Yamamoto [1998] in photon switching by quantum interference; in their theoretical work, the optical system, in the ideal limit, may absorb two photons at the same time but not single photon, and therefore functions as an absorptive optical switch with an adjustable bandwidth and with a switching energy of about one photon per atomic cross-section. In another work treating the four-level N-type EIT system, Hong, Jack, Yamashita and Mukai [2002] theoretically discerned the three major contributions to χ (3) arising from the energy shifts of the ground states induced by the probe field. In their study of such four-level EIT system with frequency-degenerate probes, quantum interference among the three contributions cannot only enhance χ (3) more effectively in comparison to a three-level system with the same characteristic parameters, but can also make the ratio between its real and imaginary parts controllable. It was also predicted that the most effective enhancement in Kerr nonlinearity generally occurs at a frequency offset determined by the atomic transition frequency difference and the coupling Rabi frequency.

§ 3. Enhanced four-wave mixing processes with induced atomic coherence Extensive studies on four-wave mixing (FWM) processes have been carried out in the recent past because of their practical applications in generating new wavelengths of light, phase conjugation, and as tools for spectroscopic techniques. Many experiments in FWM were carried out in the strong-pump and weak-probe limit (both forward and backward configurations) using alkali-metal vapors as the nonlinear medium (see list of references in Lin, Rubiera and Zhu [1995]). Theoretical modelling of FWM including inhomogeneous broadening and effects of atomic collisions for two- and three-level atomic systems have been reported by Boyd [2003]. Under the condition that the Rabi frequency of the pump field is larger than the natural linewidth of atomic transition but smaller

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than the Doppler width, two peaks were observed in degenerate or nearly degenerate FWM processes under different strong-pump and weak-probe configurations, which can be explained well by using the dressed-state picture (Verkerk, Pinard and Grynberg [1986]). In another experimental study of cw nearly degenerate FWM emission from a Doppler-broadened Rb atomic system (D2 line), three peaks were observed under the condition that the Rabi frequencies of the forward pump, backward pump and probe beams be greater than the atomic natural linewidth but smaller than the Doppler width, as the probe laser was scanned across the same Rb D2 line. This three-peak spectral structure has been interpreted as the resonances in energy structure of the dressed atomic states (Lin, Rubiera and Zhu [1995]).

3.1. Three-level atomic systems A typical three-level system used for FWM is shown in fig. 12. The pump waves with frequencies ω1 and ω2 interact with the adjacent transitions while the probe wave at frequency ωP interacts with one of the transitions (Li and Xiao [1996]). Under appropriate phase matching conditions (as discussed further), a new signal wave is generated at the frequency ωC = ω1 + ω2 − ωP . In the previous discussion (Section 2), the key idea of enhancing nonlinear optical processes by atomic coherence was that the absorption of the generated field (1) (related to linear susceptibility χD ), as well as the pump and probe fields, can

Fig. 12. NDFWM based on EIT in a three-level -system of 87 Rb atoms. Pumping waves are at frequencies ω1 and ω2 , and the probe wave is at frequency ωP . The signal wave is generated at the frequency ωC = ω1 + ω2 − ωP and experiences transparency produced by the pumping wave at frequency ω2 . (After Li and Xiao [1996].)

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be substantially reduced as a result of EIT effects, while at the same time the nonlinear coefficient χD(3) associated with the nonlinear optical process is greatly enhanced. This leads to a significant enhancement in nonlinear optical signal generation in an optically dense material, in which there would be a large absorption for the generated field without the EIT effect. An interesting effect of enhancing the nonlinearity χ (3) in an atomic medium is the observation of enhanced nondegenerate four-wave mixing (NDFWM) (Li and Xiao [1996]). The experiment was performed in a three-level -type Dopplerbroadened system of Rb atoms [fig. 12] with cw diode lasers. The two hyperfine levels Fg = 1 and Fg = 2, spaced by 6.8 GHz, of the ground state 5S1/2 serve as the two lower states of the system. The excited state 5P1/2 , Fe′ = 1 serves as the common upper state. As the weak probe wave ωP is tuned to (ω2 − 450) MHz, a new signal wave with frequency ωC is generated at a resonance frequency with the Fg = 1 to Fe′ = 1 transition, satisfying the phase-matching condition for the wavevectors. Under the steady-state condition most atoms are in the ground state Fg = 1 due to the optical pumping. In the absence of any EIT, the generated signal wave at frequency ωC will suffer a large absorption in the dense atomic medium. However, in the presence of EIT as shown in fig. 12, this atomic system is transparent to the generated signal field due to NDFWM. Owing to this transparency one can further increase the atomic density of the medium and, therefore, the total efficiency of the nonlinear optical process. The propagation equation for the amplitude of a generated signal wave in the atomic medium can be expressed as (Li and Xiao [1996]) 1 ∂EC ωN (1) ∂EC (3) + =i χD EC + χD E1 E2 EP∗ , ∂z c ∂t 2c

(3.1)

where Ei , i = 1, 2, P, C, correspond to the amplitudes of the fields with frequencies ω1 , ω2 , ωP and ωC , respectively; N is the atomic density of the medium. (1) The first term on the right-hand side of eq. (3.1) (proportional to χD ) describes the linear absorption and dispersion of the medium for the generated signal wave, (3) and the second term (proportional to χD ) describes the nonlinear FWM process. The measurement of linear susceptibility in steady state is straightforward by using standard procedure (Xiao, Li, Jin and Gea-Banacloche [1995]) in the absence of the pumping wave ω1 (E1 = 0). The steady-state solution of eq. (3.1) reads (1) (3) EC = (iω/(2c))NLχD E1 E2 EP∗ for an optically thin medium (χD N L ≪ 1). (3) Hence the measurement of χD can be done, in principle, by measuring the generated signal wave EC with low atomic density. The experimental arrangement has been described schematically by Li and Xiao [1996]. Two diode lasers provide the pumping waves ω1 and ω2 . The probe

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wave (ωP ) is produced from the side-band modulation of the pumping wave ω2 by applying an RF modulation frequency ωM = 450 MHz. The side-band intensity is 10−4 that of the pumping wave carrier (ω2 ). These three waves pass collinearly through a temperature-controlled Rb vapor cell. The generated signal wave ωC (= ω1 + ωM ), which has the same polarization as the pumping wave ω1 , is detected by the beat signal at the frequency ωM between the signal wave ωC and the pumping wave ω1 using an APD detector connected to a spectrum analyzer. In order to measure Im χD(1) , the amplitude of the RF modulation signal is set to zero (EP = 0), the intensity of the pumping wave ω1 is reduced to 50 µW/cm2 , and the field ω1 is scanned across the resonance. The absorption reduction at the resonant frequency is due to the EIT effect [fig. 13(a)]. The NDFWM signal EC E1∗ + EC∗ E1 ∝ Re χD(3) is recorded by the beat signal between the generated

(1)

Fig. 13. (a) Measured linear susceptibility (absorption) Im χD for the signal wave versus frequency

(3) detuning. (b) Third-order susceptibility Re χD for the NDFWM in an optically thin medium with an (3) (1) atomic density of 2.6 × 1010 cm−3 . Both Im χD and Re χD are in arbitrary units. (After Li and

Xiao [1996].)

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signal wave ωC and the pumping wave ω1 , when ω1 is tuned across its center (3) frequency (ω21 − 450) MHz. This gives the third-order susceptibility Re χD as (3) depicted in fig. 13(b). It is clear that the third-order susceptibility Re χD is reso(1) nantly increased while the absorption Im χD is reduced at the resonant frequency due to the EIT condition to allow the generated signal field to propagate in the dense atomic medium.

3.2. Four-level atomic systems In an earlier work (Hemmer, Katz, Donoghue, Cronin-Golomb, Shahriar and Kumar [1999]) using a four-level atomic system in a double- system, optical high phase-conjugate gain (∼50) was observed under low-intensity pump (1 W/cm2 ) and response time in microseconds. In this experiment CPT was identified as the phase-conjugate mechanism. Another interesting experiment on NDFWM is in a double- system of rubidium atomic vapor which was demonstrated using pumping fields propagating collinearly (Lu, Burkett and Xiao [1998]). By comparing different pumping configurations, direct experimental evidence of EIT-enhanced NDFWM signal was established. The double- system involves four energy levels of 87 Rb (fig. 14). The principle of FWM is similar to that for the three-level system, but the advantage of the four-level system is that the residual Doppler linewidth (0.01 MHz) is much less than the natural linewidth (∼6 MHz) of the transitions. By using copropagating laser beams, the two-photon resonance, as well as EIT, is preserved

Fig. 14. Schematic of the four-level double- system of the Rb atom. (After Lu, Burkett and Xiao [1998].)

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for all atoms, offering an ideal system for investigation of the influence of EIT in the double- system. The frequencies of the pumping fields can be scanned across the two transition lines F = 1 → F ′ = 1 and F ′ = 2. One of the pumping diode lasers (tuned to the transition F = 2 → F ′ = 1) is modulated by an 812 MHz RF signal to produce a sideband serving as the probe signal. As a result of this modulation, the carrier and one sideband of this diode laser are resonant with transition lines F = 2 → F ′ = 1 (ω1 ) and F ′ = 2 (ω2 ), as pumping and probe fields (fig. 14), respectively. Another pump field (ω1 ) is generated from a separate diode laser. All these beams propagate collinearly and are focused onto a temperature-stabilized rubidium vapor cell. In order to have an efficient NDFWM process the conditions Φ4 = Φ1 − Φ2 + Φ3 and Δ4 = Δ1 − Δ2 + Δ3 should be satisfied. Here Φi denote phase variations of the fields and Δi = ωi − ωj k are detunings of the field frequencies from corresponding atomic line centers. A reference beam of frequency (ω1 +80) MHz, generated from an upshift by an acousto-optic modulator, beats with the generated signal wave yielding 732 MHz and 892 MHz beat signals for ω4 = (ω1 ± 812) MHz, respectively. Thus the generated signal field at frequency ω4 is clearly identified. For the same four-level configuration of fig. 14 one can have several other double- or systems for generating NDFWM. The nonlinear signal is shifted either 812 MHz up or 812 MHz down from the pump field ω1 as shown in the insets of fig. 15. Figure 15 shows typical results corresponding to nonlinear signals at (ω1 + 812) MHz and (ω1 − 812) MHz, respectively. The intensity of the beat signal is proportional to the square root of the generated signal power P4 √ (IS ∝ P4 ). In fig. 15 the four peaks are attributed to four particular NDFWM processes in either or double- configurations as shown in the insets, and they are basically NDFWM in three-level systems as described earlier. The probe field ω3 in insets A and C comes from the lower sideband of the diode laser. The inset B is the same as the scheme shown in fig. 14. By observing the intensity variation of peak B with resolution bandwidth of the spectrum analyzer the estimated bandwidth of the beat note turns out to be 2 kHz, which is much narrower than the natural linewidth (6 MHz) of the atomic transition. This implies that the generated signal wave follows the phase of the pump field ω1 . The estimated bandwidths of the other three peaks are also in the order of several kHz. The dependence of generated signal power (for the double- system of fig. 14) on the power of one of the pump fields is shown in fig. 16(a). The signal power first rises and then falls beyond the pump power of 1 mW, indicating a critical balance between the two strong pumps ω1 and ω2 for an efficient NDFWM process. In contrast, other double- (fig. 15, inset D) and (fig. 15, inset A) systems show ordinary saturation behavior.

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Fig. 15. Beat signal intensity versus the detuning of the pump field ω1 . Upper trace, 732-MHz beat signal; lower trace, 892-MHz beat signal. The four insets show the relevant level configuration for each peak. The pump powers are P1 = 1 mW and P2 = 25.3 mW. (After Lu, Burkett and Xiao [1998].)

Zhu’s group has recently demonstrated resonant FWM with slow light in a cold atomic sample (Kang, Hernandez and Zhu [2004b]). The experiment was carried out in cold 87 Rb atoms confined in a magneto-optical-trap (MOT) using D1 transition lines in a four-level double- configuration under the condition that the pump and probe intensities are below their saturation levels, thus ensuring slow group velocity for the probe light and the generated signal light. In this way a high FWM efficiency of 10% was obtained (fig. 17). Another work under CPT conditions reported efficient nonlinear frequency conversion in a quadruple resonant double- atomic system using collimated laser fields with modest power densities (Merriam, Sharpe, Shverdin, Manuszak, Yin and Harris [2000]). This work is an experimental demonstration of non-

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Fig. 16. Generated nonlinear signal power as a function of pump power P1 . The dotted curves are guidelines for the experimental data. P2 = 25.3 mW. (a) Double- system of fig. 14. (b) Two other double- systems; the upper and lower traces represent the configurations of insets D and A of fig. 15, respectively. (After Lu, Burkett and Xiao [1998].)

Fig. 17. (a) Calculated group velocity of the probe light and (b) FWM efficiency vs the coupling Rabi frequency ΩC . The dots (solid lines) are the experimental data (theoretical calculations). The experimental parameters are Ω/2π = 4 MHz, ΩP /2 = 0.4 MHz. (After Kang, Hernandez and Zhu [2004b].)

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linear optics in a density-invariant Rabi-frequency matching regime. In this experiment, three resonant, arbitrarily phased fields are applied at z = 0 and, after a characteristic length, the atoms are transparent to all fields through generation of a properly phased fourth resonant field. The main reason for transparency to occur is the equality of the ratios of complex Rabi frequencies in each -channel. Also, as soon as this transparency is established then no change in amplitude or phase occurs in the system. The conversion efficiency is limited by the preparation energy requirement necessary to establish EIT in the first -channel and by power broadening in the second -channel. Overall 1% energy conversion efficiency was achieved and a vacuum ultraviolet (VUV) pulse energy of 1 µJ was generated by applying optical pulses of 10 µJ at 233 nm and 50 µJ at both 283 nm and 406 nm wavelengths. Recently, Harris’s group has reported another experimental demonstration of FWM with an EIT system in an altered double- configuration using cold atoms in a MOT (Braje, Balic, Goda, Yin and Harris [2004]). In this work a backward geometry was used in an optically thick medium, where weak Stokes and antiStokes beams counterpropagate. The energy-level schematic for 87 Rb in the altered double- configuration is shown in fig. 18. The transition |2 → |3 is driven by a coupling laser field of frequency ωC . The coupling field is responsible for the quantum interference and creates transparency for an anti-Stokes field. The interesting finding of this work is that the bandwidth of the transmission window for applied anti-Stokes field can be controlled by the magnitude of the coupling field and the optical depth of the atomic medium. A spin wave of welldefined wavevector and phase is generated by the anti-Stokes and coupling laser. Another laser (called ‘pump’) of frequency ωP (off-resonant from the |1 → |4 transition) then mixes with the spin wave to produce a traveling wave dipole moment at Stokes frequency ωS . The energy conservation is preserved in the system,

Fig. 18. Backward-wave four-wave mixing in a prototype, four-level system. In the presence of strong pump and coupling lasers, a weak anti-Stokes laser generates a counterpropagating, phase-matched Stokes beam. (After Braje, Balic, Goda, Yin and Harris [2004].)

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i.e., ωS = ωP + ωC − ωAS . The appropriate condition to obtain transparency at anti-Stokes frequency is obtained when ΩC2 (squared Rabi frequency of coupling beam) is greater than the product of the decay rate from state |3 and the dephasing rate of transition |1 → |2 . The conversion/generation efficiency of this EIT-based process is the square of the optical depth times the generation efficiency of the conventional nonlinear optical process. The photons are generated exactly at the line center of the resonant transition because of the EIT conditions. Both Raman gain and parametric gain (due to phase-matched interaction of pump, Stokes, and coupling lasers with the anti-Stokes wave) play an important role in this process of Stokes wave generation. An experimental study of resonant six-wave mixing in coherently prepared Rb atoms was recently carried out by Kang, Hernandez and Zhu [2004a]. In this experiment EIT provided control in a four-level atomic system by suppressing the linear susceptibility and enhancing the nonlinear susceptibilities, which led to resonantly enhanced, slow-photon six-wave mixing at low light intensities. The light emission in the six-wave mixing process can be viewed as resulting from diffraction of slow light off a resonant nonlinear grating induced in the four-level system by a standing-wave pump field.

§ 4. Controlled optical bistability and optical multistability with three-level atoms inside an optical cavity Another remarkable application of EIT and atomic coherence is in controlling the phenomenon of optical bistability (OB) in atomic systems which shows a hysteresis loop in the plot of cavity output intensity versus input intensity. In the 1980s OB attracted a great deal of attention, both theoretical and experimental, for systems of two-level atoms confined in an optical cavity, owing to potential applications in all-optical switching, memory, and optical transistors. Optical bistability in atomic systems has generally been divided into two classes, i.e., absorptive and dispersive (refractive) OB. Nonlinear absorption (or the saturation of the atomic transition) and an intensity-dependent refractive index are thought to be the two physical mechanisms generating absorptive and dispersive atomic OB, respectively. The literature concerning early work on OB in two-level systems is very exhaustive and hence readers can refer to the reviews by Lugiato [1984] and Gibbs [1985]. Most of the earlier experimental studies on atomic OB were devoted to two-level alkali atoms confined in an optical resonator (Lugiato [1984], Gibbs [1985], Rosenberger, Orozco and Kimble [1983] and Orozco, Kimble, Rosenberger, Lugiato, Asquini, Brambilla and Narducci [1989]). Hysteresis has

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been observed experimentally in the transmission spectrum (i.e., vacuum Rabi splitting) of a cavity filled with a collection of two-level atoms. This system under weak excitation behaves like simple harmonic oscillators, which become anharmonic as the excitation increases. The anharmonicity grows to a point where hysteresis appears in the transmission spectrum, eventually causing the two peaks to merge into one (Gripp, Mielke, Orozco and Carmichael [1996]). The theoretical models of atomic OB have considered the interaction of a collection of two-level atoms with a single-mode field in an optical cavity (Lugiato [1984], Gibbs [1985], Rosenberger, Orozco and Kimble [1983] and Orozco, Kimble, Rosenberger, Lugiato, Asquini, Brambilla and Narducci [1989]). Also, bistable behavior was studied theoretically (Harshawardhan and Agarwal [1996] and Walls and Zoller [1980]) and observed experimentally (Mlynek, Mitschke, Deserno and Lange [1984]) in three-level atomic systems inside optical cavities. The OB from coherently driven -type three-level atoms inside a Fabry–Perot resonator was observed due to a nonlinear mechanism which was attributed to transverse optical pumping and population trapping in ground-state sublevels. This system under Hanle resonance displayed dispersive or absorptive bistability (Mlynek, Mitschke, Deserno and Lange [1984]). Recently, controllability of the shape, width, threshold, and direction (rotation of hysteresis loop) of atomic OB have been experimentally demonstrated in a three-level atomic EIT system inside an optical ring cavity (Joshi, Brown, Wang and Xiao [2003], Joshi and Xiao [2003], Joshi, Yang and Xiao [2004, 2005] and Wang, Goorskey and Xiao [2002c]). An interesting extension of OB is optical multistability (OM) which was first predicted and observed in a nonlinear medium interacting with two different optical cavity field modes (see list of references in Joshi and Xiao [2003]). In an early theoretical work Kitano, Yabuzaki and Ogawa [1981] predicted optical tristability in a three-level system in -configuration under a large atomic detuning and no saturation. Later, Cecchi, Giusfredi, Petriella and Salieri [1982] observed such phenomenon experimentally. Savage, Carmichael and Walls [1982] generalized this work to include saturation in the dispersive limit. By including the effect of ground-state coherence, Arecchi, Kurmann and Politi [1983] predicted tristability as well as higher-order bistability in a Na atomic system under near resonant D1 excitation. In the following, we will mainly discuss the progress made in recent years towards the controllability of OB/OM in three-level atomic system inside an optical ring cavity. Based on exact knowledge and controllability of the linear absorption, dispersion, and Kerr nonlinearity at various coupling and cavity field parameters (such as optical powers and frequency detunings), one can control and manipu-

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late the shape and direction of rotation of OB/OM in such a composite system by simply tuning various experimentally controllable parameters.

4.1. Theoretical calculation The phenomenon of atomic OB in N homogeneously broadened two-level atoms confined in an optical cavity has been modelled extensively over the past two decades. The basic OB phenomena could be well understood by the standard model given by Bonifacio and Lugiato [1978] almost thirty years ago. Their model essentially considers a unidirectional ring cavity having four mirrors Mi , i = 1, . . . , 4, as shown in fig. 19. M1 and M2 have reflection and transmission coefficients R and T , respectively, such that R + T = 1. For the sake of simplicity we assume M3 and M4 to be 100% reflectors. The atomic sample is contained in a vapor cell of length L. In the current consideration, let us consider atoms in a three-level -type configuration, as described in Section 2.1 and depicted in fig. 1(b). The atomic dynamics of the system is governed by the density-operator eqs. (2.9). The electric field seen by the atoms can be written as

E = EP e−iωP t + EC e−iωC t + c.c. . (4.1) The coupling field EC at frequency ωC applied to the transition |3 to |2 does not circulate inside the optical ring cavity. The probe field for the EIT system at

Fig. 19. Schematic diagram of a unidirectional ring cavity having four mirrors (M1 –M4 ) and an atomic vapor cell of length L. Mirrors M3 and M4 are perfectly reflecting (R = 1 for both). The incident and the transmitted fields are represented by EPI and EPT , respectively, and the coupling field EC is noncirculating in the cavity.

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frequency ωP (interacting with atomic transition |1 to |2 ) circulates inside the cavity as the cavity field. The coupling field acts as a mere controlling field. The induced atomic polarization responsible for OB is given by P (ωP ) = Nμ12 ρ12 ,

(4.2)

where N is atom number density. The probe field EP enters into the cavity from the partially transparent mirror M1 and drives one of the atomic transitions in such EIT system. The coupling field provides further regulation of the induced polarization P (ωP ) on the probe transition via quantum coherence effect and thus modifies the absorption/dispersion profiles of the atomic medium for the cavity field. Because of the boundary conditions for the optical ring cavity, the incident field EPI , the transmitted field EPT and fields at different locations in the cavity EP (0, t)/EP (L, t) obey the following boundary conditions (Lugiato [1984] and Gibbs [1985]): √ EPT = T EP (L, t), √ EP (0, t) = T EPI (t) + Re−iδ0 EP (L, t − Δt), (4.3) where L is the length of the atomic sample and Δt = (2l + L)/c is the time taken by light to travel from mirror M2 to mirror M1 via M3 and M4 [l is the length of the side arm between M2 (M1 ) and M3 (M4 )]. The cavity detuning is defined as δ0 = (ωcav − ωP )LT /c, where ωcav is the frequency of the cavity mode nearest to frequency ωP and LT ∼ = 2(l + L) represents the total length of the ring cavity. The dynamical evolution of the probe field inside the cavity is governed by ∂EP ∂EP (4.4) +c = 2πiωP μ12 P (ωP ), ∂t ∂z with the boundary conditions defined in eq. (4.3). In order to obtain the polarization P (ωP ) one needs to first numerically solve the set of density-operator eqs. (2.9) in the steady-state limit, and then to integrate eq. (4.4) using eq. (4.2) in the steady-state limit over the length of the sample. The steady-state boundary conditions become √ EPT = T EP (L), √ EP (0, t) = T EPI + Re−iδ0 EP (L). (4.5) In the absence of the coupling field, the -system reduces to the usual two-level system (Lugiato [1984] and Gibbs [1985]). The atomic OB results from the numerical simulation procedure described above are presented in fig. 20. It is conspicuous from fig. 20 that the coupling field causes lowering of the bistability threshold due to Autler–Townes splitting

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Fig. 20. The decrease in the threshold due to the presence of coupling field for three-level atomic system in -type configuration. Here C = 400, ΔP = 1, ΔC = 0, γ21 = γ23 = 1. Curves A, B, C, D and E are for ΩC = 1, 3, 5, 7 and 10, respectively. All parameters are in units of γ1 .

and EIT, thus providing controllability of the atomic OB curve (Harshawardhan and Agarwal [1996]). Due to EIT, the absorption at the line center decreases with increasing strength of the coupling field. When the coupling field becomes too large the bistability disappears due to the modified absorption/dispersion, as well as nonlinear properties of the system. The model for theoretically studying OM in three-level rubidium atom in -configuration is basically the same as that discussed for OB above. In order to quantify the origin of OM in this composite system we consider the steadystate polarization P (ωP ) solved from the density-operator equations. In general, the polarization is a ratio of two polynomials of orders 5 and 6 in ΩP , of the form (Harshawardhan and Agarwal [1996]) P (ωP ) =

ΩP [a1 + a2 |ΩP |2 + a3 |ΩP |4 ] , b1 + b2 |ΩP |2 + b3 |ΩP |4 + b4 |ΩP |6

(4.6)

in which the complex numbers ai and bi are functions of ΔC , ΔP , PC , atomic density and decay constants. This form of P (ωP ), showing higher-order nonlinearities and complicated dependence of absorption/dispersion on system parameters, is responsible for the OM observed in this three-level atomic system inside an optical cavity. In the case of two-level atoms P (ωP ) reduces to a ratio of a linear and a quadratic polynomial in ΩP and, hence, can only show bistable behavior.

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4.2. Controllable optical bistability The basic experimental setup for atomic OB measurements is similar to that used in measuring n2 for the three-level -type configuration of rubidium atoms, as described in Section 2.2.1 (see fig. 7). The coupling field is tuned to the 5S1/2 , F = 2 → 5P1/2 , F ′ = 2 transition while the probe (cavity) field is tuned to the 5S1/2 , F = 1 → 5P1/2 , F ′ = 2 transition in 87 Rb (D1 line) near 795 nm. With the frequencies of both diode lasers locked, the length of the optical ring cavity is scanned across its resonance by applying a ramp voltage to the PZT on one of the cavity mirrors. Without applying the coupling field the cavity transmission profile is symmetric. The presence of the coupling field enhances n2 of the atomic medium, leading to an asymmetry in cavity transmission profile. As the cavity length is scanned from longer to shorter (with a scan rate of 0.45 µm/ms and n2 > 0), the transmission profile becomes asymmetric, reaching its maximum value much later than if n2 = 0 [see, e.g., eq. (2.14)]. The upper trace of fig. 21 shows this clearly for ΔC = 7 MHz, ΔP = 0 corresponding to maximal positive Kerr nonlinearity (as shown in fig. 9) (Wang, Goorskey and Xiao [2001, 2002c]). The Rabi frequencies were estimated to be ΩC = 2π × 69 MHz and ΩP = 2π × 20 MHz at the center of the Rb vapor cell. When the cavity is scanned from shorter to longer (n2 > 0) for the same parameters (Wang, Goorskey and Xiao [2002c]), the cavity transmission intensity takes the lower trace of fig. 21 which rises sharply and falls off gradually with a much longer time. This hysteresis behavior is a clear indication of OB for the intracavity field and is a consequence of enhanced Kerr nonlinearity for the intracavity field interacting with the atoms. In order to monitor the input–output intensity characteristics of the cavity field an electro-optical modulator, was used in the path of the probe (cavity) field before it entered the cavity. The optical cavity was locked with another diode laser and a triangular modulation of the cavity input intensity with a duration of 5 ms was applied using this modulator. The temperature of the atomic vapor cell was kept at about 68◦ C in this experiment. Figure 22 displays several OB curves which are experimentally controlled by coupling-beam frequency detuning (ΔC ) and power (PC ), keeping all other parameters (like atom number density, cavity detuning, and ΔP ) unchanged. Figure 22(a) shows a typical OB with ΔP = 0, ΔC = 51.4 MHz, PC = 1.5 mW, and cavity detuning of about 40 MHz (Joshi, Brown, Wang and Xiao [2003]). Figure 22(b) clearly shows how one can control the OB curve by simply changing the parameter ΔC to 25.7 MHz. Both switching thresholds (Y1 and Y2 ) and width (Y2 − Y1 ) of the OB can be controlled by changing ΔC and keeping PC fixed. The upper and lower switching threshold intensities Y1 and Y2 are defined at

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Fig. 21. Cavity transmission profile showing bistability for PPin = 30 mW, PC = 18.5 mW, ΔP = 0, ΔC = 7 MHz. Upper trace is when cavity length is scanned from shorter to longer. Lower trace is the opposite (n2 > 0 in both cases). (After Wang, Goorskey and Xiao [2002c].)

Fig. 22. The input–output intensity characteristics of the optical cavity with ΔP = 0: (a) for PC = 1.5 mW, ΔC = 51.4 MHz; (b) for PC = 1.5 mW, ΔC = 25.7 MHz; (c) for PC = 8.4 mW, ΔC = 85.6 MHz; (d) for PC = 8.4 mW, ΔC = 256.8 MHz. (After Joshi, Brown, Wang and Xiao [2003].)

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dY/dX = 0 in the OB hysteresis curve. In figs. 22(c,d) PC has been changed to 8.5 mW, with ΔC = 85.6 MHz in (c) and ΔC = 256.8 MHz in (d). Dramatic changes in the shape of the OB hysteresis curve are apparent. Thus, the bistability range and the switching intensity thresholds Y1 and Y2 can be experimentally controlled very effectively by the additional auxiliary beam parameters PC and ΔC in such three-level systems (Joshi, Brown, Wang and Xiao [2003]), which is not available in a two-level system. The OB intensity threshold values are functions of many parameters, e.g., ΔP , cavity detuning, atomic number density (or cooperativity parameter), PC and ΔC . The reason for observing OB at quite low intracavity field intensity and weak coupling intensity is a result of enhanced Kerr nonlinearity near resonance due to atomic coherence being induced by the coupling beam interacting with the nearby transition slightly off-resonance (i.e., ΔC = 0) in such EIT medium. The OB curves displayed in figs. 22(a,b) are mixtures of absorptive and dispersive kinds where the dispersion is also greatly enhanced by the Kerr nonlinearity in such systems (Wang, Goorskey and Xiao [2001]). When ΔC is raised [fig. 22(c)] the nonlinearity in the system reduces, and so the switching intensity Y2 , as well as the width of the hysteresis cycle, increases. For very large ΔC [256.8 MHz in fig. 22(d)] the dispersion is governed mainly by frequency detuning, a situation similar to two-level dispersive OB (Lugiato [1984] and Gibbs [1985]). To quantify the controllability of OB curves with variation of ΔC at given PC , the switching threshold intensity ratio Y2 /Y1 is plotted in fig. 23 as a function

Fig. 23. Ratio of switching threshold intensities Y2 /Y1 as a function of ΔC . Curve A is for PC = 8.4 mW, curve B is for PC = 1.5 mW. (After Joshi, Brown, Wang and Xiao [2003].)

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of ΔC for two different coupling intensities (curve A: PC = 8.4 mW; curve B: PC = 1.5 mW) (Joshi, Brown, Wang and Xiao [2003]). For small ΔC , Y2 /Y1 is small (indicating a lower OB threshold) because of the enhanced Kerr nonlinearity near resonance. As ΔC rises, Y2 /Y1 also rises because the nonlinearity reduces substantially. Further increase in ΔC again reduces Y2 /Y1 , which is due to the effect of dispersion caused by ΔC alone (at larger ΔC , the Kerr nonlinearity n2 decreases appreciably and the system behaves like a two-level atomic system). A similar trend is seen in the width of the OB hysteresis cycle with changing ΔC . The OB behavior observed in such three-level EIT systems is quite different from the previously studied two-level atomic systems. Due to the induced atomic coherence near EIT resonance, the absorption, dispersion and nonlinearity are all greatly altered. The advantages of such a three-level system are its simplicity of operation and controllability. By propagating the coupling and probe fields collinearly through the vapor cell containing three-level -type rubidium atoms there is no need to use an atomic beam or cold atomic samples for eliminating the first-order Doppler effect.

4.3. Controllable optical multistability Observations of multistable/multiple hysteresis behavior with nonlinear media in optical cavities were reported by several groups, and in the early experiments the optical cavity was filled with atoms having several degenerate or nearly degenerate sub-levels in the ground state and driven by linearly polarized light (Giusfredi, Salieri, Cecchi and Arecchi [1985], Giacobino [1985], Hamilton, Sandle, Chilwell, Satchell and Warrington [1983] and Mitschke, Deserno, Lange and Mlynek [1986]). In the transmitted light, polarization switching occurs and three different stable states of polarization can exist, giving rise to multiple hysteresis and symmetry-breaking bifurcation attributed to competition between hyperfine and Zeeman pumping. Most of these early experiments used magnetic fields and high-pressure buffer gases, and relied on the Zeeman coherence as an efficient mechanism for observing OB/OM. Controlling OB to OM behavior using spontaneously generated coherence a in three-level atomic medium inside an optical ring cavity was predicted (Joshi, Yang and Xiao [2003a, 2003b]). Experimental demonstration of a new kind of OM was recently reported (Joshi and Xiao [2003]) in a three-level -type configuration of Rb atoms inside an optical ring cavity. As the frequency detuning of the coupling beam and the temperature of the vapor cell (which is directly related to the atomic density) increase further from

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Fig. 24. Observed OM in the input–output intensity characteristics of the system with parameters ΔP = 64 MHz, ΔC = 43 MHz, T = 90.5◦ C and PC = 8.5 mW. (After Joshi and Xiao [2003].)

the conditions for observing OB described in the preceding section, multistable hysteresis behavior emerges (Joshi and Xiao [2003]). Figure 24 displays a typical OM curve observed in such a system under the experimental conditions of PC = 8.5 mW, T = 90.5◦ C, ΔC = 43 MHz, ΔP = 64 MHz. Two sets of three stable steady-state points on the extended hysteresis cycle can be seen. The path taken by the cavity field intensity, as the input field power increases from zero to about 4.5 mW and back down to zero again, is marked by arrows. Such multistable behavior appears only between the typical highly absorptive and dispersive parametric regions. In fig. 25, curves (i) and (ii) represent pure dispersive and pure absorptive bistability, respectively, for parameters close to the experimental conditions of fig. 24. Curve (i) [(ii)] is plotted after artificially deleting the absorption [dispersion] contribution from the cavity input–output intensity relationship. As the input intensity rises, the cavity output intensity moves on the dispersive bistable curve (i) to point A (the upper threshold), where the output intensity jumps up to the upper branch of the dispersive curve at point B and further increases on this dispersive bistable curve (i). When the cavity input intensity comes down again, the output intensity switches to the upper branch of the absorptive bistable curve (ii) where the two curves cross each other at point E. The output intensity keeps going down to point C (the lower threshold) and jumps down to D, then goes back to zero. The path taken as described above more or less

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Fig. 25. Construction of optical multistable behavior (represented by the dark lines) with the help of pure-dispersive [curve (i)] and pure-absorptive [curve (ii)] optical bistability curves co-existing in a system with similar parameters as in fig. 24. (After Joshi and Xiao [2003].)

simulates the multistable curve observed in the experiment, as shown in fig. 24. Note that the solid curve in fig. 25 remains on the stable parts of the two bistable curves (i) and (ii), hence the observed OM is a result of coexisting absorptive and dispersive optical bistabilities in this EIT system. Figure 26 shows the transition from OB to OM as the atomic number density is increased by raising the temperature of the Rb vapor cell (Joshi and Xiao [2003]). The parameters are set for the experiment at ΔP = 0, ΔC = 42.8 MHz, PC = 8.5 mW, and temperatures of (a) T = 65.4◦ C, (b) T = 74.5◦ C, (c) T = 85.3◦ C and (d) T = 92.5◦ C. It is clear that a high atomic number density is required for observing multistable behavior. A similar transition from OB to OM can be realized by fixing the temperature of the atomic cell and changing the coupling-frequency detuning ΔC . For small ΔC the intensity input–output curve has typical absorptive OB characteristics, but as ΔC increases (with all other parameters fixed), OM behavior begins to appear (Joshi and Xiao [2003]). The observed OB to OM behavior in this system of three-level atoms inside an optical ring cavity is quite different from what was observed in previously studied atomic systems. Multistability in conjunc-

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Fig. 26. Input–output intensity characteristics of the system for different atomic number densities, with parameters ΔP = 0, ΔC = 42.8 MHz, PC = 8.5 mW: (a) T = 65.4◦ C; (b) T = 74.5◦ C; (c) T = 85.3◦ C; (d) T = 92.5◦ C. (After Joshi and Xiao [2003].)

tion with some phase-coherent techniques (Mair, Hager, Phillips, Walsworth and Lukin [2002]) can be utilized to make improved optical transistors, memory elements, and all-optical logic gates. Experimental demonstrations of controlling such OB/OM behavior provide an important step towards these goals.

4.4. Controllable direction of hysteresis cycle in optical bistability/optical multistability Typical hysteresis curves have a counter-clockwise (called ‘forward’) rotation due to energy considerations. For example, in OB with two-level atoms, the counterclockwise hysteresis cycle (HC) (or forward HC) is attributed to the dissipation in the system (Gibbs [1985]). Other hysteresis phenomena in ferroic materials also have forward HCs (Vadhawan [2002]). However, in an early OB experiment with a semiconductor medium (Jewell, Gibbs, Tarng, Gossard and Wiegmann [1982]), where a 4.2 µm thick GaAs etalon was used at 80◦ C, the competition between electronic nonlinearity due to free excitons and the thermal effect resulted in the

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switching-down intensity being higher than the switching-up intensity in the OB, producing a clockwise hysteresis cycle (or ‘backward’ HC). In that experiment, the cavity detuning also played a significant role. Backward HC was also observed in a cavityless OB system based on an optically induced absorption change due to the thermal effect near the free and bound excitons in a CdS semiconductor system (Dagenais and Sharfin [1985]). The three-level EIT system discussed above for controlling OB/OM is quite versatile, and it was demonstrated recently that the width and direction of the OB/OM hysteresis cycle could be efficiently controlled by only varying the coupling-laser frequency detuning (ΔC ) while keeping all other experimental parameters fixed (Joshi, Yang and Xiao [2004]). Figure 27(a) essentially depicts the OB/OM hysteresis loop with parameters PC = 14.0 mW, T = 65◦ C, cavity detuning Δθ = 50 MHz, ΔC = 103 MHz, ΔP = 0. It represents a normal forward hysteresis cycle (the main loop) where the upward-switching threshold is higher than the downward-switching threshold. For a slightly higher value of ΔC = 137 MHz, the HC has the form of the numeral eight [fig. 27(b)]. The ar-

Fig. 27. Observed input–output intensity characteristics of the system, with parameters ΔP = 0, Δθ = 50 MHz, PC = 14.0 mW, T = 65◦ C: (a) ΔC = 103 MHz; (b) ΔC = 137 MHz; (c) ΔC = 171 MHz; (d) ΔC = 275 MHz. As clearly seen, the hysteresis cycle changes shape and rotation direction as the coupling beam detuning increases. (After Joshi, Yang and Xiao [2004].)

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rows show the path taken by the cavity field intensity when the input intensity is scanned up and then down using an electro-optical modulator. It is easy to see both the usual ‘forward’ hysteresis loop (lower loop) as well as the ‘backward’ loop (upper loop) in this composite hysteresis curve. Upon further increasing ΔC to 171 MHz, the system exhibits dramatic changes in the width and direction of the main HC [fig. 27(c)]. The width of the main HC decreases considerably, and now it moves in the backward (clockwise) direction, i.e., the upward switching threshold intensity becomes lower than the downward one. At a certain ΔC value the two threshold intensities become identical and the main HC disappears. The area enclosed within the HC is a measure of the energy dissipation in the system and one can have a controllable energy dissipation in the system with specially chosen parameters. The backward HC is expanded by further increasing ΔC to 275 MHz. Comparison of figs. 27(a,d) clearly reveals that the positions of upward and downward threshold values are reversed by changing only the coupling-laser frequency detuning (ΔC ), with all other system parameters fixed. Such behavior can be exploited to implement all-optical switching by alternating the coupling-beam detuning (ΔC ) between two values so as to switch the cavity intensity between the upper and lower branches. A similar transition from forward to backward HC can be accomplished by changing only Δθ , with all other system parameters fixed. The ratio of upper switching threshold intensity to lower switching threshold intensity is given in fig. 28 for two different temperatures (Joshi, Yang and Xiao [2004]). Clearly, the ratio of threshold intensities decreases as ΔC increases. When the ratio crosses the magnitude of 1, the hysteresis cycle reverses its direction. The shape and transition from forward to backward HC change when the atomic density (determined by the temperature of the atomic cell) is raised. The observation of backward HC in this three-level EIT system is very different from the earlier observations (Jewell, Gibbs, Tarng, Gossard and Wiegmann [1982] and Dagenais and Sharfin [1985]), since the thermal effect could not contribute to the atomic system. As the thermal effect is a very slow process, the current system will have advantages in potential applications for all-optical switching and all-optical storage devices. Also, there was a lack of control over the observed backward HCs in the previous experiments. Recently, in a four-level atomic system of 87 Rb atoms inside an optical ring cavity, the dependence of the OB curve on the nonlinear phase shift has been studied (Chang, Wu, Xie and Wang [2004]). This nonlinear phase shift is introduced by the coupling field, and the atomic system is operated in the near-CPT condition due to atomic coherence. The absorptive and refractive nonlinearities enhanced by atomic coherence are responsible for this phenomenon. This effect

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Fig. 28. Ratio of upper to lower switching threshold intensities as a function of ΔC , with ΔP = 0, Δθ = 50 MHz, PC = 14.0 mW. Curves A (squares) and B (circles) are for T = 65◦ C and T = 70◦ C, respectively. (After Joshi, Yang and Xiao [2004].)

was utilized to demonstrate an all-optical flip-flop and storage of optical signals with a low peak power of several tens of microwatts.

4.5. Dynamic hysteresis in optical bistability The HC in atomic OB has been observed in the input–output intensity plot when the cavity input field was scanned adiabatically (Lugiato [1984], Gibbs [1985]). The HC can have nonzero area even in the limit of zero sweeping rate of the input field; then such a system exhibits a static HC. As the input-field scanning frequency Ω increases, the shape and area of the HC change significantly, which gives a dynamical HC (Jung, Gray, Roy and Mandel [1990] and Goldsztein, Broner and Strogatz [1997]). This dynamical HC is due to the non-adiabatic variation of the input field which causes delay in transition to the upper state of the HC. The additional area (known as dynamical hysteresis area) acquired in the hysteresis loop is due to the instability of the system. The static hysteresis features of OB are well understood, but due to some unresolved issues dynamical hysteresis is still a very much open research topic. There are several physical situations where dynamical hysteresis has got profound importance. For example, in magnetic and optical switching devices, the area of dynamical HC provides power

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dissipation by repetitive switching at a frequency Ω. Using one-dimensional theory of dynamic hysteresis it was demonstrated that the shift of switching points and the area of HC scale as the two-third power of the switching frequency, in accordance with experimental results involving a bistable semiconductor laser (Jung, Gray, Roy and Mandel [1990] and Hohl, van der Linden, Roy, Goldsztein, Broner and Strogatz [1995]). Recently, dynamical HC was demonstrated in a warm Rb vapor using resonantly enhanced Raman generation without an optical cavity (Novikova, Zibrov, Phillips, Andre and Walsworth [2004]). Analyses of dynamic HC using a switched bistable system have been reported by Jung, Gray, Roy and Mandel [1990] and Goldsztein, Broner and Strogatz [1997]. Such a system can be described by a particle in a quartic double-well potential driven by a periodic force. The equation of motion for this system is (Joshi, Yang and Xiao [2005]) dx (4.7) = ax − bx 3 + G(t), dt where a and b are constants and G(t) = E sin(Ωt) is the control parameter, with E chosen large enough so that the system is repeatedly going past the turning points. Equation (4.7) can be realized physically in the longitudinal mode bistability of a semiconductor laser (Yamada [1986]) and for a nonlinear polarization model describing dispersive bistability (Risken, Savage, Haake and Walls [1987]). For the dispersive bistability, a and b are related to cavity frequency detuning and third-order nonlinear dispersion parameters, respectively (Risken, Savage, Haake and Walls [1987]). For an EIT system in three-level -configuration, the thirdorder nonlinear dispersion (parameter b) is given by χ (3) in eq. (2.13). The dynamical HC obeys the area scaling law A(Ω) − A(0) ∝ Ω 2/3 in the limit of Ω → 0, where A(0) is the area of the static hysteresis loop (Jung, Gray, Roy and Mandel [1990]), and matches well with experimental results on the bistable semiconductor laser system (Hohl, van der Linden, Roy, Goldsztein, Broner and Strogatz [1995]). For larger Ω, the scaling law has been predicted (Goldsztein, Broner and Strogatz [1997]) to be A(Ω) ∼ 1/Ω as Ω → ∞. The control of dynamic HC by varying the sweeping frequency Ω of the cavity driving field is shown in fig. 29 for a three-level -type system of rubidium atoms under the experimental conditions PC = 14 mW, ΔC = 80 MHz, ΔP = 20 MHz, cavity detuning Δθ = 50 MHz, T = 70◦ C, and sweeping rates Ω = 150 Hz [fig. 29(a)], 1.5 kHz [fig. 29(b)], 2.5 kHz [fig. 29(c)] and 3.0 kHz [fig. 29(d)]. The OB curves displayed in figs. 29(a,b) are basically dispersive or refractive type OB for the parametric conditions selected here. When the sweeping rate changes adiabatically or nearly adiabatically (Ω ∼ 0.1–0.2 kHz), the area of the HC changes very little. As Ω increases, the HC area also increases [fig. 29(b)] because the

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Fig. 29. Input–output intensity characteristics of the optical cavity field for different rates of triangular scan, with parameters PC = 14 mW, ΔC = 80 MHz, ΔP = 20 MHz, Δθ = 50 MHz, T = 70◦ C: (a) Ω = 150 Hz; (b) Ω = 1.5 kHz; (c) Ω = 2.5 kHz; (d) Ω = 3.0 kHz. (After Joshi, Yang and Xiao [2005].)

system displays a delay in making the transition to the upper state of the bistable curve. The increase in area approximately follows the scaling law A(Ω) ∝ Ω 2/3 (Jung, Gray, Roy and Mandel [1990] and Goldsztein, Broner and Strogatz [1997]) within experimental uncertainty. There is a decrease in the area of dynamical HC in fig. 29(c) when Ω is increased further. However, above a certain value of Ω the shape of the HC changes dramatically [fig. 29(d)] and it becomes backward as discussed earlier. The HC area then increases with Ω again [fig. 30(a)]. As the sweeping rate Ω is further increased, the shape of the dynamical HC continues to change in an interesting manner [fig. 30(b)], and then the HC area decreases following the asymptotic scaling law of A(Ω) ∼ 1/Ω as Ω → ∞ in figs. 30(c,d), i.e., the area of the dynamical HC approaches zero at very high sweeping rates (Goldsztein, Broner and Strogatz [1997]). Such behavior is due to the dynamical cutoff or the imbalance between the sweeping time period (∼1/Ω) and the mean dwell time in one of the states (Thorwart and Jung [1997]). The HC is observed

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Fig. 30. Same as fig. 29, but for: (a) Ω = 5.0 kHz; (b) Ω = 10.0 kHz; (c) Ω = 15.0 kHz; (d) Ω = 30.0 kHz. (After Joshi, Yang and Xiao [2005].)

in the bistable system because the system stays in the metastable minimum of the quartic potential. As the sweeping rate increases, the system does not get enough time to stay in either of the metastable states, so the HC area vanishes. The observed dynamical HC behavior at low and high Ω limits follows the predictions of existing scaling laws (Jung, Gray, Roy and Mandel [1990], Goldsztein, Broner and Strogatz [1997]) quite well. The controllability of dynamic HC with experimental parameters has been effectively demonstrated in this EIT system.

§ 5. Controlled optical switching in three-level atomic systems For communication networks, digital computers, and signal processing systems, efficient and fast switching is essential. In view of current developments in highdata-rate communication systems and all-optical computing, high-speed photonic switches are in demand. One of the original goals of studying OB in various systems is its potential application in all-optical signal processing. The main attraction of optical switching is that it enables routing of optical data signals without

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the need for conversion to electrical signals and, therefore, is independent of data rate and data protocol. Use of all-optical elements will increase speed and bandwidth, lower error rates, consume less power, and reduce overall cost. The essential element in all-optical communication and all-optical computation is to achieve effective and fast all-optical switching. Multi-level EIT systems are ideal for such applications, since the linear and nonlinear optical properties of the probe beam can be dramatically changed by another coupling laser beam due to atomic coherence. The possibility of using nonlinear phase shift to exploit all-optical switching was theoretically analyzed in the early 1980s by Jensen [1982], and has been extensively investigated afterwards by employing both second- and third-order nonlinearities. Those studies mainly concerned couplers based on ultrafast electronic nonlinearities, requiring switching powers which, although in short pulses, appear excessively large for practical applications. Conversely, switching devices for network reconfigurations do not need to be as fast and can be realized with materials having slower responses, such as those based on photorefractive, thermal or molecular nonlinearities. Here, we review only a few recent experiments done in multi-level atomic systems based on atomic coherence. It is clear from earlier discussions (Sections 2 and 4) that by using the two-photon Doppler-free configuration in a threelevel atomic system, e.g., two laser beams co-propagating in a -type three-level atomic medium, the first-order Doppler effect is eliminated. Therefore, one can easily observe optical bistability in an atomic vapor cell inside an optical ring cavity (Joshi, Brown, Wang and Xiao [2003]) without the need for a complicated vacuum system for atomic beams or cold atomic samples. Also, owing to the modified linear absorption, dispersion, and nonlinearity in such systems, atomic OB has been observed with lower optical powers and was controlled easily with the coupling laser beam (Section 4). These advantages provide a simple setup for demonstrating optical switching between the two steady states of a bistable curve by adding positive and negative pulses to the cavity input intensity (Brown, Joshi and Xiao [2003]). Optical switching has also been demonstrated experimentally in a three-level atomic system by changing the nonlinearity of the system, where the actual switching action happens outside the bistable region, and the switching is controlled by the coupling-laser beam frequency detuning or intensity (Wang, Goorskey and Xiao [2002d, 2002e]). Also, absorptive photon switching was theoretically predicted (Harris and Yamamoto [1998]) and experimentally demonstrated (Yan, Rickey and Zhu [2001b]) in a four-level atomic sample.

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5.1. Controlled switching between bistable states For a typical bistable curve as sketched in fig. 31(a), the cavity output intensity will stay on the lower branch when the input intensity is increased from zero or a lower value outside the bistable region until it reaches the threshold value Y2 , when it jumps to the upper branch. Similarly, when the input intensity decreases in the upper branch from higher values outside the bistable region, the output intensity will stay on the upper branch until it reaches the threshold value Y1 , when it jumps down. This property of OB can be utilized to realize an optical switch. Let the initial input intensity be set in the middle of the bistable curve and the output intensity at the lower branch (point A) initially. Then, an intensity pulse is added with its peak value higher than the threshold value Y2 , as shown in fig. 31(b). This intensity pulse brings the output intensity to the upper branch (point C) and, as the pulse ends, the output intensity will end up on the upper branch position B and stays there. Then, as another, negative pulse comes, the output intensity will be brought down to the lower branch A (through point D) again. So, the binary (high and low) values of the output intensity of the system are determined by the added positive or negative pulses on the input intensity.

Fig. 31. Schematic illustration of the switching mechanism: (a) typical input–output intensity bistable curve. A, B, C and D are output intensities for corresponding input intensities in (b), which gives a time sequence of the cavity input intensity. (After Brown, Joshi and Xiao [2003].)

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Fig. 32. Optical switching of cavity output intensity controlled by cavity input intensity. (a) Cavity input power as a function of time with positive and negative pulses, respectively. (b) Output steady-state intensities A and B controlled by the pulses in (a). Parameters are ΔP = −76.9 MHz, ΔC = −19.3 MHz, and PC = 9.5 mW. The temperature of the rubidium cell is 68◦ C. (After Brown, Joshi and Xiao [2003].)

Figure 32 shows the experimental demonstration of such optical switching (Brown, Joshi and Xiao [2003]). For a typical bistable curve the initial input power is set at 1.73 mW. The positive and negative pulses were formed by applying pulse voltages to an electro-optical modulator. The amplitudes of the intensity pulses are chosen to be just a little higher than the halfwidth of the bistable HC. The initial output intensity is in the lower branch (state A). As can be seen from fig. 32, the output intensity is brought to the upper branch [state B in fig. 32(b)] by the positive pulse [fig. 32(a)] and stays there; then the negative pulse brings it down to the lower-branch value (state A) again. The states A and B are very stable and the extinction ratio of this switching action reaches about 20:1.

5.2. All-optical switching controlled by coupling laser beam In the three-level EIT system the coupling (controlling) laser beam can be used to control the probe (switching) beam by two different methods. The first method is to use frequency detuning of the coupling laser beam while the second method employs the coupling laser power (Wang, Goorskey and Xiao [2002d, 2002e]).

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Such schemes of controlling one light by another light beam form the bases for all-optical switching. Due to enhanced nonlinearity and reduced absorption in three-level EIT systems, such processes can be very efficient. Such all-optical switching is fundamentally interesting and can have applications in all-optical communication, optical logic gates, and all-optical information processing. 5.2.1. Control by coupling laser frequency The Kerr-nonlinear index of refraction of a three-level -type atomic system can be greatly enhanced near resonance by orders of magnitude and can change sign with a small change (tens of megahertz) in the frequency detuning of the coupling beam as discussed in Section 2 (Wang, Goorskey and Xiao [2001]). Such enhancement in Kerr nonlinearity is caused by the atomic coherence induced in this EIT medium. There are two kinds of nonlinearities in such a system, that is, crossphase modulation (Hau, Harris, Dutton and Behroozi [1999], Chang, Du, Yao, Xie and Wang [2004] and Schmidt and Imamoglu [1996]) and self-phase modulation (Wang, Goorskey and Xiao [2001]), and both can be used to control the steady states of the system. The experiment described in the next paragraph makes use of the self-phase modulation (i.e., the refractive index is given by n = n0 + n2 IP ) with a nonlinear medium inside an optical cavity. With the knowledge of these dramatic changes of the Kerr-nonlinear index of refraction near resonance, one can demonstrate that the cavity field intensity can be controlled by the frequency detuning of the coupling field at a relatively low optical power. Such action constitutes controllable all-optical switching in this composite atom–cavity system. The experimental arrangement is basically the same as described in fig. 7 except that an electro-optical modulator EOM1 is placed in the coupling-beam feedback path to change its frequency (Wang, Goorskey and Xiao [2002d]) and another electro-optical modulator EOM2 is placed in the path of the probe beam before entering the cavity to see the OB. The frequency switching of the coupling beam (between ΔC = 111 MHz and ΔC = 135 MHz) is achieved by phase modulation using EOM1 . The cavity field is locked at ΔP = 123 MHz [which is midway between the two alternating frequencies of the coupling beam and serves as the operation point in fig. 33(a,b)]. The ring cavity length is scanned from longer to shorter by a ramp voltage on the PZT for ΔC = 135 MHz [= (ΔP + 12) MHz] and ΔC = 111 MHz [= (ΔP − 12) MHz], respectively. OB clearly appears in fig. 33(b) with quite a low threshold power of 0.3 mW when the Kerr nonlinear index n2 is larger (n2 ∼ = 6 × 10−7 cm2 /W) for ΔC = 135 MHz. When the nonlinear index is smaller and has a negative sign (n2 = −0.8 × 10−7 cm2 /W) for ΔC = 111 MHz, the threshold increases dramatically and bistability does not appear in the input power range (0.6 mW) shown in fig. 33(a). The cavity

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Fig. 33. Cavity transmission power vs cavity input power. These are the steady-state curves of the system for (a) n2 = −0.8 × 10−7 cm2 /W [ΔC = (ΔP − 12) MHz] and (b) n2 = 6 × 10−7 cm2 /W [ΔC = (ΔP +12) MHz], respectively. The cavity input power of the switching beam is set at 0.39 mW. (After Wang, Goorskey and Xiao [2002d].)

input power is set at PPin = 0.39 mW and the coupling frequency detuning is switched between ΔC = (ΔP + 12) MHz and ΔC = (ΔP − 12) MHz, as shown in fig. 34(b), the cavity output intensity is then switched between two distinct steady-state values, as shown in fig. 34(a). The average cavity output peak power of the “on” state is about 1.2 mW and the average power of the “off” state is less than 0.034 mW, which gives a switching ratio of better than 30:1 (Wang, Goorskey and Xiao [2002d]). One can see (from fig. 33) that changes in ΔC cause changes in the steady-state curve of the system and force the cavity field to operate at different intracavity intensities. One can also consider this switching action as a frequency-to-amplitude signal conversion since by modulating the frequency of one laser beam, the intensity of another laser beam is controlled. Such

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Fig. 34. Optical switching of the cavity field controlled by the controlling beam. (a) ‘On’ and ‘off’ states of the cavity field controlled by the voltage applied on EOM1 , shown in (b), for two frequency detunings ΔC = 111 MHz (upper voltage level) and ΔC = 135 MHz (lower voltage level), respectively. (After Wang, Goorskey and Xiao [2002d].)

frequency-to-amplitude conversion could have important applications in optical communication and optical information processing. This all-optical switching is quite efficient since it only needs to switch the frequency detuning of the coupling beam by a small amount (24 MHz in this case) which, in turn, gives a switching ratio of better than 30:1 for the cavity output intensity. 5.2.2. Control by coupling laser intensity Similar control of cavity field was also achieved by changing the intensity of the coupling beam instead of its frequency detuning in the system of three-level atoms inside an optical ring cavity (Wang, Goorskey and Xiao [2002e]). In this method the switching occurs between the two distinct steady states with and with-

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out the coupling beam. These two steady states are formed as a result of absorption change and enhancement of Kerr nonlinearity similar to the case in the preceding section (Section 5.2.1). This kind of all-optical switching is also different from the absorptive photon switching reported for a four-level atomic system (Yan, Rickey and Zhu [2001b]) as will be discussed in the next section. The experimental arrangement is also similar to what was used and discussed for the experiment in Section 5.2.1, except that EOM1 is now placed in the path of the coupling beam before it enters the atomic cell (Wang, Goorskey and Xiao [2002e]). The two different steady states are created with two different couplinglaser powers (PC ) of 0 and 14.3 mW, respectively, with ΔP = 0, ΔC = 7 MHz. When the optical cavity is locked at Δθ = 4.6 MHz and PC is switched between 0 and 14.3 mW, the cavity transmission power switches between two steady-state values with a switching ratio of 20:1. This is because the change of PC causes a change of steady-state curve and forces the cavity to operate at different intracavity intensities. The experiment was performed at an intracavity power of PPin = 0.24 mW. 5.3. Controlled optical switching in four-level N-type atomic system The basic idea of absorptive photon switching by quantum interference was proposed by Harris and Yamamoto [1998] who considered a four-state N-type atomic system that will absorb two photons but will not absorb one photon. This occurs as a result of quantum interference or EIT that prohibits single-photon absorption. The system may function as an optical switch where a pulse of light at one frequency will cause the absorption of light at a second frequency. Since quantum interference is very fragile, such an optical switch is sensitive. In the ideal case, a pulse of energy equal to that of a single photon is sufficient to open and close the second channel. In particular, the two-photon nonlinearity described by Harris and Yamamoto [1998] is the absorptive analog of the (reactive) Kerr nonlinearity of Schmidt and Imamoglu [1996]. Experimental demonstration of such absorptive photon switching in a cold four-level atomic system was reported by Yan, Rickey and Zhu [2001b] who observed switching ‘off’ and ‘on’ of the probe absorption for both a cw probe laser and a pulsed probe laser at relatively low optical powers. 5.3.1. Theoretical calculation The four-level atomic system considered by Harris and Yamamoto [1998] is an N-type configuration (see fig. 11). All fields interacting with the respective transitions are considered to be monochromatic, and atoms are in their ground states.

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After solving Schrödinger’s equation for the probability amplitudes of all four states, the susceptibility at the probe frequency in steady state can be written as (Harris and Yamamoto [1998]) χ(ωP ) =

C Δ 24 ) M(|Ω|2 − 4Δ , 2 P Δ C Δ 24 − |ΩC | Δ 24 − |Ω|2 Δ P 4Δ

(5.1)

in which M = N|μ13 |2 /h¯ ε0 and ΩP , ΩC , Ω are the probe-, coupling- and switching-field Rabi frequencies, respectively. The complex detunings are defined C = (ωP − ωC ) − (ω2 − ω1 ) + iγ12 and Δ 24 = P = ωP − (ω3 − ω1 ) + iγ13 , Δ as Δ (ωP − ωC + ω24 ) − (ω4 − ω1 ) + iγ24 , with γij the dephasing linewidths for the respective transitions. χ(ωP ) describes the resistive nonlinearity, and also the associated Kerr nonlinearity (Schmidt and Imamoglu [1996]) in the appropriate limit. Using Maxwell’s equations along with polarization P (ωP ) = ε0 χ(ωP )E(ωP ) one can obtain expressions for the power loss 2αL, the phase shift, and the groupvelocity delay time for the probe pulse of frequency ωP as (Harris and Yamamoto [1998])

(W24 + 2γ12 )(WC + W24 + 2γ12 ) + 4Δ2P 2αL = NLσ13 , (WC + W24 + 2γ12 )2 + 4Δ2P

ΔP WC βL = N Lσ13 , (WC + W24 + 2γ12 )2 + 4Δ2P

WC (WC + W24 + 2γ12 )2 − 4WC Δ2P , TD = N Lσ13 (5.2) [(WC + W24 + 2γ12 )2 + 4Δ2P ]2 where σij = ωij |μij |2 /ε0 ch¯ γij (with ωij = ωi − ωj ) are the power absorption cross-sections of respective transitions and WP = ΩP2 /2γ13 , WC = ΩC2 /2γ23 and W24 = Ω 2 /2γ24 are golden rule transition rates. If one takes power loss at the probe frequency to be unity, i.e., 2αl = 1, and assumes WC to be large compared to W24 and γ12 , then crit W24 =

WC , NLσ13

TD =

N Lσ13 , WC

(5.3)

where ΔP and Δ24 are neglected compared to respective linewidths. The required energy per unit area at frequency ω24 for switching is then (Harris and Yamamoto [1998]) Energy 2π h¯ ω24 hω ¯ 24 . = = Area σ24 3 λ224

(5.4)

So, a pulse at frequency ω24 with the energy of single photon and focused to a spot size of half a wavelength is efficient to make this medium opaque to a photon

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of frequency ωP . The duration of the pulse of frequency ω24 must be equal to the group delay time TD . The speed of switching is decided by the golden rule transition rate WC . Thus, the switch can operate at an energy cost of one photon per switching event. 5.3.2. Experimental demonstration The energy-level diagram for the 87 Rb atoms used in the experiment (Yan, Rickey and Zhu [2001b]), done in a magneto-optical trap (MOT), is depicted in fig. 11. A coupling laser (at frequency ωC ) drives the D1 , F = 2 → F ′ = 1 transition at 795 nm and creates dressed atomic states |+ and |− [|+ = √1 (|3 + |2 ) 2

and |− = √1 (|3 − |2 )]. A weak probe laser (at frequency ωP ) drives the D1 , 2 F = 1 → F ′ = 1 transition and forms a standard -type configuration for EIT. Another switching laser couples the D2 , F = 2 → F ′ = 3 transition at 780 nm. All these lasers can be turned on and off by acousto-optic modulators according to the desired time sequence. The probe laser and the switching laser are linearly polarized parallel with each other and perpendicular to the linearly polarized coupling laser. The transitions induced among the magnetic sublevels by the three lasers can be grouped together according to the selection rules and form a manifold of four-level systems. To a good approximation, this system can be viewed as equivalent to the generic four-level system discussed in Section 5.3.1 (Harris and Yamamoto [1998]). The photon switching in such a four-level atomic system is based on the interference-enhanced nonlinear two-photon absorption and inhibited linear one-photon absorption discussed in the preceding section. By using the dressed-state picture it can be shown that the single-photon absorption experiences a destructive interference and vanishes at the line center of the probe transition while the two-photon absorption in the dressed state interferes constructively, leading to enhancement at the line center. The probe absorption coefficient is now governed by the switching-laser intensity so the probe absorption can be turned ‘on’ and ‘off’ by the switching laser, giving rise to absorptive photon switching as shown in fig. 35. The bandwidth of such two-photon absorption is determined by the EIT width, which limits switching speed. In another novel experiment, Ham and Hemmer [2000] demonstrated coherence switching in a four-level solid (Pr:YSO) system using an enhanced nondegenerate four-wave mixing technique. The mechanism involved in such switching is based on simultaneous suppression and enhancement of two-photon absorption and could be useful for high-speed optical switches.

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Fig. 35. (a) Calculated Iout (t)/Imax of a pulsed probe laser versus time. The top dotted curve is Iout (t) = Iin (t) (no switching pulse). Imax is the amplitude of the input probe pulse. The two lower curves show the output probe pulse Iout (t) when the switching pulse is turned on. The relevant parameters are Ω ′ = 3Γ3 , g = 0.2Γ3 , Ω = 2Γ3 (solid line) and Γ3 (dot-dashed line), respectively. (b) Measured output probe pulses, Iout (t), versus time under the conditions given in (a). (After Yan, Rickey and Zhu [2001b].)

§ 6. Controlled optical instability with three-level atoms inside an optical cavity In systems with atoms inside an optical cavity, under certain conditions, the cavity field can become unstable due to the nonlinear interaction between the atoms and the cavity field. Such optical instability was observed in the upper branch of the OB curve in a system consisting of two-level atoms (atomic beam) inside an optical ring cavity (Orozco, Rosenberger and Kimble [1984]). This instability is classified as Ikeda instability (Ikeda, Daido and Akimoto [1980]), which appears under the condition of medium response time much faster than the cavity round-trip time. Multimode instabilities as well as oscillatory instabilities culminating in optical turbulence in a two-level atomic system were also discussed

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previously (Segard, Macke, Lugiato, Prati and Brambilla [1989]). In the mixed absorptive–dispersive OB situation, the off-resonant-mode instability was studied and it was shown that a portion of the lower transmission branch could also be unstable (Asquini, Lugiato, Carmichael and Narducci [1986]). In a bistable system of a homogeneously broadened two-level active medium in a cavity, dynamic instability was observed due to the onset of the cavity side modes (see, e.g., the special issue on instabilities edited by Abraham, Lugiato and Narducci [1985] and Segard, Sergent, Macke and Abraham [1989]). Other kinds of self-oscillation and instability were also observed using different two-level atomic systems inside optical resonators (Firth, Harrison and Al-Saidi [1986]). In a three-level system consisting of a cold cloud of Cs atoms inside an optical cavity a dynamic instability was observed in the transmitted field (Lambrecht, Giacobino and Courty [1995]). In this Cs-atomic cloud the degenerate Zeeman sublevels of the 6S1/2 , F = 4 and 6P3/2 , F ′ = 5 states interact with two cavity fields pumped by one input circularly polarized laser beam. There is competition between optical pumping to the state 6S1/2 , F = 4, mF = 4 from all other Zeeman sublevels and optical saturation of the transition from the state 6S1/2 , F = 4, mF = 4 to the state 6P3/2 , F ′ = 5, mF ′ = 5, leading to instability in the cavity field. The oscillatory characteristics in the cavity output field were qualitatively explained with a simplified (quasi two-level) theoretical model. Although the phenomena related to optical dynamic instability in various atomic systems have been studied over past two decades, there still is a lack of experimental control in systematically investigating such dynamic effects. The three-level EIT system inside an optical ring cavity provides a good system for studying this interesting phenomenon, with many advantages. The most important advantage is the ability to independently control the frequencies and intensities of the coupling beam and the probe beam over large parameter ranges. The other advantages include the reduced absorption (Wang, Goorskey and Xiao [2002c]), increased dispersion (Xiao, Li, Jin and Gea-Banacloche [1995]), and greatly enhanced nonlinearity (Wang, Goorskey and Xiao [2001]) in such three-level EIT system, so that the optical dynamic oscillation can be observed at much lower intensities and with a much more sensitive dependence on the frequency detunings and intensities of the laser beams. Also, recent advances in demonstrating EIT-related effects have fueled a renewed interest in optical dynamic effects in multi-level atomic systems, which could have potential impacts on applications of EIT in all-optical switching, all-optical buffering, and optical soliton generation. In view of these advancements the controllability of such optical dynamic effects becomes particularly important.

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6.1. Theoretical calculation A system of N three-level -type atoms inside an optical ring cavity, as shown in fig. 1(b), is considered here. The theoretical model can be constructed by using the density-matrix equations [see eq. (2.9) in Section 2.1] for three-level atoms together with the modified Maxwell equation for the cavity field as τ

dαP = t2 αpin − γcav αp + iΦcav αP , dt

(6.1)

where for simplicity a quantity α (with |α|2 representing the average photon flow, expressed in units of number of photons per second) is defined such that it is related to the field strength E by (Yang, Joshi and Xiao [2004b]) 2nǫ0 cSP EP , αP = (6.2) hω ¯ P 2nǫ0 cSC EC , αC = (6.3) hω ¯ C with ǫ0 the free-space permittivity and c the speed of light in vacuum; SP and SC are the areas of the transverse sections of the probe and coupling light beams, respectively. Only probe laser circulates in the cavity and αP can be written as a complex variable, αP = α1 + iα2 . The coupling laser does not circulate inside the cavity, so αC can be assumed real. Equation (6.1) describes the change of the intracavity probe field αP on a round trip time duration τ due to the driving field αPin entering through the laser mirror M2 (fig. 7) with a transmission coefficient t2 , to the cavity decay γcav (due to the losses of the intracavity medium and the finite transmissions of the mirrors), and to the round-trip phase shift Φcav (Yang, Joshi and Xiao [2004b]). The total round-trip phase shift in the cavity has four contributions: The first, Φ0 , is due to the phase shift of the field propagating in the free space of the cavity, and depends on the geometrical length of the cavity as 2π (6.4) (L0 + vcav t), λ where L and L0 are the instant and initial cavity lengths, and vcav is the cavity scanning speed. Two more contributions, due to the presence of the atomic medium inside the cavity, are a linear phase shift ΦL and a nonlinear phase shift ΦNL , proportional to the first-order and third-order susceptibilities, respectively, of the atomic system given by eqs. (2.12) and (2.13). From eq. (2.12) one can see that χ (1) and χ (3) depend on three parameters ΔC , ΔP and ΩC Φ0 = kL =

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(which latter depends on the coupling-laser power PC ). For simplicity, we define ΦL ≡ e(ΔC , ΔP , PC ) and ΦNL ≡ f (ΔC , ΔP , PC )|αp |2 , where e and f are functions of the parameters ΔC , ΔP and PC . When the condition of strong coupling field is not satisfied, the general density-matrix eqs. (2.9) are used to evaluate the nonlinear susceptibility required for this calculation. The fourth contribution to the round-trip phase shift ΦP originates from the change in the population of the participating levels, i.e., 2 (1 + wC + wP )ΦL , 3 with wP = ρ22 − ρ11 and wC = ρ22 − ρ33 . Hence the total phase shift involved in the cavity round trip is given by ΦP =

(6.5)

Φcav = Φ0 + ΦL + ΦNL + ΦP

2π 2 = vcav t + 1 + (1 + wP + wC ) e + f |αP |2 . λ 3

(6.6)

In the above equation only the dispersive interactions of fields with atoms are considered and the absorption effect has been neglected (Yang, Joshi and Xiao [2004b] and Lambrecht, Giacobino and Courty [1995]). For the EIT system under small frequency detuning the absorption coefficient seen by the probe laser beam is nearly zero (Wang, Goorskey and Xiao [2002c]), while in the large detuning limit the phase shift due to absorption is ΦA ∝ 1/Δ2P (Hiloco, Fabre, Reynaud and Giacobino [1992]), which is very small compared with the linear and nonlinear phase shifts ΦL ∝ 1/ΔP and ΦNL ∝ αP2 /Δ3P and can be neglected. The atomic equations (2.9) and field equations (6.1) constitute the total model of the system with N three-level atoms inside the optical ring cavity. Solution of these equations produces oscillations in the cavity transmission profile as shown in fig. 36 for initial conditions wP (0) = −1, wC (0) = 0, α1 (0) = 6 × 106 , α2 (0) = 0, γcav = 0.03 Hz, γ21 = γ23 = 3 × 106 Hz, γ31 = 1.1 × 106 Hz, τ ≈ 1.2 ns, t2 = 0.17 (corresponding to 3% intensity transmissivity) and vcav = 200 µm/s. These coupled differential equations show extreme stiffness for the experimental conditions so Gear’s third-order method with a variable step size is used to solve these equations. The appearance (and disappearance) of such dynamic oscillation and the oscillation period can be controlled by changing the system parameters, such as the powers of the coupling beam (PC ) and the cavity input field (PP ), as well as frequency detunings of the coupling and the probe beams (ΔC and ΔP ). In the experiment involving a cold Cs-atomic cloud a similar kind of oscillations was observed (Lambrecht, Giacobino and Courty [1995]). The model described by Lambrecht, Giacobino and Courty [1995] is a limiting case of the

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Fig. 36. Experimentally observed and theoretically calculated cavity transmission profiles from an optical ring cavity containing three-level -type atoms for three different values of coupling frequency detuning. Experimental observations are in the left-hand column, corresponding theoretical calculations in the right-hand one. The parameters used in the experiment and theoretical calculation are: PC = 11 mW, PPin = 2.8 mW, ΔC = 0, and (a), (d) ΔP = 25 MHz; (b), (e) ΔP = 35 MHz; (c), (f ) ΔP = 45 MHz. (After Yang, Joshi and Xiao [2004a].)

generalized model discussed above. Suppose only two-level atoms are considered such that the frequency detuning of the input field is very large compared to the atomic decay rate, i.e., ΔP ≫ γ , then terms related to the linear and nonlinear

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phase shifts e and f in the above model become e= f =

Ng 2 , ΔP −2Ng 4 Δ3P

(6.7) ,

(6.8)

where g is the coupling constant of the atoms with the field, g2 =

2 ω d21 P . 2ǫ0 hS ¯ Pc

(6.9)

These expressions agree with the model used by Lambrecht, Giacobino and Courty [1995]. The generalized theoretical treatment of the system with threelevel atoms inside an optical ring cavity allows the direct comparisons with experimental studies and can be useful for investigating other dynamic effects, such as chaos and tunneling in this system.

6.2. Experimental measurements The experiments to observe dynamic instability with three-level atoms inside an optical ring cavity were carried out using basically the same experimental setup as shown in fig. 7 (Wang, Goorskey and Xiao [2002c], Yang, Joshi and Xiao [2004b]). Without the coupling beam, the cavity-field transmission profile is basically symmetric in shape. In the presence of the coupling field, depending on parameter values, the cavity transmission profile can be asymmetric or oscillatory due to dynamical instability. The dynamic oscillations in the cavity transmission profile are very sensitive to the experimentally controllable parameters, such as intensity and frequency detuning of the coupling beam, and intensity and frequency detuning of the probe (cavity input) beam. Figure 36 shows comparisons of experimentally measured cavity transmission profiles (left) with theoretically calculated results (right) for three different probe frequency detunings ΔP = 25 MHz [fig. 36(a,d)], ΔP = 35 MHz [fig. 36(b,e)] and ΔP = 45 MHz [fig. 36(c,f)]. The other parameters used were PC = 11 mW, PPin = 2.8 mW and ΔC = 0. There is very good agreement between the theoretical calculations and the experimental results. The oscillation time period decreases with increasing probe frequency detuning and at a certain value the oscillation finally stops. The controllability of the oscillation period as is shown as a function of ΔP and of ΔC in figs. 37(a,b), respectively. The experimentally measured data and the theoretically calculated results agree quite well. For these plots the parameters were

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Fig. 37. Comparison between experimentally measured and theoretically calculated oscillation period versus (a) probe frequency detuning ΔP and (b) coupling frequency detuning ΔC . The parameters used for the plots are: (a) ΔC = 0, PC = 11.0 mW and PPin = 3.0 mW and (b) ΔP = 0, PC = 11.0 mW and PPin = 3.0 mW for both the experiment and theoretical calculation. (After Yang, Joshi and Xiao [2004a].)

PC = 11.0 mW, PPin = 3.0 mW, ΔC = 0 [fig. 37(a)] and ΔP = 0 [fig. 37(b)], for both experimental and theoretical results. The atomic number density used for the theoretical curve was N = 1010 cm−3 , corresponding to T = 70.3◦ C. Other ways

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Fig. 38. Comparison between experimentally measured and theoretically calculated oscillation period versus (a) coupling power and (b) cavity (probe) power. The parameters used are: (a) ΔC = 0, ΔP = 35 MHz and PPin = 3.0 mW and (b) ΔC = 0, ΔP = 35 MHz and PC = 11.0 mW for both experiment and theoretical calculation. (After Yang, Joshi and Xiao [2004a].)

of controlling the oscillation period are by changing the power of the coupling beam or the probe beam as shown in figs. 38(a,b), respectively. The parameters used for these plots were ΔC = 0, ΔP = 35 MHz, T = 70◦ C, PPin = 3.0 mW [fig. 38(a)] and PC = 11.0 mW [fig. 38(b)].

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[2, § 7

The instability described here is caused by two competing dynamic processes in the system, i.e., optical pumping from state 5S1/2 , F = 2 to state 5P1/2 , F ′ = 2 by the coupling field, and the nonlinear saturation effect in the transition from state 5S1/2 , F = 1 to state 5P1/2 , F ′ = 2 due to the cavity field. The oscillation time period is determined by the relative strengths of these two dynamic processes, which are affected by the intensities and frequency detunings of the coupling and probe fields. Both frequency detunings play the same role, while the probe and coupling powers play opposite roles, i.e., larger probe power increases the effect of nonlinear saturation while larger coupling power increases the effect of optical pumping.

§ 7. Summary and outlook Various nonlinear optical processes, such as four-wave mixing, harmonic generation, and optical bistability/multistability, have been shown to be greatly enhanced by making use of atomic coherence effects in multi-level atomic systems. Owing to the unique controllability of absorption, dispersion, and nonlinear coefficients in these EIT-related atomic systems, nonlinear processes of one laser beam can be controlled all-optically by another laser beam. For simple threelevel atomic systems, when two-photon Doppler-free configurations are used for atoms in an atomic cell both linear absorption (Gea-Banacloche, Li, Jin and Xiao [1995], Li and Xiao [1995a]) and dispersion (Xiao, Li, Jin and Gea-Banacloche [1995]) of the probe beam can be tuned by using a low-power cw coupling laser beam. Similarly, by using the property of nonlinearity-induced asymmetry in the cavity transmission, the Kerr-nonlinear index of refraction n2 was experimentally measured in a three-level EIT system (Wang, Goorskey and Xiao [2001, 2002a, 2002b]) to show large variations and enhancement when compared to the twolevel atomic system. It was shown that four-level atomic systems (such as N-type configuration) can further increase the nonlinear index by several orders of magnitude (Schmidt and Imamoglu [1996] and Kang and Zhu [2003]) and give rise to large nonlinear phase shift with optical power in single-photon level (Harris and Yamamoto [1998]). Due to the reduced absorption and enhanced nonlinearity in multi-level atomic systems induced by atomic coherence, resonant and near-resonant four-wave mixing processes in three- and four-level atomic systems can be greatly enhanced (Li and Xiao [1996] and Lu, Burkett and Xiao [1998]). Rich physical phenomena appear when such three-level atoms are kept in an optical ring cavity. Not only can one observe the intensity effects of optical bistability and optical instability, as are also exhibited by a two-level atomic system inside an optical cavity,

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but other, novel effects such as optical multistability, instability, backward hysteresis cycle, and dynamical hysteresis, were also demonstrated (Wang, Goorskey and Xiao [2002c], Joshi, Brown, Wang and Xiao [2003], Joshi and Xiao [2003], Joshi, Yang and Xiao [2004, 2005], Yang, Joshi and Xiao [2004a]). More interestingly, control of these phenomena was achieved simply by tuning the auxiliary coupling-beam frequency detuning or its power. Such systems with controllable linear and nonlinear dynamical properties are ideal for studying nonlinear dynamical properties and their applications. One such example is controlled all-optical switching. By adjusting the frequency or intensity of the coupling laser beam, the intensity of the cavity (probe) field can be tuned “on” or “off” with high efficiency (Wang, Goorskey and Xiao [2002d, 2002e]). Looking forward, there are still many interesting areas in nonlinear optics which can be explored further with multi-level atomic systems having atomic coherence. One of the most profound ones will be to achieve efficient nonlinear optical processes with single photons, e.g., single-photon switching (Harris and Yamamoto [1998]), photon blockade (Imamoglu, Schmidt, Woods and Deutsch [1997]), and single-photon quantum networking (Duan, Lukin, Cirac and Zoller [2001]). With the controllable linear and nonlinear coefficients, and better designed energy-level systems with optimized atomic coherence and quantum interferences, such goals of implementing efficient nonlinear optics with single photons should be achieved in the near future. With entangled or correlated photon pairs generated from such coherently prepared media (van der Wal, Eisaman, Andre, Walsworth, Phillips, Zibrov and Lukin [2003] and Kuzmich, Bowen, Boozer, Boca, Chou, Duan and Kimble [2003]) and sufficiently efficient nonlinear interactions at single-photon level to achieve logic gates, photon storage, and buffering, one can expect to have quantum information processing and quantum networking. Another interesting direction of research in such multi-level atomic systems with induced atomic coherence is nonlinear dynamics, such as chaos and tunneling. Very recently, chaos via the frequency-doubling route was observed in a system with three-level atoms inside an optical ring cavity (Yang, Joshi and Xiao [2005]). The chaos was observed in the region with enhanced dispersion and nonlinearity, which is unique to this three-level EIT system. With the controllability of experimental parameters, one should be able to explore the possibility of controlling chaos and related effects. Another possibility is to observe tunneling in an optically bistable system. Although an optically bistable system with a double-well potential should be ideal for studying tunneling, such interesting effects have not been observed in two-level atoms inside an optical cavity. The main reason is the extremely long tunneling time predicted for such systems. However,

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[2, § 7

as demonstrated earlier, the bistable curve, and therefore the double-well potential, can be controlled by the coupling-beam parameters, and one can expect to modify the barrier between the wells and significantly reduce the tunneling time.

Acknowledgement Funding support from the National Science Foundation is gratefully acknowledged.

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E. Wolf, Progress in Optics 49 © 2006 Elsevier B.V. All rights reserved

Chapter 3

Photonic crystals by

Henri Benisty Laboratoire Charles Fabry de l’Institut d’Optique, Centre Scientifique, Bât 503, 91403 Orsay cedex, France

Claude Weisbuch Laboratoire de Physique de la Matière Condensée, Ecole Polytechnique, 91128 Palaiseau cedex, France

and Materials Department, University of California at Santa Barbara, Santa Barbara, CA 93106, USA

DOI: 10.1016/S0079-6638(06)49003-X

ISSN: 0079-6638 177

Contents

Page Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179

Main variables and notations . . . . . . . . . . . . . . . . . . . . . . . . .

180

§ 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

180

§ 2. Basics of periodic dielectric media and theoretical tools . . . . . . . .

184

§ 3. Three-dimensional photonic crystals . . . . . . . . . . . . . . . . . .

214

§ 4. Two-dimensional photonic crystals . . . . . . . . . . . . . . . . . . .

229

§ 5. Photonic crystal fibers . . . . . . . . . . . . . . . . . . . . . . . . . .

271

§ 6. Conclusion and perspectives . . . . . . . . . . . . . . . . . . . . . . .

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Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References and special issues . . . . . . . . . . . . . . . . . . . . . . . . .

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Nomenclature AlOx ARROW bcc BZ CAIBE CMT CROW DBR DFB DOS f.c.c. FDTD FBZ FIB FSS ICP-RIE Hn LED PBG PCF PhC PML PWE QD QED QW RIC RIE SEI SEM

aluminum oxide obtained from AlAs by lateral oxidation antireflecting resonant optical waveguide body-centered cubic (Bravais lattice) Brillouin zone chemically assisted ion beam etching coupled mode theory coupled resonator optical waveguide distributed Bragg reflector distributed feedback (laser) density of states face-centered cubic (Bravais lattice) finite difference time domain first Brillouin zone focused ion beam frequency selective surfaces inductively coupled plasma RIE hexagon-shaped cavity (missing holes) with n holes per side light-emitting diode photonic bandgap photonic crystal fiber photonic crystal; “membrane” and “substrate” approaches, see fig. 17 perfectly matched layer plane wave expansion quantum dot quantum electrodynamics quantum well refractive index contrast reactive ion etching spontaneous emission inhibition scanning electron microscope 179

SER SoI TMM WDM Wn

spontaneous emission rate silicon-on-insulator transmission matrix method wavelength domain multiplexing straight waveguide in photonic crystal, made of n missing

Main variables and notations exp(−iωt) RIC = n/n ε εav βm , β Fm K G a η = 1/ε

conventional time dependence of complex fields refractive index contrast relative dielectric constant average dielectric constant modal (general) propagation constant of a guided mode profile of a guided mode reciprocal lattice wavevector in a 1D case reciprocal lattice wavevector in a 2D or 3D case period (in the 1D case) inverse of relative dielectric constant, impermeability

§ 1. Introduction The concept of photonic crystals originates at the intersection of wave optics and solid-state physics. They extend the multidimensional periodicity, so ubiquitous in nature’s atomic and molecular edifices, into the realm of optics. The 1930s and 1940s have been termed “the golden age” of wave mechanics as it was then first applied to electrons in periodic crystals. However, it took five decades for the band concepts to be extensively applied to periodic structures in electromagnetism, a domain now identified by the words “photonic crystal”. It is worth underlining Brillouin’s work on wave propagation (Brillouin [1953]), as it pioneered many of the concepts involved. The demanding man-made elaboration of most photonic crystals (with some noticeable exceptions: opals in 3D, sea-mouse pins in 2D and, to some extent, iridescent feathers and butterfly wings) explains to some degree this delayed emergence. The recent advent of submicron-, hence wavelength-scale technology capable of synthesizing, depositing, molding or etching an ever-increasing number of materials has been a triggering factor in the fruitful quest for artificial materials with novel properties that address some of the fundamental aspects of 180

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181

light–matter interactions, in particular, the control of spontaneous emission. Not only deterministic technology, but more generally, nanofabrication has made remarkable contributions to the domain, including bottom-up avenues to photonic crystals such as the ever-improving self-assembly of opals. More fundamental motivations were also maturing, notably prompted by the idea of assessing Anderson-type localization in the area of electromagnetism. This requires a full understanding of the fate of light in disordered media when increasing the “photonic strength” of the constituent scatterers. Triggered by the progress in atomic physics in the early 1980s, the control of spontaneous emission in solids, especially in semiconductors, became a fundamental motivation. The pioneering papers by Yablonovitch [1987] on control of spontaneous emission and John [1987] on limits of localization, which were diverse in origin and in approach, and also previous work by Ohtaka [1979], clearly demonstrated that the concept of photonic crystal was on the verge of blossoming. The domain flourished owing to several factors: its theoretical appeal, the technological demands of the semiconductor optoelectronic community, and the endless interest in improving solid-state emitters, be it lasers or spontaneous emission sources. Many concepts of photonic crystals originate in the modified density of propagative photon states by a strong-index-contrast periodic medium. For this reason, practitioners in optoelectronics were quick to grasp this topic: the description of electron and hole bands fully relies on k-space, and modulated-index structures in semiconductor sources date back to the proposal of the DFB (distributed feedback) laser in 1964 (see Kogelnik and Shank [1971]). After almost two decades of expanding investigations, it is a challenge to cover the whole field of photonic crystals (PhCs). For various collected works, books and reviews we refer the reader to Soukoulis [1993, 1996, 2001], Rarity and Weisbuch [1996], Joannopoulos, Meade and Winn [1995], Rarity and Weisbuch [1996], Joannopoulos, Villeneuve and Fan [1997a, 1997b], Benisty, Gérard, Houdré, Rarity and Weisbuch [1999], Sakoda [2001], Busch, Lölkes, Wehrspohn and Föll [2004] and Lourtioz, Benisty, Berger, Gérard, Maystre and Tchelnokov [2005]. Special issues and two reviews are listed in Table 1. During the writing of this chapter on photonic crystals, a number of issues evolved swiftly, which we have been unable to deal with completely. Nevertheless, the reader will be provided with many opportunities to grasp a large set of effects, mostly based on the simple “linear” properties of PhCs (see comments on nonlinearities and PhCs in the conclusion). Let us make a last comment on the many approaches to photonic crystals. With the benefit of hindsight, we may say that the transversal aspect of this concept was key to its success. This is underlined by the fact that it was never limited

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Photonic crystals

Table 1 Review papers and special issues of regular journals on photonic crystals, in chronological order Year

Publication

Title

Reference

1994

J. Mod. Opt. 41, 171–404

Special Issue on Photonic Band Structures Review paper: Photonic bandgaps in periodic dielectric structures

Kurizki and Haus [1994]

Special Issue of the PECS I Conference (Laguna Beach, CA) Review paper: Photonic crystals in the optical regime – past, present, future

Scherer, Doll, Yablonovitch, Everitt and Higgins [1999] Krauss and De La Rue [1999]

Krauss and Baba [2002]

Opt. Quantum Electron. 34, 1–310

Special Issue of the PECS III Conference (St. Andrews, Scotland) Special Issue on Microcavities and Photonic Crystals Special Issue on Photonic Crystals

Photonics Nanostruct. Fundamentals Appl. 2 (2), 57–159

Special Issue of the PECS V Conference (Kyoto, Japan)

Benisty, Kawakami, Norris and Soukoulis [2004]

Prog. Quant. Electr. 18, 153–200 1999

J. Lightwave Technol. 17, 1928–2207 Prog. Quant. Electr. 23, 51–96

2002

IEEE J. Quantum Electron. 37, 724–926 C. R. Phys. 3, 1–102

2004

Villeneuve and Piché [1994a]

Weisbuch and Benisty [2002] De La Rue [2002]

to well-defined segments of physics journals, but spanned from Physical Review A, B, E, to most optical and quantum electronics related reviews and journals, also extending largely into microwave publications connected with the field of Frequency Selective Surfaces (FSSs) (Chan [1995]). The motivations for considering PhCs as part of optics core are several-fold. While no sub-domain can as yet claim today that PhCs have revolutionized its fate or created killer applications, the domain’s status is clearly frontier research and technology. Thus the teaching of PhC physics and optics is an attractive way to motivate young students and scientists in disciplines ranging from engineering to fundamental physics, all the more because the concepts they will acquire are universal. This chapter is organized as follows: In Section 2 we start with a reminder of 1D periodic dielectric stacks; this is a simple geometry that nevertheless displays a complex behavior of gaps and waves. We then describe the converse case: perfect boundaries (ideal metal) but confinement in more dimensions. We notably remind the concept of density of states (leading to local density of states and the Purcell effect). We then move on to periodic structures with 2D or 3D periodic-

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ity, discussing their k-space structure, the lattice choice, etc. We further detail the principle of 3D band calculation. A special subsection outlines the birth of photonic crystals and of the related seminal microwave experiments on homogeneous (but finite) crystals and on crystals with defects; it is concluded by a brief account of 2D pioneering work. Finally, we examine the various modeling tools and the data they provide, such as modes, “external” properties (reflection, transmission, diffraction), emission properties, etc. Within the scale of a book chapter, we attempt to guide the reader towards PhC concepts in Sections 3–6, as follows: Section 3 is dedicated to 3D PhCs. In spite of their obvious theoretical merits, the 3D systems realized in practice are still unsatisfactory. After an introductory subsection, we devote a particular section to fabrication which deals with sphere assemblies (with f.c.c. or diamond lattices), opals, inverse opals, and “woodpile”-type PhCs in either the optical or the microwave regime. Some alternative systems are examined, for example variants of macroporous Si or the so-called autocloning Si/SiO2 assemblies, that possess 3D periodicity. The next subsection gives some ideas as to measurements. The difficulties encountered in attempting to demonstrate the control of spontaneous emission in these systems are emphasized. The last subsection gives a taste of the special case of metallodielectric 3D PhCs and discusses the related issues of gaps for plasmons, subwavelength hole properties and “left-handed” materials. Section 4 makes use of the simpler 2D system to bring the reader closer to the many complex issues of theory and modeling as well as reality. It is also a good playground to provide examples of avenues that can be explored. After an introductory subsection on theoretical aspects, a large subsection deals with the general issue of implementing 2D systems on a dielectric guide (heterostructure or membrane). This implementation raises the issue of the light line and the “leaky modes” or resonances above this light line. We next discuss the corresponding realizations. At that point there is a good opportunity to outline the issues involved in applying PhC concepts to a variety of miniature ultra-compact devices aimed, for example, at routing and (de)multiplexing, in which basic building blocks such as waveguides and cavities are made to interact. PhC-based sources, i.e. lasers and LEDs, are also discussed. Section 5 deals with photonic crystal fibers, a topic that would require a review of its own. Here we highlight the main ideas, as the beauty of the results on these novel fibers is a source of inspiration for the more standard non-fiber photonic crystals. Finally, Section 6 presents perspectives on the tasks and challenges to be addressed in the forthcoming decade(s).

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§ 2. Basics of periodic dielectric media and theoretical tools 2.1. 1D Bragg structure The periodic stack of two (or more) materials of alternate index is apparently a simple system (Yariv and Yeh [1984], Baets, Bienstman and Bockstaele [1999]). Its use is well known in interference systems or as microcavity mirrors (Stanley, Houdré, Oesterle, Ilegems and Weisbuch [1993], Stanley, Houdré, Oesterle, Gailhanou and Ilegems [1994]). It also serves as a satisfactory model accounting for the numerous DBR and DFB-type devices with 1D index periodicity inside a waveguide (Tamir [1990], Kogelnik and Shank [1972], Kazarinov and Henry [1985], Baets, David and Morthier [1993], Coldren and Corzine [1995]). However, this system loses part of its simplicity when issues such as the control of spontaneous emission are considered (Rogers, Deppe and Streetman [1990], Rikken [1995], Tocci, Scalora, Bloemer, Dowling and Bowden [1996], Savona [1999]). Figure 1 shows such a stack. The conventional labeling of layers L and H, with indices n1 < n2 is adopted, with the outer medium having an index no . The basic physical behavior of this stack is described by the phases of successive reflections (Fresnel reflections in the words of optics), or by the coupling of incoming photon states with contrapropagating free-photon states by the periodic

Fig. 1. (a) Scheme of a periodic Bragg stack with period a. Reflection occurs when all rays are in a phase constructive condition. (b) k-space picture for the case of a vertical wavevector: the periodic component of the dielectric constant, of wavevector K scatters the incident wave into a reflected wave.

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“potential” (the dielectric constant) in a solid-state physics analogy. The former view has the obvious advantage of directly revealing the trends expected for nonnormal incidence, with the two s, p polarizations playing distinct roles, and it also conveys the idea that waves in the L (and even H) layers may be evanescent rather than propagative. A comparison of the 1D Helmholtz equation with the 1D Schrödinger equation shows the limitation of the solid-state analogy. Schrödinger’s equation, −h¯ 2

∂ 2ψ

+ V (z) − Em ψm = 0, ∂z2

(2.1)

represents a particle of wavefunction ψ(z) in a potential V (z), of eigenenergy Em . The Helmholtz equation (Saleh and Teich [1991]), 2 2 ∂ 2 Fm n ω 2 − (2.2) − − β m Fm = 0, ∂z2 c2 represents the complex electric field profile Fm at frequency ω for an index profile n = n(z), where the full electric field is given by E(x, z, t) = Fm (z) exp(iβm x) exp(−iωt).

(2.3)

Inside the H medium, and in terms of a plane wave, the x propagation constant is βm = (n2 ω/c) sin θm , with θm the angle with respect to the normal. A comparison of the solutions of the two equations shows that there is no simple analogy between the electronic and photonic systems: the Helmholtz equation yields 2 (i.e., “eigenangles” of the propagation ray) at fixed ω rather eigenvalues of βm than eigenvalues of the photon energy h¯ ω. Another point to note is that the “potential” of the Helmholtz equation scales with ω2 . Let us stress that this difference between the two systems comes in part because we do not consider, as is usual, any propagation in the x–y directions for the electron case (in a quantum well [QW], the 1D calculus just gives the subband edges, improperly called “levels”) while it is obviously required for an off-normal ray in optics. Nevertheless, the HLHL. . . infinite stack at fixed ω is a canonical wave problem fully similar to historical ones (Kronig and Penney [1931]) and to the popular superlattice model of electrons in heterostructures. It produces corresponding bands and gaps not only when the effective index n2eff = β 2 c2 /ω2 lies between the two extreme dielectric constants n21 and n22 , but also below. It is somewhat obvious that by varying ω, while remaining at given H-angle θ , frequency bands and gaps can also be found rather than “angular gaps”. However, even at normal incidence θ = 0, the simple picture of phases along optical paths only gives the central location of such gaps, not their width. Namely, for an optical

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path equivalent to a 2pπ phase (p integer) on a round-trip through an LH unit cell, reflections of successive LH cells tend to interfere constructively at the expense of power in the transmitted beam. Hence the expression 2pπω (2.4) = pλ c gives a frequency ω in the pth gap (Russell, Birks and Lloyd-Lucas [1995]). Let us exemplify the use of the solid-state picture to calculate the 1st gap width ω of a canonical infinite HLHL. . . structure in the limit of small refractive √ index contrast (denoted RIC = n/n = (n2 − n1 ) εav ). In this view, the weakly modulated dielectric profile ε(z) = n2 (z) is written 2(n1 d1 + n2 d2 ) =

ε(z) =

1 p

2

εpmod exp(ipKz) + exp(−ipKz) ,

K=

2π , a

(2.5)

and we start from free photons for which we write the dispersion relation √

ω εav . (2.6) c Here, εav ∼ (ε1 + ε2 )/2 is the average dielectric constant. It results in principle from the full homogenization procedure (Halevi, Krokhin and Arriaga [1999a, 1999b], Datta, Chan, Ho and Soukoulis [1993], Economou and Zdetsis [1989]). As shown in fig. 2, the periodic index modulation couples preferentially any two states differing by 2π 2π k = p (2.7) for p = 1 . = a a k=

This results from the predominant coupling between two states by the periodic perturbation (the “matrix element” of the perturbing “potential”; see Kittel [1976], Ashcroft and Mermin [1976]), which is a momentum conservation expressed by products of the following type 1

exp(ik1 z) εpmod exp(ipKz) + c.c. exp(ik2 z) . (2.8) 2 The coupling is efficient only if frequency conservation is simultaneously allowed (see fig. 2). The simplest case of energy conservation is k1 = π/a = −k2 , for which the unperturbed frequency is πc kc = √ = ω0 ω= √ εav a εav

(≡ ωBragg ),

similar to eq. (2.4) when d1 ≈ d2 ≈ a/2 and

√ εav ≈ (n1 + n2 )/2.

(2.9)

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Fig. 2. (a) Dispersion of free photons and coupling by wavevector K. (b) Reflectivity spectrum (left) and dispersion relation (right) for a one-dimensional Bragg system. (c,d) Intensity patterns at band edges of the fundamental gap are stationary waves with maxima in either the H medium (lower band edge) or the L medium (upper band edge).

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The extra information obtained here is based on fig. 2: the two linear combinations of the coupled states, reading exp(ik1 z) ± exp(ik2 z), are the eigenfunctions of the perturbed “Hamiltonian” and just read cos(Kz/2) and sin(Kz/2). From here on, these two modes preferentially sample the H and L media for cos(Kz/2) and sin(Kz/2) respectively, having a maximum in one layer and a zero in the other layer. Defining ε as the expectation value of ε(z) when weighted by any field profile Fm (z), we have a ε(z)F 2 (z) dz . ε = 0 a 2 m (2.10) 0 Fm (z) dz

Using this definition for the two states, cos(Kz/2) and sin(Kz/2), we find that the 1D frequency gap at θ = 0 appears by writing the two band edge frequencies naively as 1 K ωval, con = c (2.11) , 2 εval, con

where ωval relates to the cos(Kz/2) state (“val” stands for “valence”; as for the valence band of semiconductors, and we have εval > εav , meaning that the field is concentrated in the H layer). Conversely, ωcon relates to the sin(Kz/2) state (“con” stands for “conduction”, as for the conduction band, and the field is concentrated in the L layer1 ). We leave it to the reader to check that the gap width ω = ωcon − ωval is basically controlled by the first Fourier component εpmod of ε(z), so that for our example of the square-shaped ε(z) profile, the following approximate relation holds 2 ω ≈ RIC. (2.12) ω π In this exercise, a classic in the electron theory of crystals (Kittel [1976], Ashcroft and Mermin [1976]), we have encountered two important band-related features which will be generalized in 2D and 3D: • At the fundamental gap of simple structures, states (modes) of the lower (resp. higher) frequency edges are preferentially located on the higher (resp. lower) index medium. • The gap size in a given direction is directly related to the Fourier components of ε(r) able to scatter a wave in this direction to a wave scattered into another allowed direction (not necessarily along the incident direction!).

1 They are also often called dielectric (val) and air (con) band edges, respectively.

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A last feature of importance is to what extent a wave penetrates in the lattice for frequencies inside the gap and how much of its group velocity dω/dk is slowed down when approaching band edges. From the simple Fresnel reflection coefficient or from the Coupled Mode Theory (CMT), it can be inferred that the same quantity, the refractive index contrast denoted RIC, is again crucial. The Fresnel reflection coefficient for normal incidence is r = (n1 − n2 )/(n1 + n2 ) = √ (n1 − n2 )/(2 εav ) = RIC/2. In the situation of constructive interference of L and H reflections, the result is a reflection coefficient equal to the RIC per period. From this remark, the attenuation length of the incident “pump” beam may be estimated because it is given by the number N of periods such that reflection has properly built up. A basic estimate is N = 1/RIC, yielding naively a cumulated reflection of 0.5. Hence a third feature is: • The wave penetration at mid-gap is of the order of a/RIC, with an exponentially decaying pattern along z [zero and maxima also shift from those of fig. 2(c) to those of fig. 2(d) across the gap]. A coupled-mode approach not developed here provides more quantitative insight for band edges, giving the typical trend for group velocity: • The group velocity drops to half its unperturbed value when approaching the band edge by an amount ω = ωd −ωcon given, for example on the conduction side, by the following approximation ωd − ωcon ≈ 0.05 RIC ω0 .

(2.13)

From this solid-state picture we have grasped the basics of the 1D gap at θ = 0. We can now extend it to account for the evolution of gaps with θ , essentially obtained from the behavior of round-trip phases and Fresnel coefficient evolutions. The resulting overall picture of the effect of a 1D periodic stack is unfolded in fig. 3. This picture is made in 3D k-space. The role of the 1D periodicity along the z-axis is to couple waves whose kz differ by K = 2π/a (or multiples thereof . . .), and some allowance is made around this Bragg value that is proportional to the RIC (but this allowance is angle-dependent and of course polarization-dependent). Thus, taking all elements from solid state and optics into account, a slab of k-space centered around K/2 and −K/2 along the z-axis is an appropriate way to visualize the forbidden gap due to z-periodicity in 3D k-space [see fig. 3(b)]. This is detailed for increasing frequencies in figs. 3(c–g) (the exercise could also be worked out by plotting spectral evolution for increasing angles). For frequencies well below the Bragg frequency, only some shape birefringence occurs. The phase and group velocities differ in different directions with respect to z (Datta, Chan, Ho and Soukoulis [1993], Yeh, Yariv and Hong [1977]). For a

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Fig. 3. (a) Axis used shown in real space. (b) Scheme of reciprocal space and “forbidden slab”. (c–g) Three-dimensional picture of the evolution of the stopband of a one-dimensional periodic system as a function of frequency: (c) low frequency; (d) about mid-gap frequency at normal incidence; (e) conduction (“air”) band edge at normal incidence; (f) conduction band frequency a little above the gap; (g) far above the gap, forbidden regions correspond to kx ≫ kz . (h) Three-dimensional view of (f) showing a ring-shaped forbidden band. (i) Constant-energy surfaces corresponding to cases (f) and (h), see fig. 9 and Section 2.6.

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frequency ωa just above the normal-incidence low-frequency band-edge ωval [but less than the Bragg frequency ω0 ∼ = (ωval + ωcon )/2], the directional gap around z is a narrow cone. Its angular extent will rapidly grow as ωa goes towards the Bragg frequency ω0 . The angular extent of this gap is maximal when ω reaches ωcon , the upper-frequency band-edge. For ω slightly above ωcon , a gap exists only between two cones, but none at normal incidence. The angular extent of this gap is all the more polarization-dependent as the angle increases. It even disappears for p (TM) polarization around the L/H Brewster angle. Conversely, for an s (TE) wave, this gap becomes more and more efficient towards grazing angles since rs , the s (TE) polarization Fresnel coefficient, steadily increases in this regime, until one reaches the regime of evanescent waves in the L layers (similar to electron states of the superlattice below the barrier level, but above the well level). Guidance by sandwiching a flat slab between two such multilayer stacks has often been considered in the literature (Yariv and Yeh [1984]). One popular name for some such systems is the “ARROW” waveguide (AntiResonant Reflective Optical Waveguide) (Duguay, Kokubun, Koch and Pfeiffer [1986]).

2.2. Density of states, local DOS Global treatment From fig. 3, one can see that at a given frequency ω there are always angular windows with propagating photon states allowed, that take nearly 4π steradians. This means that the density of states (DOS) of such a 1D system by itself will not be much different from the usual bulk 3D value (given here per polarization), 1 n3 ω 2 (2.14) . V 2π2 c3 In other words, a slab of material sandwiched between 1D layered mirrors would exhibit 2D-type behavior only in a moderate fraction of the solid angle [see, for instance, fig. 3(f)]. Let us remind ourselves from fig. 4 of what are the ideal forms of the DOS of lower-dimensional systems for simplified scalar waves, as shown by Brorson [1995], Brorson and Skovgaard [1996], Ho, Chu, Zhang, Wu and Chin [1996] and Ho, Wang and Park [1999]. In a 2D system, say a slab of material of thickness L and index n, between two perfect mirrors with zero-field boundary conditions, the electric field of the guided modes with TE polarization has the simple expression ρ3D (ω) =

E(x, z, t) = E(x, z)e−iωt = Fm (x) exp(iβm z)e−iωt ,

(2.15)

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Fig. 4. Left to right: schematic density-of-states ρ(ω) of photonic systems of dimensionality 3, 2, 1 and 0.

with z one of the two free directions, with mπx (m) ω Fm (x) = sin , βm = neff , (2.16) c L and with the effective index [whose definition (2.16) is general] given by 2 mπ 2 nω mπc 2 (m) 2 , βm + = . neff = n 1 − (2.17) Lω L c The second relation reflects the separation of the overall momentum between confined and free directions. Each branch m adds, beyond its cutoff frequency ωm = mπc/L, the same contribution to the photonic DOS, linear in ω, ρ3D (ω) =

0
n2 ω . 2πc2

(2.18)

This is useful in the context of resonant-cavity light-emitting diodes (RCLEDs, see Section 4.4 on LEDs) (Benisty, De Neve and Weisbuch [1998a, 1998b]) as it implies that each mode will carry away the same spontaneous emission. In an ideal 1D system confined in two directions, similar equations can be written. However, since the dispersion relation ω(β) ≡ ω(k ) is 1D and verifies ∂ω 0, each branch cut-off gives rise to a singularity for the DOS, scal∂β |β=0 = o , where ωo ing as 1/ ω − ωm,p m,p is the cut-off frequency of the mode indexed m, p similar to eqs. (2.16), (2.17) (Kleppner [1981]).

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In a 0D system, such as a photonic box, ideally the DOS is a series of Dirac o ). Of course, in practice, any photon loss mechanism smooths peaks δ(ω − ωm,p,q out these singularities and makes the system’s quality factor Q finite rather than infinite. Local photon DOS and spontaneous emission Predicting the spontaneous emission modification in a PhC amounts primarily to evaluating how many photon states interact with a given dipole d (an emitter). The perturbative regime of “weak” light–matter interactions is relevant in most experiments, as opposed to strong coupling, such as for atoms and excitons with confined photons (Baba, Hamano, Koyama and Iga [1991], Haroche [1992], Yokoyama, Nambu and Kawakami [1995], Yokoyama and Ujihara [1995], Burstein and Weisbuch [1995], Savona [1999], Peter, Senellart, Martrou, Lemaitre, Hours, Gérard and Bloch [2005], Weisbuch and Benisty [2005]). The concept of irreversible spontaneous emission with a given spontaneous emission rate (SER) is thus valid. However, we can see from Section 2.1 and fig. 2 that for some bands or band edges, the intensity is completely redistributed inside the unit cell or in the present case inside the photonic box (Lee, Xu and Yariv [2000], Busch, Lölkes, Wehrspohn and Föll [2004], Megens, Wijnhoven, Lagendijk and Vos [1999a, 1999b]). In the description of the interaction by the Fermi golden rule relevant to the “weak coupling” regime, the photon DOS is associated to an overlap factor | ψexc (r)|E(r) · r|ψground (r) |2 between the relevant electronic wavefunctions ψexc |, |ψground and the field of the electromagnetic mode E(r), a matrix element in the language of quantum mechanics. For this reason, modes are seen with a position-dependent strength, leading to a summation that depends on the position of the dipole (often a delta-function compared to the extent of the field mode). Hence the summation over modes leads to a heavily position-dependent result, such as inside the unit cell of a PhC. This basic aspect of the interaction affects coupling from, say, inside a PhC or a cavity to the outside world, and hence the outside spatial distribution of light. This follows earlier related findings for planar microcavities (Björk, Heitmann and Yamamoto [1993]) and slab waveguides (Wittke [1975], Brorson, Yokoyama and Ippen [1990]). The overall topic is complex in spite of the relatively simple idea. We refer the reader to the various special issues and books introduced in Section 1, each shedding its own light on the matter. We would like to emphasize that the principle is classical, and not quantum-mechanical: the quantum-mechanical presentation has the advantage of a seamless proximity with textbook presentations of spontaneous emission in vacuum.

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On the contrary, the classical treatment requires the use of a Green dyadic (Hughes, Rammuno, Young and Sipe [2005]) and its imaginary part, and would often be classified as “advanced electromagnetism”. The basis goes back to an Ansatz known as Sommerfeld treatment of spherical waves (Born and Wolf [1999]) and how to handle field discontinuities at a source as shown in works by Ford and Weber [1984], Kurizki and Genack [1988], Benisty, Stanley and Maier [1998], following the pioneer work by Lukosz around 1980. It would take us too far to follow this track, although it relates to so many basic ideas in optics (Huygens–Fresnel principle, Gouy phase shift). The field coined “nanophotonics”, especially with regards to near-field, local tip and aperture effects (see Section 6), addresses this topic nowadays. See Section 3 for a few words on local DOS in 3D PhC. The Purcell effect A manner in which to address changes in Spontaneous Emission Rate (SER) was proposed for cavities in 1946 (Purcell [1946]) and revived in the 1990s for microcavities in the optical regime (it had been established meanwhile in atomic physics with microwave transitions between Rydberg levels of alkali atoms, see Haroche [1992]). For a mode that occupies a cavity, the modal volume Vmode may essentially be defined as the volume of an equivalent perfectly confined cube. For the situation of a dipole at the main antinode of the field, and parallel to it, the SER is easily compared to that of vacuum. The density of states of the single mode may be seen as the main ingredient of the SER modification: ρ3D (ω) peaks approximately at 1/Q, where Q is the mode quality factor. A detailed calculation (Gérard, Sermage, Gayral, Legrand, Costard and Thierry-Mieg [1998]) gives a SER modification by the “Purcell factor”, (λ/n)3 3 Q Q ∼ . 2 Vmode Vmode 4π Obtaining higher Purcell factors and making optimum use of the accelerated SER is one of the holy grails of microcavity research (Barnes, Björk, Gérard, Jonsson, Wasey, Worthing and Zwiller [2002], Lourtioz, Benisty, Berger, Gérard, Maystre and Tchelnokov [2005]). FP =

2.3. Generalizing to 2D and 3D periodic crystals One of the main desires when considering PhCs is to generalize fig. 2 to achieve 0D and 1D systems that behave like the ideal confined photonic wire and even more like the photonic box of fig. 4.

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This desire is natural in order to avoid losses of metallic cavities at optical frequencies (UV to near infrared). Simple dielectric shapes lack some of the properties desired for ideal photonic wires or boxes, although some very interesting realizations exist: liquid droplets, silica microspheres, microdisks, semiconductor-based wires (Chang and Campillo [1996]), micropillars (Gérard, Sermage, Gayral, Legrand, Costard and Thierry-Mieg [1998]), etc. In brief, these relatively simple shapes do not achieve adequate confinement in the dimensions concerned primarily because of large mode volumes. However, this could be compensated by the very high Q-values reached (Kippenberg, Spillane and Vahala [2004], Vahala [2003], Cai, Painter, Vahala and Sercel [2000]). Alternatively, we may hope that owing to their multidimensional periodicity, the simplified realspace square and cubic lattices of fig. 5 could exert the desired confining role in k-space, if the sets of forbidden slabs of k-space associated to each periodicity (similar to those of fig. 2), overlap sufficiently. If frequencies exist such that their associated spheres (3D) or circles (2D) lie inside these k-space slabs, electromagnetic waves at these frequencies would feel a confining effect similar to that of ideal metallic layers, and one could exert a similar control of light–matter interaction through the DOS, the mode profile, etc. Let us note at this stage that this achievement is in principle straightforward in 3D (fig. 6): a 3D forbidden gap means ρ3D (ω) = 0 for the corresponding frequencies (the photonic bandgap, or PBG), and thus, complete inhibition of spontaneous emission for an emitter inside the system, as originally proposed by Yablonovitch. However, in 2D, one has at least two intersecting cases: one can inhibit modes propagating “horizontally”, i.e., normally to the invariant axis of the PhC (fig. 5, the situation is a two-dimensional photonic bandgap, often denoted 2D PBG), or one can [similar to fig. 3(g)], suppress all modes having some nonzero β value along this axis (then the property will hold for some β interval). This basically amounts to considering that the intersection of the “k-space slabs” with a sphere |k| = const. is a surface that can contain a stripe of “parallels” of the sphere (a thin set of “latitudes”). These two extreme cases (gap for β = 0 or for some finite β) correspond to two types of applications examined further in this chapter: • β = 0 corresponds to the quest for “planar PhCs” that can achieve in-plane control of light by 2D periodicity, relying on some other kind of control for the third dimension. This is notably through the classical refractive index guiding scheme, the workhorse of integrated optics and photonic integrated circuits (Coldren and Corzine [1995]). • |β| ≈ β0 (where β0 is the light wavevector in the uniform core medium) corresponds to photonic crystal fibers (Section 5), whereby the PhC surrounds a

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Fig. 5. A picture of the real space and reciprocal space of structures with periodicity along one to three dimensions. In 1D, one has only one pair of “forbidden slabs”. In 2D, one has four pairs for a square lattice but six for a triangular lattice. The gap may be created in an omnidirectional manner in the plane of periodicity provided that it is thick enough. This is easier for the triangular lattice. Note the complex shape of the out-of-plane behavior suggested by the sphere with two square calottes. In 3D, there are three pairs of slabs for the simple cubic lattice. They may now comprise the whole sphere and allow an omnidirectional gap.

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Fig. 6. Schematics of a photonic box with confinement by a three-dimensional periodic structure, and an emitter inside.

central (core) region where confined light may propagate as it does in classical, refractive index-based optical fibers, but with a new series of avenues opened.2 After this brief account of the main ideas leading to the concept of photonic crystals, two arguments appear in moving from 1D to higher-dimensional dielectric systems. Firstly, insofar as the dielectric pattern of each unit cell of the lattice (the photonic “atoms” if we take the picture of the electron periodic potential) remains simple, there is an obvious advantage in looking for lattices having the roundest possible First Brillouin Zone (FBZ). Based on the visualization of the fundamental gap in k-space through the associated “forbidden slabs” described above and derived from the Fourier components ε˜ (G), a pair of slab arises normally to each reciprocal vector G. Then, in 2D, a triangular lattice with a hexagonal FBZ is more suitable to fit a circle into the set of six slabs than a square lattice and the associated set of four slabs. In 3D, the best solution in terms of roundness is the FBZ of the face-centered cubic (f.c.c.) lattice (the same as that of diamond, Si and zinc-blende, since these lattices only have an extra translation in real space). A natural measure of roundness is the pseudo-eccentricity of the (hexagonal FBZ) first Brillouin √ zone ξ = kmax /kmin . In 2D, the triangular lattice √ has ξ = 2/ 3 ≈ 1.15, whereas the square one features ξ = 2 =√1.41. This is shown in fig. 5. In 3D, while a simple cubic lattice features ξ = 3 = 1.73, 2 Two obvious cases are the use of an “air” core to minimize nonlinearities at high transmitted power, at the entrance of a long-haul link, and fibers with special dispersion properties – eternally monomode fibers, zero-group-velocity-delay-shifted fibers, etc. – on account of the peculiar PhC behavior, e.g., phase of reflection vs. incidence angle in the photonic gap, as discussed in Section 5.

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√ the FBZ of an f.c.c. crystal has the much lower value ξ = 5/3 = 1.29. The refractive index contrast (RIC) of many materials with air should be sufficient to have overlapping gaps even though such values of ξ are not so close to unity. By simple reasoning, it can be suggested that the gap overlap arises when the upper frequency of the gap in the kmin direction and the lower frequency of the gap in the kmax direction meet halfway (with respect to the central gap frequencies of each direction, which scale like k’s). This means, with reference to fig. 2, a relative halfgap opening (ω2 − ω0 )/ω0 of (ξ − 1)/2 in a given direction. This (ξ − 1) scaling provides a good basis for understanding the behavior of simple fundamental gaps. Apparently, this gap opening only requires a modest dielectric constant modulation εmod ∼ (ξ − 1)εav , that is 30% or so in relative terms. However, an important point to note is that a much higher RIC than described by this simple argument is needed. For any practical application, the need for very high index modulation is a strong limitation. In the real world of dielectrics and at a given frequency, the range of available relative dielectric constants ε is bounded to upper limits of typically ε ∼ 15 in the near IR and 8 in the blue wavelengths range (λ ∼ 400 nm). Specifically, one reason to request large modulation is that when going to 2D and 3D periodicity, the large ε1mod that was achieved in 1D (e.g., in a Si/silica or Si/air stack), cannot be achieved simultaneously for the two or three independent wavevectors of interest (the averaged index difference is scaled down by the 3D structuring). For example in the simple cubic case, the three Fourier components εmod (π/a, 0, 0), εmod (0, π/a, 0) and εmod (0, 0, π/a) have to share the available total ε variation, being reduced typically by a factor 1/3. More mathematically, the simplest hypothetical crystal that can be thought of, in 3D, would be the sum of three identical piecewise-constant functions: ε(x, y, z) = 1 + ε S(x) + S(y) + S(z) , (2.19) where S(x) is a unit square waveform of period a so that ε has a minimum value of 1 and a maximum value of 1 + 3ε. Hence, even for this hypothetical crystal (remarkably easy to treat because ε is separable), each directional gap dictated by ε would be divided by 3: one has to match 1 + 3ε with the maximum available ε (constrained by the choice of the material) instead of 1 + ε in the 1D case. To attain the required modulation for two or three directions of space in a feasible geometry, generally very far from separable, and piecewise constant, it turns out that the dielectric constants need to be typically ranging from ε ∼ 7 to ε ∼ 14 if the other material is air. To summarize the picture resulting from this comparison between 1D and 3D structures, any clever 3D lattice of silicon and air cannot reach, in a single direction, a fraction of the strong confining features of a Si/air 1D stack, be it the large

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gap spectral width or the short penetration length. This emphasizes the difference when comparing with the perfect metal confinement. However, as we will see, some 3D lattices can extinguish the DOS over the entire Brillouin zone whereas the 1D lattice is fundamentally unable to achieve this effect. Let us finally signal two caveat(s) in considering the merits of gaps and bandstructures in various dimensions: (i) Some excitation arose around the possibility that 1D stacks could be “perfect mirrors”. This was obtained, however, by considering, a truncated k-space associated with waves impinging from a low-index surrounding medium for example (see Chigrin, Lavrinenko, Yarostky and Gaponenko [1999], Winn, Fink, Fan, Chen, Michel, Joannopoulos and Thomas [1998]). (ii) In 2D and 3D, arbitrarily low group velocities appear in photonic band structures (Section 2.4) even in the case of “free photons”. This can firstly be seen as the result of mapping all farther Brillouin zones into the first zone. Multiple equivalent segments in these farther zones sample several directions. The vector addition of the reciprocal wavevector G and the Bloch wavevector k then results in this somehow misleading appearance. In a complementary view, this means that the eigenmode belongs to a degenerate set of two (or more) plane waves. Its global group velocity is the projection of the individual velocities on the wavevectors bisector. The same situation of field superposition arises in the textbook waveguide exercise of determining the field between two perfect metal slabs, but associated in this case with a genuine physical boundary condition.

2.4. Basics of band calculation We briefly outline here the plane-wave expansion method, which is the basic method for calculating band structures, and is the generalization of fig. 2, taking the form ω = ω(k) in 2D or 3D. In this method, all fields E, H are Fouriertransformed and explicit use is made of the Floquet–Bloch theorem. This has been treated by Kurizki and Haus [1994] in a special issue, Zhang and Satpathy [1990], Leung and Liu [1990a], Sakoda [2001], Joannopoulos, Meade and Winn [1995], Lourtioz, Benisty, Berger, Gérard, Maystre and Tchelnokov [2005], Busch, Lölkes, Wehrspohn and Föll [2004]. See also the MIT code at the Internet site http://abinitio.mit.edu/mpb/ . For example, it states that the H field at a given frequency ω can be written as a superposition of eigenmodes of the form H(r, t) = Hk (r)eik·r e−iωt ,

(2.20)

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where Hk (r) is periodic. These eigenmodes of infinite structures may have a complex relationship with outside waves in a finite or semi-infinite PhC geometry. Denoting by G the reciprocal lattice vectors (G[n1 a1 + n2 a2 + n3 a3 ] is a multiple of 2π for any integers nj ), Hk can thus be rewritten h1G e1G eiG·r + h2G e2G eiG·r , Hk (r) = (2.21) G

and h2G are the scalar coefficients of e1G and e2G with these two latwhere ter vectors forming an orthogonal triad with (G + k). The divergence theorem, ∇ · H = 0, ensures that there is no component of H along G + k, hence the two vectors e1G and e2G suffice. The plane waves form a convenient basis. But individually, they are generally not solutions of Maxwell equations for any effective uniform medium. So these constituent plane waves should be considered with care. The most direct way to proceed is to obtain the eigenfrequencies as functions of k by using the following form of the Maxwell equations in a medium with permeability μ ≡ μ0 and nonuniform ε(r), h1G

ω2 ∇ × ε −1 (r)∇ × H = 2 H. (2.22) c The curl operator is diagonal in k-space. The impermeability, defined as η(r) = ε−1 (r), is written as ηG eiG·r η(r) = ε −1 (r) = (2.23) G

with the reciprocal relation 1 η(r)e−iG·r d3 r, ηG = V

(2.24)

where V is the unit cell volume. When investigating novel PhC structures, e.g., different unit cells in a given Bravais lattice, only these “form factors” change. For some complex shapes such as the Yablonovite, the calculation may be cumbersome. Inserting eqs. (2.23), (2.24) and (2.21) into eq. (2.22), and projecting onto the eiG·r functions, one gets an infinite matrix eigenvalue problem G′

G′

h2 = M11 h1 + M12 GG′ G GG′ G

ω2 1 h , c2 G

h2 = M21 h1 + M22 GG′ G GG′ G

ω2 2 h , c2 G

(2.25)

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where the matrix elements are given by

′

′ Mλλ = (k + G) eλG ηG−G′ eλG′ k + G′ . GG′

201

(2.26)

The electric field is obtained from the magnetic excitation by the curl relation. The 2D version of this calculation, and related discussion, can be found in works by Villeneuve and Piché [1992a, 1992b, 1994a], Plihal and Maradudin [1991], Plihal, Shambrook, Maradudin and Sheng [1994], Meade, Rappe, Brommer and Alerhand [1993], Cassagne, Jouanin and Bertho [1995a, 1995b, 1996]. An important feature in the pure 2D case is the polarization separation, allowing either field H or E to be scalar. In practice, one has to truncate eq. (2.25) to a finite number N of plane waves G = {G1 , . . . , GN }, and it is safe to assume zero for all those matrix elements having their difference G − G′ outside this set. The set is usually chosen with |G| < Gmax , that is limited to some disk of k-space. One delicate issue is how to reduce the numerical error due to truncation Sözüer, Haus and Inguva [1992]. Ho, Chan and Soukoulis [1990] obtained η’s Fourier components as follows: one first truncates the Toeplitz-type matrix εG−G′ , and then inverts it. Plihal and Maradudin [1991], on the other hand, directly Fourier-transformed η. Both methods lead to different convergence as will be dealt with in the Section 2.7 on theoretical methods and grating computation problems. The usual method to display the band structure [the eigenvalues ω(k)] is taken from crystal physics: a circuit is chosen along the first BZ that samples all distinct symmetry lines of the irreducible BZ, ensuring fairly enough that extrema met along the circuit for any band index are the actual extrema over the whole k-space. Such examples will be given in Sections 3 and 4.

2.5. The discovery of PBG Although the concept of photonic bands was envisioned many decades ago, its application to multidimensional structures first appears in the papers by Ohtaka [1979] and Ohtaka and Numata [1979]. However, it was not until the 1987 paper by Yablonovitch [1987] and the simultaneous work by John [1987] that interest arose for these challenging systems and particularly for the concept of photonic bandgap (PBG). Whereas the localization concept put forward by John did not lend itself to straightforward experimental verification, the zero-photon-DOS between photonic bands clearly implies zero transmission (and thus total reflection into allowed diffracted orders) in any direction. The spontaneous emission inhibition (SEI), also directly related to zero-photon-DOS, was, and still is, delicate

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to observe: It requires monitoring the lifetime of an atom or molecule embedded as deeply as possible in a photonic crystal, hence, with a smaller and smaller coupling to the outside detector. An extra requirement to keep in mind is that the radiative channel rate modification of SEI should not be masked by nonradiative decay channels. The years 1987–1991 were loaded with a number of controversies, though: following Yablonovitch and Gmitter [1989] and their first experimental results, the band calculation was scalar (Satpathy, Zhang and Salehpour [1990], Leung and Liu [1990a]). The attempts to check band calculation were somewhat imprecise, and hence not really conclusive: a set of air spheres suitable for microwave transmission experiments (cm-sized spheres) was fabricated in an f.c.c. arrangement, by a tedious layer-by-layer stacking method, and the transmission band edges were traced at varying incidence angles. See fig. 7 for some basic 3D crystals and their band structures (Haus and Sözüer [1994]). An intense controversy developed (Maddox [1990], Yablonovitch [1991], see also Ho, Chan and Soukoulis [1993]) when the vector calculation (Leung and Liu [1990b], Zhang and Satpathy [1990]) established that basic compact arrangements had an unavoidable degeneracy at the W point of the first BZ, preventing the expected gap opening. The same year, Ho, Chan and Soukoulis [1990] proved that a diamond-type arrangement of dielectric and air spheres could lift the degeneracy and easily provide the desired gap opening. Bouncing on these ideas, Yablonovitch, Gmitter and Leung [1991] (see also Yablonovitch and Leung [1991]) proposed a novel way to build a microwave-suited crystal with adequate diamond symmetry [see fig. 7(e)]. They drilled in a cube three intersecting sets of cylindrical holes at a 35◦ tilt from normal and a 120◦ azimuth from each other onto a square lattice of sites on one cube face. This mimics the (111) galleries of the Si or diamond crystal as seen from a (100) face. The measurements confirmed the band position. This structure is now often named “Yablonovite” [see fig. 7(e)] and was the first confirmed “PBG material”, that is, a PhC with an omnidirectional gap, in agreement with predictions by Chan, Ho and Soukoulis [1991]. It was shown that the f.c.c. lattice has a full bandgap, albeit in the inverted opal geometry (f.c.c. air spheres packed in a dielectric), between the 8th and 9th bands (Sözüer and Haus [1993] and references therein, Yablonovitch [1993], Meade, Rappe, Brommer and Joannopoulos [1993]; see also Section 3). In 1994, the so-called woodpile-type structures [fig. 7(f)] was proposed by Ho, Chan, Soukoulis, Biswas and Sigalas [1994] in order to match to the diamond lattice. The period comprises 4 layers, the 3rd and 4th being similar to the 1st and 2nd, but shifted by one half of the rod pitch. This idea was then successfully tested in the microwave regime (Özbay, Abeyta, Tuttle, Tringides, Biswas, Chan, Soukoulis and Ho [1994]), between 12 and 19 GHz and later at higher frequencies

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Fig. 7. Basic 3D photonic crystals and some characteristics of their band structure (from Haus and Sözüer [1994], work performed at Rensselaer Polytechnic Institute): (a) cubic lattice of square rods; (b) band structure, with first Brillouin zone in inset; (c) band structure of close-packed inverse air spheres in an f.c.c. lattice (in ε = 13 dielectric constant); (d) same for overlapping air spheres in a diamond lattice, 81% air in ε = 3.62 dielectric constant; (e) principle of “Yablonovite”, the three drilling axes mimicking the set of (111) galleries of silicon or diamond lattice; (f) “woodpile” or “layer-by-layer” structure.

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Fig. 8. Progress towards higher operating frequencies of photonic crystals as a function of time during the period 1991–2000 (after Soukoulis [2001]).

(Özbay, Tuttle, Sigalas, Biswas, Soukoulis and Ho [1995]). Another kind of structure amenable to planar processing steps of deposition and masking/etching was proposed by Fan, Villeneuve, Meade and Joannopoulos [1994]. Figure 8 shows the time progression towards smaller features and higher frequencies for the various 3D structures that will be described in more detail later. Point defects, surface modes The ability of PhCs to confine light is most spectacularly demonstrated when it comes to single-mode point defects. The defects fabricated by adding or removing material to the unit cell have, in simple cases, specific properties (see Sakoda [1997] for classification of symmetry properties). Just like donors in semiconductors (which have one extra electron and more importantly one extra proton, e.g., P in Si), extra material in PhCs means an extra potential well, and a localized state appears with its energy emerging in the PBG from the upper air band downwards (fig. 9). Conversely, removed dielectric results in an acceptor-type photonic defect mode lying just above the dielectric band edge. Defect energies shift away from their “native” bands as the strength of the perturbation (size, ε) increases. A finite size may be required to have the defect frequency in the gap: a defect state is derived from a given band (think of semi-

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Fig. 9. First examples of defects in “Yablonovite”. (a) Cross-sectional view of the crystal in the 1, −1, 0 plane, showing the nonspherical atoms centered on the black dots and the unit cube cross-section (dashed rectangle); donor defects consist of inserted dielectric spheres of variable sizes (bottom right) and acceptor defects are slices cut as shown, of variable width. (b) Schematic microwave spectral measurement result for a deep defect; note the gap and in the middle, the transmission peak, much smaller than unity in general. (c) Frequencies of defect modes vs defect volume, as measured (open and solid circles) and as calculated (curves).

conductor dopants). This band may differ from the band of the nearest band edge if the band structure is somewhat complex. Energy-wise, when situated in the gap, defect “levels” have an actual linewidth dictated by the tunneling-out of photons for finite crystals, and by dielectric losses (Özbay and Temelkuran [1996]). Spatially, defects can lead to a high degree of confinement of optical waves, depending on the refractive index contrast (RIC) and the defect design. In a 1D system, the λ/4 defect layer in a DBR can be shown to lead to the highest localization, as can be expected from the resultant λ/2 cavity (Stanley, Houdré, Oesterle, Ilegems and Weisbuch [1993]).

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Such defect states, either shallow or deep in the PBG (Meade, Brommer, Rappe and Joannopoulos [1991a]), were readily demonstrated by experiments on “Yablonovite” structures (Yablonovitch, Gmitter, Meade, Rappe, Brommer and Joannopoulos [1991]) (fig. 7) and provided the final boost to launch the topic of photonic bandgaps (PBGs) and photonic crystals (PhCs) further towards either fundamental phenomena or applications. It is also worth noting that the mere truncation of the periodicity at a PhC surface is also able to produce localized surface modes, with their frequency possibly in the photonic gap of the bulk. These photonic states are the analogue of surface states familiar in the area of semiconductor interfaces. The surface periodicity is of course of lower dimensionality (Meade, Brommer, Rappe and Joannopoulos [1991b], Ramos-Mendieta and Halevi [1996], Robertson, Arjavalingam, Meade, Brommer, Rappe and Joannopoulos [1993a]). 2D Photonic crystals To briefly conclude this history of the discovery of PBG, it is interesting to note that although they do not provide spontaneous emission inhibition (SEI), the properties of 2D structures, with k restricted to a plane and the invariance normal to this plane, were investigated between 1990 and 1994 (Robertson, Arjavalingam, Meade, Brommer, Rappe and Joannopoulos [1993b]). A number of useful concepts could be developed without the complexity of 3D structures. Additionally, the understanding of 2D structures is simpler because the two polarizations decouple. One simple rule, well explained by Joannopoulos, Meade and Winn [1995] and related to the field topology (see also Section 4), states that as far as one works around the fundamental gap, arrays of dielectric columns tend to have a gap for the TM polarization (E columns) while arrays of air columns in a dielectric matrix have a preferred TE gap (H columns) [figs. 10(a–d)]. As for the lattice, the triangular/hexagonal lattices have the roundest BZ. Note that the TE gap opening [figs. 10(d,e)] is not very demanding in terms of photonic strength (filling factor or refractive index contrast). The quest for a PBG common to both polarizations was however initially considered as more important. It has led to two preferred simple structures: • The first is an array of air columns in a high-index matrix, with very high airfilling factors [fig. 10(e), right-hand side]. Hence, the structure has both the “veins” that allow the electric field to circulate in high-index material as for TE polarization, and the necessary material left between the three air columns to play the role of dielectric columns crucial for obtaining a gap for TM polarization.

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Fig. 10. Basics of band gaps in two-dimensional periodic structures, for “horizontal” propagation: (a) generic system; (b) dielectric column version; (c) basic air hole version; (d) first Brillouin zone and TE band structure of a triangular array of hole with 30% air filling factor where ε = 11.3 for the dielectric; (e) gap map for TE and TM modes; (f,g) equifrequency contours in (f) valence (dielectric) band and (g) conduction (air) band for a 35% air filling factor and εr = 11.3, TE polarization.

• The second is a graphite-type arrangement of dielectric columns which are sufficiently close to each other to play the role of “prisms” between the three holes in the array of air columns, and which do not have genuine veins. Nevertheless they also exhibit a full gap (see Section 4). Kurizki and Haus [1994] have edited a special issue of the Journal of Modern Optics devoted to photonic band structures (see, e.g., fig. 7) where a number of useful results on gap size and occurrence are given for the various 2D and 3D structures with representative indices (e.g., bulk silicon).

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2.6. Bloch waves Band structures and wavevector diagrams Each point of a band structure diagram represents a Bloch wave, i.e., a solution of eq. (2.21) or a simpler scalar version in 2D. The specific optical behavior of these waves cannot be summarized by the 1D picture of fig. 2. Even for a onedimensional periodic stack, the picture of evolving gaps that we have drawn in fig. 3 misses several aspects of Bloch waves. A “wavevector diagram” in the same k-space as fig. 3 will reveal more information, at the expense of more complexity. It is a plot of the equifrequency surface in k-space defined as the locus of wavevectors k such that ω(k) = ω0 , where ω0 is some frequency of interest; the principle is similar to that of the Fermi surface of electron energy bands in solidstate physics. This diagram has the same periodicity as the Brillouin zone (Ward, Pendry and Stewart [1994]). For example, it is one-dimensionally periodic for the cases of fig. 3 [see fig. 3(i)] and two-dimensionally periodic for the 2D case of fig. 10 [see figs. 10(h,g)]. In both cases, equifrequency contours restricted to the plane of periodicity suffice to describe most phenomena of interest. The basic physics of gap formation starting from a uniform medium is the “attraction of gaps”: As seen in fig. 2 for the lower “valence” band, at constant ω, k appears to be larger than it would be without a gap [the dispersion ω(k) is flattened]. Thus, in comparison to lower frequencies ω0 , for which such equifrequency surfaces are circles (homogenized medium, see Halevi, Krokhin and Arriaga [1999a], Datta, Chan, Ho and Soukoulis [1993], Johnson and Joannopoulos [2003]), when ω0 increases towards the gap, they protrude towards Brillouin zone edges. The rest of the evolution of gaps is complex, and has several entries: multiplicity of Brillouin zones just reflects the presence of multiple Fourier components of the periodic part of the Bloch wave. The strongest Fourier components may govern one problem, and other components may govern other problems, making a unique choice of the Brillouin zone too restrictive. The group velocity of a Bloch wave, on the other hand is well defined, independently of the zone: It is by definition vg = ∇k (ω), normal to the equifrequency surface, along the steepest line of the ω(k) level lines. Visually, peculiarities of this group velocity are obvious. Near the band edge, its direction varies widely. Close to full bandgaps, this equifrequency contour shrinks to a small circle. Russel and co-workers (Russell, Birks and Lloyd-Lucas [1995], Russell and Birks [1996] and Atkin, Russell, Birks and Roberts [1996]) have given various accounts of the use of such diagrams and their significance with respect to the underlying photonic states.

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What are the specific effects and dispersion observed? Bloch waves with rapid variation of group velocity as a function of k or ω can lead to a number of effects and applications. These effects are either directional effects or spectral/temporal effects. In the category of directional effects, “anomalous” refraction may occur at an interface between, say, a uniform medium and a PhC: the group velocity may be either widely variable, or on the contrary very stationary for small variations of the incident angle or frequency at the interface. This will be discussed in Section 4.4 as the “superprism” and “supercollimator” effects. When playing with graded photonic crystals, one may even envision the concept of “Hamiltonian optics”, with the possibility for light beams to follow peculiar two-dimensional paths (Russell and Birks [1999]). In the spectral/temporal domain, the group velocity variation affects pulse propagation. “Slowed-down” light is expected. Slowed-down light experiences a stronger light-matter interaction. This means that a critical parameter such as the “half-wave-voltage” Vπ of an electro-optic modulator could be much reduced if it could operate with such waves. If a cw experiment is performed, e.g., transmission across a slab of PhC of N periods (each period of thickness a as in Labilloy, Benisty, Weisbuch, Smith, Krauss, Houdré and Oesterle [1999]), and insofar as one order in the Bloch wave is dominant, the effect will be seen on the Fabry–Perot fringes due to round-trips inside the slab, themselves caused by reflection between Bloch waves and the uniform outside medium. As a basic rule, the state of interference of the transmitted beam is dominated by the round-trip phase, φ = 2kN a being an even or odd multiple of π. Thus, maxima, associated with φ = 2mπ with some integer m, appear at those frequencies for which k(ω) = mπ/Na. They are a at fixed location in k-space. For a flatter dispersion and a smaller group velocity, their frequency will be more closely spaced. These expected narrow fringes may be difficult to observe in practice, since the light spends more time in the structure and is prone to loss and scattering mechanisms (Hughes, Rammuno, Young and Sipe [2005]). From these examples, one observes that various complex features should be analyzed in detail whenever finite structures are studied. For this, we need to turn to the required theoretical tools.

2.7. Theoretical tools: a brief overview In addition to the simple list of preferred methods below, much more can be added on the complexity of electromagnetism in 3D, and on the conceptual efforts to

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blend it with light-matter interactions, towards spontaneous emission inhibition (SEI) in its simplest form, and quantum electrodynamics (QED) in general. The list below only marginally addresses these aspects. Also, for the sake of brevity, we will leave aside a number of valuable efforts in the field, especially a number of approaches based on various aspects of the Green function (Leung [1993], Algul, Khazhinsky, McGurn and Kapenga [1995], Hughes, Rammuno, Young and Sipe [2005]), or approaches from solid-state physics, such as the novel use of Wannier functions (Albert, Jouanin, Cassagne and Bertho [2000]), or tight-binding parametrization (Lidorikis, Sigalas, Economou and Soukoulis [1998]). Finite-difference time-domain (FDTD) This popular method was established in the microwave regime. It carries out direct evolution in the time domain with sufficient accuracy to require only a reasonable meshing of the metals and dielectrics involved, typically on the order of λ/20 (Qiu and He [2000a, 2000b]). Yee (Yee [1966], Kunz and Luebbers [1993]) established a clever algorithmic scheme using staggered time and space grids, which ensures very acceptable convergence (see Chan, Yu and Ho [1995]). For example, to obtain the transmission spectrum of a given finite photonic crystal, one illuminates numerically a piece of this crystal with a wave generated by a short pulse, having a sufficiently broadband spectral content to cover the expected features (gaps, etc.). The pulse source may lie on a surface, or a line, or at a point. A detector located on the other side of the modeled structure samples the transmitted fields. A Fourier analysis of the time-dependent detector signal retrieves the transmission spectrum of interest. Boundary conditions must be paid special attention to minimize spurious reflections. To date, the best known solution is the so-called “Perfectly Matched Layer” (PML) which involves surrounding the simulation area by successive purely absorbing layers (Berenger [1994], Taflove [1998]). In particular, the absence of reflection can be obtained by the mathematical trick of introducing a complex thickness. This works better than gradual absorption, whereby an imaginary part of the dielectric constant is added in the “walls”. One interesting aspect of this method is that the computation time, for a given accuracy, scales with the size of the system (“order-N method”, Chan, Yu and Ho [1995]). However, many spectral features of interest also shrink when the size is increased, hence trends should be examined with care. In practice, for 3D structures, only parallelepipeds with a volume of about 105 cells can be handled reasonably. An interesting finding in the case of strongly-index-modulated structures is the optimal selection of an ε value for calculation cells lying at the boundary between

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211

two piecewise constant media, as reported by Meade, Rappe, Brommer and Alerhand [1993]. This was actually in the context of improved band calculations using plane waves by a Carr–Parinello-type scheme that avoids writing down the full matrix in reciprocal space by using the space (real or reciprocal) where the operator is diagonal, together with fast Fourier transform of the eigenvector sought between the two spaces. It is interesting to note that systems with highly modulated index have prompted this kind of advances. Nearly the same reason explains the need for improved factorization rules mentioned in Section 2.4. Finally, for the purpose of using a single cell when calculating the band structure and eigenmodes of an infinite crystal, FDTD calculations can be combined with Bloch-type boundary conditions with an imposed exp(ik · r) phase factor in one or several directions. Transmission matrix method (TMM) The closest relative to the familiar matrix methods applied to multilayers in optics is the Transmission Matrix Method (TMM), originally derived by Pendry [1994, 1996a, 1996b] after he had originally applied the same principle to model lowenergy electron diffraction (LEED). TMM works in the real space and frequency domain, by advancing the wave through successive planes that cut the structure. It is generally recognized that TMM is relatively heavy computationally (Bell, Pendry, Martin and Ward [1995]) and scales poorly with the size of the structure. It should also be distinguished from a plane-wave-type method derived by Sakoda to calculate the transmission through finite PhC samples (Sakoda [1995a, 1995b, 1995c]). Finite-elements methods Solving Maxwell equations in the harmonic regime as a set of partial differential equations can be achieved by a finite-element scheme. This is closely related to the above transmission matrix method. The interest of finite elements is of course to map naturally any detailed shape that arise in the models. It exists in particular in the form of commercial packages. It is practiced most for photonic crystal fibers discussed in Section 5, as an extension to mode solvers for channel waveguides and fibers. The specific finite elements that manage field discontinuities at dielectric boundaries are called “edge elements” or “Nédelec finite elements”, after their inventor.

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Plane wave expansion with a supercell The plane wave expansion seen in Section 2.4 gives a good picture of the eigenmodes at moderate expense. For this reason, it has been adapted through the use of supercells to model localized structures that are either extended in one or two dimensions (channel or planar waveguides), as well as finite defects, such as cavities formed in a PhC by removing a number of spheres, cylinders, etc. Nothing changes in the fundamental mathematics and related expansions. However, the use of the wavevector k now obeys different constraints. To study a straight guide for example, an artificial periodicity is introduced, normally to the guide axis. The dispersion found for k in this artificially periodic direction can be used to assess the unwanted parasitic coupling between adjacent waveguides, whereas the dispersion in a given guide is well rendered when varying k along its axis. Following this line of thinking, the localized character of some modes can be checked by varying the simulation cell and asserting which are asymptotically stable when increasing spacing between the elements of interest. Of course, the scaling of supercell PWE (as opposed to Carr–Parinello type) is relatively disastrous, as the matrix memory size scales as the power 2d of the real space size in dimension d. In practice, N ≈ 1000 plane waves can be used on any computer in order to provide a satisfactory insight up to a 50-cell system. It is safest in any case to revisit the optimal representation of the unit cell pattern, as done in the study on super-Gaussians by Villeneuve and Piché [1994a, 1994b], or to improve the factorization rules (Li [1996], Lalanne and Morris [1996], Nevière and Popov [2003], David, Benisty and Weisbuch [2006]). Korringa–Kohn–Rostocker method The original method was derived in solid-state physics in 1954. It is based on the diffraction matrix of a unit cell, as will be described for the method below, and makes use of an expansion of the Green function on spherical harmonics. One significant advantage of this function is that one directly obtains crucial quantities such as the global or local density of states. This method is more stable than the plane wave expansion, especially with respect to the Gibbs phenomenon; see Moroz [1995] for a detailed description, and Ohtaka’s early work (Ohtaka [1979]). Multiple multipole methods This method has been applied in two main areas. For the first kind of application, multiple scattering can be cast into a simple matrix form when the scatterers are spheres or cylinders, and the waves are of well-known mathematical forms

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(Bessel–Hankel–Neumann functions for cylinders, for example). The relationship with incoming plane waves has to be made from adequate projection coefficients. The interest lies in the fact that the scaling is not too severe with the number of “atoms” involved. It is parent to the KKR method and it is also called a scattering matrix method, although this naming is too generic (let us recall that a scattering matrix in general relates incoming and outgoing waves of any structure). Rigorous coupled-wave analysis and Fourier-modal methods Indeed, scattering matrices also appear in the so-called Rigorous Coupled Wave Analysis (RCWA, also known as Fourier-modal method). This class of frequencydomain methods originated in the area of grating diffraction (Petit [1980], Moharam and Gaylord [1981], Popov [1993], Nevière and Popov [2003]) to overcome the limitations of the classical Rayleigh expansion. The key point of such methods is to avoid full 3D discretization by exploiting, notably, an analytic approach in one direction. The mathematical difficulties that considerably slowed convergence were related to factorization rules again. They relate to the discontinuities of the field components at dielectric boundaries. In essence, a proper treatment is obtained by considering the continuous component at these boundaries. This proves crucial for convergence. Li [1996, 1997] and Lalanne and Morris [1996] solved this issue in the framework of RCWA. But the conclusions also apply to the plane wave expansion and explain the differences between, e.g., the method of Ho and that of Plihal and Maradudin. Photonic crystal slabs were subsequently treated with the same class of method by Whittaker and Culshaw [1999], under the name of scattering matrix method (which is admittedly confusing, their emphasis was on the distinction with transfer-matrix methods). Besides scattering problems, the method also enables the determination of Bloch modes by looking for the poles of the scattering matrix Of particular significance are the imaginary parts of the poles of this matrix (infinite determinant) which relate to the important topic of modal losses for “resonant modes” or “leaky modes”. These modes, to be introduced in Section 4, arise in PhC slabs and in periodically modulated waveguides above the light-line. A clear discussion of leaky modes is notably given in Tikhodeev, Yablonskii, Muljarov, Gippius and Ishihara [2002]. One public code of interest for several generic structures is CAMFR from P. Bienstman (see http://camfr.sourceforge.net). Sources can also be included, a topic that has been worked out in the frame of light extraction and directivity in devices such as light-emitting diodes or more generic ones (Delbeke, Bockstaele, Bienstman, Baets and Benisty [2002], Rigneault, Lemarchand and Sentenac [2000], Fehrembach, Enoch and Sentenac [2001], Pottage, Silvestre and Russell [2001]).

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Generalizations of RCWA to nonperiodic structures have been treated with remarkable success, notably as regards finite-height 2D structures (Silberstein, Lalanne, Hugonin and Cao [2001], Palamaru and Lalanne [2001], Lalanne and Benisty [2001], Lalanne [2002], Sauvan, Lalanne and Hugonin [2005]). In addition to the RCWA, a number of “semianalytical” methods (which all avoid 3D discretization) have been proposed and often applied to photonic crystals, such as the differential method, the integral method, the C-method, and the method of modal expansion. See Nevière and Popov [2003] for a general discussion and Gralak, de Dood, Tayeb, Enoch and Maystre [2003] for an application to “woodpile” photonic crystals. After this short tour of basic physics and history of PBG materials, we will deal in the next section with the many efforts devoted to modeling, fabricating and characterizing 3D structures.

§ 3. Three-dimensional photonic crystals 3.1. Introduction The main expectation of 3D photonic crystals is to provide a full multidirectional photonic bandgap. At frequencies inside the bandgap, Bragg diffraction of light is strong in all directions and prevents any propagative solution of Maxwell equations in any direction. For this reason, 3D PhC could inhibit spontaneous emission, as originally proposed by Yablonovitch [1987], or more generally play an important role in creating new quantum-electrodynamic (QED) or quantum-optics effects (John [1987, 1991], John and Quang [1994], Kurizki, Sherman and Kadyshevitch [1993], Kofman, Kurizki and Sherman [1994], Yokoyama and Ujihara [1995], Huang, Lu and Zhu [1998], Zhu, Li, Yang and Li [2003], Soukoulis [2001]). This is because the absence of modes also means the absence of vacuum fluctuations, these fluctuations being key to spontaneous emission, parametric generation, etc. Further, ultimate control can result from defect modes, which could serve as cavities with arbitrarily high quality factors. Leakage would be dictated only by the size of the crystal (as long as the material is lossless). How much do all these expectations suffer from disorder is another point of interest, both fundamental and applied. Anderson localization has been predicted to occur for controlled disorder, in the contribution by John [1987]. See also Genack and Garcia [1993] for this topic. A series of novel quantum-optical dynamic effects have been predicted in the framework of 3D photonic crystals with localized photon states, albeit not always in realistic models (see criticisms in Zhu, Yang, Chen, Zheng and Zubairy [2000]).

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In practice, fabrication by design, and proper measurements are still major challenges in this field (Romanov, Fokin, Butko, Alperovich, Johnson and Sotomayor-Torres [1996], Romanov [1998], Wijnhoven and Vos [1998], Megens, Wijnhoven, Lagendijk and Vos [1999a, 1999b], Megens, Schriemer, Lagendijk and Vos [1999], Petrov, Bogomolov, Kalosha and Gaponenko [1998], Petrov [1998], Vlasov, Deutsch and Norris [2000], Blanco, Chomski, Grabtchak, Ibisate, John, Leonard, Lopez, Meseguer, Miguez, Mondia, Ozin, Toader and van Driel [2000], Schriemer, van Driel, Koenderink and Vos [2000], Wong, Kitaev and Ozin [2003], Lodahl, van Driel, Nikolaev, Irman, Overgaag, Vanmaekelbergh and Vos [2004], Koenderink, Lagendijk and Vos [2005]). We shall therefore build this section around these two main themes, adding to the second theme of measurement a discussion of spontaneous emission modifications. As for the theory of 3D photonic crystals, until recently most experimental studies have focused their effort on obtaining a gap, and on the transmission– reflection–diffraction properties of a finite sample. The bottom-up approach was therefore rapidly prompted to substantiante the initial calculations and quests of gaps (Anderson and Giapis [1996]). As the obtainment of the fine details and accurate structures was longer than in 2D structures, a large part of the fine predictions remains to be fully assessed. Let us only remind the reader that most structures of interest possess the diamond symmetry (fig. 7), with the exception of the simpler f.c.c. symmetry for the case referred to as “inverse opal”, built from the classical compact stacking of spheres, but inverting spheres and vacuum, as detailed below. Three basic types (figs. 7 and 11) are (i) the classical diamond stack of spheres (also lattice of Si), (ii) the so-called woodpile stack, whereby a four-period stacking of rods at 90◦ is used, and (iii) the Yablonovite type with a set of three drilled galleries that mimic the [111] tunnels of the diamond structure. The desire to make 3D PhC a reality at all frequencies (see fig. 8) has finally resulted in a multitude of proposals applicable to fabrication that depart from these basic structures in various respects: lower symmetries and chirality, more complex unit cells (Toader and John [2001], Toader, Chan and John [2004], Campbell, Sharp, Harrisson, Denning and Turberfield [2000]), multiplicity of material, etc. (see Busch, Lölkes, Wehrspohn and Föll [2004] or the special issues), quasicrystals (Chan, Chan and Liu [1998], Hagelstein and Denison [1999]), anisotropic materials (Li, Gu and Yang [1998]). However, few of these have been demonstrated. The same is true for unusual light–matter interaction effects such as Cerenkov radiation in PhC (Luo, Ibanescu, Johnson and Joannopoulos [2003], Garcia de Abajo, Pattantyus-Abraham, Zabala, Rivacoba, Wolf and Echenique [2003]).

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Fig. 11. Examples of 3D photonic crystal realizations with (a) c.f.c. stack of spheres with a white lozenge underlying a specific crystallographic plane (C.M. Sotomayor, Univ. Wuppertal); (b) TiO2 inverse opal based on a latex opal template: note the voids at contact between original spheres and also at partly filled initial voids, and the view of the layer beneath (D. Mc Comb, Imperial College); (c) silicon inverse opal of the kind used for demonstration of a complete gap (A. Blanco, G.A. Ozin, F. Meseguer, S. John); (d) “woodpile” arrangement, also displaying a complete gap (Univ. Kyoto); (e) “autocloning” method (scheme); (f) micrograph of a cross section (over 120 layers) and a top view of a Nb2 O5 –SiO2 stack grown on a “pixellated” grating (Univ. Tohoku, Sendai); (g) modulated-pore macroporous silicon (principle); (h) micrograph and (i) zoom on the pores with a 6 µm period (MPI Halle).

3.2. Fabrication of 3D photonic crystals The submicron scale of a photonic crystal in the optical regime makes 3D PhC fabrication a challenge. The first PhC structures having a PBG were relatively easy to fabricate at microwave frequencies (cm scale) and they provided, during the 1990s, an efficient testbed for the basic electromagnetic properties and the clarification of issues such as the role of crystal termination (Brown, Parker and McMahon [1994]). Two main approaches to fabricating 3D periodic structures are self-assembly and layer-by-layer deterministic assembly. We first discuss the former approach.

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Self-assembly Initially, the solution of self-assembly for optical 3D PhC was seen as the easiest way to attain small dimensions and large samples. One starts from an assembly of spheres of submicron size and as monodisperse as possible. It is obtained by chemical methods, from colloid chemistry (Xia, Gates and Park [1999], Balakirev, Bogomolov, Shuravlev, Kumzerov, Petranovskii, Romanov and Samoilovich [1993]). When undergoing sedimentation in a liquid, a solution of such spheres naturally creates a compact stacking [fig. 11(a)]. Unfortunately, a compact arrangement is not a sufficient criterion for perfect order (f.c.c. vs hexagonal): starting from a perfect (111) basal plane, the sequence of subsequent planes has to be ABCABCABC. . . , if the f.c.c. order is desired. Randomness in the sequence leads to some disorder along the 111 axis. For neutral spheres, no long-range order exists to prevent random stacking. A clever improvement was to use charged colloids at this crucial settling stage. With more long-range interaction, stacking faults are much reduced (Vos, Driel, Megens, Koenderink and Imhof [2001]). This brief example gives a flavor of the physico-chemical issues that arise in attempting such synthesis. Nature has achieved a natural assembly of this kind in the semi-precious stone called “opal”, which is, microscopically, an assembly of silica spheres with voids between them. It is, on the contrary, not monocrystalline: it is the diverse crystalline orientation of the domains that give opals their desired multicolor aspect. Once a 3D edifice is built, it is clear from Section 2 that strong requirements arise for obtaining the PBG, such as strong index contrast, sufficient fraction of air, and adequate geometry. Bare silica opals, with relative index contrast RIC = 1.5, are far from these requirements. Even with higher RIC, n = 3.5 say, there is no way to obtain an omnidirectional bandgap with a compact stacking of spheres. If an edifice of spheres is to be selected, it should then be a noncompact diamond lattice, with half the sphere filling factor (fsphere = 0.34) compared to the compact arrangement. Sedimentation is thus unlikely to be compatible with the “holey” diamond sphere arrangement. Let us mention that recently, and outside self-assembly, a tour de force in nanorobotic manipulation (García-Santamaría, López, Meseguer, López-Tejeira, Sánchez-Dehesa and Miyazaki [2001], Miyazaki, Ohtaka and Sato [2000]) was accomplished by stacking one-by-one a diamond-type crystal of spheres onto a 2D template under an electron microscope. Among the possible routes based on such patient manipulations, one may stack alternate silica and latex sphere layers, and dissolve selectively every other sphere by a solvent or hydrofluoric acid, provided some care has been taken to microscopically enforce sphere–sphere adhesion (Meseguer and Miguez [2004]).

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The key to a self-assembled PhC with a true gap is then to “invert” the opals, by filling the voids and getting rid of the spheres [Figs. 11(b,c)]. Even in this case, since the 3D bandgap lies between higher bands (8th and 9th, fig. 7) it is highly sensitive to imperfections, and hence demanding on fabrication. 2% disorder is enough to destroy the bandgap (Li and Zhang [2000], Koenderink, Lagendijk and Vos [2005]). The filling step can take place in either vapor phase or liquid phase, taking care that reactants reach the inner layers (say at least the first ten layers) before the reaction takes place. Silicon can be infiltrated by chemical vapor deposition. Titania (TiO2 ), other oxides and some chalcogenides can be precipitated or condensed from various precursors. The control of the various steps is critical. In fig. 11(b), one can observe air holes in the veins between the opal spheres (incomplete infiltration) and at the points where the spheres touch each other. Variations in these can lead to large variations in the bandgap (Busch and John [1998]). To get rid of the spheres, one can just make use of calcination, as long as the deposited infilling is a mineral that withstands heating to several hundreds of ◦ C. Another way is to use hydrofluoric acid for silica spheres. This was the case for the first convincing demonstration of a 3D bandgap at λ = 1.5 µm by this method (Blanco, Chomski, Grabtchak, Ibisate, John, Leonard, Lopez, Meseguer, Miguez, Mondia, Ozin, Toader and van Driel [2000], PalaciosLidon, Blanco, Ibisate, Meseguer and Lopez [2002]). A number of other inverted opal studies have taken place in the last few years (Vlasov, Bo, Sturm and Norris [2001]). A compromise has been sought between control of fabrication and the value of the refractive index, by using titania (TiO2 ) or other systems with relatively modest index. Another interesting variant introduces a small deterministic seed by causing sedimentation on a hard patterned substrate (Xia, Gates and Park [1999], Ferrand, Egen, Griesebock, Ahopelto, Müller, Zentel, Romanov and Sotomayor-Torres [2002], Romanov and Sotomayor-Torres [2000]). One creates for example ordered macro-channels of PhC with lateral dimensions of a few sphere diameters, where disorder is limited by the imposed boundary conditions. However, care has to be taken that prior (capillarity) or posterior (calcination) phenomena play no spurious role. Deterministic fabrication In deterministic procedures, use is made of a mastered 2D patterning process and careful steps are taken to stack an assembly of similar 2D patterns. The woodpile structure was actually conceived with this requirement in mind. Here the 2D pattern is simply a series of aligned square grooves with about 1:1

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mark–space ratio. In early microwave and THz attempts, such grooves traversed the Si wafer, and one could mechanically stack the grids in the required sequence with proper mechanical alignment means (Özbay, Michel, Tuttle, Biswas, Ho, Bostak and Bloom [1994]). In the optical regime, processes are more difficult. One clever way proposed by Noda, Yamamoto and Sasaki [1996], Yamamoto, Noda and Chutinan [1998] and Noda, Yamamoto, Imada, Kobayashi and Okano [1999] was to split a patterned wafer and fuse the two parts together at 90◦ to form a half period. By carefully removing the top substrate, one can iterate the process of splitting and sticking at 90◦ to get one period, and so on [fig. 11(d)]. Alignment by holographic means is needed to align the parts together, based on the feedback provided by the diffraction pattern of laser light of adequate photon energy. Another alternative is to start the sequence by deposition of a high-index material (e.g., poly-Si), followed by patterning (e.g., by lift off), and then adding a second deposition of some low-index or sacrificial material. This latter is then “planarized”, again very carefully, by a sequence of adequate technology steps that leaves exactly one half period built up. Alignment on the previous step is then optimized, as described in relevant papers (Fleming and Lin [1999], Lin, Fleming, Hetherington, Smith, Biswas, Ho, Sigalas, Zubrzycki, Kutz and Bur [1998]). Similar to this method, but better suited to defects and waveguides, the MIT team has proposed a layer-by-layer stack, whereby a 2D pattern is defined instead of groove, in every other layer fabricated (Fan, Villeneuve, Meade and Joannopoulos [1994]). It is not simple to address the relative merits of the different proposals. This is witnessed by the fact that recently, both approaches succeeded in incorporate defects with unambiguous photonic signatures (Ogawa, Imada, Yoshimoto, Okano and Noda [2004], Qi, Lidorikis, Rakich, Johnson, Joannopoulos, Ippen and Smith [2004]). Alternative fabrication methods We address here fabrication methods that are rather deterministic, but exploit uncommon aspects of deposition or electrochemical dissolution, or resist patterning processes. Therefore, they belong to a special class and are difficult to classify along with the “top-down” or the “bottom-up” approaches. As will become apparent later, this also brings some freedom from other view points. We do not intend to be exhaustive here, but we cover the more successful techniques. Autocloning. “Autocloning” is a term introduced to denote a particular use of sputtering techniques involving two different materials such as silica and silicon, or another high-index oxide. Sputtering is rather seen as an engineer’s high-throughput deposition technique, with delicate structural relations between

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sputtered targets and deposited material. However, when alternatively depositing the two materials on a patterned substrate (say silicon) in adequate conditions, a typical V-groove geometry arises, which becomes stable and reproducible from period to period [figs. 11(e,f)]. Even better, in the case of a complex pattern of the substrate, there is little spatial “cross-talk”, so that adjacent regions with different pitches and different photonic properties may be designed. Similarly, large changes in vertical period are allowed. With all these possibilities and with a good knowledge of basic optical properties, not only can regular 3D photonic structures be elaborated (Sato, Miura, Ishino, Ohtera, Tamamura and Kawakami [2002]), but also slab and channel guides, polarizing devices, etc., based on “heterostructured” photonic crystals. The spectral domain covers ranges from visible to infrared (1.5 µm). Another application based on epitaxial growth can be found in Sabarinathan, Battacharya, Zhu, Kochman, Zhou and Yu [2001]. Resist patterning by photons: X-ray patterning and molding, holography, confocal microscopy. The idea to obtain a 3D structure by dissolving a positive resist that has been cross-linked by optical means dates back to the beginning of photonic crystals. The idea is that a resist template can be further inverted using soft processing, in a manner similar to the inversion of opals. The result may be used again as a mold to obtain the desired shape and constituent material. One early success using this technique was the use of X-ray patterning through a mask, in an attempt to reproduce the drilling of Yablonovite in the microwave resin originally used in 1990. X-rays are directional and have no sizable diffraction nor too much absorption if care is taken to operate below the energies of the relevant atomic absorption threshold for the length scale of interest. PMMA resist templates with 6 µm thickness and good optical properties have been obtained, and from them, replica in copper have been cast (Cuisin, Chelnokov, Lourtioz, Decanini and Chen [2000]). Another possibility is to operate using holography in the visible. This follows the initial proposal by Berger, Gauthier-Lafaye and Costard [1997a, 1997b]. N +1 beams define an N-dimensional fringe pattern. The process works well for N = 2 with 3 beams. Since then, various teams have explored the possibilities offered by this approach. In principle, control of the relative phase and absolute intensities of the interfering beams gives a lot of freedom to the unit cell that is expected to result (Campbell, Sharp, Harrisson, Denning and Turberfield [2000]). However, practical realization has to circumvent a number of difficulties that arise in developing a resist with only modest gradients in local exposure dose, and with sizable residual absorption. The improvements are well traced in Campbell, Sharp, Harrisson, Denning and Turberfield [2000] and in Wang, Xu, Su, Zeng, Chen, Wang,

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Pang and Tam [2003], and the principles in Toader, Chan and John [2004]. The main limitation of this approach is of course that, without extra processing steps, it is limited to purely periodic, defect-free structures. The use of exposure by confocal microscopy may overcome this difficulty, at the expense of a long-lasting, serial writing process. Difficulties include focusing inside thick, weakly absorbing resist and the limit to the total dose that can be imposed from accumulated light passing out of focus. One elegant way to circumvent this is to use two-photon photopolymerization, whereby the exposure dose is the square of the local intensity (Kaneko, Sun, Duan and Kawata [2003]). Only the focal volume receives a sufficient photon density to undergo polymerization, whereas all out-of-focus regions are virtually unexposed. This method is promising for producing complex template shapes, beyond regular photonic crystals. Macroporous silicon technology, modulation. Macroporous silicon technology is treated later in this chapter as it was the only method capable of producing veryhigh aspect ratio holes in silicon for 2D PhCs a decade ago (Grüning and Lehmann [1996]). Continued work has shown that it is also possible to substantially modulate the pore diameter through modulation of the etching current density. This leads to a three-dimensional structure as well (Busch, Lölkes, Wehrspohn and Föll [2004], see also figs. 11(g–i)). Other ways to go beyond 2D with macroporous silicon include the use of focused ion beam (FIB) etching to add the extra series of holes needed to reach the Yablonovitch-type arrangement (Wang, Chelnokov, Rowson, Garoche and Lourtioz [2000], Chelnokov, Wang, Rowson, Garoche and Lourtioz [2000]). The subject is crystallographically involved and not detailed here. The good news is that the FIB etching process works much better when operating on an already porous structure because sputtered material is easily evacuated sideways rather than merely through the hole’s mouth. Controllable photonic crystals. Ways to control the gaps are still in infancy. Infiltration by liquid crystals is a promising concept (Busch and John [1999]) that has been demonstrated in 2D PhCs by Schuller, Klopf, Reithmaier, Kamp and Forchel [2003]. Optical control is practiced in microwave experiments (Chelnokov, Rowson, Lourtioz, Duvillaret and Coutaz [1998]). Mechanical control, as is done in microcavities (Dantec, Benyattou, Guillot, Spisser, Seassal, Leclercq, Viktorovitch, Rondi and Blondeau [1999]), has not been much investigated. We briefly comment on nonlinearities in PhCs (a form of control of light by light) in the conclusion Section 6. We terminate this review of 3D PhC fabrication by acknowledging that other important processes have been neglected. The imperfect nature and/or delicate

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structural characterization of 3D PhCs is a key point to remember. Due to these imperfections, assessing the photonic properties of 3D PhCs requires a lot of care in experiments and interpretation, as we will see below.

3.3. Measurements Measurements from outside Essential studies are reflectivity and transmission measurements, looking for stopband spectral positions and to the Bragg attenuation length LB , the basic parameters after Bragg peak position (Thijssen, Sprik, Wijnhoven, Megens, Narayanan, Lagendijk and Vos [1999]). The quest for photonic gaps started by looking at directional gaps, especially in opals (Astratov, Vlasov, Karimov, Kaplyanskii, Musikhin, Bert, Bogomolov and Prokofiev [1996], Vlasov, Astratov, Karimov, Kaplyanskii, Bogomolov and Prokofiev [1997], Vos, Sprik, van Blaaderen, Imhof, Lagendijk and Wegdam [1996], Vos, Sprik, Lagendijk and Wegdam [1996]). Since the initial studies were made with direct opals, with modest photonic strength (small index contrast, and non-optimal unit cell for this contrast), these gaps could be taken approximately as those of just 1D periodic structures, notably the gap along the ŴL (111) stacking direction. This has proven to be of limited validity. When probed with beams impinging off the (111) direction, the angular reflectivity of opals and furthermore of inverse opals with larger “photonic strength”, the (111)-related gap does not follow a regular evolution as it would for a 1D system, due to band repulsion between multiple Bloch waves. This is known as the “avoided crossing” phenomenon (van Driel and Vos [2000]). The shape of the peaks themselves also carries noticeable information, as it does in X-ray crystallography (Zachariasen [1945]). For samples with disorder, such as stacking faults or small disorientations among adjacent domains, etc., the secondary lobes typical of stopband edges were smoothed out. An effort to attain single-crystal measurements with adequate microscopic means (see fig. 12) showed that ordered PhC slabs had the proper optical response, with all side lobes. This type of “clean” measurement shows clearly how averaged measurements over many crystal orientations distort understanding, leading in that particular case to a drastic overvaluation of the bandgap (Vlasov, Deutsch and Norris [2000]). In particular, it might indicate that through extrapolation based on this apparent photonic strength, one may assume to have an omnidirectional bandgap while that is not the case. Even so, as will be seen below, such careful measurements still leave room for other important disorder effects in the crystal. Opals and inverted opals typical

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Fig. 12. Spectroscopy of a single opal crystal (after Vlasov, Deutsch and Norris [2000]): (a) multidomain sample; (b) single-domain zoom; (c) macroscopic (thin line) and microscopic (thick line) reflection spectra.

suffer from a (1–5)% sphere size site-to-site disorder. Some low-order Fourier harmonics responsible for some stop gaps may be quite immune to this disorder, but the scattering length ℓ (measuring how long a Bloch wave is “ballistic”) can drop to quite small values with this limited amount of disorder, especially in inverse opals, when increasing the photonic strength to large value and large index contrast (Koenderink, Lagendijk and Vos [2005]). Transmission experiments are also used. They are performed with most convincing results on deterministic structures. For example, the intricate GaAs layerby-layer structures by Noda, Tomoda, Yamamoto and Chutinan [2000] have shown an impressive 40 dB attenuation at normal incidence, and 7 dB over the full range of the expected bandgap. Imperfections in opal structures tend to broaden and affect more the stop-gap transmission than the reflectivity because a transmitted beam probes all layers, not just the top ones. A number of other effects are present in pass bands in somewhat thick samples, due to disorder at various scales: Rayleigh scattering and defect scattering. For example, point defects can be resonant defects. A comparison of the mean free path to the sample thickness is a prerequisite in the full interpretation of 3D PhC transmission data. To finely probe disorder with “usual” light, the original enhanced backscattering experiment (Koenderink, Megens, van Soest, Vos and Lagendijk [2000], Huang, Eradat, Raikh, Vardeny, Zakhidov and Baughman [2001]) provides an unambiguous tool. Typically, in the good samples probed, a mean free path of ℓ = 10 µm is found, which is five times the Bragg attenuation length LB , showing that disorder leaves a fair margin for basic wave-periodicity interaction. However, for more complex optical wave processing, as envisioned in optical microchips, disorder is a serious issue especially in 3D systems where the promises of technology are not as mature as for 2D patterning (Koenderink, Lagendijk and Vos [2005]).

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Difficulties in the control of spontaneous emission Firstly, as we mentioned in Section 2.2, in frequency regions of nonzero DOS, the propagating fields may have a substantially localized structure, for example being much more intense in dielectric than in vacuum. As a result, the concept of a local DOS (LDOS) can be used in the “weak coupling” regime (irreversible spontaneous emission) to describe the optical effects of the structure on an atom situated at a particular point. As for the lifetime, any 3D photonic crystal with large Fourier components of the dielectric constant is expected to have its photon density of states altered to a large degree. In other words, even partial gaps should lead to sizable reduced DOS with respect to the average material, while band-edge singularities could also prove important (although not divergent or even discontinuous at all in 3D). When going beyond the weak coupling regime, theoretical difficulties are also to be considered. For example, a simple model based on an isotropic discontinuous DOS has been used by some teams, leading to striking possibilities of modifying the light–matter interaction (John and Quang [1994]). The spurious presence of a divergent DOS in this model is the main reason for the strength of the alleged effects. Great care has to be taken for this topic, caveats being addressed in Zhu, Li, Yang and Li [2003] and Huang, Lu and Zhu [1998]. It is worth emphasizing at this point that the overall effect on an emitter is not only to change the lifetime, but also to drastically change the emission pattern, the change being also emitter-position dependent. Experimentally, one of the main expected manifestations is the inhibition of spontaneous emission (Martorell and Lawandy [1990], Whittaker [2000], Lodahl, van Driel, Nikolaev, Irman, Overgaag, Vanmaekelbergh and Vos [2004]). To demonstrate this phenomenon, internal light sources such as fluorescent molecules, quantum dots (these latter providing gain, see Vlasov, Luterova, Pelant, Hönerlage and Astratov [1997]) or even atoms are placed inside photonic crystals, in its voids or incorporated in one of the constituents. Most convincing experiments to date concern self-assembled inverse opals. Only limited total DOS effects (see fig. 13) have been convincingly demonstrated until now, due to the difficulty of having a suitable reference system, with exactly the same physico-chemical environment of the molecule as the sample under test, and also to have all-solid-angle light collection: otherwise, the observed effects could stem from local DOS only, and most missing photons would just be uncollected. Again, if the emitter is not sampling the unit cell uniformly, the local DOS, will create important variations with respect to the average material. The angular redistribution of emission is commonly observed, with a clear attenuation when a stop gap overlaps the emission spectrum: The directional stop

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Fig. 13. (a) Emission in a partial band gap in the presence of disorder where the light emitted inside may be seen even in the bandgap direction due to appreciable scattering. (b) Emission above the Bragg wavelength, where light is seen as well on top, so no distinct limit to the gap can be assigned with the stopband + disorder case in (a). (c) Ideal case of emission inside a full PBG. (Inspired by Megens, Wijnhoven, Lagendijk and Vos [1999b].)

gaps repel emission outside some cones. However, the attenuation is much less than, for example, in transmission experiments (Koenderink, Johnson, GalisteoLopez and Vos [2002]). This is attributed to disorder and multiple scattering [see fig. 13(a)]. The key factor limiting the attenuation is the ratio of Bragg length LB to photon mean free path ℓ. Only with minute disorder and large ℓ is the attenuation that of the perfect finite crystal. In the most recent experiments reported (Lodahl, van Driel, Nikolaev, Irman, Overgaag, Vanmaekelbergh and Vos [2004]), only a limited change in lifetime (∼20%, either lengthening or shortening) is reported.

3.4. Metallodielectric systems and microwave regime This is a major topic in itself. It is not our ambition to describe this field in any detail from its origin, either in the microwave or the optical regime (Kuzmiak, Maradudin and Pincemin [1994], Nicorovici, McPhedran and Botten [1995], Moroz [2002], Sigalas, Chan, Ho and Soukoulis [1995], Sigalas, Soukoulis, Chan and Ho [1994]). The case of true metal edifices is beyond our scope, because of the high dispersion of the dielectric constant, and also because of the role of the associated losses in the optical regime (Takayama and Cada [2004]), which make predictions such as those based on a lossless plasma in Moroz [1995], of theoretical interest but of modest applicability.

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Some systems, however, are acquiring the status of model systems. They can thus be useful bridges between the communities. The field is presently driven by concepts derived from plasmon and plasmon polariton studies rather than PBG and PhC studies, although much is common. Starting from small metal particles and advancing to more connected structures one goes continuously from an electrostatic regime, where the field tends to be localized (e.g., at tips or between spheres or bow-tie-shaped “antennae”), to a propagative regime well documented for plasmons. Let us remind the reader that the plasmon, a wave localized at a metal–dielectric planar interface, has a TM(−p) type polarization, with a dispersion that is photon-like (group velocity ∼c/n) at low and high frequencies, with √ an evolution towards flat dispersion around the frequency ω ≈ ωp / 2, where ωp is the metal plasma frequency (see fig. 14, refer to the Drude model of metals) (Maier and Atwater [2005], Zayats, Smolyaninov and Maradudin [2005], Dionne, Sweatlock, Atwater and Polman [2006]). The difficulty in harnessing plasmons for the modification of spontaneous emission in the optical regime is to a large extent a matter of spatial scale. At small

Fig. 14. Typical shape of plasmon dispersion curves, resulting from the metal plasma oscillation interaction with the surface. The lower branch lies entirely below the light line of slope c/n, where n is the index of the dielectric (upper medium, see inset). Metal losses are neglected. If a structure of period a is superimposed on the metal surface (holes, . . .) a gap appears in the indicated region. The field profile is peaked (TM polarization, and negative value of the metal dielectric constant), and the fraction of the field in the metal decreases at low frequencies.

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scale (wavevector k ≫ 2π/λ = ω/c), the plasmon field is intense, but a large continuum of electron–hole pair excitations can also capture electromagnetic energy and dissipate it (Ford and Weber [1984], Huang, Lin and Deppe [1993], Drexhage [1974]). At larger scale (with a wavevector k now nearly of the same order as k = 2π/λ), the plasmon field decreases, the mode profile being expelled in the dielectric with an increasingly long-reaching tail, and the resulting SER modifications are not so large. Recently, strong coupling with plasmon as one of the oscillators has nevertheless been reported (Bellessa, Bonnand, Plenet and Mugnier [2004]). Concepts for plasmons on periodically structured surfaces and their use in emitters are now well documented in the following references: Kitson, Barnes and Sambles [1996], Barnes, Murray, Dintinger, Devaux and Ebbesen [2004], Bozhevolnyi, Erland, Leosson, Skovgaard and Hvam [2001], Vuckovic, Loncar and Scherer [2000], Barnes [1999], Christ, Linden, Zentgraf, Schubert, Nau, Thikodeev, Gippius, Kuhl, Schindler, Holleitner, Stehr, Crewett, Lupton, Klar, Scherf, Feldmann, Dahmen, von Plessen and Giessen [2004]. A specific interest has arisen in subwavelength holes in relatively thin metallic films, since “giant” transmission results were reported by Ebbesen, Lezec, Ghaemi, Thio and Wolff [1998]. In simple terms, the normal incidence transmission of these tiny holes is much larger, at specific spectral peaks, than their surface fraction. Thus, the old Bethe–Bouwkamp prediction of rapidly decaying light transmission in the subwavelength regime (Bethe [1944]) is clearly violated. The array of holes in the metal brings a complex physics. While it is recognized that the array acts to form surface plasmons on the metal film, the further physical steps are still controversial, such as whether the air holes simply assist plasmon tunneling between both sides of the film as other scatterers would, or they do channel inside them the electromagnetic energy, with only a plasmon-mediated “capture”. The detailed relation of all these mechanisms with poles and zeros of the scattering matrix is indeed still a challenging issue. Still, it is doubtless that simple arrays of “dimples” of modest corrugation do act as antennas for the far field, either in emission or reception mode, as indicated by recent results (see, for example, Lezec and Thio [2004]). Microwave systems can of course serve as a test for transmission–reflection– diffraction experiments, be it a dielectric or a metal. Some concepts, such as impedance matching, are also crossing the frontiers between the microwave and optical communities (Boscolo, Conti, Midrio and Someda [2002]). Microwave systems have, however, a specific behavior when it comes to metals. Properties

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of a metal-based photonic crystal are quite distinct from that observed at optical frequencies, as losses become minute, as does the skin depth. Connected metallic structures tend to reflect incident waves in the low-frequency regime. This creates a so-called “plasmonic gap” starting from ω = 0 on (McGurn and Maradudin [1993]). By “cutting” metallic bonds, e.g., by means of diodes, with a bias-controlled conductivity, a controlled bandgap arises (Lourtioz, de Lustrac, Gadot, Rowson, Chelnokov, Brillat, Ammouche, Danglot, Vanbésien and Lippens [1999]). For these bands, it has been proposed, by J. Pendry in particular, that the physics of thin metal wires in connection to photon bands is controlled by the enhanced “inductance” of such small wires (Pendry, Holden, Robbins and Stewart [1999]). More recently, they have been used in the context of “left-handed materials”, which feature simultaneously negative effective permittivity ε and negative permeability μ (Shelby, Smith and Schultz [2001], Garcia and Nieto-Vesperinas [2002]). In practice, there is a magnetic resonance in adequate metallic structures such as split-rings, and storage of magnetic energy occurs in a small frequency band (typically in the 5–10 GHz regime, but limits towards optical regime are not so clear, and the THz regime is certainly in reach). In microwave experiments, a lot of care should be taken with certain aspects. Here, we give only two examples: firstly, coherence of an electromagnetic beam is usually perfect, so that speckle in experimental data is an issue, and can blur faint peaks (e.g., defect modes, see fig. 9). Secondly, diffraction should be taken care of, partly because detectors in microwave are often smaller than the diffraction limit and they are commonly in some near-field-type situation. A typical problem is to introduce a large superperiod when cutting a crystal at a small angle off a symmetry axis, for example to fabricate a prism-type sample for negative refraction experiments. A last topic of recent interest comprises metallic photonic crystals in the nearto-mid-infrared range. The black-body radiation of such a system displays, upon modest heating, a spectrally and directionally specific pattern (Lin, Fleming and El-Kady [2003], Marquier, Joulain, Mulet, Carminati, Greffet and Chen [2004]). While the potential could be large, allowing to engineer radiation coupling in everyday life (think of directional heating from a glowing filament!) and in photothermovoltaic generators for example, the difficulties of fabricating and properly measuring the corresponding objects make the first conclusions still ambiguous and contraversial.

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§ 4. Two-dimensional photonic crystals 4.1. Theoretical aspects Owing to the relative simplicity of 2D lattices, and the less demanding computational load, an exhaustive scan of the parameter space has been achieved compared to 3D systems. A first set of studies has been along the band structures and gaps vs. lattice and “atoms” (shape inside unit cell) (Padjen, Gérard and Marzin [1994], the special issue edited by Kurizki and Haus [1994], Gérard, Izraël, Marzin, Padjen and Ladan [1994], Villeneuve and Piché [1992a, 1992b, 1994a, 1994b], Baba and Matsuzaki [1995, 1996]). These studies have assessed the privileged status of the triangular lattice, closest to a round Brillouin zone, in ensuring the largest gap for a given index contrast. A recent approach to the issue of maximizing gaps capitalizes on inverse problem methods (Burger, Osher and Yablonovitch [2004]). To begin with, it is interesting to consider an array of cylinders (cylindrical “atom”), in the context of the quest for the absolute bandgap in 2D, i.e., the achievement of a bandgap common to both polarizations. We recall that for triangular lattices of identical atoms, this absolute gap is found only for arrays of holes (“air columns”), and for air-filling factors a little below the close-packing condition, that is for thin dielectric veins between the holes. So, what happens for the related graphite lattice? This lattice has two “atoms” per cell, say two cylinders. These cylinders could also be made different (in diameter or dielectric constant), having in mind the analogy with the lamellar boron nitride compound. Of interest to us is the fact that the graphite structure of identical dielectric columns with a small dielectric filling factor also exhibits an absolute gap (Cassagne, Jouanin and Bertho [1997], Barra, Cassagne and Jouanin [1998]). A closer look at this structure (fig. 15) shows that this property is well explained by its topological similarity with the triangular array of holes at large air fillingfactor. In this case, the veins between two holes become thin, and the dielectric is left mainly between sets of three adjacent holes. The lattice of these dielectric regions is almost identical to a graphite lattice, and this explains the similarity. Insofar as the gap is obtained between low-lying bands, the physical meaning is that the fine details of these dielectric regions are unimportant, justifying that the topological similarity of the lattices of dielectric regions translates into the presence of similar gaps (Gadot, Chelnokov, Lustrac, Crozat, Lourtioz, Cassagne and Jouanin [1997]). This is one of dozens of examples showing that an important merit of twodimensional structures is to easily exemplify which avenues can be explored. Be-

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Fig. 15. (a) Graphite lattice of dielectric rods and (b) high-air-filling-factor triangular lattice of round holes. The arrangement of remaining dielectric between holes is of the graphite type, as shown by the six gray superimposed disks at top left.

yond the case of plain crystals, the understanding of defects in 2D lattices has been a powerful lever for the community in grasping many of the concepts of photonic crystals, especially that of defects with different dimensionalities (point defects and line defects) and the complexities arising upon coupling exterior plane waves to such localized defect modes.

4.2. 2D photonic crystal with vertical guidance Since it was found impractical to fabricate self-standing arrays of tall columns or very long straight pores in a dielectric (with the two notable exceptions of macroporous silicon and alumina), 2D structures of finite height were given more attention. Given the ubiquitous role of guided optics in the present-day optoelectronics and optics world, it was an obvious approach to put together 2D photonic crystals with guiding stacks in the third dimension, z. The paper by Meade, Devenyi, Joannopoulos, Alerhand, Smith and Kash [1994] launched the idea in a rather optimistic manner, proposing several combinations of “TIR” (Total Internal Reflection) guiding in one or two dimensions and PhC guiding in the other ones. An open question at the beginning of the story was to what extent the dielectric z confinement would survive the perforation by holes. Of course, for the guided-optics community that had spent so much energy in limiting scattering or leakage of guided waves by careful design and processing techniques, it was natural to fear a very high degree of loss. On the other hand, newcomers to the field had an open eye for how many of the PhC properties could be manifested by such structures, such as spontaneous emission inhibition (SEI) or the possibility to implement very sharp bends. Since so few cells are required in the bandgap

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to display these properties, the concern of insufficient vertical confinement was initially deemed of secondary importance. A topic closely related to vertical guidance is the behavior of off-plane propagation, in a system of infinite uniform extension in the third dimension (Feng and Arakawa [1996]). This will be dealt with further for macroporous silicon (Section 4.3, see also Jamois, Wehrspohn, Andreani, Hermann, Hess and Gösele [2003]) and for photonic crystal fibers (Section 5). The light-line issue The primary concept in dealing with the issue of vertical confinement in such structures is the light line in the dispersion diagram (see fig. 16). For propagating waves, uniform cladding dielectric regions of index nclad bounding the guide have the simple dispersion relation ω=

c nclad

|k p + kz z|,

(4.1)

p being an in-plane unit vector. The 2D modes that we expect to be supported by the guiding region have a specific dispersion relation, which may be restricted to the first Brillouin zone due to periodicity. This is given by ω = ω(k ).

(4.2)

It is important to stress that we deal at this stage with 2D structures of infinite in-plane extension. The main issue is whether a guided mode can be coupled to a radiation mode of the surrounding medium, leading to a “leakage”. In the reciprocal view, an impinging plane wave from the outside medium would feed a z-localized resonant/guided mode. The phenomenon is reminiscent of the Wood anomaly of gratings and of corrugated waveguides which have been largely documented since then, see Tikhodeev, Yablonskii, Muljarov, Gippius and Ishihara [2002], Astratov, Culshaw, Stevenson, Whittaker, Skolnick, Krauss and De La Rue [1999], Whittaker and Culshaw [1999], Netti, Harris, Baumberg, Whittaker, Charlton, Zoorob and Parker [2001], Andreani and Agio [2002] and Andreani, Gerace and Agio [2004], Nevière and Popov [2003]. In the case of linear waveguides with one remaining direction of periodicity, this issue has been well treated by Johnson, Villeneuve, Fan and Joannopoulos [2000], and later by Andreani and Agio [2003]. Both kinds of modes, guided and radiating, can be coupled only if they are phase matched over the entire space, that is if they share the same k . The existence of lossless guided modes can then be permitted if they have a k larger than any of those of the outside media (which are limited to k < nclad ω/c), otherwise the modes are evanescent in the z-direction and do

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Fig. 16. (a) Schematic of a slab supporting guided modes and a continuum of radiation modes (b) with a periodic modulation, a guided mode can acquire a small component onto a radiating mode, thanks to an in-plane wavevector G of the reciprocal lattice. (c) Corresponding wavevector schemes in the vertical plane. (d) One-dimensional fundamental gap with superimposed light lines. (e) Two-dimensional band structure diagrams with extreme light lines superimposed.

not radiate. The highest radiating k value is k = nclad ω/c corresponding to an outside grazing ray (kz = 0). The line k = nclad ω/c defines the light line in the in-plane dispersion diagram. Of course, the symmetric line k = −nclad ω/c plays the same role.

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In the region of the in-plane ω(k ) dispersion diagram outside the “light cone”, where larger wavevectors verifying nclad ω (4.3) c are encountered, a truly guided mode can exist. The other region, with smaller wavevectors, |k | < nclad ω/c, lies inside the light cone, and coupling of an inplane mode to outside modes is possible. Strictly speaking, there is only a mode continuum above the light line. But the energy may be strongly concentrated in the in-plane guide around certain frequencies: “Modes” found above the light line are called “leaky modes” or “resonances” (see above). They can be described by a complex k at constant ω, the imaginary part of which would describe the spatial decay of guided energy that would be launched at a given point in the infinite structure. Alternatively, an imaginary part of the frequency for a real k would describe the lifetime of energy density in such a leaky mode (Whittaker and Culshaw [1999], Tikhodeev, Yablonskii, Muljarov, Gippius and Ishihara [2002]). Notably, such complex quantities are found when examining the zeroes and poles of the scattering matrix S(k , ω) relating plane waves of given k on both sides of the guiding structure (see Section 2). The calculation of these imaginary parts, in simple words, the strength of the coupling, is a delicate task. Pole “hunting” is complex when several resonances (photonic bands above the light line) intervene and tend to mix. Furthermore, the region of the light line itself tends to be singular. One “geometrical optics” reason for this peculiarity is the peaked density of photons associated with the critical angle where the outside rays have grazing incidence. Snell’s law for a planar interface shows that around this condition, the ratio of outside and inside coupled elementary solid angles, dΩout /dΩin , diverges. |k | >

Polarization Strictly speaking, a perforated guiding structure has lost the invariance that allowed polarizations to separate in Maxwell equations: the modes have irreducible six-components fields. (We deal with 2D in-plane patterns. For simple grooves/1D texture in the waveguide, the polarizations separate only if the plane of incidence contains the grating wavevector G.) However, an approximation often holds and majority polarization components are used to label modes. Here, due to the merging of two different communities, some ambiguity has arisen in the naming of “2D PhC + guide” modes. Stemming from the pure 2D structures, infinitely extended along z, one has modes that are called E and H [see figs. 17(h,i)], while stemming from the planar waveguide, one has the TM and TE

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Fig. 17. Two-dimensional photonic crystal with vertical waveguiding based on (a) the “substrate” approach and (b) the “membrane” approach. (c) Equivalent 2D picture of both situations. (d)–(i) Comparison of naming conventions and field components for the various cases: (d,e) TE polarization without and with PhC; (f,g) TM polarization without and with PhC; (h) 2D picture for (e) and the “opto” and “theory” names; (i) 2D picture for (g) and the “opto” and “theory” names.

modes, respectively [see figs. 17(d,f)]. For the perforated guide (or generally in any periodically textured guide, see Andreani and Agio [2002]), we will adopt the latter naming convention here. Hence we call TE the modes whose H field may be seen as a quasi-scalar Hz component, and TM those whose E field may be seen as a quasi-scalar Ez component. This view is best suited to the case of modest vertical confinement. A vertically extended waveguide mode tends to mimic a plane wave with kz = 0. The vanishing kz components translate into a negligible in-plane H component for what

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we call the TE mode, which will in this limit approach H modes of the infinitely extended 2D case. “Membrane” and “substrate” Having set this frame, it can be said that the options taken experimentally until now are divided into two extremes that we will call the “membrane” and the “substrate” approaches [see figs. 17(a,b)] (Weisbuch, Benisty, Olivier, Rattier, Smith and Krauss [2001], Weisbuch, Benisty, Rattier, Smith and Krauss [2001]). In the case of the membrane, the outer media have an index as low as possible, e.g., vacuum or air, and the system has to be supported by bridges on the side. Although this is technologically possible, it has several drawbacks for applications, thermal sinking issues and coupling to the outside world being the two foremost ones. As a partial remedy, one can use a low-index layer of silica (n ≈ 1.47) or alumina (n ≈ 1.7) as the bottom cladding. This still leads to light cones which are fairly narrow when the guided modes lie in a high-index material (n ≈ 3). Such cases thus tend to be classified as “membrane”. The meaningful criterion is actually whether the experiments of interest address modes above or below the light line. It turns out that the fundamental TE-polarized gap of the triangular lattice of holes in materials such as semiconductors still retains a sizable fraction below the light line with a silica or alumina bottom cladding. Hence their classification into the “membrane” category. The “substrate” case relates to guides with modest vertical index contrast, typically n < 1, as happens in semiconductor heterostructures, most notably for the two influential compound systems of the (opto)electronic industry, GaAlAs/Ga(In)As/GaAlAs and InP/GaInAsP/InP. For this case, the light line is very low, and the area corresponding to the PhC gap falls completely above it. In an average-index approach, for instance, the perforated PhC waveguide has an average index below that of the cladding layer. All its modes are therefore above the light line (a more detailed discussion is needed if the bottom cladding is perforated very deeply as well). There are obvious advantages to this approach in terms of electrical/optical device integration. It was explored by the authors and many of their collaborators for this reason (Benisty, Olivier, Rattier and Weisbuch [2001], Benisty, Rattier and Olivier [2002], Benisty, Weisbuch, Labilloy, Rattier, Smith, Krauss, De La Rue, Houdré, Oesterle and Cassagne [1999]). The merits of both approaches are diverse. With respect to the ease of etching holes, the membrane is best as the hole aspect ratio (height/diameter) can be limited to unity. However, for the subcase of silica bottom cladding, for example, it has been shown that it would be better to pursue the etching of the

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holes into the silica layer (Sondergaard, Bjarklev, Kristensen, Erland and Broeng [2000]). This situation notably arises when considering a silicon-on-insulator (SoI) basis. For the “substrate” approach, the criterion for the aspect ratio entails that the holes have to overlap the guided mode profile as completely as possible, up to 99.9% practically (D’Urso, Painter, O’Brien, Tombrello, Yariv and Scherer [1998], Ferrini, Lombardet, Wild, Houdré and Duan [2003], Benisty, Lalanne, Olivier, Rattier, Weisbuch, Smith, Krauss, Jouanin and Cassagne [2002], Lalanne and Benisty [2001], Lalanne [2002], Kafesaki, Agio and Soukoulis [2002]). The penalty of not etching below the core is apparent in Chen, Haus, Fan, Villeneuve and Joannopoulos [1996]. This large overlap criterion translates into a challenging demand for the “deep etching” techniques, with an aspect ratio of around 10 and with very vertical sidewalls (at most ∼10 mrad ∼0.5◦ off vertical). Losses must be examined for each specific application as real-world structures are not infinite. Considering in particular the use of modes localized in microcavities carved into a PhC, as is sought for filtering and telecom applications, there is no simple rule for obtaining the losses of each approach. The mode localization in the PhC plane breaks down the light-line argument, and the losses must be specifically calculated and engineered. This will be discussed in more detail below in view of recent results. The main message to retain is that for modes above the light cone (and thus localized modes at cavities), it is delicate and calculationintensive to predict the losses. Finally, be it leaky or genuine guided modes, it is interesting to compare in these PhCs based on vertical waveguides the band structures calculated from the exact 3D model and from a 2D model (Qiu [2002a, 2002b]). The preferred choice for the matrix index in 2D is the effective index of the guided mode (one essentially assumes monomode behavior in the vertical direction, although multimode could be considered, but would lead to many complex mode couplings, see for example David, Meier, Sharma, Diana, DenBaars, Hu, Nakamura, Weisbuch and Benisty [2005]). This is a natural way to proceed as the effective index neff = k ω/c is the index of the wave seen “from top”. The comparison shows (Qiu [2002a]) that all low-lying bands have their structure preserved, but that frequencies may be underestimated by this method, especially in the “membrane” approach, depending on the band under consideration. 2D + ε′′ We devote special attention here to a phenomenological modeling of the losses that is specific to 2D slab photonic crystals and for that reason was not included in Section 2. The essential idea is that one may account for the power radiated by a leaky mode outside the plane of guidance through the introduction of a power

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dissipation mechanism in the holes, i.e., a fictitious current in phase with the electric field. In the frequency domain, such a dissipation is best introduced through an imaginary part of the dielectric constant in the holes, ε ∗ = 1 + iε′′ (Benisty, Labilloy, Weisbuch, Smith, Krauss, Béraud, Cassagne and Jouanin [2000]). In the time domain, it is advantageous to introduce this “ohmic” aspect through an adequate conductivity (Qiu, Jaskorzynska, Swillo and Benisty [2002]). This model is expected to give trends (Ferrini, Houdré, Benisty, Qiu and Moosburger [2003]) that apply in a restricted frequency domain due to its underlying assumptions (on mode shapes, etc.). Its microscopic origin can be seen as the lack of separability of the dielectric landscape for a perforated waveguide. The separability of variables was used in Section 2.3. By contrast, a separable system can be checked to be lossless (Ctyroky [2001]). An obvious advantage is that only a 2D calculation needs to be performed instead of a 3D one. This facilitates establishing a power budget for a somewhat complex structure or optical function, for which browsing through alternatives in 3D is most often out of reach. Conversely, in its simplest form, a single parameter such as ε ′′ suffices to account for the losses of a waveguide, a cavity, a PhC semitransparent mirror, etc. The approach is most sensible to the “substrate” case. In this case, losses are present for all modes but are essentially modest because, even for a submicronic vertical extent, a guided mode in the substrate approach has weak leaky k-components. Conversely, this approach with ε is unable to render the existence of the light line and, thus, of lossless modes of membranes. The essence of a lossless mode is the coherence of the Bloch wave in the membrane, enabling a phase shift between “scatterers” (holes) too large to match any propagative mode in the air cladding. In the 2D + ε ′′ approach, dissipation can mimic scattering as long as the overall scattering is not heavily dependent on coherence. Fortunately, this is the case in the substrate approach. While coherence does exist for the actual scattering mechanism, its main role is to finely tune the directions of leakage (which are nothing but the radiation diagram for out-of-plane losses) but it does not significantly change the scattered intensity, i.e., the total losses (see “recycling” effects in Lalanne, Hugonin and Gérard [2004], Lalanne and Hugonin [2003]). This discussion is germane to the discussion of the Purcell effect in planar microcavities, whereby a modest lifetime change (Purcell effect) is obtained, but huge directionality effects also arise. In the next subsection, we give an overview of the many various realizations of 2D PhC structures that have flourished in the last decade.

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4.3. 2D realizations Macroporous Si Among the remarkable properties of silicon, its ability to be cast into new shapes is impressive. Porous silicon is obtained from anodic oxidation in HF-containing electrolyte. In its microporous form, it was discovered to exhibit sizable luminescence, leading to a number of investigations prompted by its potential in the optoelectronics industry. The macroporous shape of interest here has much bigger pores and no special nanostructures-related electronic properties. It is obtained upon photoelectrochemical anodic oxidation of n-type Si, whereby the holes (minority carriers) needed in the electrochemical reaction are generated on the backside of the wafer by illumination. They migrate a long distance to reach the electrochemical interface. This latter then develops a stable pattern of holes, a stability mainly due to the electrostatic “tip” effect that focuses line currents at hole bottoms while the anisotropy of the crystal itself favors the formation of straight pores. The aspect ratios obtained are remarkably large. One decisive trick is that the holes can be seeded at the desired positions by a first lithographic step combined with an alkaline etching process that digs etch pits along the lithographically defined lattice. The pioneering experiments by Grüning et al. (Grüning and Lehmann [1996], Grüning, Lehmann, Ottow and Busch [1996]) led to macroscopic samples (100s of microns) with micronic periods but nevertheless thin (∼0.2 µm) and stable veins between the holes on the whole sample height. Etched pieces could be micromachined into shapes suited to infrared Fourier-transform spectrometers, where the absolute gap in the 3–5 µm region clearly appeared. Further progress was achieved mainly in two places, Halle (Birner, Grüning, Ottow, Schneider, Müller, Lehmann, Föll and Gösele [1998], Leonard, van Driel, Busch, John, Birner, Li, Müller, Gösele and Lehmann [1999]) and Orsay (Rowson, Chelnokov and Lourtioz [1999], Rowson, Chelnokov, Lourtioz and Carcenac [1998]), which led to gaps at telecom wavelengths (λ = 1.55 µm) (see fig. 18). The introduction of defects into these PhCs is feasible and leads to a string of interesting results (Leonard, van Driel, Birner, Gösele and Villeneuve [2000]). One impediment however relates to the hole distortion around hole-free defects (Agio, Lidorikis and Soukoulis [2000]) due to the limits of the current streaming effect mentioned above. It is impossible in practice to have missing hole defects that are more than two lattice periods wide, for example. On the other hand, because the period is relatively large, photolithographic masking on a large sili-

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Fig. 18. Electron micrograph of a macroporous silicon sample with pitch ∼0.68 µm, leading to a bandgap at λ = 1.55 µm (work done at Institut d’Electronique Fondamentale, Orsay).

con wafer yields thousands of elementary structures (guides, cavities, etc.), which helps study many structure variations. Sufficient vertical regularity is needed to approximate an infinitely extended structure probed by a quasi plane wave (beam waists ∼ several tens of λ). This condition is quite demanding for defect cavities, as is the flatness of the mirrors in high-finesse Fabry–Perot cavities. Near-field tools have revealed this aspect (see a string of results in Busch, Lölkes, Wehrspohn and Föll [2004]). An interesting prospect that emerged recently is to vary the electrochemical current during the etching process, leading to vertical variations of the pore diameter [figs. 11(g–i)]. This could lead to structures with a vertical guiding capability due to a locally more dense average medium at narrower pore sections [fig. 19(a)], or even to structures with complex three-dimensional shapes and properties still to be determined and understood. 2D PhCs with a guide: membrane type One advantage of membranes is the easier assessment of their geometric parameters obtained by simple scanning microscopy inspection. The height is limited so that the holes may be fully inspected from the top [fig. 19(b)] and the roughness,

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Fig. 19. Various 2D photonic crystal realizations with confinement in the third dimension: (a) modulated current etching of macroporous Si (work done at IEF, Orsay); (b) free-standing InP membrane with a single hole defect (work done at Caltech); (c) PhC etched by chemically assisted ion beam etching (CAIBE) in an InP-based heterostructure; the waveguide is represented as dotted lines and a sketch of the mode profile is given; the smaller hole on the left of the hole array is intentional (taper studies, work done at LPN, Marcoussis and KTH, Kista, see Talneau, Mulot, Anand and Lalanne [2003]); (d) GaAs on “AlOx ” PhC structure (work done at Sandia National Laboratories, Albuquerque).

residual slope, etc. can be assessed with fair accuracy. The materials of interest are the Si, GaAs and InP semiconductor systems due to their mastered heteroepitaxy. Before 2D PhCs, the earliest microcavities based on “air bridges” were actually defects in a structured system of one-dimensional nature (Foresi, Villeneuve, Ferrera, Thoen, Steinmeyer, Fan, Joannopoulos, Kimmerling, Smith and Ippen [1997], Fan, Winn, Devenyi, Chen, Meade and Joannopoulos [1995]). They can clearly be considered as striking achievements. A full 2D bandgap on a mem-

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brane was demonstrated by Kawai, Inoue, Carlsson, Ikeda, Sugimoto, Asakawa and Takemori [2001]. Another system approaching the membrane situation of a high-index layer sandwiched between two low-index layers is provided by silicon which has been available for many years in crystalline form over a silica layer in so-called SoI (silicon-on-insulator) substrates on large wafers for the microelectronics industry. For this reason, silicon-on-insulator (SoI) is of foremost interest for PhC applications. It is also considered as a platform for high-index-contrast ridgewaveguide-based integrated optics (Manolatou, Johnson, Fan, Villeneuve, Haus and Joannopoulos [1999]). Investigations into the potential of SoI for integrated optics without PhCs have highlighted difficulties that would also occur in the case of PhCs. The comparison will be instructive (Bogaerts, Wiaux, Taillaert, Beckx, Luyssaert, Bienstman and Baets [2002]). The small mode size in waveguiding structures results in a delicate coupling to the outside world, and also in a very high penalty for any residual sidewall roughness compared to conventional low-index-contrast integrated optics structures (e.g., doped silica on pure silica, δn/n ≈ 1% or less). Fabrication-wise, the etching of membrane structures is performed first by standard microelectronic techniques (the pattern definition being carried out by e-beam lithography). The main requirement is the capability of performing a sacrificial etching of those layers underneath the to-be membrane. In the case of silicon, HF dissolves the silica underlayer. The reader may consult the literature for the specialized etchants used for (In,Ga)(As,P) and (In,Ga,Al)As compounds. After this stage, one obtains free-standing membranes. Three ways to realize membranes resting on low-index layers are as follows: • One can use the selective lateral oxidation of Al-rich Ga(Al)As layers into the low-index Alx Oy insulator, generally named “AlOx ” [fig. 19(d)] (Chow, Lin, Johnson, Villeneuve, Joannopoulos, Wendt, Vawter, Zubrzycki, Hou and Alleman [2000]). • For SoI, where the 2D PhC hole array is etched into the silicon, the silica may stay in place. Whether silica is perforated or not depends on the etching method used to define the holes (Bogaerts, Wiaux, Taillaert, Beckx, Luyssaert, Bienstman and Baets [2002], Baba, Motegi, Iwai, Fukaya, Watanabe and Sakai [2002]). • For InP or GaAs based heterostructure, the membrane can be bonded on another substrate (glass or silica on silicon) and the initial wafer used for growth is completely sacrificed. It incorporates an etch-stop layer on which the substrate etchant will virtually not act, so that the layers grown on this first substrate are safely protected from etching while the substrate dissolution can be

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completed. An advanced variant is the ion-implantation-based process known as “Smart Cut™” making the wafer brittle a few microns beneath its exposed surface. At the end of the day, one has some PhC-patterned membrane, possibly with some photonic defects of interest. For a rapid calculation of bands and defect frequencies, one can use as a first step a 2D calculation with the effective index of the membrane. The difficulty of launching light in these systems is an important issue when they are not resting on a substrate and stand in air: one has to go from a classical wafer guiding structure to the PhC structure by a “bridge” that starts at a poorly defined location. Another solution is to cleave across the membrane, at higher risks of course, but with a fair probability to lock the cleavage plane on the PhC symmetry plane (row middle axis). These problems have been circumvented to a large extent by generating light directly in the structures of interest through photoexcitation of electron–hole pairs, as was done in the pioneering membrane experiments of the Caltech group (Lee, Painter, D’Urso, Scherer and Yariv [1999], Lee, Painter, Kitzke, Scherer and Yariv [1999]), followed by those of the LEOM group in Lyon (Letartre, Seassal and Grillet [2001]). For this purpose, as had been shown earlier on substrate-type PhCs (see below), active layers such as quantum wells or quantum dots are grown in the middle of the membrane (where they best interact with the guided mode), and they channel the electron–hole pairs photoexcited in the surrounding material due to their smaller gap. The recombination of electron–hole pairs at an etched interface or even at the horizontal semiconductor–air interface is a very common obstacle to achieving good luminescence. This is even more true for lasing action, for which a prerequisite is an optimal channeling of carriers (both spatially and energetically) into lasing modes possessing high useful radiative recombination efficiency. It is less severe in the InP system for physico-chemical reasons. The recombination on the slab interfaces can be limited by growing epitaxial carrier barriers (high-energy electronic gap) as the last layers in the membrane. Limiting the recombination at the vertically etched hole surfaces is more delicate, and is a general problem for both the “membrane” and the “substrate” approaches. 2D PhC with a guide: substrate type The most successful ideas for allowing the operation of PhCs on heterostructure with modest vertical index contrast, the “substrate” approach, appeared in the work by Krauss, De La Rue and Brand [1996] and Krauss and De La Rue [1996a, 1996b], following early efforts in etching technology (Krauss, Song,

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Thoms, Wilkinson and De La Rue [1994]). They established that it was necessary to etch the holes to such a point that the whole profile of the guided mode would “feel” the holes, and they also devised that a moderate air-filling factor (unfavorable to the formation of the TM gap) was an unavoidable condition for attaining acceptably low losses (Krauss, Painter, Scherer, Roberts and De La Rue [1998], Krauss, Smith, Vögele, Murad, Wilkinson, Grant, Burt and De La Rue [1997]). These were the key ingredients for effective 2D behavior in spite of the leaky nature of the modes involved (Labilloy, Benisty, Weisbuch, Krauss, De La Rue, Bardinal, Houdré, Oesterle, Cassagne and Jouanin [1997], Labilloy, Benisty, Weisbuch, Smith, Krauss, Houdré and Oesterle [1999]). The present authors and collaborators further substantiated these requirements in more quantitative terms with the help of the 2D + ε ′′ approach (Section 4.2), by making a more explicit connection between the PhC and heterostructure parameters [especially the air-filling factor f , the vertical index step ε, the hole depth and the parameters describing precisely the tail of the mode profile ζ (z) in the substrate] and the effective loss parameter ε ′′ (Benisty, Lalanne, Olivier, Rattier, Weisbuch, Smith, Krauss, Jouanin and Cassagne [2002], Ferrini, Houdré, Benisty, Qiu and Moosburger [2003], Ferrini, Lombardet, Wild, Houdré and Duan [2003], Lalanne and Benisty [2001]). A special interest of classical heterostructures is the ease of obtaining a cleaved facet. As we will see below, measurements imply the use of either an internal or an external light source. For coupling an external source, the integrated-optics and laser-diode community has used a cleaved facet since the very beginning. In the case of an internal source, much practiced by the authors (Labilloy, Benisty, Weisbuch, Krauss, Houdré and Oesterle [1997], Labilloy, Benisty, Weisbuch, Krauss, Smith, De La Rue, Cassagne, Jouanin, Houdré and Oesterle [1999]), the cleaved facet ensures a reproducible transmission to the outside, a welcome feature for calibrated measurements (precision in the 5% range). Let us now detail each of the two semiconductor systems for which PhCs on a substrate were investigated. GaAs. The emitters of interest when starting from a GaAs substrate have wavelengths between 620 nm (AlInGaP quantum wells) and 1250 nm (InAs dots), the latter value moving further away when nitride-containing compounds (GaInNAs) are considered, as shown since 2000. It is natural to focus on the near gap region, 850–1070 nm, due to the use of Si-based detectors and CCD matrices. Creating a photonic gap for these wavelengths while respecting the criterion of relatively low air-filling factors results in periods ranging between 200 and 350 nm, and hole diameters ranging from 140 to 250 nm. For vertical monomode behavior, the

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GaAs waveguide core thickness is around 300 nm, and the guided mode extends a few hundreds of nm down into the Al-rich bottom cladding (the index contrast typically features n > 0.4 for this cladding). An upper cladding, for instance 200 nm thick, is also desirable to bury the mode as much as the etch depth allows, and minimize losses by avoiding the tail of the mode to extend into air. With these requirements, the etch depth for reduced-loss operation should be in the 700–800 nm range, which means aspect ratios (height/diameter) of at least 5–6. For the development of e-beam lithography and RIE etching technology of such structures, see Smith, Olivier, Benisty, Rattier, Weisbuch, Oesterle, Houdre and Krauss [2001], Ferrini, Leuenberger, Mulot, Qiu, Moosburger, Kamp, Forchel, Anand and Houdré [2002], Krauss and De La Rue [1999], Krauss, Smith, Vögele, Murad, Wilkinson, Grant, Burt and De La Rue [1997]. A number of pioneering studies have been performed in these systems (see below). They were also used to extensively benchmark losses in the 2D + ε′′ approach (Ferrini, Houdré, Benisty, Qiu and Moosburger [2003], Olivier, Benisty, Weisbuch, Smith, Krauss and Houdré [2003], Qiu, Jaskorzynska, Swillo and Benisty [2002], Rattier, Benisty, Smith, Béraud, Cassagne, Krauss and Weisbuch [2001], Schwoob, Benisty, Olivier, Weisbuch, Smith, Krauss, Houdre and Oesterle [2002]) before transferring the approach to the InP system. The use of InAs quantum dots (QDs) as broadband probes has been particularly fruitful (Labilloy, Benisty, Weisbuch, Krauss, Smith, De La Rue, Cassagne, Jouanin, Houdré and Oesterle [1999]). Lasing of confined in-plane round microcavities proved easier to obtain in QW-based structures (Shaw, Roycroft, Hegarty, Labilloy, Benisty, Weisbuch, Krauss, Smith, Stanley, Houdré and Oesterle [1999], see also Raffaele, De La Rue, Roberts and Krauss [2001] for microlasers with straight mirrors) due to the larger material gain. InP. Emission at the long-haul telecom wavelength λ = 1550 nm is the main asset of epitaxial growth on InP substrates. The vertical index contrast compatible with heteroepitaxy requirements on InP is at most n = 0.2 at 1550 nm, and even less at 1300 nm, the short-haul telecom wavelength. Indices are slightly lower than for GaAs (core index of ∼3.3 instead of ∼3.55). The thickness for a monomode waveguide increases considerably compared to GaAs: a core thickness of 400–600 nm may remain monomode (the wavelength is also larger, of course), while a bottom cladding thickness of well over 1 µm is needed. The top cladding may be thin (300 nm) unless metallic layers (electrodes) lie on top of it. Then, much larger thicknesses are required to keep absorption losses by the metal at a reasonable level. As for the parameters of the 2D PhC, and focusing on the case of 1550 nm emission, the PhCs period of interest ranges from 380 to

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550 nm and the diameter from 200 to 400 nm. The far larger vertical extent of the guided mode into the bottom cladding means that etching is required to go much deeper: typically twice the GaAs depth (∼2 µm), with an associated aspect ratio of at least 7–8. In the quest for such elongated holes, it was found that etching often results in conical holes, i.e., holes that have a rather straight upper section, but a conical bottom section, with a rather well-defined cone angle at those depths where the mode profile has not yet completely decayed. This presented an opportunity to quantify the penalty of conicity again in terms of the imaginary ε ′′ in the 2D + ε ′′ approach (Ferrini, Houdré, Benisty, Qiu and Moosburger [2003]). Adapted etching techniques proved to be either dense-plasma techniques such as inductively coupled plasma reactive-ion-etching (ICP-RIE) or chemically-assisted ion-beam etching (CAIBE). For ICP-RIE examples see Ferrini, Lombardet, Wild, Houdré, Olivier, Benisty, Djoudi, Legouézigou, Hubert, Sainson, Chandouineau, Fabre, Pommereau and Duan [2002]; for CAIBE examples see Mulot, Anand, Ferrini, Houdré, Wild, Moosburger and Forchel [2004] and Mulot, Anand, Swillo, Qiu, Jaskorzynska and Talneau [2003]. For InP the inclusion of emitters was mainly restricted to quantum wells. There is no common equivalent to the layers of InAs dots in terms of both broad spectral distribution and small re-absorption coefficient, although some InAs-based “elongated box” or “quantum dash” epitaxial structures should certainly be considered (Salem, Benyattou, Guillot, Bru-Chevallier, Bremond, Monat, Hollinger and Gendry [2002]).

2D PhC with a guide: lower indices Lower-index materials forming vertical waveguides are typically oxides (including LiNbO3 ), nitrides, and organic materials. Some of them have been investigated for PhC, with interesting results, even though the omnidirectional bandgap is much restricted or absent due to the low RIC with air. Oxides are generally very difficult to etch in a perfectly anisotropic manner, in particular LiNbO3 . Fabrication of PhCs in the intermediate case of silicon nitride (n > 2) on silica (n = 1.48) has been actively pursued (Charlton, Parker and Roberts [1997], Charlton, Parker and Zoorob [1999], Charlton, Zoorob, Parker, Netti, Baumberg, Cox and Kemhadjian [2000]). Since internal emission is virtually impossible in these systems, characterization requires an end-fire coupling technique. However, this is made delicate by the quasi-absence of a cleavage plane (Netti, Charlton, Parker and Baumberg [2000]).

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Organic materials have also been pursued. In relative terms, an index contrast of 0.1 or 0.2 around n ≈ 1.5 is an interesting contrast. The actual potential of this approach is still unclear, but the flexibility of the underlying material is of course an advantage for envisioned applications (Augustin, Böttger, Eich, Etrich, Fuchs, Iliew, Hübner, Kessler, Kley, Lederer, Liguda, Nolte, Meyer, Morgenroth, Peschel, Petrov, Schelle, Schmidt, Tünnermann and Wischmann [2004]). It turns out that, perhaps due to the absence of cleavage and to the resulting difficulties in coupling extended waves, a majority of published work on either kind of PhCs relates to waveguides, etc. This is treated in the next subsection.

4.4. Applications and characterization methods In the period 2000–2004, a number of impressive realizations have been made using PhCs on membranes and substrates (see Soukoulis [2001], and the IEEE Journal of Quantum Electronics special issue edited by Krauss and Baba [2002]). Broadly speaking, an approximate classification could be made according to light source. The use of an external light source (white light or a tunable laser, typically fed to the waveguide by an end-fire technique) probes only the modes excited by this source. Conversely the use of an “internal” light source, exploiting the photo- (or electro-)luminescence of embedded species (dyes, quantum wells, quantum wires or quantum dots, etc.), possibly excites all modes (Ferrini, Leuenberger, Mulot, Qiu, Moosburger, Kamp, Forchel, Anand and Houdré [2002]). Here, the detection process may ignore some internal modes or conversely include unwanted ones, depending on whether their outside radiation pattern ignores or feeds the detection optics (Labilloy, Benisty, Weisbuch, Krauss, Smith, De La Rue, Cassagne, Jouanin, Houdré and Oesterle [1999]). We will see that some applications were demonstrated using one of these measurement methods or both. So it is advantageous to keep these two schemes in mind (figs. 20 and 21). However, further in this subsection, we find it more rational to base our description on the type of application. We start with the case of cavities, a basic building block for wavelengthselective applications. We then go to lasers, typically formed from such cavities, but also formed from simple crystals used at a band edge thanks to the low associated group velocity. The topic of straight waveguides then follows. It comprises a special paragraph on waveguide losses because such losses are a well-known benchmark in integrated optics, where the loss in dB per unit length (say cm) is a convenient yardstick. Next we examine waveguide bends. The coupling of two systems such as cavities and waveguides is one further complexity, remarkable for its many interesting proposals.

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(a)

(b) Fig. 20. (a) The classical integrated optics method, also called “end-fire” method is adapted here to the case of a PhC waveguide, with ridge access waveguides. Note the different scales, and the rather exaggerated hole size in the PhC block, typical diameters and periods usually being 200–400 nm. Note that the method is difficult to apply without a cleavage plane. Measurements are exemplified in (b), for a three-missing-rows (“W3”) waveguide whose specificities are discussed later [see figs. 24(c–f)]. Here different etching processes A and B (B, the best) result in a variable shape of the resonant transmission dip feature. Note the dynamic range, well over 30 dB, needed to properly carry out this characterization example.

We further turn to systems that exploit the full 2D character of bands: superprisms and supercollimators. The last application studied is LED and light extraction, which requires a 3D exploitation of 2D photonic crystals, hence providing a different point of view.

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Fig. 21. The internal light source method: (a) spectrum of InAs/GaAs quantum dot emission grown with a broad size distribution in the waveguide, seen by the spectrometer CCD; (b) reference measurement using guided photoluminescence (PL, open arrow), taken for a spot at distance d from a cleaved edge; (c) transmission of a PhC structure, using the same distance d; (d) use of laser excitation for probing a PhC-based microcavity, for example, the diffracted front PL containing the spectral information on the cavity modes being further collected [note that a combination of (c) and (d) is feasible for more complete studies]; (e) typical result of the measurement of 15-rows-thick crystals (inset) with seven different periods from a = 180 nm to a = 300 nm, at a constant air-filling factor. The spectra are stitched together on a single spectrum: the resulting spectrum shows the photonic band gap (here along ŴK) while the oscillation in transmission, clearly visible in the high-frequency range, testifies that Bloch modes created in the crystal undergo interferences due to multiple reflection between its edges (see Labilloy, Benisty, Weisbuch, Smith, Krauss, Houdré and Oesterle [1999]).

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Cavities The search for cavities in photonic crystals (Yablonovitch [1995]) is multifold: (1) they provide an excellent testbed for PhC modeling: specific calculation techniques can be applied (supercell calculations, etc.); (2) they provide a very useful structure for evaluating PhC losses: measured Q’s provide a value for leakage integrated over all possible lossy modes, in-plane or out-of-plane. Their measurement is rather straightforward: one simply relies on any internally emitted light leaking out the plane to assess the cavity Q (see the example of “radial” modes in disk-shaped cavities, where physics of resonance and leakage is easy to grasp, in Labilloy, Benisty, Weisbuch, Krauss, Smith, Houdré and Oesterle [1998] and in Ochoa, Houdré, Ilegems, Benisty, Krauss and Smith [1999]), a method much easier to implement than the internal light source method observed on cleaved edges or end-fire methods (Noda, Chutinan and Imada [2000], Baba, Fukaya and Yonekura [1999]). Cavities are also essential for many types of lasers or emitters, as the electric field build-up into cavity modes allows increased light–matter coupling (i.e., lower thresholds) and can lead to mode-selective spontaneous emission, which is of particular interest when one desires directional emission. PhC microcavities are in high demand if one wishes to reach ultimate physical limits such as for single-photon emitters or the strong coupling regime of light–matter interaction (Reithmaier, Sek, Löffler, Hofmann, Kuhn, Reitzenstein, Keldysh, Kulakovskii, Reinecke and Forchel [2004], Yoshie, Scherer, Hendrickson, Khitrova, Gibbs, Rupper, Ell, Shchekin and Deppe [2004]). In both cases, the important parameters, essential to the structure performance, are Q and Vmode , where Vmode is the mode volume. The Purcell enhancement factor in the spontaneous emission rate is directly proportional to Q/Vmode (see Section 2.2). Almost 1/2 similar is the factor Q/Vmode relevant in attaining the “strongly coupled” character of the light–matter interaction, where a dipole and the cavity field into which it emits form a hybrid quasiparticle such as the cavity polariton (Weisbuch, Nishioka, Ishikawa and Arakawa [1992]). It is therefore no surprise that a large number of studies on PhC-based microcavities has been performed. In practice, seminal work on Fabry–Perot-type microcavities [fig. 22(a)] and fully confined “2D” microcavities in 2D PhCs was done on a GaAs substrate (Krauss, Vogele, Stanley and De La Rue [1997], Labilloy, Benisty, Weisbuch, Krauss, Bardinal and Oesterle [1997], Smith, Krauss, De La Rue, Labilloy, Benisty, Weisbuch, Oesterle and Houdré [1998], Smith, Krauss, De La Rue, Labilloy, Benisty, Weisbuch, Oesterle and Houdré [1999], Smith, Krauss, Benisty, Rattier, Weisbuch, Oesterle and Houdré [2000]). Ultimate cavities (one missing hole) were reported by the Caltech group on InP-based membranes. The

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Fig. 22. (a) Fabry–Perot-type cavity between two PhC slabs (assuming that the diffraction cut-off is not reached). (b) Generic cavities in a triangular geometry (array of holes, say) and their nicknames “H3”, “H2”, and below them “H1” for the single-hole-missing case, “lozenge” (9 missing holes) and “L5” or “dash” or “closed-W1” cavity. (c) Mode profile (absolute value of the magnetic field) for a symmetric mode of the H5 cavity (f ≈ 0.28, ε ≈ 11.5). (d) Two almost degenerate modes of a lozenge cavity (f ≈ 0.43, ε ≈ 11.5). (e) Cavity with smaller holes supporting a monopole mode in the PhC gap. (f) “Shallow donor” cavity supported by a region of slightly smaller holes, possibly with a gradual variation.

Caltech group continued the race to obtaining high-Q cavities with a value of Q = 2,800 (Yoshie, Vuckovic, Scherer, Chen and Deppe [2001]). They also proposed inverse-problems approach to the issue (Geremia, Williams and Mabuchi [2002]). Clever radiation-cancellation mechanisms where discussed by the MIT group (Karalis, Johnson and Joannopoulos [2004]). The UCSB group successfully demonstrated relatively high Q for InAs QDs in GaAs in a membrane as well (Reese, Becher, Imamoglu, Hu, Gerardot and Petroff [2001]). A number of cav-

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ities and lasers have also been investigated by the KAIST group in Korea (Park, Hwang, Huh, Ryu, Lee and Kim [2001], Ryu, Kim, Park, Hwang, Lee and Kim [2002], Shin, Kim, Hwang, Ryu, Park, Song and Lee [2002], Ryu, Kwon, Lee and Lee [2002]) [fig. 22(b)]. It is not clear yet what makes these cavities not as good as the theoretical prediction, although suspicion is directed towards residual fluctuations. A breakthrough occurred in 2002 by the Kyoto group led by S. Noda, working on air-bridge membranes. They reached Q’s of 6,000 and then 45,000 in 2003 with cavities derived from the closed-W1 type [see fig. 22(b) and fig. 25(f)], making use of subtle improvement (“shifting holes”) of the terminations (Song, Noda and Asano [2003], Akahane, Asano, Song and Noda [2003a, 2003b]). This was followed by the announcement of Q = 600,000 at the time of writing of this chapter (Song, Noda, Asano and Akahane [2005]) using a new “waveguide heterostructure” design. The physics behind obtaining such high Qs is still somewhat controversial (Sauvan, Lalanne and Hugonin [2004], Sauvan, Lecamp, Lalanne and Hugonin [2005], Sauvan, Lalanne and Hugonin [2005], Vuckovic, Loncar, Mabuchi and Scherer [2002a, 2002b], Englund, Fushman and Vuckovic [2005]), but many properly tuned ingredients are certainly needed (light cone, low group velocity, modal matching, disorder effects). An unexpected use of simple Fabry–Perot-like cavities built on a line defect arises in connection with assessment of the photonic crystal quality, especially along the “substrate” approach. Such microcavities are very sensitive probes as one expects unity transmission in the perfect case with symmetric mirrors (e.g., each of four rows of PhCs) due to the absence of diffraction for the geometry of interest (Rattier, Benisty, Smith, Béraud, Cassagne, Krauss and Weisbuch [2001]). The peak transmission and to a lesser extent the linewidth are then very sensitive to the PhC losses, and for example, if using the 2D + ε ′′ simulation, to the PhC loss parameter ε′′ . A highly desired application of fully localized modes in microcavities is the realization of a strong Purcell effect. Valid experiments to determine the Purcell factor FP = τ0 /τ in PhC-based cavities seem difficult to achieve, compared to microdisks or micropillars (Gérard, Sermage, Gayral, Legrand, Costard and Thierry-Mieg [1998], Gérard and Gayral [1999], Vahala [2003]). First limited results were reported by Happ, Tartakovskii, Kulakovskii, Reithmaier, Kamp and Forchel [2002]. More recent results really exploit this potential (Badolato, Hennessy, Atatüre, Dreiser, Hu, Petroff and Imamoglu [2005], Englund, Fattal, Waks, Solomon, Zhang, Nakoaka, Arakawa, Yamamoto and Vuckovic [2005], Baba, Sano, Nozaki, Inoshita and Kuroki [2004]) and also reveal more complexities, in particular concerning the detailed behavior of InAs quantum dots.

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Photonic crystal lasers Lasers have played a major role in photonics in relation with confined structures (Zhang, Chu, Wu, Ho, Bi, Tu and Tiberio [1995]), periodic structures, noise (Henry [1986]), etc. The same is true for photonic crystals. The use of PhCs for lasers can be classified in five points: (i) One uses a homogeneous PhC pattern, which entirely overlaps the active region. Then, lasing is found to occur on modes corresponding to low group velocities at the edges of the PBG, the PhC providing distributed feedback at Bragg resonances, like in DFB lasers (see the work of Dowling, Scalora, Bloemer and Bowden [1994] in 1D). The most peculiar property of such lasers is the possible occurrence of lasing in a set of directions equivalent in k-space, e.g., three equivalent directions in a triangular lattice (Monat, Seassal, Letartre, Regreny, Rojo-Romeo, Viktorovitch, Le Vassor d’Yerville, Cassagne, Albert, Jalaguier, Pocas and Aspar [2002], Imada, Noda, Chutinan, Tokuda, Murata and Sasaki [1999], Noda, Imada, Okano, Ogawa, Mochizuki and Chutinan [2002], Noda, Yokoyama, Imada, Chutinan and Mochizuki [2001], Notomi, Suzuki and Tamamura [2001]). Work on organics also took place, sometimes in the “weak photonic strength” regime of classical (1D) DFB structures (Berggren, Dodabalapur, Slusher, Bao, Timko and Nalamasu [1998], Meier, Mekis, Dodabalapur, Timko, Slusher, Joannopoulos and Nalamasu [1999]). Some care should be taken in defining what really is a low group velocity in these investigations (Sakoda, Ohtaka and Ueta [1999]). We signaled for this purpose in Section 2 that apparent low group velocities show up in photonic band structures even in the case of “free photons”, for which ambiguities must be carefully lifted. Conversely, an unambiguous enhancement of optical gain (+40%) was observed in a low-group-velocity region in a waveguide (Schwoob, Benisty, Weisbuch, Cuisin, Derouin, Drisse, Duan, Legouézigou, Legouézigou and Pommereau [2004]). (ii) One uses cavities carved in the 2D PhC, which define localized modes into which the modal gain is largely increased compared to nonlocalized states. In this case, lasing occurs inside the PBG spectral range. PhC waveguides can also be used, even if they are open (Inoue, Sasaki, Ishida, Sugimoto, Ikeda, Tanaka, Ohkouchi, Nakamura and Asakawa [2004]). Note that cavity-type modes also exist at singular points of waveguide such as bends (Inoshita and Baba [2003], Mekis, Fan and Joannopoulos [1998]). (iii) One can also use PhC structures as in-plane end mirrors for lasers, as pioneered by O’Brien, Painter, Lee, Cheng, Yariv and Scherer [1996]. This

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has the remarkable advantage that one can obtain high-reflectivity mirrors, which can then allow ultrashort cavities. Additional advantages are that the mirror is distributed, i.e., less prone to catastrophic optical damage (COD) and to fabrication fluctuations than cleaved edges or homogeneous etched facets. More complex structures using cascaded in-plane cavities, such as coupled-cavity lasers using an extra semi-transparent PhC intermediate mirror, have also been successfully demonstrated, leading to singlelongitudinal-mode operation (as is done for each laser of fig. 23, see references below). In such structures, PhC boundaries can be used to define lateral mode confinement.

Fig. 23. Example of an application of photonic crystals to define two lasers and monolithically integrate them. (a) Scheme of the chip where each laser is composed of two cavities of slightly different length, coupled together; this produces monomode lasing at the coincidence of the two “combs” of cavity resonances; the exit facet of each laser channels this light into a combiner for coupling to a single fiber; tuning of the two currents in the front and rear sections, contacted separately, provides tuneability. (b) Micrograph of the device, where the left shows a zoom on the back mirror and intermediate mirror. Note the use of two different PhC orientations to avoid diffraction effects at mirrors and boundaries.

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(iv) One can define lasers with a more distributed structure, like those using CROW waveguides based on a periodic array of coupled cavities, as in Happ, Markard, Kamp, Forchel and Srinivasan [2001] (see next subsection on waveguides). (v) Another variant is inspired by the so-called α-DFB, familiar to power laser diode practitioners, and is envisioned for application to quantum cascade lasers in the mid-infrared (Bewley, Felix, Vurgaftman, Bartolo, Lindle, Meyer, Lee and Martinelli [2001], Vurgaftman and Meyer [2001]). More advanced laser structures, incorporating several PhC functions, have been produced such as MOPAs (Master-Oscillator Power Amplifiers, see Benisty, Weisbuch, Olivier, Houdré, Ferrini, Leuenberger, Wild, Lombardet, Qiu, Anand, Mulot, Karlsson, Swillo, Jaskorzynska, Agio, Kafesaki, Soukoulis, Talneau, Kamp, Forchel, Moosburger, Happ, Duan, Cuisin, Chandouineau, Drisse, Gaborit, Legouezigou, Legouezigou, Lelarge, Poingt, Pommereau and Thedrez [2004], and Weisbuch, Schwoob, Olivier, Benisty, Talneau, Duan, Krauss, Smith, Houdré, Ferrini and Agio [2004]). Similarly, a monolithic combination of two lasers, each of them being a tunable two-section PhC laser, has been reported, as shown in fig. 23 (Kamp, Happ, Mahnkopf, Duan, Anand and Forchel [2004], Mahnkopf, Arlt, Kamp, Colson, Duan and Forchel [2004], Mahnkopf, Kamp, Forchel, Lelarge, Duan and März [2004]). Comparing the benefits of the various approaches is a delicate task; it would request to weigh all the expected performances in given applications. One should, however, keep in mind that devices injected through deep-etched PhC structures will always suffer from some etching-induced damage of the active region, with a carrier surface recombination rate dependent on the materials used (InP-based materials are much less sensitive than GaAs-based ones). Whenever power is needed, the design will unavoidably require large-area lasing regions. Examining specific results, lasing was reported in one-missing-hole cavities by the Caltech group (Painter, Husain, Scherer, Lee, Kim, O’Brien and Dapkus [2000], Painter, Lee, Scherer, Yariv, O’Brien, Dapkus and Kim [1999], Painter, Vuckovic and Scherer [1999], Painter, Husain, Scherer, O’Brien, Kim and Dapkus [1999]), but the mode involved in the first papers was, for fabrication reasons, a rather extended “acceptor” mode, rather than a properly confined cavity mode. Since then, the team has refined its understanding of these ultimate cavities and has carried out extensive simulations to reach designs with lower threshold (Loncar, Yoshie, Scherer, Gogna and Qiu [2002]) and higher quality factor Q (Vuckovic, Loncar, Mabuchi and Scherer [2002a, 2002b]). Among these designs, the so-called “fractional dislocation” has achieved some success. It breaks the

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hexagonal symmetry along one direction in order to lift the degeneracy of the two degenerate “dipole” modes of the H1 cavity. More recently, a number of cavities based on GaAs on AlOx have also shown lasing action by photopumping at the KAIST Center (Korea) (Hwang, Ryu, Song, Han, Lee and Jang [2000], Ryu, Kim, Park, Hwang, Lee and Kim [2002], Shin, Kim, Hwang, Ryu, Park, Song and Lee [2002]). The same group also introduced another approach to attract the “monopole” mode from the conduction band towards the mid-gap position, namely by diminishing the diameter of the first holes around the cavity (Park, Hwang, Huh, Ryu, Lee and Kim [2001], see fig. 22(c)), a continuous path whose limit is the “H2” cavity. The recent demonstration of an electrically pumped laser diode resonating in this kind of membrane-based quasi-ultimate cavity (Park, Kim, Kwon, Ju, Yang, Baek, Kim and Lee [2004]) suggests that microlasers based on photonic crystals could be interesting devices for physics and applications in the future. This kind of laser probably holds the record “β factor”, the fraction of spontaneous light channeled feeding the lasing mode, with values around β = 0.1. A large Purcell factor is one of the ingredients for this high value. While there is still a long way to attain a so-called thresholdless laser (β → 1) (Lourtioz, Benisty, Berger, Gérard, Maystre and Tchelnokov [2005]), it is nevertheless expected that a laser or LED with a high β factor and a narrow spectrum will offer unprecedented possibilities for quantum optics and the generation of uncommon states of light. Waveguides and waveguide losses This field has been blossoming since the turn of the millennium, marked by a race towards low propagation losses. It is delicate to compare figures that relate to different wavelengths, different guide widths and heterostructures, so that the relevance of a given performance depends on the precise context of integrated optics (see Marcuse [1974] for fibers and channel waveguides in general). Some of the initial papers depicted “spot detection” at cleaved edges by cameras (Baba, Fukaya and Yonekura [1999]), which is not quantitative. Quantitative data appeared in work by Smith, Benisty, Olivier, Rattier, Weisbuch, Krauss, De La Rue, Houdré and Oesterle [2000] and Smith, De La Rue, Rattier, Olivier, Benisty, Weisbuch, Krauss, Houdré and Oesterle [2001], and then many others (Talneau, Le Gouezigou and Bouadma [2001], Sugimoto, Tanaka, Ikeda, Tanaka, Asakawa, Sakaki and Ishida [2004], McNab, Moll and Vlasov [2003]). A commonly accepted naming of the waveguides is “Wn” for a canonical n-missing-rows waveguide (see W1 in fig. 24; Olivier, Rattier, Benisty, Smith, De La Rue, Krauss, Oesterle, Houdré and Weisbuch [2001]), with extra specifications for the crystallographic orientation etc.

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Fig. 24. Photonic crystal waveguides in 2D photonic crystals: (a) the one-missing-row waveguide, nicknamed “W1”; (b) dispersion relation and vertically collected luminescence reflecting the DOS of photon states (see Letartre, Seassal and Grillet [2001]); (c) multiple-missing-row waveguides and mode profiles in the metallic-wall approximation; (d) 2D dispersion relation of a “W3” waveguide (three missing rows) in TE polarization (H polarization); (e) zoom on one of the anticrossings around u = 0.26 [arrow in (d)] between the fundamental mode (larger slope, “fast” mode) and a higher-order mode (smaller slope, a slow, Fabry–Perot-like mode); (f) magnetic field maps at the two points marked “a” and “b” in (e); (g) waveguide bend and electric field map in a PhC consisting of dielectric pillars (work from MIT); note the larger width of the field lobes (half-wavelength) compared to the PhC period; (h) waveguide bend realization in a 3D PhC (Kyoto University, see Chutinan and Noda [2000a]).

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It is worth mentioning also that the nature of modes in straight waveguides was not clarified at once. Although the band structure had been calculated early enough by the supercell method (Benisty [1996]), and “phononic” crystals with similar phenomena had appeared in simulations (Kafesaki, Sigalas and Garcia [2000]), it was not until experiments showed mini stopbands in the transmission of the fundamental mode of a “W3” waveguide that the phenomenon of mode coupling in PhC waveguides was fully clarified. It is well described in the work by Olivier, Rattier, Benisty, Smith, De La Rue, Krauss, Oesterle, Houdré and Weisbuch [2001], Olivier, Benisty, Smith, Rattier, Weisbuch and Krauss [2002], Qiu, Azizi, Karlsson, Swillo and Jaskorzynska [2001]. Actually, any corrugated waveguide (of period a) supporting multiple modes is liable to induce coupling between two of the modes it supports whenever their (1) (2) wavevectors k , k differ by a multiple of 2π/a. What is novel in the case of PhC waveguides is the fact that this coupling never leads to in-plane radiation leakage, as the surrounding crystal is used in its forbidden frequency gap. Hence, energy can be properly manipulated between modes with the help of the corrugation, while ideally, avoiding, energy losses. Dispersion can also be broadly modified (Davanço, Xing, Raring, Hu and Blumenthal [2005]) in relation with gain (Schwoob, Benisty, Weisbuch, Cuisin, Derouin, Drisse, Duan, Legouézigou, Legouézigou and Pommereau [2004]) and this should likely influence lasing (Sugitatsu, Asano and Noda [2004]). At the present stage, a modification of the spontaneous emission (lifetime, Purcell effect and also edge emission) has been demonstrated in W3 waveguides (Viasnoff-Schwoob, Weisbuch, Benisty, Olivier, Houdré and Smith [2005], Viasnoff-Schwoob, Weisbuch, Benisty, Olivier, Varoutsis, Robert-Philip, Houdré and Smith [2005]). Of course, the existence of the third out-of-plane dimension and its related leakage blurs the ideal picture of perfect 1D singularities with, e.g., divergent DOS (Kleppner [1981]), but the extent of this could be limited by clever engineering, using a similar successful approach as that of very high-Q cavities. In general, the lower the losses, the more difficult the measurements are: given the modest length of samples (50–500 µm typically), limited overall losses are incurred in this case. The cutback method familiar to fiber-optics practitioners has to be adapted by fabricating a set of samples of different lengths. Even if the experiment gives access to small losses, the losses incurred by the coupling to the outside (fibers, lenses), which is rarely below 2 dB per coupler (see below the discussion on PhC tapers), may dominate the overall loss. Thus, a lot of care should be taken when assessing losses; for example, more than two lengths should be used for the information to be made sufficiently redundant. Good knowledge of the possible variations (e.g., facet imperfections, be it cleaved or sawn and pol-

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ished facets) is required to assess losses in the sub-dB range. For a 50 dB/cm loss and a length of 100 µm (250 periods of 400 nm), a 0.5 dB drop has to be assessed. For a 5 dB/cm loss (not reached in first attempts, but now a more accessible value) and a length of 500 µm, a 0.25 dB loss has to be assessed. A top view of the scattered light is sometimes used, and here also (Loncar, Nedeljkovic, Doll, Vuckovic, Scherer and Pearsall [2000], Loncar, Nedeljkovic, Pearsall, Vuckovic, Scherer, Kuchinsky and Allan [2002], Lombardet, Ferrini, Dunbar, Houdré, Cuisin, Drisse, Lelarge, Pommereau, Poingt and Duan [2005]), the method is valid only against several checks. Much work has been directed towards demonstrating and exploiting the “lossless”, leakage-free modes which are theoretically expected when their dispersion lies below the light cone (see Section 4.1). The canonical “W1” waveguide offers such a region, albeit in a range much narrower than the large TE-polarized crystal PBG. The dispersion of the guided mode there tends to be very flat, which makes direct coupling difficult. Good, reflection-free coupling between two different waveguides depends on matching the mode profiles and the “impedance” of the modes (as explained by Boscolo, Conti, Midrio and Someda [2002], Sanchis, Bienstman, Luyssaert, Baets and Marti [2004], Sauvan, Lalanne and Hugonin [2005] and Xu, Lee and Yariv [2000]). The group velocity plays a role in these parameters (see also coupled cavities and superprisms). The concerned region of the W1 waveguide has both a low group velocity (measured down to the mode cutoff, and found to be ∼c/100, see Notomi, Yamada, Shinya, Takahashi and Yokoyama [2001]; even higher values were reported at the PECS VI conference by the IBM group and the NTT group) and a special mode profile with more in-plane extent than in the “refractive” part of the dispersion. For the purpose of physical characterization, this difficulty can indeed be circumvented by the internal excitation: Measurements of the scattered photoluminescence of a “closed” waveguide [see fig. 24(b)] directly reveal the underlying band structure in some detail (density of states, quality factors of resonances, etc., see Letartre, Seassal and Grillet [2001]). In view of the intrinsic limitations in the operational range of W1, investigation of alternative geometries with extra holes, larger/smaller holes in the row adjacent to the guide, etc., have been pursued by several teams (see, for instance, Notomi, Shinya, Yamada, Takahashi, Takahashi and Yokohama [2002]) with some success. Conversely, for measurement of PhC waveguides based on external sources, it has been clearly shown in the “substrate” approach that when the guided mode of the PhC waveguide is acceptably matched to that of a standard ridge waveguide, established integrated-optics methods apply successfully. Notably, the measurement of the variation of fringe contrasts between ridge waveguide cleaved facets

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provides rich data on losses of various kinds, either at transitions or distributed along guides (Talneau, Le Gouezigou and Bouadma [2001], Talneau, Lalanne, Agio and Soukoulis [2002], Talneau, Mulot, Anand and Lalanne [2003]). Another novel technique demonstrated by Galli, Belotti, Bajoni, Patrini, Guizzetti, Gerace, Agio, Andreani and Chen [2004] exploits a silicon hemisphere to get signatures of propagative as well as evanescent lossless modes. It generalizes the well-kown prism coupling technique of planar waveguides toward probing nanophotonic structures. A question that remains open is whether coupling can attain the low-loss regime exemplified by silica/doped-silica based integrated optics, a domain where the circuit mode size is nearly matched to classical glass fiber modes (8 µm mode diameter). Adaptation of the mode size between smaller guides (e.g., those of laser diodes, typically 1–2 µm in size) and fibers is often achieved through adiabatic tapers (Xia, Menon and Forrest [2005]): they force the mode to be de-confined and to swell into the cladding by a gradual reduction of the core size. A detailed discussion of PhC tapers (adiabatic, interferometric, . . .) is beyond the scope of this chapter, but it is worth mentioning that the combination of Bloch modes (periodicity) and gradual variations has triggered an exciting and demanding quest. Some clues can be found in work by Talneau, Mulot, Anand and Lalanne [2003], Lalanne and Talneau [2002], Happ, Kamp and Forchel [2001] and Mekis and Joannopoulos [2001], and more theoretical but more general considerations in Johnson, Bienstman, Skorobogatyi, Ibanescu, Lidorikis and Joannopoulos [2002]. Waveguide losses. Waveguide losses for PhC waveguides are currently under intense investigation. Values have been obtained, notably, for λ = 1.55 µm. The position with respect to the light line, as discussed in fig. 16, should be borne in mind for comparison. In summary, losses are presently lowest in waveguides based on membranes for regions below the light line. The polarization of interest is TE-like, although some interesting results in TM have also been measured (Arentoft, Sondergaard, Kristensen, Boltasseva, Thorhauge and Frandsen [2002]). Loss figures in the sub-10 dB/cm have been reported for W1 or near-W1 systems. Above the light line, on the contrary, losses are huge, far exceeding 1000 dB/cm, for these membranes. The best result, until 2004, was that by Notomi, Shinya, Yamada, Takahashi, Takahashi and Yokohama [2002]. They achieved a 60 dB/cm loss value in a “W0.7” waveguide even narrower than W1. Results further presented at PECS V (Kyoto, March 2004, ∼7 dB/cm; see Sugimoto, Tanaka, Ikeda, Tanaka, Asakawa, Sakaki and Ishida [2004]) and PECS VI (Aghia Pelaghia, June 2005) were impressive (around 1 dB/cm). This prompted the theoretical work underlined above

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such as by Hughes, Rammuno, Young and Sipe [2005] (and references therein) or furthering Andreani’s approach (Andreani and Agio [2003]) toward these systems. This is at variance with the substrate approach, where the modes are always above the light line. As hinted in Section 4.2, the losses are then akin, in a 2D top view, to dissipation in air holes having a fictitious imaginary part of the dielectric constant, ε = 1 + iε′′ . This dissipative contribution has a simple scaling with hole depth, notably (Ferrini, Lombardet, Wild, Houdré and Duan [2003]). For perfect infinite holes, this phenomenological parameter ε ′′ reaches an “intrinsic” value, depending only on the heterostructure parameters. Using slightly extended vertical guiding (500–900 nm wide heterostructure core on InP) combined with highquality deep etching, losses in the 60 dB/mm range for W1 are feasible. Lower losses, say tens of dB/cm, can easily be obtained in a wider waveguide such as W3, where the modal overlap with the PhC is much lower (see Kamp, Happ, Mahnkopf, Duan, Anand and Forchel [2004], Kotlyar, Karle, Settler, O’Faolain and Krauss [2004]). Even though very low losses are difficult to achieve, a large freedom in design is recovered, as well as the possibility to integrate PhC structures in combination with real-world electrically pumped optoelectronic devices that require a substrate. How low the losses should be is also unclear because of the simultaneous miniaturization of a whole device, reaching, e.g., the sub-100 µm range. Also, when going to devices, the straight waveguide losses are not the only element – losses at all singular points (bends, coupled cavities, etc.) are equally important. These remarks provide a natural transition towards the consideration of these elements in the next subsection. Bends, splitters, combiners The hope to miniaturize integrated optics components with PhCs after the initial proposal by Meade, Devenyi, Joannopoulos, Alerhand, Smith and Kash [1994] was made more quantitative in the simulations by Mekis, Chen, Kurland, Villeneuve and Joannopoulos [1996] in the MIT group. They suggested that bends in PhCs would work well (no reflection, high transmission) with relatively simple designs. However, the PhCs used in MIT 2D simulations consisted of dielectric pillars rather than air holes [fig. 24(g)], not lending themselves to feasible integrated optics structures. Microwave demonstrations started with Lin, Chow, Hietala, Villeneuve and Joannopoulos [1998] continued by Temelkuran and Özbay [1999] and by Bayindir, Özbay, Temelkuran, Sigalas, Soukoulis, Biswas and Ho [2001]. Applying this concept to air-hole PhCs with missing-row waveguides proved much more difficult. The first results showed a narrow pass band in basic designs

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(Lin, Chow, Johnson and Joannopoulos [2000]). With a more general view, as the guide index is now larger than the average index of the PhC (contrary to air guides of these first models), the guided mode “sees” the wall period as a large period compared to some effective λ/n value in the guide. The guided light, being above the corresponding cut-off, may thus undergo diffraction. In straight sections, this diffraction is frustrated by waveguiding, as the diffracted wavelets cannot generally excite a coherent beam within the discretized modes of the straight waveguide (absence of phase matching). One could say that diffraction is “tamed” in straight waveguides and the modes that result are the Bloch modes (see the discussion above for Bragg diffraction between modes of a straight guide). However, this management of diffraction no longer holds at bends, resulting for example in narrow pass bands (Chow, Lin, Wendt, Johnson and Joannopoulos [2001], Olivier, Benisty, Rattier, Weisbuch, Qiu, Karlsson, Smith, Houdré and Oesterle [2001]). A lot of work has been carried out to avoid performance degradation induced by the possible light leakage channels at a bend (Olivier, Benisty, Weisbuch, Smith, Krauss, Houdré and Oesterle [2002], Benisty, Olivier, Weisbuch, Agio, Kafesaki, Soukoulis, Qiu, Swillo, Karlsson, Jaskorzynska, Talneau, Moosburger, Kamp, Forchel, Ferrini, Houdré and Oesterle [2002]). These channels are: out-of-plane losses (that occur at a bend even if the access straight guides are operated below the light line); strong reflection of the impinging guided mode (Chutinan, Okano and Noda [2002], Chutinan and Noda [2000b]); and when the guide is multimode and its fundamental mode launched on one side, mode conversion, i.e., radiation into the excited waveguide modes. Of course, for a bend in a 3D PhC, although the detail of the geometry may be more involved [fig. 24(h)] (and certainly very delicate to realize), only reflection is an issue for monomode access guides (Chutinan and Noda [2000a]). An apparent trend in 2D is that reducing the mode reflection most often translates into a smoother bend. In one limit, such a bend acts more like a corner mirror. In another limit, a smooth bend called “adiabatic” bend is also obtained by a continuous curvilinear distortion of the straight waveguide. This approach has also been worked out (Borel, Frandsen, Torhauge, Harpoth, Zhuang, Kristensen and Chong [2003], Benisty, Olivier, Weisbuch, Agio, Kafesaki, Soukoulis, Qiu, Swillo, Karlsson, Jaskorzynska, Talneau, Moosburger, Kamp, Forchel, Ferrini, Houdré and Oesterle [2002]). One proposed approach involves impedance matching by a few extra holes before the bend that act in analogy with “stubs” on microwave lines familiar to microwave practitioners (Boscolo, Conti, Midrio and Someda [2002]). It can also be seen as a weak matched cavity (see further). What can be said in general is that the design of a PhC bend with adequate properties is a challenging inverse problem, being actively tackled (Borel, Harpoth, Frandsen,

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Kristensen, Shi, Jensen and Sigmund [2004], Smajic, Hafner and Erni [2003b]). Zeroing radiation into unwanted modes, e.g., reflection, is by no means trivial. To the authors’ eyes, this issue is an inverse problem (finding a structure with given diffraction properties) somewhat reminiscent of the electromagnetic problem addressed when elaborating the “stealth” airplanes that cancel radar echo in crucial directions. As for the work carried out on couplers/splitters, it turns out to be very dependent on the waveguides and bandwidth involved (Mekis, Dodabalapur, Slusher and Joannopoulos [2000], Sugimoto, Tanaka, Ikeda, Yang, Nakamura, Asakawa, Inoue, Maruyama, Miyashita, Ishida and Watanabe [2003]). Excellent achievements by the Japanese FESTA team in the frame of switching in optical networks have resulted in good control of multiple Mach–Zehnder interferometers on membranes operating at around 1.3 µm (Sugimoto, Nakamura, Tanaka, Ikeda, Asakawa and Inoue [2005]). This is a good example as it shows that mastering photonic design of bends and splitters as well as advanced materials and technology opens the road toward real-world devices. To finish this tour, we note that instead of being a penalty, the spectral dependence can be indeed exploited to provide a desired selectivity (Qiu, Mulot, Swillo, Anand, Jaskorzynska, Karlsson, Kamp and Forchel [2003]). This is an alternative to the use of cavities as selective elements, discussed below. Coupled cavities, cavity-waveguide coupling Cascading or coupling the above elements – cavities, waveguides and bends – appears as a natural step in elaborating photonic circuits able to treat optical signals in a very compact way. A string of coupled cavities can be treated in a simple way analogous to the tight-binding approximation of textbooks on solid-state physics. The electromagnetic field (Bloch states) is written as a linear combination of the individual cavity elements. The dispersion relation of such a system consists of minibands that span frequencies around the isolated cavity frequencies, of the form ω = ω0 + ω cos(kx). The central frequency ω0 is only slightly shifted from the isolated cavity value, while ω is dictated by coupling strength. Demonstrations of the power of this concept proposed by Stefanou and Modinos [1998] and more explicitly by Yariv, Xu, Lee and Scherer [1999] have first been made in microwave in 3D (Bayindir, Özbay, Temelkuran, Sigalas, Soukoulis, Biswas and Ho [2001], Bayindir, Temelkuran and Özbay [2000]) and next in 2D PhCs on Ga(Al)As guides (Olivier, Smith, Rattier, Benisty, Weisbuch, Krauss, Houdré and Oesterle [2001], Olivier, Smith, Benisty, Weisbuch, Krauss, Houdré and Oesterle [2002]); see figs. 25(a,b). One obvious field of application is that of “slow waves”

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with a low group velocity which is a direct consequence of a low ω, even far from miniband edges (see first attempts in Karle, Chai, Morgan, White and Krauss [2004]). Another envisioned application is the possibility to bend, with little penalty, such strings of cavities, for instance at 60◦ if their original mode has six-fold symmetry. Owing to the resonant nature of each cavity transmission, the light loses memory of its previous direction and will exit naturally through the next thinner cavity wall. Coupled cavities and waveguides in photonic crystals can be implemented either in an “axial” fashion, whereby the situation is analogous to inserting a Fabry–Perot resonant cavity onto the light path of an ordinary beam, or in a lateral fashion, in which case only frequencies close to the resonant cavity frequency are likely to interact, all the others going through the guide undisturbed. In “axial” coupling, mode matching is a critical issue, as different mode profiles will certainly lead to reflection and poor transmission (Moosburger, Kamp, Forchel, Oesterle and Houdré [2002]). Poor matching also favors indirectly but efficiently the out-of-plane radiation loss channel by unbalancing the underlying competition (Olivier, Smith, Benisty, Weisbuch, Krauss, Houdré and Oesterle [2002], Mulot, Qiu, Swillo, Jaskorzynska, Talneau and Anand [2003]). The issue of “vertical” mode matching between the guide section that often forms the cavity and the structured guide section that forms the reflector has also received attention: Lalanne and Hugonin [2003] suggested that short tapered transitions could be of foremost importance in this matter. A continuation may be found in the work by Sauvan, Lalanne and Hugonin [2005]. In the horizontal directions, the coupling of “Wn” waveguides to various cavity/defect shapes has been addressed in the works mentioned above, but a general conclusion cannot easily be drawn, mainly due to the delicate issue of accommodating the crystal lattice constraints. In the case of lateral coupling, energy goes to the cavity and is shared between out-of-plane losses and a back flow to the waveguide, which translates into reflection and diminishes transmission (it is a coherent process). There is no directivity of the backflow, a topic that will be addressed later. It is important to highlight here that it is in this side-coupling geometry that the team of S. Noda demonstrated very-high-Q cavities recently (Q ≈ 45,000, Akahane, Asano, Song and Noda [2003a]; Q ≈ 600,000, Song, Noda, Asano and Akahane [2005]; this last result showing that near-ideal matching of mirror modes to the cavity mode is much successful, see fig. 25). With a good knowledge of the coupling (a delicate issue, the guide being for instance bounded by reflectors formed at the facets as depicted in Fan [2002]), the relationship between the measured “loaded Q” (with energy backflow into the waveguide) and the unloaded isolated cavity Q can be

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Fig. 25. (a) Principle of the coupled-cavity waveguide in a photonic crystal, with a string of nine “H2” cavities. A light source at the input is shown. (b) Measured transmission and dispersion relation, with the fundamental-like mode shown by full circles, and folding twice at zone edges (see Olivier, Rattier, Benisty, Smith, De La Rue, Krauss, Oesterle, Houdré and Weisbuch [2001]). (c) Side coupling between a large “H7” cavity and a W3 waveguide. The top spectra are for thicknesses of the coupling wall of 4 and 2 rows, the bottom spectrum is the W3 guide transmission with a mini-stopband. (d) Sketch of the particular mechanism used for this coupling, akin to the mini-stopband coupling between the fundamental mode and the fifth mode of the W3 waveguide (Smith, De La Rue, Rattier, Olivier, Benisty, Weisbuch, Krauss, Houdré and Oesterle [2001]). (e) Side coupling from a W1 waveguide to a cavity (point defect here), and resulting spectrum of light collected out-of-plane. (f) Side coupling to a “donor”-type cavity, based on three missing holes, exhibiting a very high Q for adequate termination modifications (“hole shifting”, see Akahane, Asano, Song and Noda [2003a, 2003b]).

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assessed. Anyway, the large progress since the first demonstrations of all-PhC cavity-guide coupling by Noda, Chutinan and Imada [2000] or by Smith, De La Rue, Rattier, Olivier, Benisty, Weisbuch, Krauss, Houdré and Oesterle [2001] and by Seassal, Désières, Letartre, Grillet, Rojo-Romeo, Viktorovitch and Benyattou [2002] is impressive. However, the main thrust is to selectively drop or add a wavelength channel to those circulating in the guide, in the framework of the now famous WDM transmission scheme of optical networks. This actually requires some selective coupling element to sit between two guides. The mini-stopband makes use of the higher-order mode of a multimode waveguide to play this frequency-selection role in the theoretical proposal of Olivier, Weisbuch and Benisty [2003], and in the more experimental approach of Viasnoff-Schwoob, Weisbuch, Benisty, Cuisin, Derouin, Drisse, Duan, Legouézigou, Legouézigou, Pommereau, Golka, Heidrich, Hensel and Janiak [2005]. Rules for directivity of the extracted/inserted signals require elaborate symmetry considerations, which we shall not deal with in detail (Villeneuve, Fan and Joannopoulos [1996], Fan, Villeneuve, Joannopoulos and Haus [1998], Fan [2002]). Briefly, coupling by two somewhat separated and identical cavities leads, for proper spacing, to a pattern of interference that is destructive in one direction and constructive in the other. The scheme used in conventional integrated optics for such a function relies on “whispering-gallery modes” of rings or micro-rings which are side-coupled to waveguides, whereby the two modes of the two cavities alluded to above now transform into the two contrapropagating modes of the ring. This specialized topic is intriguing and also puts a high demand on technology to reach tiny couplings and high Q in a reproducible manner. Results from leading Japanese teams can be found in the work of Akahane, Asano, Song and Noda [2003a, 2003b], Notomi, Shinya, Mitsugi, Kira, Kuramochi and Tanabe [2005] and Notomi, Shinya, Mitsugi, Kuramochi and Ryu [2004]. Superprisms, supercollimators One fascinating concept recently put forward by Japanese teams is that of the “superprism”. Due to the highly singular nature of the dispersion relation of Bloch waves near critical points, and particularly of the equal-frequency contours in reciprocal space near those points, the fate of a plane wave impinging onto a PhC, e.g., its angle of propagation in the crystal or its further refraction at another tilted interface of a prism-shaped crystal (see fig. 26), can vary by huge amounts for small variations of its angle or frequency. Actually, early findings in this field can be found in Russell’s work (as reported in Russell, Atkin, Birks and Roberts

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Fig. 26. Use of equifrequency surfaces for superprism and supercollimator effects: (a) generic equifrequency contours in a square Brillouin zone, according to the acoustic phonon-type dispersion relation indicated, note the superprism wavevector input (left) and the supercollimator beams input (bottom); (b) superprism effect: zoom on the rapidly evolving group-velocity direction near zone edge; (c) use of the phenomenon to separate frequencies; (d) generic layout for experiments; (e) pseudo-guiding, or apparent reduced diffraction, in the supercollimator effect and the possibility of guiding crossing beams in such a macroscopically homogeneous PhC piece.

[1996], Russell and Birks [1996] and Russell, Birks and Lloyd-Lucas [1995]) and a few related ones in the 1980s. However, the thrust for miniaturizing wavelength multiplexers and demultiplexers for telecom WDM deployment has attracted attention on this novel principle. Most of the fundamentals lie in the conservation of the k-vector component parallel to the interface between two media. This fact already had some importance for the simple diffraction properties of photonic crystals (Labilloy, Benisty, Weisbuch, Krauss, Cassagne, Jouanin, Houdré, Oesterle and Bardinal [1999]).

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It should be borne in mind that these media have nontrivial dispersion relations (Gralak, Enoch and Tayeb [2000]) and thus nontrivial group-velocity dispersion. Indeed, the nontrivial dispersion arises from a structure as simple as a one-dimensional periodic stack (see fig. 3), but nevertheless, our example will be in 2D. The parallel wavevector conservation condition, together with the conservation of frequency, limits the Bloch waves in the illuminated medium to a few, progressive and evanescent. Each of these Bloch waves may have (infinitely) many Fourier harmonics, but with a unique relation between them. To grasp the basic effects, the most useful tool is the wavevector diagram, which is the locus of allowed k’s at constant frequency, discussed earlier (figs. 3 and 10). Normal to this diagram is the group velocity ∇k (ω) of the wave. Once a modematching condition has been found, locating the mode in a diagram extended to adjacent Brillouin zones may be useful (Notomi [2000]). Although this diagram is formally the same in every replica of the Brillouin zone, it is a useful guide to identify where such a mode “really” is (see Section 2.6): (i) what would be its location without the folding, or in other words, (ii) which is the reciprocal vector G, within the set {k + G} of the wavevectors in its Fourier transform, that carries the largest amplitude AG . This sheds much light on the subsequent behavior of this mode (Lombardet, Dunbar, Ferrini and Houdré [2005]). Physics and devices attempt to exploit the special phenomena expected near the critical points of band dispersions. At these points, the topology of constantfrequency contours varies singularly for infinitesimal changes in frequency. The term “superprisms” and “supercollimator” have been coined to designate structures exploiting this phenomenon since the two pioneering papers of the group of Kawakami in Sendai (Kosaka, Kawashima, Tomita, Notomi, Tamamura, Sato and Kawakami [1999], Kosaka, Notomi, Tamamura, Sato and Kawakami [1998]). Their demonstration was based on “autocloning” PhC (Section 3.2). They evidenced how the important changes in group velocity or wavevector could be put to good use, especially to separate different frequencies. Of course, if such a device is used with a finite beam, this implies some k-dispersion and limits the exploitation of these singularities. As recently explained by Baba and Matsumoto [2002] and Baba and Nakamura [2002], there are actually two ways to exploit the superprism effect. The first is based on differences in Poynting vector direction vs. frequency, which leads to walk-off and separation between two beams of two different frequencies. The other is the “wavevector superprism”: the two beams now have similar Poynting vector directions and little walk-off but a widely different k, therefore a distinct phase portrait. When exiting the PhC through a face cut at a different angle from

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the input face, the two beams emerge at almost the same location, but they will propagate at different angles in the uniform surrounding medium. From this point on, a lens is needed to focus different frequencies onto different detectors in real space. A converse effect (see fig. 26) experimentally pioneered by the same team is the supercollimator effect. Upon exciting a bunch of Bloch modes around a point for which isofrequency contours have no curvature (inflexion points), the bunch propagates in the same direction. This is particularly striking when applied to a spatially confined beam (a beam focused at the entrance of the proper PhC), which necessarily possesses several k-components (reciprocal space/real space). Such a confined beam subsequently propagates with apparently no diffraction for a long distance (Prather, Shi, Pustai, Chen, Venkataraman, Sharkawy, Schneider and Murakowski [2004], Pustai, Shi, Chen, Sharkawy and Prather [2004]) without the special spatial requirements of the Bessel beams of classical diffractive optics, obtained from classical optical beams using so-called axicons. However, this apparent suppression of diffraction is only obtained in a narrow bandwidth. Nevertheless, there are fascinating possibilities of beam crossings in special photonic circuits based on PhCs operating in this regime, and these perspectives have triggered the interest of several groups (Wu, Mazilu, Gallet and Krauss [2003], Lupu, Cassan, Laval, El Melhaoui, Lyan and Fideli [2004], Smajic, Hafner and Erni [2003a]). LEDs Light emitting diodes have changed status in the recent years, going from simple signaling tasks (“power on”) towards a more ubiquitous role as a generalpurpose source for lighting, optical communication, medicine, etc. So-called high-brightness LEDs constitute the most important industrial success of the opto-III–V semiconductor industry, especially since the advent of GaN-based blue-green-white LEDs. These latter notably would present huge opportunities for energy saving in general lighting if light extraction in excess of 50% could be achieved at low cost. The authors have been involved in the topic of microcavity LEDs, an elegant way to extract more light and increase brightness. By placing the active region in a planar microcavity, interferences “shape” the angular emission diagram into lobes that concentrate the optical power (Benisty, De Neve and Weisbuch [1998a, 1998b]. One such lobe, if properly tuned, may extract 10 times more light compared to an unstructured system. The fundamental mechanism was described in a very simple fashion by Kastler [1962] addressing the issue of spontaneous emission of atoms placed inside a Fabry–Perot cavity.

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It is natural, today, to think of photonic crystals for gaining still more extraction thanks to the stronger confinement. Let us indicate some avenues. • PhCs can act to supplement microcavity LEDs. These can at best directly extract 25–40%, depending on structure and material (Benisty, De Neve and Weisbuch [1998a, 1998b], Delbeke, Bockstaele, Bienstman, Baets and Benisty [2002], Weisbuch, David, Fujii, Schwach, DenBaars, Nakamura, Rattier, Benisty, Houdré, Stanley, Carlin, Krauss and Smith [2004]). The light still missing is however recoverable. It mainly consists of guided modes (Stringfellow and Craford [1997], De Neve, Blondelle, Van Daele, Demeester and Baets [1997], usually dissipated (absorbed . . .). Photonic crystals can then be used to extract these modes (Rattier, Benisty, Stanley, Carlin, Houdré, Oesterle, Smith, Weisbuch and Krauss [2002], Rattier, Krauss, Carlin, Stanley, Oesterle, Houdré, Smith, De La Rue, Benisty and Weisbuch [2002]), as discussed below. Texturing the surface with a virtually random pattern is one way to achieve part of this task without relying on periodicity (Schnitzer, Yablonovitch, Caneau, Gmitter and Scherer [1993], Windish, Heremans, Knobloch, Kiesel, Döhler, Dutta and Borghs [1999]), however at the cost of a large extraction length. • Radiation inhibition inside the solid is the basic mechanism that can be devised around the original PhC bandgap concept. It was calculated by Fan, Villeneuve, Joannopoulos and Schubert [1997] in a model case, suggesting almost perfect extraction efficiency for emission from inside a PhC slab, in that case due to mode losses out-of-plane. Experimentalists had to face many difficulties (Baba, Inoshita, Tanaka, Yonekura, Ariga, Matsunami, Miyamoto, Koyama and Iga [1999]). Only recently were laboratory demonstrations reported along this line (Fujita, Takahashi, Tanaka, Asano and Noda [2005]). It remains to be seen whether a room-temperature real-life device can be based on a suppressed radiative transition, thus diminishing the internal quantum efficiency. • Encouraging first experiments in rather another direction were reported by Boroditsky, Krauss, Coccioli, Vrijen, Bhat and Yablonovitch [1999] and Boroditsky, Vrijen, Krauss, Coccioli, Bhat and Yablonovitch [1999]. The aim was to first have light coupled to a slab-type lateral waveguide, and then feed a photonic crystal membrane. In the latter, one uses the capability of some modes to efficiently scatter into air. This is reminiscent of the “leaky mode” issue in a photonic crystal slab (Section 4.2). • It is important in quantitative experiments to have a proper reference, where the excitation of carriers is unambiguously quantifiable. The absorbed fraction in photoexcitation can be greatly modified by the photonic environment. Many

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reports are plagued by ambiguities in this respect (Erchak, Ripin, Fan, Rakich, Joannopoulos, Ippen, Petrich and Kolodziejski [2001]). • The extraction by a grating for light with extended angular and frequency range is not an easy task (compare to simpler solutions for guided laser light in Hagberg, Eriksson and Larsson [1995], based on blazed outcoupling gratings). One attempt to capture all angles was proposed in Rattier, Benisty, Schwoob, Weisbuch, Krauss, Smith, Houdré and Oesterle [2003], by means of “orientationally disordered” PhC, based on a tiling called the “Archimedean” tiling, proposed initially to improve bandgaps (David, Chelnokov and Lourtioz [2000]), in the spirit of quasicrystals (Zoorob, Charlton, Parker, Baumberg and Netti [2000]). Whatever the in-plane geometry, the depth of the grating is both a photonic and a material dependent issue. Shallow gratings not penetrating the active layers do not spoil the radiative recombination efficiency, but mitigate the extraction performance. • Plasmons may help, but care should be taken of losses they induce (Vuckovic, Loncar and Scherer [2000], Barnes, Björk, Gérard, Jonsson, Wasey, Worthing and Zwiller [2002], Barnes [1999]). • Ultimate cavities with strong confinement of photons in all three directions pose different challenges. Ideally, the strong Purcell effect should channel carriers in the associated radiative mode. However, sidewall nonradiative recombination is a first generic challenge that has often spoiled photonic improvements, all the more at room temperature, where even quantum dot capture is insufficient. Nevertheless, the recent report by Baba, Sano, Nozaki, Inoshita and Kuroki [2004] shows convincing data at room temperature. It also points out an additional caveat: inhomogeneous carrier distribution, in real space and in energy. Finally the far field emission pattern also matters. In the present status of PhC microcavities, the emission diagram is much less favourable to collection than that of micropillars. Undoubtedly, both systems are avenues for single photon sources. • For the GaN-based device structures of foremost importance for lighting applications, the situation is complex as thick layers may carry the guided light in multiple modes not easily outcoupled by the PhC (David, Meier, Sharma, Diana, DenBaars, Hu, Nakamura, Weisbuch and Benisty [2005]). Overall, the application of PhCs concepts to light extraction for high-efficiency LEDs is by no means a simple challenge. So far, no convincing report of high extraction efficiency extraction in a LED implementation exists. However, there certainly is potential in PhCs to reach the performance (in terms of lumens/watt and cost) expected for sources in solid state lighting.

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§ 5. Photonic crystal ﬁbers Issues of fibre world Almost everybody in optics holds a tribute to the beautiful invention of the optical fiber. Demands on fibers mainly stem from the requirements of extreme powers and extreme wavelengths for some applications, whereas low losses were the leading requirement for telecom applications. As far as physics was concerned, once the low-loss issue was properly addressed by good silica-based fibers for operation around 1550 nm, telecom applications next met limitations related to nonlinearities. This phenomenon induces various penalties to routed signals, against which a multitude of strategies have been devised in the laboratory. The best strategies are somewhat complex. The present WDM format, synonymous with very high data rates (tens of Terabit/s), has some distance to go before fully exploiting the huge available fiber bandwidth (∼50 THz). Erbium amplifiers and more recently Raman amplifiers have added new requirements to dispersion when attempting to exploit their full advantage. Furthermore, there are various practical issues (deemed as unavoidable in usual fiber formats), such as bending penalties, polarization mode dispersion, etc., that can be revisited for new fiber concepts. With this in mind, let us look at the novelty brought by the so-called “photonic crystal fibers” (PCFs). This term is applied here to any fiber having a non-trivial cladding-core structure, with some regularity, be it only “azimuthal” regularity, and assuming that voids (air) participate in this cladding structure. The reader may find most references in two recent reviews by Knight [2003] and Russell [2003]. For the theory of classical fibers and dielectric channel waveguides, see Marcuse [1974].

The playing field of photonic crystal fibers In order to lift the simplest part of the mystery for the reader, we briefly mention fabrication of PCFs which usually involves two stretching steps carried out in drawing towers with carefully controlled ovens (∼1800◦ C). The first step converts a macroscopic assembly of glass capillaries or rods of a few tens of cm long to a “preform”, of 1 meter length or so, having a correspondingly contracted hole pattern. At this point, the quality of a first stage of “coalescence” of capillaries can be checked so that only the desired set of voids remains. The second step produces the final diameter, a few hundred microns, and a “usual” length (meters to kilometers). For some tests such as dispersion, a few cm are enough. The

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Fig. 27. Evolution of basic quantities between the in-plane bandgap and the case of photonic crystal fibers: (a) position of related waves in k-space, on the sphere of radius k = nω/c; (b) real-space configuration for the in-plane bandgap; (c) real-space configuration for the photonic crystal fiber bandgap.

homogeneity has been tremendously improved by the teams and small companies producing these fibers. The interest around PCF initially arose because they could offer guidance based on a clever generalization of the initial photonic bandgap concept. It has blossomed in other directions since then. It is seen in the illustration below that 2D photonic crystals were devised for operation with a small or zero momentum in the invariant direction [figs. 27(a,b)]. To get guidance in a silica based system, this was of no interest. Rather, in a fiber, one may look at what happens for a large momentum β = kz along the invariant direction [figs. 27(a,c), see also fig. 3, where general off-plane propagation was introduced]. Actually, there is no difficulty in imposing a β = kz = 0 value, and carrying out a mode calculation relating the frequency ω/2π to the in-plane wavevector k , as was done above, scanning a circuit along the symmetry axis of the irreducible first Brillouin zone. Then, one is looking for a bandgap that is “shifted” in k-space, covering an annulus near the “poles” of the sphere rather than around the equator [fig. 27(a)]. A marvelous effect of this shift is that the underlying Fresnel reflections now correspond to large angles of incidence. So, even for the modest index contrast of air and silica (n ≈ 1.5), a strong interaction arises, and the possibility exists (Birks, Roberts, Russell, Atkin and Shepherd [1995]) to obtain an omnidirectional gap, i.e., a gap for all directions of k . This is illustrated in a typical calculation in fig. 28.

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Fig. 28. Band structure of a typical silica/air fiber (triangular lattice of air holes – filling fraction 34%), for a given value of the propagation constant along the fiber (the βΛ product is 10.06 here). X and J are alternate names for the M and K points of the first Brillouin zone. In addition to the absence of modes below the normalized frequency value ωa/c = 7.4 (a value related to the effective index of the structured silica for propagation purely along the hole axis), note the existence of a narrow gap around ωa/c = 9.

What happens in the bandgap is that light propagation for the fixed β = kz value is impossible in a range of frequencies. Then, if a defect is introduced such that the mode localized by this defect reaches the same β = kz value, this is a guided mode, because it cannot escape the defect, at least not through an infinite crystal around. As one operates with large Fresnel reflections, in practice 4–10 rows suffice as was the case for semiconductor PhC. Therefore, the essential parameters defining a PCF of this kind are the geometry of the lattice (triangular, honeycomb, . . .), its pitch Λ (used in the PCF field rather than the symbol a used for the pitch in planar PhCs), its hole diameter d, and the defect size, e.g., a single missing or added hole of the same diameter. The difference with the in-plane case appears again through the periods Λ; these are typically on the order of 1–3 µm for near-infrared light, instead of λ/2n ≈ 0.5 µm for an in-plane first-order gap. This is related to the smaller transverse k (denoted k⊥ ) due to operation near the “pole” of the sphere k = nω/c in k-space. Note also that the gap is relatively tiny, showing that there is not much room in targeting such a bandgap.

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Fig. 29. The playing field of photonic crystal fibers: the plot has the air-filling fraction as the abscissa and the modal index as the ordinate, with the silica line (an obvious upper bound) and the air line n = 1 (not a lower bound, think of hollow waveguides in general). The typical appearance of fibers of interest is shown, with their familiar name.

As for the required defect, that is the fiber core, all possibilities are available in principle. It could be air, or silica, or a different lattice, or a different material (a fluid). It could have some privileged direction in its shape to act just as a polarization maintaining fiber. Let us look at the symmetric, air/silica PCFs only (fig. 29). Convenient parameters are the air-filling fraction and the modal index (that is the ratio βc/ω ≡ kz c/ω for the lowest frequency of the plot in fig. 28). The latter tells how the fundamental mode samples the defect and the surrounding photonic crystal cladding. The following paragraphs thus briefly comment on each of the limits. The top left figure corresponds to a high-index core, so that the guiding mechanism need not be the bandgap. The bottom left figure is the pioneer one, with a core effective index lower than that of the cladding and a honeycomb lattice around the core. The bottom right figure is the “air-guiding” fiber, relying very clearly on the bandgap effect at the first look.

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The top right figure, the “high-” fiber, shows one more possibility not obvious in the basic PCF concept. The gray area of “allowed modes” is obtained assuming a cladding medium with an adequate effective index. To cover the wide range of parameters shown, this effective index must be computed numerically by evaluating the propagation constant β of the “fundamental” cladding mode and defining this effective index as neff = β/(ω/c). Such computations can be done by using approximate analytical methods or by exact calculations using for instance the plane-wave method, f being the air-filling factor. Let us discuss them in sequence, before concluding by a complement to Section 2.7 on theoretical calculation methods for PCFs. Single-mode fiber The kind of PCF discussed here has a bulk silica core of index nsilica . Considering the cladding (air-filling factor f ) as an effective lower-index medium (neffective < nsilica ) the situation clearly suggests the existence of index guiding rather than Bragg guiding. This is actually the case. But, at variance with a standard fiber, monomode guiding can be guaranteed through a virtually infinite frequency interval. In a standard core/cladding fiber it is well known that at high frequencies, the onset of multimode behavior arises when the “V number” familiar to practitioners in the field exceeds 2.405. This V number is given by V = V = (πD/λ) n2core − n2cladding , where D is the diameter. The square root term describes how “deep” the potential well is for the guided mode profile. The simplest explanation for the very wideband monomode behavior is that the average index seen by the guided-mode profile depends on the frequency, and increases towards high frequencies to reduce the “well depth” for the guidedmode profile. This is because the guided-mode profile in the “cladding” displays a mixed decaying and oscillating behavior in the periodic lattice. At very low frequencies, however, the modal index is about the average index of the cladding, and the mode tails are slowly decaying like in a classical fiber. Thus, they cannot carry large k⊥ components and they sample the cladding homogeneously. The behavior is fully similar to that of a large-index-contrast fiber, n ≈ 0.1 for a 20% air filling factor. As the frequency increases, the mode usually tends to be squeezed in the core. This trend is much moderated here because of the increased ability of the mode profile to mold into the air/silica pattern. While not being a full standing wave pattern reminiscent of fig. 2, the mode profile in the cladding may partly avoid air holes. This raises the equivalent cladding index neffective towards nsilica , as

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the wave tends to be localized in the silica, and decreases the risk to become multimode (limiting the V number). This trend continues endlessly towards the “blue” regions of the spectrum, being limited only by the silica intrinsic losses and the increased Rayleigh-type scattering due to residual roughness. In other words, coming back to fig. 27, even if the confinement stems from some effective index, the reflection comes from a lattice whose response is peaked in terms of wavevector, and can be said to act as a “modal sieve”. This is the key to this “endlessly monomode” behavior (Birks, Knight and Russell [1997]). Considering applications, avenues are thus opened towards incoherent light– light interaction between widely separated frequencies, as such an interaction is maximized when both frequencies are carried in similar guided modes. One simple application is to retain the monomode operation for a very-largecore fiber without the delicate minute index steps that would be needed in classical fibers. High power transport, e.g. for CO2 laser machining, could thus be easier. Dispersion is dealt with below for high- fibers. For the present “single-mode” fiber, design of dispersion-flattened fibers has been developed over the last few years. However, losses are still too large to deploy such fibers in idealized highbit-rate, dispersion-free optical networks. Dispersion compensation fibers, having some 10–100 times the dispersion of silica fibers, are also being designed and investigated. The issue of birefringence is not dealt with in detail here, but it goes without saying in this particular example that any core that privileges one transverse direction can yield huge birefringence values that can be exploited for various purposes (Ortigosa-Blanch, Knight, Wadsworth, Arriaga, Mangan, Birks and Russell [2000]).

True PC guidance The “holy grail” of full PhC (i.e., photonic bandgap) guidance was first achieved in a system with 95% silica and 5% air. A honeycomb lattice was adopted for the cladding, with an extra hole in the center of the hexagon as a defect (fig. 29, bottom left) (Knight, Broeng, Birks and Russell [1998], Cregan, Mangan, Knight, Birks, Russell, Roberts and Allan [1999]). It is not straightforward to provide “gap maps” akin to fig. 10(e) to explain this result. This is because kz is now a third parameter. As a matter of fact, one common way to present a “gap map” among PCF specialists is to fix the air-filling factor f (say 45%), and plot the frequency gaps (those of fig. 28, using the dimensionless ordinate ωΛ/c) as a function of kz Λ = bΛ. One variant (see fig. 29) is to look at the inverse of slopes of this

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ω(β) diagram, which have the meaning of refractive (phase) indices, and thus take βΛ/(ωΛ/c) = βc/ω as the ordinate. If one turns to solutions with much less silica and much more air, a guided mode relying on the bandgap effect may occur with a modal index close to unity or even smaller. The bandgaps in question are the thin dark regions drawn inside the continuum of allowed modes. With a honeycomb lattice, the situation resembles more thin strands of silica held by still thinner veins, as illustrated in fig. 29 and modeled in fig. 30 in one particular case. The presence of the strands could be seen as a possible channel for light, with an index high enough to capture any light in the vicinity. However, this is not the case, as long as the strands remain reasonably thin. One reminiscent situation is that of “ARROW” waveguides, where guiding in a low-index region is obtained by a cladding of thin layers of high index, but working in such a fashion that reflection is enhanced (Duguay, Kokubun, Koch and Pfeiffer [1986]).

Fig. 30. The density of states of the “cobweb fiber” (structure shown in the inset) is displayed as variable gray levels in the plane of propagation constant (abscissa) and effective modal index (ordinate); the gap is the light region, and goes below unity modal index. The defect mode is shown as a gray solid line with dashed ends, lying just below the air modal index in this gap.

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Figure 30 shows various data in the βΛ–βc/ω plane (see caption). The main data is the density of states which is zero above the effective index of the cladding. Note also zeroes in the gap that forms and that goes below the n = 1 light line. The line crossing this gap is a guided mode. Such a dispersion relation is essentially that of a “hard wall” waveguide, meaning that the mode is tightly confined in such a “large” defect. This defect is still a good compromise: it allows coupling with a “normal” NA of 0.17, and the higher-order modes it supports are much more lossy, so that after a few meters only the fundamental mode is left. Applications of such fibers are very promising. First, one could think of getting rid of nonlinearity up to unprecedented bounds. One can also dream of losses even lower than those of silica fiber, given the minute overlap with air. This would even mean that the wavelength of minimum attenuation, a compromise between Rayleigh scattering of silica (short wavelength) and onset of Si–O absorption harmonics (long wavelengths) would red-shift because of the different scaling of the two phenomena. In actual experiments, it is also possible to inject fluids in such a large hollow core, and observe the corresponding interaction with light. Gases showing large Raman scattering cross-sections (hydrogen) or fluid with large nonlinearities (CS2 ) would offer a unique behavior with a guided beam in the core in confinement conditions not available otherwise. Channeling solid particles along the fiber by optical trapping is yet another possibility. High-Delta fiber A high- fiber (fig. 29, top right) is very much like a silica strand. It is however held by thin veins. The term “cobweb fiber” is sometimes used. When kilometers of such fibers are drawn these veins are even more extraordinary than the core. They retain their 100 nm thickness over a kilometer, which makes them the most anisotropic man-made object at this scale. Guiding relies here on the silica/air index step with no optical role of the surrounding medium. One obvious interest of such a system is its highly nonlinear behavior. It arises at powers much below those of classical fibers due to the tighter air confinement. We describe in the next paragraph the specific dispersion properties of such fibers. Special dispersion We mentioned earlier the possibility to engineer the dispersion in “single-mode” fibers. Dispersion relates to the second derivative of ω(β). It describes group delay and its frequency variations, while the first derivative essentially describes

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group velocity. The acronym “GVD” is used here for “Group Velocity Dispersion”, the definition is D ≈ (λ/c)(∂ 2 n/∂λ2 ), and the usual unit is ps/nm/km. A reference value at 1550 nm is +13 ps/nm/km (a positive “anomalous” value since it is on the low-frequency side of the zero-GVD frequency). The dispersion has two additive origins in fibers: (i) the intrinsic one of silica, which goes through zero around 1300 nm, and (ii) the modal dispersion. Only for “exotic” radial profiles can a large shift of the zero-GVD frequency arise with the doped silica/silica system. The physics behind obtaining a modified GVD is complex in “single-mode” PCF because the waveguide dispersion is now larger and complex. It is simpler to describe the main trends in high- fibers. The much larger waveguide modal dispersion is due to the much larger index step. Figure 31 illustrates the dispersion of a classical silica/doped silica fiber compared to that of a mere silica strand. A twenty-fold increase in waveguide dispersion can readily be seen with values reaching some about −200 ps/nm/km. It is easy to imagine that such huge waveguide dispersion makes it possible to design the zero-GVD at any desired frequency. If weak material dispersions are to be compensated, it can already be said from fig. 31 that one possible zeroGVD working point will be close to the ratio λ/r ≈ 2.8 (zero-GVD of the strand. Note that this is not the zero-GVD point used in most nonlinear experiments, but the so-called “second” zero-GVD point.). Hence, provided extremely small radii can safely be obtained (<1 µm), zero-GVD fibers can be fabricated in the visible down to 550 nm.

Fig. 31. Typical group-velocity dispersion in a high- fiber, compared to that of a typical conventional step-index fiber (index step around 0.01).

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Special applications: nonlinear optics, supercontinuum generation, etc. We devote most of the end of this section to selected nonlinear optics application in photonic crystal fibers. It will be noted by the reader that, in this chapter, no account is given of the various studies investigating the nonlinear behavior of “ordinary” PhCs, especially 2D slab PhCs. Although some common phenomena may be sought, such as phase matching in sum and difference frequency generation, the main difference lies in the maturity with respect to applications. PCFs are almost “ready-to-use” optical components, and for this reason are already commercially available through some start-up companies. Let us propose some reasons why this is so: Small series are of direct interest to users; their specifications can be tightly defined (though not obeyed fully deterministically from design yet); light coupling (NA) and connectors are within standards; manipulation is straightforward; no cooler is needed even for watts injected; etc. No such statement holds for nonlinear “slab” PhC devices as yet. The strength of the nonlinear optical behavior of PCFs can be exemplified by the phenomenon of soliton frequency downshift. It is well known that a soliton forms as a stable solution in a fiber as a result of two compensated effects: Kerr nonlinearity (increase of index at high intensity) and waveguide dispersion. The former induces a self-phase modulation inside the pulse and globally slows it down, allowing the low-intensity front to travel faster. The red frequency components are ahead, in the region of smallest index shift, where phase fronts are most separated (see, for example, the book by Agrawal [2001]). Conversely, anomalous waveguide dispersion (the negative GVD) causes the blue part of a pulse to advance on its red components. At some “magic” power, the two effects cancel each other, and a stable soliton appears. In experiments conducted on PCFs at 850 nm (Reid, Cormack, Wadsworth, Knight and Russell [2002], Wadsworth, Knight, Ortigosa-Blanch, Arriaga, Silvestre and Russell [2000]), it appeared that the pulse remained relatively narrow well above this magic power (16 pJ/pulse, 100 fs). Furthermore, a downshift of the soliton frequency then appeared, as illustrated in fig. 32(a). This shift stems from another nonlinear effect, the Raman gain (a nonparametric effect). The highfrequency tail of the soliton can indeed pump the low-frequency tail, distant by a few THz, even though, for such small shifts, the Raman gain in silica is much smaller than it is at its peak frequency of 12–14 THz. This continuous pumping in the frequency domain causes the pulse to shift down in frequency, since the instantaneous high-frequency tail is deemed to decay when pumping the Raman gain. Such a shift is in the range λ ≈ 100 nm. This complexity goes to a larger level if the power is pushed up further. We reproduce in fig. 32(b) results, from the Bath group, showing how a virtual

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(a)

(b) Fig. 32. (a) Shift of the soliton frequency in the work of Wadsworth, Knight, Ortigosa-Blanch, Arriaga, Silvestre and Russell [2000]. (b) Formation of light supercontinuum for increasing power injection in the PCF, see text (courtesy University of Bath, J. Knight).

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continuum arises in such conditions. We shall not go through the list of them here. We concentrate of the fact that a large part of these phenomena are “parametric”, relying on frequency sums and differences. Then, it can be easily intuited that the zero-GVD frequency plays a special role in the final spectra. This special role is related to the relaxed requirement for phase matching. The easiest situation that can be grasped is degenerate four-wave mixing with a pump at ωp : the two frequencies ω1 and ω2 satisfying 2ωp = ω1 + ω2 have to satisfy also 2np ωp = n1 ω1 + n2 ω2 (the n’s being phase indices here) for phase matching. Both conditions are automatically satisfied around the zero-GVD wavelength when the curve n(ω) is linear. This is actually what happens around the zero-GVD point at least in a reasonable neighborhood. Hence, in fig. 32(b), corresponding to injection of ultra-short pulses at 850 nm, not only are red-shifted components generated, but also blue-shifted ones up to the visible and in a wideband manner. The output of the PCF is thus an extremely bright source of apparently white light, albeit with a somehow irregular spectrum macroscopically. Such a bright white source is already a blessing per se in numerous optics applications, even if the spectrum is not “flat”. But things are even better than that. As the visible light generation is essentially parametric it retains the phase information of the original pulse trains. These usually stem from a mode-locked laser and have perfect pulseto-pulse correlation. The spectrum of the source is a comb of frequencies spaced by flaser = 1/τlaser , the inverse of the laser repetition rate τlaser (usually tens or hundreds of MHz, for meter-sized cavities). Upon creating novel spectral content in the supercontinuum generation process the comb of frequencies with the same spacing flaser is retained even up to the visible radiations. Thus, the spectrum of supercontinuum can be seen as the full bunch of harmonics of flaser from infrared to visible! This supercontinuum generation process has thus “heterodyned” the laser frequency with a visible spectrum. This gives wonderful possibilities for metrology. They are already used to simplify metrology chain and port frequency standards to end users with much less demand than in the classical heterodyning chains classically used for this purpose. Just while finishing the manuscript of this chapter, we learned that two of the three winners of the 2005 Nobel Prize in Physics (Glauber, Hall and Hänsch) were rewarded, among other things, for the exploitation of this “comb frequency technique”. This could be a welcome conclusion to the extraordinary and unexpected possibilities brought about by PCFs in many fields of optics. It would be unfair, however, to minimize the task of modeling these fibers, without which no safe

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understanding of these phenomena could occur. The last subsection provides an introduction to this topic (involving only linear dispersion).

Brief overview of PCF modeling tools One usually starts from the wave equation for the magnetic field in the frequency domain (to favor a Hermitian operator), and assumes the z-dependence to be of the exp(iβz) form in all fields. The main methods are as follows: • The plane-wave method, applied to transverse coordinates, with the use of a supercell. Variants to impose the frequency and find the wavevector can be devised (see the specific chapter by Ferrando and colleagues in Busch, Lölkes, Wehrspohn and Föll [2004]), which allows to introduce dispersion n(ω) more naturally. • The localized function method. The orthogonal basis functions are Hermite– Gaussian functions in the transverse coordinates. • The finite-element method. Meshing issues are crucial in this method. Note also that for the solution of the vector field problem, the so-called “Nedélec finite elements” have to be used, which inherently fulfill the boundary conditions for the discontinuous electric field components and the related magnetic field derivatives. • The multipole method. It was described earlier, in relation with PhC in general, and applies in particular to PCF. It is the preferred method for round holes as it represents these structures efficiently. Other methods have been adopted (beam propagation, FDTD, . . .) with more or less success.

§ 6. Conclusion and perspectives The hopes for the photonic crystal concept were high: rethinking the distribution of the dielectric constant to obtain a full control of optical waves and modes. We are now reaching some level of understanding of the field, having demonstrated some of the concepts, understanding of the limitations and seeing the compromises to be made in designing real-world structures. The most difficult phase might be ahead, that of reduction to reality. Further progress will likely proceed in the future, as in the past, through a series of mutual fertilizations of nanotechnology and physics. Both phases of this interaction result in a broadening of the research community for this area. The example of Burger, Osher and Yablonovitch

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[2004] and others given in Section 4 could further lead to massive computing for optimal design incorporating fabrication constraints. We hope that the reader feels the visions and thoughts that have marked the first two decades of this story, and that he may also add his own contribution to the understanding of the photonic crystal concept and to its application. Importantly, a number of open questions, pending for a long while, have recently received fairly satisfying answers when confronted with the limits of present techniques: 3D photonic crystals pose severe fabrication problems. Studies of elementary defects in 3D PhCs have just begun in the optical regime, and may be difficult. 2D photonic crystals can perform many of the roles of 3D PhCs, provided a lot of care is exercised at the design stage. High-Q microcavities, and their impact on spontaneous emission control, provide an everyday confirmation of this first-order analysis in the academic arena. However, in spite of these first successes, the impact of photonic crystals on everyday life or even on advanced technical achievements is still very tiny, compared to, for instance, the impact of quantum wells in optoelectronic devices. Nevertheless, they have attracted interest across many subfields and, indeed, in many diverse fields. We hope this is apparent even though we have limited our emphasis to “linear” properties of PhCs. We omitted the topic of nonlinear photonic crystals, reported in particular in the seminal work of Berger [1998, 1999]. The companion topic of nonlinearities in “linear” photonic crystals by other teams is also absent from the present chapter. In our view, the first topic is evolving into a clever field of advanced science and engineering, but heavily relying on a single material, periodically poled lithium niobate (PPLN) (Chowdury, Staus, Boland, Kuech and McCaughan [2001]). The second field, the operation of nonlinearities in PhCs, is certainly promising as well. It does not yet easily lend itself to quantitative evaluation of the benefit of modified Bloch waves (e.g., slowed down or confined) because many waves are involved (harmonics), not all in photonic gaps, so that enhancements detected in harmonic generation, some very large (Mondia, van Driel, Jiang, Cowan and Young [2003]), nevertheless appear to be limited by losses at the different frequencies. Detailed and specific analysis is therefore required to study each structure (Villeneuve, Abrams, Fan and Joannopoulos [1996], Sakoda and Ohtaka [1996], Mingaleev and Kivshar [2002], Soljacic, Luo, Joannopoulos and Fan [2003]). In a few years results with fully mastered losses will undoubtedly emerge as many teams worldwide endeavor to achieve large nonlinear effects in PhCs, with either fast or slow nonlinearity mechanisms (Bristow, Wells, Fan, Fox, Skolnick, Whittaker, Tahraoui, Krauss and Roberts [2003], Raineri, Vecchi, Yaco-

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motti, Seassal, Viktorovitch, Raj and Levenson [2005], Notomi, Shinya, Mitsugi, Kira, Kuramochi and Tanabe [2005]). Similarly, the impact on spontaneous emission control and quantum optics effects has only been hinted at in our treatment. Fundamentally, the manipulation of photons for long enough times or with a strong enough light-matter interaction requires some kind of photon mode confinement for which photonic crystals are, with little doubt, among the best candidates given the recent breakthroughs on microcavities. To put things in perspective in the present era of ever-improved technological achievements, a lapse of about ten years was necessary between the first vertical cavity lasers (VCSELs) at the end of the 1980s and the experimental evidence of enhanced spontaneous emission (Gérard, Sermage, Gayral, Legrand, Costard and Thierry-Mieg [1998]) from the germane micropillar structures in 1998. A similar lag can be expected between the present high-Q cavities and their good use for controlling ultimate quantum-optics phenomena for example, or for other fancy ideas such as atom chips, atom lasers, etc. As far as chips are concerned, the successful operation of 2D structures and microcavities certainly ensures that genuine chips exploiting high parallelism can be made and that large-scale applications can be sought. While a 3D bandgap was originally seen as a must for physics and chip architecture, restriction of propagation to less dimensions (2D) does not preclude such large-scale applications. For most desired effects, and especially when it comes to controlled propagation, what is required is 3D confinement as opposed to 3D bandgap, a requirement provided by 2D + 1D with modest mitigation. In microelectronics, in the age of giga-transistor chips, one still uses a single plane for all the active electronic circuits. Among the possibilities in so many domains, let us attempt the delicate exercise of selecting those ideas that should develop through the forthcoming decades. The idea of an omnidirectional gap is still the holy grail of photon control. Even though partial gaps and leakage channels are often encountered, the quest for omnidirectional control is always present. This continues the evolution of the community of confined photon systems. The topic of planar systems was quite well known at the turn of the millennium. It will be soon amusing to remind that indeed “microcavity” was once used almost exclusively in the context of planar cavities. Today, going to multi-dimensional structures, with strong index contrast (thus, also away from holography), is a general option to take advantage of an additional set of modified physical properties. The importance of photonic Bloch waves is an unforeseen result. Their proper use outside the bandgap is key to design and understanding. The wavevector diagrams in 2D and 3D pioneered by P. St J. Russell provide a deep insight into many essential properties (existence of couplings to the outside, at PhC bound-

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aries and edges, etc.) that are a touchstone also for the obtainment of desired device-oriented operations such as superprism and supercollimator. Not surprisingly, the role of disorder (imperfection, tolerance to differences from design, etc.) is a matter of continuing debate (Fan, Villeneuve and Joannopoulos [1995], Bogaerts, Bienstman and Baets [2003], Bogaerts, Bienstman, Taillaert, Baets and Zutter [2001], Kaliteevski, Cassagne and Albert [2002], Veronis, Dutton and Fan [2004], Hughes, Rammuno, Young and Sipe [2005]). In spite of the short lengths expected in good photonic gap, any ultimate property such as 100% reflection, resonances due to minute couplings between confined Bloch waves in a slab and radiation modes, etc., is perturbed by disorder to a large extent. It seems to be the price to be paid for dealing with singularities. The importance of this topic is echoed in new forums, such as ongoing European Networks of Excellence (to which one author, HB, belongs), that emphasize various aspects of this issue. As an example, while it has been taught for decades that Fabry–Perot mirrors have to be flatter than “usual” λ/10 mirrors to attain the Finesse expected from their apparent (but incoherent) large reflectivity, it seems that setting up the analogue criterion on structural disorder when operating with Bloch modes (in 3D in slabs or in channel waveguides) or with cavity modes brings challenging issues. Disorder cannot be easily averaged. Its effects on modal redistribution of power are delicate to handle, and this is notably the case for the vertical/in-plane fate of scattered light in 2D. As there are no actual in-plane propagative channels due to the bandgap, disorder-induced coupling to confined modes certainly plays a major role. Sophisticated tools such as the Green function are required, although simpler tools are also crucial to everyday practice. These last two remarks show how careful one must be when dealing with the electromagnetic issues raised by photonic crystals. The reader can find more detailed treatments in the bibliography. Let us give a more personal view to this difficulty. Starting with the analogy with the electron Schrödinger equation, photons should present a simpler case, as, at small intensities, they are in a linear, interaction-free regime, as Bosons should. The present status of the field just reveals the contrary. As the structures are man-made there is no “reference crystal”. Each team has to “requalify” its fabrication or revalidate its modeling tools separately. Then, the properties measured are wave properties, not the population properties most often measured for electron devices. Electrons do not leak out of wires. They form a gentle cloud of relaxed thermalized particles (fermions) that ignore most wave aspects, e.g., when it comes to disorder (except in the area of mesoscopic physics). On the contrary, refined measurements of photonic crystals are always delicate, being, for example, easily plagued by speckle, or its equivalent for near-field probing, etc.

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Another interesting way to poise the impact of photonic crystals is to identify any likely breakthrough device in key domains of optics and electromagnetism. Lasers, and specifically laser diodes, are most likely to benefit from an impact of improved modal control, and from a bunch of new solutions to “shape” their modes (lasing and parasitic secondary ones, etc.). Whether a full gap is needed is not clear. Again, anything featuring one more degree of control of the photon modes than a usual laser resonator involves some of the photonic crystal concepts. This is true also for mirrors of gas lasers at the upper limit and, more obviously, for the example of structured VCSEL mirrors. This field could illustrate a productive cooperation and competition with diffractive optics concepts (Silberstein, Lalanne, Hugonin and Cao [2001], Lalanne and Hugonin [2003]). Near-field optics is a blossoming science and engineering field thanks to new probes and sources. The basic shapes are small cones and apertures enabling leaps into the superresolution regime (Volkov, Bozhevolnyi, Borel, Frandsen and Kristensen [2005]). Undoubtedly, the concepts can be cross-fertilized with those of tight light confinement or periodicity developed for PhCs, developing a “nanophotonic” palette. Recent witnesses are the time-resolved results by Gersen, Karle, Engelen, Bogaerts, Korterik, van Hulst, Krauss and Kuipers [2005a, 2005b]. The related topic of the control of single very localized emitters is at the heart of many biophotonic research projects making use of fluorescent markers, to investigate into cells, neurons, proteins, etc. This allows probing at the macroscopic scale events occurring at the molecular level. In metallo-dielectric structures for millimeter to centimeter wave applications the fate of the concepts generated by photonic crystals is uncertain. The existence of many “universal” tools (impedances, . . .) and the capability to control generators and sub-wavelength details of the structure make the cycle time towards applications rather shorter. Hence the way the photonic crystal concept may play a irreplaceable role is not obvious. Control of spontaneous emission is not an issue there. Conversely, mastering of high-Q modes can exploit several existing solutions. The interesting common future is maybe to answer the issues recently raised around metamaterials and so called “left-handed materials” (displaying negative refraction, superresolution for “flat” lenses, etc.). How far can a resonance of magnetic energy response (the ingredient to obtain negative permeability, not to be confused with the well-known technique of magnetic resonance in a static field) be pushed in the THz regime is a question with no clear answer to our knowledge. Plasmons (or “plasmonics”) are certainly the key concept to take advantage of metallo-dielectric structures in the optical range. They offer many facets with which to play: low-loss regimes; field enhancement at textured sur-

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faces as known from Raman scattering studies. How periodicity can best become an asset in this area remains to be fully established. Finally, it is possible that the broadest dissemination of some concepts around photonic crystals could arise through the huge application field of PCFs (photonic crystal fibers). Their relatively short time to market is, of course, an advantage. The issue is not whether those that have worked on “regular” (i.e., nonfiber) PhCs should feel unhappy about it. The good use of novel manners of shaping the matter is clearly a collective process with, fortunately, fuzzy boundaries.

Acknowledgements We first thank two persons who kindly helped in putting up this chapter, J. Knight – whose tutorials have been enlighting to the authors and form the basis of most of the material of Section 5, – as well as J.W. Haus for providing some figures of Section 2. We also warmly thank our (young) colleagues at Laboratoire PMC in Ecole Polytechnique, Palaiseau and at Institut d’Optique, Orsay for carrying every now and then informal discussions prompted by the needs of the authors’ writing process (S. Olivier, E. Viasnoff-Schwoob, M. Rattier, A. David, L. Martinelli). The authors’ work was made possible through a string of European projects (SMILED, PCIC, FUNFOX). Lively discussions with the many collaborators involved (at EPFL in Lausanne, Glasgow University, Trinity College in Dublin, Ghent University, St Andrews University, Würzburg University, France Telecom in Paris, FORTH in Crete, KTH in Sweden, Alcatel in Marcoussis, IREE in Prague, University of Udine, PhotonDesign in Oxford, HHI in Berlin) led to the many demonstrations of the unusual or remarkable properties of 2D photonic crystals. We also have special thoughts for T.F. Krauss, C.J.M. Smith and D. Labilloy who shared with us the exciting time when photonic crystal concepts became tangible (1995–2001), and made it, with their energy, simply unforgettable. This work has been partially supported by the European FUNFOX project.

References and special issues Special issues of regular journals on photonic crystals, in chronological order J. Mod. Optics 41 (1994) 171–404, Special Issue on Photonic Band Structures, edited by Kurizki, G. and Haus, J.W. J. Lightwave Technol. 17 (1999) 1928–2207, Special Issue of the PECS I Conference (Laguna Beach, California), edited by Scherer, A., Doll, T., Yablonovitch, E., Everitt, H.O. and Higgins, J.A.

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IEEE J. Quantum Electron. 37 (2002) 724–926, Special Issue of the PECS III Conference (St Andrews, Scotland), edited by Krauss, T.F. and Baba, T. C. R. Physique 3 (2002) 1–102, Special Issue on Microcavities and Photonic Crystals, edited by Weisbuch, C. and Benisty, H. Optical and Quantum Electronics 34 (2002) 1–310, Special Issue on Photonic Crystals, edited by De La Rue, R.M. Photonics and Nanostructures, Fundamentals and Applications 2 (2) (2004) 57–159, Special Issue of the PECS V Conference (Kyoto, Japan), edited by Benisty, H., Kawakami, S., Norris, D.J and Soukoulis, C.M.

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E. Wolf, Progress in Optics 49 © 2006 Elsevier B.V. All rights reserved

Chapter 4

Symmetry properties and polarization descriptors for an arbitrary electromagnetic waveﬁeld by

Christian Brosseau Laboratoire d’Electronique et Systèmes de Télécommunications and Département de Physique Université de Bretagne Occidentale, CS 93837, 6 avenue Le Gorgeu, 29238 Brest cedex 3, France e-mail: [email protected]

Aristide Dogariu College of Optics and Photonics/CREOL, University of Central Florida, Orlando, FL 32816, USA

DOI: 10.1016/S0079-6638(06)49004-1

ISSN: 0079-6638 315

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 2. A brief survey of the interplay between polarization concepts and geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 3. Density matrix and the convexity property of the states of polarization 332 § 4. Polarization of a plane wave . . . . . . . . . . . . . . . . . . . . . . .

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§ 5. Polarization of an arbitrary wave . . . . . . . . . . . . . . . . . . . .

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§ 6. Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 7. Summary and prospects . . . . . . . . . . . . . . . . . . . . . . . . .

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Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix A: Lie groups in polarization optics . . . . . . . . . . . . . . . .

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Appendix B: Madison convention for the density matrix of massive spin-1 particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix C: Degree of polarization of a field consisting of a superposition of an ensemble of evanescent waves of random amplitude and of black-body radiation . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 1. Introduction Polarization of light abounds with dichotomies: it is known since more than three centuries, and there is no question it has generated a considerable amount of excitement among researchers in the last decades; it is an aspect of the visual world that is detected by insects and by many vertebrates other than mammals but is hidden from us, yet its origins are rooted deep in statistical physics and electromagnetism; its applications involve areas as diverse as photonics, information technology and biology, yet its understanding is still incomplete. Several important, but not completely answered questions remain in polarization optics: can an ideal (plane) polarizer fully polarize an obliquely incident wave? What is the polarization descriptor if the direction of propagation cannot be well defined, or if the wave front is not planar as in optical systems where focusing elements are involved? The concentrated efforts of many scientists over the past 330 years (since Bartholinus’ observation of the double refraction of light by calcite in 1669) have resulted in a rich store of valuable information. Many famous names, such as Stokes, Poincaré and Wolf, are associated with the theory of polarization. Listing the major papers is now a daunting task even if we restrict the description to theoretical aspects. Much of the work reported thus far in this area has been discussed almost exclusively in terms of plane wavefields. This model has guided our thinking about polarization since its introduction. Such discussions may indeed be relevant, but the growing interest in electromagnetic fields in unconventional materials and structures has prompted many scientists to reconsider this modeling paradigm and ask the right questions: What really entails the concept of polarization? What symmetries have polarization states? What are the algebraic and geometric descriptors of polarization? The study of the polarization properties of a general (nonplane) electromagnetic field has received rather little attention from a theoretical perspective, with the notable exception of work by Fainman and Shamir [1984]. Only in the last few years have experimental reports appeared which have stimulated many theoretical studies. This effort is largely driven by developments in numerous areas of physics where such fields are encountered, from near-field microscopy to medical imaging and biological physics. Recent developments in nano-optics have 317

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sparked a number of experimental and theoretical studies (see Carozzi, Karlsson and Bergman [2000], Setälä, Shevchenko, Kaivola and Friberg [2002], Lindberg, Setälä, Kaivola and Friberg [2004], Setälä, Lindfors, Kaivola, Tervo and Friberg [2004], Lindfors, Setälä, Kaivola and Friberg [2005], Vahimaa and Tervo [2004], Dennis [2004], Nieto-Vesperinas and Garcia [1996], Kreibig and Vollmer [1995], Takahara and Kobayashi [2004], Barnes, Dereux and Ebbesen [2003], Ellis and Dogariu [2005a] and references therein). These represent just a few of the many examples that have recently prompted the optics community to investigate the case of an electromagnetic wave with a nonplanar wavefront. Thus an overview of our present understanding of the theoretical aspects of polarization of an arbitrary electromagnetic wavefield is particularly timely. Thinking about polarization descriptors has proved enormously fruitful in optics, as can be seen from the historical perspective developed in Section 2. And, we argue here, it continues to provoke questions that are – by any reasonable definitions – quite fundamental. After some unavoidable and presumably fruitful initial controversies (Setälä, Shevchenko, Kaivola and Friberg [2002], Lindberg, Setälä, Kaivola and Friberg [2004], Setälä, Lindfors, Kaivola, Tervo and Friberg [2004], Lindfors, Setälä, Kaivola and Friberg [2005], Ellis, Ponomarenko, Dogariu and Wolf [2005], Ellis and Dogariu [2005a]), the key understanding was found by recognizing that complex wavefields must display some organizing principles which should at some level be encoded in their topology. But if the topology of these fields indeed deviates from the plane symmetry, we need to develop tools and measurements to capture the underlying organizing principles in quantitative terms. Fundamental physical laws are generally subject to a variety of symmetry transformations.1 Quite independent of the detailed content of the basic laws, 1 All fundamental laws of Nature are statements about symmetry. Conservation of energy, momen-

tum, and charge are direct consequences of global symmetries. The forms of the fundamental interactions are dictated by local gauge symmetries. Symmetry unifies apparently different physical phenomena into a common framework. For example, electricity and magnetism were discovered independently and viewed as completely different phenomena before the 19th century. Maxwell’s theory and the underlying relativistic symmetry between space and time unified the electric field and the magnetic field into a common electromagnetic field. This specific unification showed that electricity and magnetism share a common microscopic origin and can be transformed into each other by going to different inertial frames. In physics, a key experimental manifestation of symmetry is the emergence of new particles or new collective modes. For example in particle physics, Gell-Mann used the SU(3) symmetry of the strong interaction to predict the Ω − resonance. Similarly, the electroweak unification based on the SU(2) × U(1) symmetry led to the prediction of the W ± and Z bosons. In condensed matter physics, the SO(5) symmetry of antiferromagnetism and superconductivity predicts a new class of collective excitations, called the π resonance, which are the Goldstone modes of spontaneous symmetry breaking. Reflecting upon the historical developments of physical theories, it seems worthwhile to carry out the symmetry approach to the problem of polarization optics.

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their symmetry properties alone impose restrictions on the possible properties of light and matter. Motivated by the recent developments, many new concepts and measures have been proposed and investigated in the past few years. In this chapter we attempt to show the unity and logical development of the subject from the very beginning. Special emphasis is given to calculations performed utilizing the density matrix representation of polarization states. The derivation of many key results distinguishes this review from the usual presentation in that it is suggested that the polarization properties of electromagnetic fields have a natural explanation through the convexity and symmetry properties of the density matrix. There are a number of reasons for the success of a group-theoretical analysis of the symmetries of the polarization structure of the electromagnetic field: (1) it provides a very elegant geometric interpretation of polarization descriptors; (2) it has a unifying character that enables one to relate Jones vectors, Mueller matrices, the Poincaré sphere, Stokes parameters, and the coherency matrices into a single, efficient formalism, (3) it allows one to identify real measurement quantities that remain invariant under complex transformations; and perhaps more importantly, (4) it reasserts the importance and fruitfulness of geometric ideas in physical theories. Some aspects of the ideas (Brosseau [1998], Bhandari [1988], Urbantke [1991], Schlichter [1990], Meier and Zakharchenya [1984], Allen and Eberly [1975], Baylis, Bonenfant, Derbyshire and Huschilt [1993] and Zapasskii and Kozlov [1995]) to describe a polarization state as a pseudospin are discussed. The great attraction of a pseudospin treatment is that it offers us different perspectives on the same problem and it provides us with a geometrical overview of the dynamic evolution of N-level systems. This review, which is somewhat selective due to the limited space available, focuses on a small number of points that in our opinion reflect the leading perspectives for potential applications. In the following sections we attempt to provide a rough guide to the current state of this emerging and exciting field, whence the advanced student or the researcher can proceed to forefront research papers. From the outset, this program introduces the reader to viewpoints and methods that both generalize and unify much of this previous background. The review deals essentially with pure theory, and no experimental results are discussed. The body of the review is divided into six sections. Section 2 covers some basic historical and background material in polarization optics. Although it is impossible to give a substantial history of polarization optics here we start with some brief (if rambling) historical comments, in particular about the interplay between polarization concepts and geometry over the centuries, which continues today. Section 3 outlines ideas from geometric algebra that are used in attacking the issue of describing the polarization of an arbitrary electromagnetic field. In particular,

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we will argue for an implementation of the density matrix which encodes all that can be known about a mixed (partially polarized) state. Section 4 shows that this formulation provides a visual aid and a computational tool for studying the polarization of a plane-fronted electromagnetic wave. In Section 5 we move beyond the planar limit, and address the important question of what a three-dimensional polarized wave means operationally; in so doing, we endeavor to explain physically why a geometric approach can be useful for dealing with the analysis of the polarization of a nonplane wave. This allows us to define new polarization descriptors due to the possibilities of nontrivial topology. Section 6 is concerned with remarks on the polarization of photons. We conclude in Section 7 with a summary of the results and avenues for future studies. For completeness and for the reader’s convenience, we have also included three short appendices which summarize important technicalities on the theory of Lie algebras, the Madison convention for the density matrix of spin-1 particles, and an illustration of how the degree of polarization for a three-dimensional field can be calculated.

§ 2. A brief survey of the interplay between polarization concepts and geometry Polarization optics has a long and distinguished history. As history evolved, so did optical physicists’ views about polarization. Most students, and many of their mentors, pay little attention to the history of the problems they are trying to solve, assuming that it is of more interest to historians than to working scientists. In reality, nothing is more helpful to the novice scientist than a deep understanding of how his or her subject has evolved.2 Although this earlier literature is extremely interesting, the authors do not advocate that all new workers in polarization theory 2 Does one need to know anything about the history of optics (or the lives of individual scientists) in

order to appreciate the subject matter? Authors of textbooks or scientific articles are under no obligation to say anything about the history of science. Yet we all know that the subject matter of science cannot be entirely divorced from those who practice it, and that scientific enquiry sometimes takes particular directions as a result of the interest or advocacy of particular individuals. Aside from the human interest involved in biographical studies, there may be some intellectual value in retracing the way physical ideas and ground-breaking works have developed. The development is often complicated, however. Occasionally good ideas emerge prematurely in obscure places and are forgotten for a while, only to be rediscovered independently. It should also be noted here that historians of science are now skeptical of the concept of discovery as a discrete event for which a particular person or persons should get credit or priority. It is not the job of the historian to decide who should get credit for a discovery, but rather to study who does credit and why. This sort of recourse to the history of optics is typical of the philosophical contributions, in that facts are carefully selected in order to construct a narrative of progress, or a “sense of history”.

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set out to acquaint themselves with the minute details of these papers. The chain of thought extends over roughly three and a half centuries from the late 1660s to nowadays. We describe some of the milestones, of the past three centuries, along the road towards increased understanding of polarization optics. As we shall see, the germ of many novel ideas in polarization optics can be traced back directly or indirectly to Stokes, Poincaré and other physicists from times long gone. The story is divided into four main steps. In so doing, we hope to give the reader a taste of the conceptual issues suggesting that geometric algebra provides a unifying conceptual framework from which to view different descriptions of the polarization states. A more comprehensive and detailed discussion (the “full story”) of the main developments in the concepts of polarization optics is contained in a monograph by Brosseau [1998].

2.1. From Bartholinus (1669) to Stokes (1852): polarization is a geometric property of light Let us begin our excursion by reviewing a few facts about man’s investigations of the nature of polarized light. The story encompasses a large cast of characters interacting with all facets of the history of mankind. Although the human eye is polarization insensitive3 (Haidinger [1846, 1847], Cronin, Shashar, Caldwell, Marshall, Cheroske and Chiou [2003], Wehner [2001], Waterman [1981], Shurcliff [1962] and Swindell [1975]) the first scientifically documented report on a polarization phenomenon did not come until 1669, when Erasmus Bartholinus, a Danish-born mathematician at the University of Copenhagen, observed the double refraction of light by calcite (the rhombohedral form of calcium carbonate). This observation eventually led to the discovery of the polarization of light by Huygens c. 1690. Huygens found that light which had passed 3 It is worth observing that even though the sun itself produces fully depolarized light, partially polarized light is abundant in natural scenes. In the sky and under water, scattering of incoming light produces partial polarization that varies with solar position and direction of view, and reflection of light from the air–water interface, and from shiny surfaces, like leaves or animal skin, produces strong polarization. For terrestrial animals with polarized-light vision, the sky presents a reliable pattern useful for navigation, but the more chaotic and unpredictable pattern of polarized-light reflection can mask or taint the “true” colors of objects. Consequently, photoreceptors in some animals that would normally be sensitive to the polarization of light are structurally modified to destroy polarization sensitivity, while other animals may evaluate viewed objects using combined spectral and polarizational cues. It is thought that the evolution of color vision was favored because of its huge utility in segregating scenes and in fostering the recognition of objects of special interest, e.g. food, individuals of the same species, etc.

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through a piece of calcite behaved differently than ordinary light. But, although he was able to describe the phenomenon of double refraction in terms of his wave construction, he was unable to account for the definition of polarized light. Much of this earlier activity was discussed in ordinary language without symbols, but some of the problems were discussed in geometric terms with diagrams. Thus the interplay between polarization optics and geometric problems has its roots in the 17th century, essentially in Europe. At that time, particle theorists or “emissionists” believed that a beam of light comprises many rays, which are in turn composed of streams of “corpuscles” (particles) traveling in accordance with dynamical laws. In this scheme, the various phenomena of light, e.g. refraction and reflection, are understood in terms of Newtonian forces acting among these particles. Newton’s universal law of gravitation defined an ideal model for classical physics, which was consciously emulated in the early study of electricity, magnetism, and speculative atomic theories. On the other hand, while the “selectionists” and “wave theorists” such as Thomas Young and Christiaan Huygens still believed the ray to be physically real and ultimately composed of particles, the physical structure of the ray was not essential to optical analysis. It is worth observing that the rays of wave theory were mathematical abstractions: they were geometrical lines, rather than physical entities. The wave theory of light was largely neglected throughout the 18th century because it did not appear to explain polarization phenomena. The problem was that it supposed that light waves – like sound waves – are purely longitudinal. However, support for the theory was revived at the start of the 19th century by Thomas Young who realized that polarization could be best explained if light has both a transverse and a longitudinal component. Augustin Fresnel’s concept of light in terms of purely transverse vibrations of tiny ether particles within each wavefront, when applied to diffraction and polarization, produced a new interpretation: the wavefronts are physically real. Consequently beams, rays, polarization, refraction, and similar concepts all take new meanings, and experimental results need to be reinterpreted. Fresnel unquestionably deserves the credit for creating a theoretical framework within which the matter could be pursued. In the years 1812–1815 an important milestone was set by the French physicist Dominique François Arago at the Paris Observatory. On the theoretical side, his principal contribution was the discovery of four interference laws governing the interference of polarized light, published in a joint paper with Fresnel, which played a key role in the demonstration of the transverse nature of lightwaves propagating in free space. In 1852 George Gabriel Stokes introduced the set of Stokes parameters: four measurable quantities (intensities) for describing the polarization properties of

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light (Stokes [1852a, 1852b]). He also formulated a mathematical treatment of unpolarized light, i.e., in that case the intensity is not affected by any rotation of the axes of the reference coordinate system or by any phase change introduced in one of the wave components. The name of Stokes, a contemporary of Maxwell, has become well known to generations of scientists, mathematicians and engineers through its association with various physical laws and mathematical formulae. In standard textbooks of mathematics, physics and engineering we find Stokes law, Stokes theorem, Stokes phenomenon, Stokes conjecture, and the Navier–Stokes equations. It is interesting to note that Stokes, while primarily a mathematical physicist, made major advances in the wave theory of light, examining the properties of the ether which he treated as an incompressible elastic medium. The pure mathematical results of Stokes arose mainly from the needs of the physical problems he and others studied. The first half of the 19th century saw great advances in the understanding of polarization. It was a heroic age, in which polarization was essentially a “static” and “qualitative” concept.

2.2. From Stokes (1852) to Poincaré (1892): polarization is an electromagnetic property of light Electromagnetism sparked the interest of a myriad of groups throughout 19thcentury Europe. Leading British, French, and German experimental Natural Philosophers4 grappled with electric and magnetic phenomena in the hope of explaining the century’s most exciting branch of physics in terms of current theories of optics and gravity. Among the enduring legacies of 19th-century science, James Clerk Maxwell’s equations of electrodynamics have long held a preferential place in the hearts of physicists. In the words of one of today’s more outspoken physicists, Steven Weinberg, the equations constitute a noncontingent fact, without which contemporary physics would be unimaginable. Maxwell’s main body of work consists in his kinetic theory of gases and his theory of the electromagnetic field, better known as Maxwell’s equations (Maxwell [1864, 1962], Harman [1990]). In optics, Maxwell also established the 3-color theory of color vision, i.e. the assertion that red, green and blue light in an appropriate mixture can give any color. What is 4 Natural Philosophy was defined by Joseph Fourier [1878] in the following terms: “Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the subject of them being the object of Natural Philosophy”.

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not sufficiently known is that Maxwell was not a mere theoretician but that he also was able to devise telling experiments. His penetrating insight into the nature of reality was primarily geometrical. Maxwell’s equations (published in 1872) were preceded by an attempt to put Faraday’s lines of force into mathematical language (in 1861) and by a fanciful model (in 1865) in which the whole of space was occupied by tiny cells capable of carrying charge and rotation (whence the magnetic part of the field). Maxwell’s equations were famously found to contain the conclusion that electromagnetic waves travel with the speed of light. Maxwell (and his contemporaries) saw what a breathtaking result that was. At that time, three theories of electrodynamics were already in place (roughly, those of Maxwell, Weber and Franz Neumann). Each theory was associated with a distinct picture of its physical significance – the traditional concept of force as action at a distance; a mixed picture in which such forces were supplemented by the actions of an intervening medium (the ether); and Maxwell’s pure picture, in which action at a distance was replaced completely by a field. Sceptical of action at a distance, the cell-like structure with which he had endowed all space was replaced by the doctrine of the luminiferous ether. The conceptual advance of Maxwell’s theory facilitated the more effective use of new mathematical techniques for analyzing optical phenomena. In 1865 Maxwell wrote: “. . . we have strong reason to conclude that light itself (including radiant heat, and other radiation if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field”. Maxwell’s electromagnetic theory was confirmed by Heinrich Hertz’s discovery of electromagnetic waves, which in turn led to remarkable advances in physics, astronomy and technology. Serious scientific interest in the geometric-algebra description of light polarization began with the work of Jules Henri Poincaré. A first-rank mathematician, Poincaré was one of France’s greatest scientific geniuses of the 19th century, the range of his interests and achievements being hard to conceive. Poincaré made many contributions to mathematics and to other sciences including celestial mechanics, fluid mechanics, the special theory of relativity and optics, to cite but a few; he is often described as the last universalist in mathematics.5 In polarization optics, Poincaré made a number of profound contributions. He introduced the 5 He was described by his mathematics teacher as a “monster of mathematics”. Much of his research involved interactions between different mathematical topics, and his broad understanding allowed him to attack problems from different angles. Still today, his work is a fruitful subject for historical enquiry as he left behind him a large archival trail, and with his polymathic interest, he has attracted much attention. Poincaré can be said to have been the originator of algebraic topology, and he claimed that his researches in many different areas such as differential equations and multiple integrals had all led

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sphere that now bears his name and the complex plane (stereographic) representations to specify the state of polarization. The use of the sphere was popularized in Poincaré’s book Théorie Mathématique de la Lumière (Poincaré [1892]). One of the conveniences of the Poincaré sphere is that it provides a geometric view to the transformation of polarized light when it interacts with optical devices in terms of rotations. A major experiment was performed by Wiener [1890] demonstrating that the electric field vector of an electromagnetic wave exerts a greater force on electrons than the magnetic field does (provided one considers nonrelativistic speeds). Thus the electric field is responsible for changes of polarization state upon the interaction of light with matter. This discovery is at the basis of the convention to choose the direction of the electric field as the polarization direction of the lightwave. During this developmental second step in the history of the concept of polarization, theorists realized that polarization is an electromagnetic property of light. In addition, this period marks the shift from geometrical to algebraic techniques in optical analysis: polarization became a “dynamical” and “quantitative” concept. However, while the mathematical apparatus had the benefit of increasing the accuracy of the analysis, more importantly it allowed a certain vagueness in the understanding of the nature of a beam of light. This field remained dormant until Norbert Wiener and his colleagues pointed out that the key to understanding optical coherence and polarization theory is in its statistical formulation for describing correlations of random processes in different fields.

2.3. From Poincaré (1892) to Wolf (1954): polarization is a statistical property of light It is interesting to remember that by the end of the 19th century theorists believe that polarization optics was a mature field and that the electromagnetic character of light waves was familiar knowledge. Many researchers, e.g. von Laue, van Cittert, Zernike, were trying at the time to develop mathematical formalisms for describing radiation fluctuations, in particular the correlation between the electric field at two space points. The breakthroughs of modern physics around 1900 forced polarization optics into the background until the rebirth of interest in the

him to topology. The more we study the work of Poincaré, the more we are impressed by his unrivaled genius as a theoretician and natural philosopher. When we consider the magnitude and extent of his discoveries and their influence on the progress of science there is no honor too great to pay to the memory of Henri Poincaré – one of the greatest scientific discoverers of all time.

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mid-20th century. Progress had to wait for the application of probability distributions to physics. Early uses of probability arguments in coherence theory focused on ways to classify the statistical behavior of optical fields. Prior to that time there had been no clear recognition of the fact that the variation of the electric field vector of light is a random process in both time and space. By the 1930s the leadership in physics passed from Europe to the USA for many reasons. Interestingly, the 1930s saw a major breakthrough in polarization optics. Norbert Wiener [1930], at the MIT Mathematics Department, introduced a rigorous mathematical basis for the theory of the coherence of light. Although Wiener gave real grounds for the concept of coherence, his ideas based on “generalized harmonic analysis” did not gain general recognition. The close relation to the mathematical description of polarization states with spinors goes back to a paper by Jordan published in 1927, thus indicating the possibility of using the wealth of mathematical techniques developed in quantum mechanics for the treatment involving partially polarized electromagnetic radiation, although Ugo Fano (Fano [1949, 1954, 1957], Fano and Fano [1980]) is recognized as being the first to have popularized the connection between the quantum treatment of polarization and the classical description of the polarization of light. Fano also anticipated the use of unitary transformations, called by mathematicians the unitary unimodular SU(N ) in N dimensions, for the analysis of the polarization properties of light. The density matrix was conceived by John von Neumann in 1927. This was a major conceptual advance, and it forms the core of statistical approach to polarization and coherence optics. In 1954, Emil Wolf [1954a, 1954b, 1959] published a seminal paper “Optics in terms of observable quantities” in Nuovo Cimento, which became the cornerstone upon which scientists now base their theoretical studies. That paper, in addition to being recognized as a ground-breaking work, has served as a catalyst for the first complete theory of polarization and coherence properties of light. There are several reasons for this fruitfulness. Perhaps the most profound is related to a comment made by Wolf [1954a, 1954b, 1959]: “. . . correlation functions of optical fields, not the fields in themselves, provide a description of optical phenomena in terms of observable quantities”. The big step in Wolf’s approach to the statistical properties of light was the introduction of a 2 × 2 coherence matrix in the space–time domain, now referred to as Wolf’s coherency matrix, the components of which contain information about correlations between two mutually orthogonal components of the fluctuating electric field vector at one point in space at the same instant of time. Note that Wolf’s coherency matrix differs from that of Wiener in that Wolf chose to take as his fundamental quantities temporal covariance functions, while Wiener used integrated power spectra. Mathematically these two ap-

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proaches are functionally related through a Fourier–Stieljes transform. In a series of subsequent seminal papers which would be interesting reading for anyone who is about to embark upon his/her first approach to the subject, Wolf demonstrated a unified theory of coherence and polarization of random electromagnetic beams by introducing a cross-spectral density matrix related to the space–frequency representation (Wolf [2003]). Further important theoretical contributions to the subject were made by P. Roman (Roman [1959a, 1959b], Parrent and Roman [1960]) and Barakat [1963] who developed the spectral coherency matrix from a somewhat different point of view than the one introduced by Wiener. Thus in the early 1960s, the conceptual basis of the coherency (density) matrix was thoroughly formulated. The application of linear algebra to the description of polarized wave interactions was realized by Robert Clark Jones and Hans Mueller who introduced matrix treatments for handling the transformation of a polarized plane wave by a deterministic optical device or scatterer (Jones [1941], Mueller [1948]). In the Jones formalism, which applies to coherent addition of polarized waves, actual electric field components forming two-dimensional complex vectors describe states of purely polarized light (Shurcliff [1962], Swindell [1975], Barakat [1963], Jones [1941], Mueller [1948], Soleillet [1929]). A scatterer or a linear optical system is then described by a 2 × 2 complex Jones matrix. The Mueller formalism allows treatment of general partially polarized waves but assumes incoherent addition by describing incident as well as scattered waves by the Stokes parameters. The polarization properties of an optical system are then characterized by a real 4 × 4 Mueller matrix acting on the incident light (Mueller [1948]). It is little appreciated, even in polarization optics books, that Paul Soleillet had in fact anticipated what is known as the Mueller matrix formalism (Brosseau [1998]). The extension of these polarization matrices to nondeterministic optical systems was realized later (Brosseau [1998], Kim, Mandel and Wolf [1987]). For many reasons the history of science in the former Soviet Union is not generally and completely known in the West. For example, G.V. Rozenberg [1946, 1960] and D.G. Stamov [1953] were pioneers in developing the study of polarized light scattering in the atmosphere. Concerning the subject at hand, Fedorov [1965, 1966, 1976] introduced a three-dimensional beam tensor in a form that is covariant on the group of three-dimensional rotations SO(3) for quasi-monochromatic light, and therefore has universality.6 These works were overlooked in other countries for several years, but their importance has since been appreciated. 6 Tensors provide a useful way to write physical relations in an invariant or a covariant form that is independent of the coordinate system. In that case physical relations become much more descriptive,

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The quaternion algebra7 (introduced by William Rowan Hamilton in 1843) has been also used, by Richartz and Hsü [1949], to compute rotations and retardations of Jones vectors (pure states of polarizations) in a way that does not depend on a specific matrix representation of the optical devices. Parenthetically, we note that Pellat-Finet [1990, 1992] proved that quaternion algebra (Berezin, Kurochkin and Tolkachev [1989]) can be applied to cases of partially polarized radiation. Pauli algebra to describe the polarization of a plane wave was used by Hestenes [1966, 1990] to show that the electromagnetic theory takes a simple form in the geometric (Clifford) algebra of real three-dimensional space (Jancewicz [1988]). Although not well known, Whitney [1971] also made an important contribution to this problem by providing applications of the Pauli spin matrices as base to expand the coherency (density) matrix and the polarization Jones matrices. Looking back fifty years later, we find that the coherence and polarization theory put forward by Wolf in the 1950s has developed into a sophisticated theory in which the degree of polarization has an algebraic (second-order statistics of the field) as well as a geometric significance. So far, this measurable parameter has proven to be useful for characterizing scattering and interaction of light beams with optical media. But is this single parameter telling the full story about random electromagnetic fields?

since a physical quantity is represented by the single symbol of a vector or a tensor instead of numerous equations in projections. 7 Fascinated by the applications of complex numbers to 2D geometry, Hamilton had been struggling unsuccessfully for many years to invent a bigger algebra that would do something similar for 3D geometry. The quaternion can be regarded as an object composed of a scalar part which is a real number and a vector part. Moreover, the vector part may be represented, in magnitude and direction, by a line joining two points in 3D space. Many mathematical terms in common use today – including scalar and vector – were introduced by Hamilton, as he developed the theory of quaternions. Interestingly, Maxwell applied the theory to problems involving heat propagation and electromagnetism. Indeed, Maxwell presented the basic equations of electromagnetism using quaternion notation in his Treatise on Electricity and Magnetism published in 1873 (Maxwell [1962]). The quaternions have now “sunk” into obscurity. The reason is that the geometry and physics that Hamilton and his followers did with quaternions is now mostly done using the dot product and cross product of vectors invented by Josiah Willard Gibbs in 1880. But it should be mentioned that quaternions are crucial to some distinctly modern mathematics, e.g. “hyperkähler manifolds” and physics. The properties of quaternions have been described in detail by Katz [1997]. In addition to having fewer components than a rotation matrix, the quaternion makes it easier to interface with computer graphics. Most computer graphics programs specify rotations using quaternions.

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2.4. And now: geometric algebra is back Whatever one’s view may be on the role of the history of optics in teaching or research, probably most people will agree that it is more challenging to deal with the 20th century than with the immediately preceding centuries. The fact that optics is now being advanced more rapidly, in more places and by more people, makes it more difficult to trace the development of an idea. What new ideas have emerged in the past 20 years? A great lesson from 20th-century physics is that fundamental symmetries of physical laws are the cornerstone of modern physics. It is a concept that is familiar to many. Theories with the greatest amount of symmetry are considered to be the most beautiful or desirable.8 Symmetry considerations have been used to study a wide range of phenomena in science and engineering, and the literature is rich with examples of their applicability. In the present context, two developments are worth noting. In a seminal 1977 paper, scalar (rotational) invariants of a wavefield were introduced by Barakat [1977, 1983]. A second advance was made by Barakat and Brosseau [1993] who worked out the SU(N ) expansion of the density matrix to calculate the von Neumann entropy of polarization which is a measure of the degree to which an optical field is polarized. This concept of entropy was further used by Brosseau and Bicout [1994] to introduce depolarization lengths which characterize elastic multiple scattering of polarized light by a dense random collection of uncorrelated and noninteracting spherical dielectric particles. Monte Carlo simulations indicated that the amount of depolarization generated from multiple scattering depends on such factors as the size and shape distribution and index of refraction (Bicout, Brosseau, Martinez and Schmitt [1994]). In a medium containing particles small compared to the wavelength (Rayleigh

8 It is a symmetry in the sense of Herman Weyl’s definition because we can think of moving the experiment forward in time as an operation that leaves the laws of Physics unchanged. These symmetries are useful: they make the task of doing physics possible. But they also have observable consequences. Associated with each of these symmetries is what is called a conserved quantity: some physically measurable quantity that cannot be created or destroyed, but only transferred from one object to another. Momentum is the physical quantity associated with the invariance of physical laws under translations in space. Angular momentum is the conserved quantity that arises from the invariance of physical laws under rotations. One important practical consequence of all this is that symmetry (applied to physical laws) is useful. The more symmetries a theory has, the more physical quantities can be identified as conserved (neither created nor destroyed), and the easier it is to actually solve the mathematical equations and make predictions. It is partly for this reason that theories with more symmetries are thought to be more desirable. However, apart from this practical consideration, the presence of symmetries makes theories more beautiful in the sense that it can incorporate or synthesize more diverse phenomena into simpler structures.

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regime), the characteristic length of depolarization for incident linearly polarized light is found to exceed that for incident circularly polarized light, while the opposite is true in a medium composed of particles large compared to the wavelength (Mie regime). In this context, depolarization by multiple scattering is connected with entropy production. It should be emphasized that the group of 2 × 2 special unitary transformations SU(2) can be found in the optics literature (Pancharatnam [1956], Ramaseshan and Nityananda [1994]). As a result of these and other considerations, the general expectation in the optics community is that the Jones-matrix calculus associated with the coherency matrix formalism, on the one hand, and the Mueller-matrix calculus associated with the Stokes formalism, on the other hand, provide two efficient formulations of the theory of polarization in algebraic terms. An interesting connection has been established with the geometric phase. The geometric phase as encountered when guiding light in an optical fiber is a special case of the Berry phase (Berry [1984, 1987a, 1987b], Zwanziger, Koenig and Pines [1990], Ramaseshan and Nityananda [1986] and Arvind, Mallesh and Mukunda [1997]) which results when any parameter describing a property of the system under concern – here the wave vector characterizing the propagation of the light field – undergoes a cyclic evolution in state space, e.g. a geodesic triangle on the Poincaré sphere. The geometric phase depends solely on the amount of solid angle enclosed by the evolution path. It is worth noting that Pancharatnam anticipated the geometric phase in a classic paper on phase shifts in polarized light (Pancharatnam [1956], Ramaseshan and Nityananda [1994]). Pancharatnam’s 1956 and Berry’s 1984 papers were followed by a great number of papers in polarization optics, e.g. Bhandari [1988], Aravind [1992], Wagh and Rakhecha [1995], Zwanziger, Koenig and Pines [1990], Ramaseshan and Nityananda [1986] and Arvind, Mallesh and Mukunda [1997]. In 1956, Pancharatnam studied the problem of determining the phase change undergone by polarized light after passing through a sequence of polarizers such that its final polarization is the same as its initial polarization. Certain topological properties of vector fields have also been studied using a space curve formalism (Balakrishnan, Bishop and Dadoloff [1990]). In recent decades, the description of wave singularities led to new applications of polarization and coherence theory. As Nye [1983, 1987, 1999] has pointed out, light whose state of polarization varies with position can exhibit different generic singularities associated with the vector nature of light.9 For example, since for 9 Phase singularities are now recognized as important features of all waves. The foundations of wave singularities were laid in papers published as early as the 1830s, i.e., W. Whewell in 1833

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circular polarization the electric field vector never passes through zero, there is no obvious time origin for phase reference. These polarization singularities are lines, on which the electric (or magnetic) polarization ellipse is purely circular (C lines) or purely linear (L lines). These singular lines of polarization optics were detected experimentally by Hajnal [1990], using the modulated scatterer technique. The local structure of these singularities is very rich, and provides information about the polarization structure of the rest of the field. Berry and Dennis [2002] and Berry [2000] have shown that the patterns of ellipse fields are different for purely paraxial and fully three-dimensional fields. In many situations the electromagnetic field does not meet the paraxially requirements of a beam. Friberg and colleagues (Setälä, Shevchenko, Kaivola and Friberg [2002], Lindberg, Setälä, Kaivola and Friberg [2004], Setälä, Lindfors, Kaivola, Tervo and Friberg [2004], Lindfors, Setälä, Kaivola and Friberg [2005]) tried to calculate a degree of polarization for an arbitrary electromagnetic field, based on the 3 × 3 spectral coherency matrix. But in the process, they had to face up to a curious feature, i.e. the value of the three-dimensional degree of polarization for an unpolarized transverse wave is not equal to zero but to 21 , which is explained based on the fact that being a two-dimensional field the electric field vector at every point in space lies in a plane and consequently this wavefield is not random in the direction orthogonal to that plane. Dennis [2001, 2002] and Berry and Dennis [2001] proposed a geometric interpretation of the 3 × 3 coherency (density) matrix for nonparaxial polarization by emphasizing its invariance with respect to spatial rotations. Under these transformations, the coherency (density) matrix decomposes into a real symmetric positive definite matrix, interpreted as the moment of inertia of the ensemble of electric field vectors, and a real axial vector, corresponding to the mean angular momentum of the ensemble. Related lines of questioning have been followed for a number of years, the goal being mostly to characterize and control the statistical properties of nonplane electromagnetic fields. It should also be noted that Agarwal [2005] presented an analysis of some difficulties associated with the 3 × 3 density matrix for partially polarized transverse electromagnetic fields. Ellis, Ponomarenko, Dogariu and Wolf [2005] and Ellis and Dogariu [2005a] calculated a degree of polarization in a completely and G.B. Airy in 1838. Polarization singularities were discovered by W.R. Hamilton in 1832 as an unexpected consequence of Fresnel’s theory of the optics of crystals. In a general anisotropic material, two waves propagate in each direction, with different speeds and polarizations. There are, however, two singular directions, optic axes, where the speeds are the same. As functions of directions, the two speeds can be represented by surfaces forming a double cone at each optic axis. Hamilton deduced that at such “diabolical points” the wave direction corresponds to a cone of rays. This conical refraction was soon observed, confirming that light is a transverse wave.

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different way by considering the eigenvalues of the spectral density matrix. Now that intense studies of this problem have begun, another exciting experimental direction has spun off. This is the physics of nano-optics where the properties of the optical field reveal the complex interaction between light and matter. Based on a seminal idea of Soleillet [1929], Morales and Navarro [2003] also recently introduced a description of polarized (plane) wave and (ideal) polarizers based on the Minkowski space–time formalism of special relativity, leading to a “kinematic” interpretation of the degree of polarization of the wave. Hence the Minkowskian geometry can be relevant to dealing with the properties of a partially polarized wave and of the changes of its degree of polarization. In this brief historical survey, we have spanned the exciting three-century period in which a large number of physicists, astronomers, electrical engineers, and mathematicians have been involved in these efforts. These scientists teach us that, in coming to grips with the meaning of abstract theories that deal with matters far removed from everyday experience, it is an important discipline to try to visualize the physical behavior the theories predict under different conditions. There does seem to be (at least) a consensus that a geometric approach to polarization optics is important. However, the details thereof have not been completely disentangled. Because of the intellectual challenge of understanding the rich diversity of polarization phenomena and because of its importance for characterizing near fields or tightly focused waves, theorists have helped to bring polarization optics into the center of development of optics. Studies of polarization phenomena in optics may not only reveal new optical phenomena, but they also cast “light” on our basic understanding of light–matter interaction on a subwavelength scale. Now that we have seen in this preamble that polarization of light is still a very modern subject, let us turn to the main idea of using the convexity property of the density matrix which represents the states of polarization. A basic consequence of the convexity property is that the polarization characteristics for an arbitrary electromagnetic wavefield can be obtained through an algebraic specification of pseudospin states. At first sight this might seem to be a rather complicated characterization of the polarization state. However we shall see that this fits naturally into the historical progression of ideas in this subject, and is not so complicated compared to some other ideas in use.

§ 3. Density matrix and the convexity property of the states of polarization In order to provide the reader with a comprehensive and detailed view of the contents and methods of this chapter, we must start from the basic concepts. It will

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be assumed in the succeeding sections that the reader is familiar with the material of this section. Preparatory to a discussion of the polarization characteristics of a nonplanar electromagnetic wavefield, we first recall how the convexity property of the density (coherency) matrix encodes all relevant information for describing the polarization state for arbitrary wavefields. This formulation extends the geometric structure of the set of polarization states in the spirit close to Poincaré’s algebraic topology description of light. In addition, this geometric-algebra approach renders the theory of polarization versatile and transparent. Density matrix and state of polarization are terms that we take as almost synonymous. Throughout this article, boldface quantities represent vectors, matrices, or tensors as the context indicates. While we have tried to make this article self-contained, we have also assumed that the reader is familiar with polarization and coherence concepts at the level of Brosseau [1998] and Born and Wolf [1999], and density matrix theory at the level of Blum [1996]. In describing the randomness of the optical field we must regard the electric field components as random processes (Born and Wolf [1999], Mandel and Wolf [1995]). Usually the most we can state about these processes can be expressed through a hierarchy of N-fold joint probability density functions. Such a complete description is rarely possible or even desirable. In most practical applications only a finite-order (especially first- and second-order) description of the stochastic field is needed for calculations of the quantities of physical interest. Once this probability density function is known, we can compute any observable function of the random electromagnetic field E by evaluating ensemble averages. When the random electric field E is represented by means of a statistical ensemble which is stationary at least in the wide sense, the full characterization of E within the context of second-order statistics requires knowledge about the correlations between field components. Throughout our analysis, we assume that the field E is represented by its complex analytical signal. We restrict ourselves to light whose spectrum is a narrow band around the central frequency and we exclude from consideration those situations where the spectrum expands dramatically.

3.1. Polarization and the density matrix As shown by Wolf [1954a, 1954b, 1959, 2003], the statistical analysis of a partially polarized wave is based on a 2 × 2 Hermitian coherency matrix N = E ⊗ E+ , where the subscript labels the dimension of the matrix, the superscript “+” stands for a Hermitian conjugate, and “⊗” denotes the direct product of matrices. This picture can also be reformulated in terms of a crossspectral coherency matrix of the fluctuating electric field (different from the

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original concept of Wolf ).10 Various theorems were quoted, and physical arguments presented, to motivate the claim that the cross-spectral coherency matrix does capture the essence of a unified theory of coherence and polarization (Wolf [2003]). To proceed further, we shall use the preferred dimensionless density matrix, DN = N / tr(N ), with tr standing for the trace, rather than the coherency matrix representation of the electric field, and pay no further attention to proportionality factors depending on units. To keep this article to a reasonable length, and hopefully readable, we are a bit sketchy in places, but there is no difficulty filling in the details. As we said above, the polarization states are identified with the density operators in the associated complex Hilbert space H of dimension N. The representation of a Hermitian operator in this space will be an N × N matrix of complex numbers; hermicity of this matrix restricts the number of independent coefficients to N 2 . It follows that determining any such operator on H requires N 2 real numbers. In addition, the density operator, conventionally constrained to have a unit trace, requires for its determination N 2 − 1 real numbers. The density matrix DN is a normalized, Hermitian and positive semidefinite linear functional on H and can therefore always be diagonalized by a unitary transformation; its eigenvalues are all nonnegative (Fano [1949, 1954, 1957], Blum [1996]). There are many ways to express the universality of the density matrix representation in the case of electromagnetic wavefields. For example, it is particularly convenient to express the density matrix for the field in terms of the N 2 − 1 Hermitian, trace-orthogonal and linearly independent matrices ON j : 2

N −1 1 N N Θj Oj , DN = N

(3.1)

j =0

where

N ΘjN = ON j = tr Θj DN

(3.2)

denote the normalized Stokes parameters. We set aside ON 0 to be a multiple of the identity matrix. This decomposition is known as the special unitary SU(N ) expansion of the polarization density matrix (Brosseau [1998], Barakat [1977, 1983], Jacobson [1979]). Whether the polarization state is pure or mixed, it is uniquely characterized by its density operator DN . Note that we need a system of N 2 − 1 10 It is important to understand that this coherency matrix represented in the space–frequency domain is fundamentally different from the space–time coherency matrix to which one is accustomed in the conventional theory of optical coherence (Wolf [2003]).

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observables to generate N 2 − 1 equations like eq. (3.2) for a determination of the state of polarization. A real difficulty is to determine what sets of observables will in fact yield N 2 − 1 independent equations for our purpose. Take for instance the von Neumann entropy of a density matrix, defined as SN (DN ) = − tr(DN ln DN ).

(3.3)

When DN is diagonal this reduces to the Shannon entropy of the diagonal entries (Brosseau [1998], Barakat and Brosseau [1993, 1991], O’Neill [1963], Wehrl [1978], Gyftopoulos and Çubusku [1997]). The entropy SN (DN ) is zero if and only if DN is a pure state, or, equivalently the density matrix DN satisfies D2N = DN . Thus the minimal entropy can be regarded as a measure of the extent to which the density matrix is close to a pure, or fully polarized, state. It is worth observing that eq. (3.3) can be also written as − i λi ln λi , where the λi are the eigenvalues of DN ; they are real, nonnegative and sum to unity [because tr(DN ) = 1]. 3.2. Scalar invariants Closely connected to this decomposition we can define scalar invariants of DN which are scalar functions of DN that are invariant with all unitary transformations DN → UDN U−1 (Barakat [1977, 1983]). The Cayley–Hamilton theorem implies that these invariants can be obtained by evaluation of the traces of powers of DN .

3.3. The convex set of polarization states Density matrices (states of polarization) can be discussed rather generally and particularly illuminating in terms of the convexity property since such states are elements of a closed bounded convex finite-dimensional Hilbert space H (Kelly and Weiss [1979], Eggleston [1963], Bloore [1976], Segal [1947]). Let us begin by recalling some standard results of algebraic geometry. S is a convex set if, given any two points x, y ∈ S, the points λx + (1 − λ)y, and any λ with 0 λ 1, of the line segment between them all ∈ S. In other words, S contains the linear combination of any two points in S, provided the coefficients in the linear combination sum to one. A point z ∈ S is called extremal if it has no such decomposition into two other points x, y of S. Extremal points of S always exist. As a corollary, ∀x ∈ S may be expressed as a convex combination of extremal points, i.e. x = ri=1 λi zi with λi 0 and i λi = 1. Note that

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this expression is not generally unique. Such convex sets are stratified by rank. The rank of a point of a convex set is the smallest number r of the zi needed; it is also defined as the minimal number of extremal points of which it is a convex combination. Thus extremal points have rank one, and the rank function divides the set into strata which are connected subsets of points of equal rank (Bloore [1976], Eggleston [1963]). Since it can be shown that a set is convex if and only if it contains every convex combination of its points, any convex sum of density operators is another density operator, so the states of polarization form a convex set. This means that, if D1N , D2N are distinct density operators and λ1 , λ2 are real positive numbers such that λ1 + λ2 = 1, then λ1 D1N + λ1 D2N is again a density operator. Of course, this no tion of convex combination can be extended to more than two states: i λi DiN , with the λi being real positive numbers summing to unity, i λi = 1 denotes a generic convex combination. Notice that the decomposition as a convex combination need not be unique (Bloore [1976], Eggleston [1963]). It may happen that a state cannot be written as a convex combination of other distinct states: in this case it is called a pure state and is defined by 2(N − 1) real parameters. Since DN is Hermitian, it may be diagonalized into a sum of projectors |λi λi | onto its orthogonal eigenspaces according to the reduction principle (Eggleston [1963]), and the density matrix can be viewed as describing a statistical ensemble of pure states DN = (3.4) λi |λi λi |, i

where the coefficients are the eigenvalues λi . Since DN is positive and of unit trace, this is a convex decomposition and these projectors form the extreme points of the convex set of states (pure states of polarization). The positivity of DN entails the positivity of the coefficients λi , and the condition tr(DN ) = 1 entails that the sum of the λi is 1. No projection operator can be further decomposed, so they form the extremal points of the convex set: projectors are pure states of polarization. Any interior point of the convex set is a convex combination of the extreme points of that set, which lie on a 2(N − 1)-dimensional hypercurve. The states that are convex combination of others are then called mixed states. In light of these considerations, it is worth observing that any density matrix DN of dimension N can be written as a convex combinations of no more than N pure density matrices, corresponding to the eigenvectors of DN . As a result, any interior point can be obtained from at most N extreme points. This result is quite similar to Carathéodory’s theorem (Kelly and Weiss [1979], Eggleston [1963])

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Table 1 Comparison of polarization and pseudospin descriptors. The symbol “T ” denotes the transpose. The SU(N ) algebra arises naturally when one adopts the pseudospin description of N-level systems Pseudospin

1 2

1

Type of wave Jones vector Symmetry Degree of polarization Scalar invariants ON j Density matrix Number of normalized Stokes parameters: ΘjN

plane (TLS) [ E1 E2 ]T SU(2) P2 tr((D2 )2 ) Pauli matrices D2 3: σj

nonplane (ThLS) [ E1 E2 E3 ]T SU(3) P3 tr((D3 )2 ), tr((D3 )3 ) Gell-Mann matrices D3 8: O3j

which states that any interior point of an arbitrary convex set of dimension d can be obtained as a convex combination of d + 1 (or fewer) extreme points of that set. A basic consequence of this convexity property is that the polarization characteristics can be obtained via an algebraic specification of pseudospin states.11 Weaver [1978] has shown that many problems involving aspects of spin-1 matrix algebra may be handled in a way analogous to the spin- 21 algebra. Table 1 summarizes the important analogies between polarization and pseudospin descriptors.

3.4. Summary of geometric algebra In summary, the reader must keep three points in mind in order to follow the discussion on the geometric description of polarization states that follows shortly: (1) States of polarization are specified by density operators characterized by a rank; they form a convex set. (2) Pure (completely polarized) states have rank 1 and are the extremal points of the set. (3) In a convex set, the set of maximal rank has maximal dimension and forms the interior of the set. The interior is surrounded by a hierarchy of hypersur11 If we consider a pseudospin s (and ignore its spatial degrees of freedom) then with respect to any orthonormal basis the density matrix becomes a (2s + 1)-dimensional matrix. Since the density matrix is Hermitian, there are in all (2s + 1)2 independent parameters characterizing it. It is also worth observing that classical wave systems resemble their electronic counterparts in many ways. Among the significant differences one may count the lack in electronic systems of a true analog to dissipation, and the lack in most classical systems of significant inelastic processes in which waves may change their frequency or lose temporal phase information.

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faces of all lower ranks down to 1. The dimension of a stratum of a given rank drops by 1 as the rank drops by 1. (4) The dimension of the set of states of rank r, 1 r 2s + 1, of a pseudospin s is r[2(2s + 1) − r] − 1, which is a nonlinear function of r. We are now equipped to investigate the polarization of waves using the convexity property of the density matrix.

§ 4. Polarization of a plane wave We first discuss the polarization of plane waves (pseudospin- 12 , or single fermionic degree of freedom) and the parameters which describe them. In the case of a two-level system (TLS),12 the Hilbert space is a real four-dimensional space (Kelly and Weiss [1979], Bloore [1976], Eggleston [1963], Feynman, Leighton and Sands [1965], Feynman, Vernon and Hellworth [1957], Allen and Eberly [1975]). The time-varying electromagnetic field is assumed transverse and the representation of plane waves may be expressed in terms of a complex electric field vector E = [ Ex Ey ]T which has no Ez component for waves traveling along the z-axis, the magnetic field components being uniquely determined by those of E. Ultimately, the transversality condition is entirely related to the vanishing of one excitation degree of a photon (as a zero-mass particle) and is the consequence of the homogeneity of three-dimensional physical space and time. The transversality of an optical field in free space is usually taken into account by the choice of the coordinate system with one axis aligned with the light propagation direction. This is the conventional simplifying assumption made in optics (Born and Wolf [1999]) and is adequate for most purposes. In a paraxial experiment, z would be the axis of the system: all the plane waves travel in directions close to the axis. Two remarks are in order. On the one hand, the two-element complex electric field is formally equivalent to a spinor (Urbantke [1991], Deschamps [1951], Cartan [1966], Penrose and Rindler [1984]). For a monochromatic wave field of angular 12 As pointed out by Feynman, Vernon and Hellworth [1957], the mathematical structures of the theories of a two-level atom interacting with radiation and of an electromagnetic wave interacting with a dielectric medium in which it is propagated as a plane wave are similar. TLS have been extensively used as prototypes for the quantum information carrier (qubit) in quantum computing. The qubit is represented by a normalized vector in R2 , with 0 and 1 represented by a fixed pair of orthogonal basis vectors (Nielsen and Chuang [1996]). It is worth noting that there are many physical embodiments of such TLS, including the spin of a spin- 12 particle, two hyperfine states of a trapped atom or ion, two neighboring levels of a Rydberg atom, and the presence or absence of a photon in a microcavity.

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frequency ω, it is convenient to split off the time dependence and represent the Jones vector, at any chosen point r = [ x y ]T , as E(r, t) = p(r) + iq(r) exp(−iωt), (4.1)

where the vectors p(r) and q(r), real two-dimensional vectors normal to Oz, are the conjugate semidiameters of the polarization ellipse. It may also be useful to consider unitary field vectors, which then have the following parametrization, at a general point r, in terms of two angles θ and δ: E = [ cos θ sin θ exp iδ ]T . The angles are related to the geometrical parameters of the polarization ellipse by a spherical triangle construction on the Poincaré sphere (Brosseau [1998], Shurcliff [1962], Wagh and Rakhecha [1995]). On the other hand, the plane wave extends to infinity in the transverse direction.

4.1. Density matrix and degree of polarization By way of introduction, for N = 2, the convex set of states has two strata, of ranks 1 and 2, and dimensions 2 and 3, respectively. In an orthonormal basis of 4 the complex Hilbert space H = C2 (or equivalently the ! real " vector space R ) the ∗ most general density matrix takes the form D2 = ac cb with a, b real, and c complex. The density matrix obeys the trace condition a + b = 1 and the positivity conditions: 0 a 1, 0 b 1, and det(D2 ) = ab − |c|2 0. In attacking polarization problems, it is useful to introduce the following density matrix expressed in terms of the normalized Stokes parameters:

1 1 σ 2 − i σ 3 1 + σ 1 = (σ 0 + u · σ ), D2 = (4.2)

+ i σ

1 − σ

σ 2 2 2 3 1 where the above mentioned positivity and trace conditions imply that −1 σ j 1

(4.3)

and |u|2 =

3 j =1

σ j 2 1,

(4.4)

σ 1 so that the Stokes vector u = tr(D2 σ ) = σ 2 lies inside or on the Poincaré σ 3 0 ball of center 0 and radius 1 in Cartesian three-dimensional space. In eq. (4.2), 0

σ 0 denotes the 2 × 2 unit matrix. Note that the direction of u gives the type of polarization. The reader is reminded that the constraints on the Stokes vec-

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tor components may be interpreted geometrically using a Lorentzian terminology (Morales and Navarro [2003]); they represent the points that are within or on the positive shell of the “Stokes cone”. Each point of this ball gives a density matrix. The mixed (partially polarized) states have rank 2 and are the points inside the three-dimensional Poincaré (Euclidean) ball S 3 , with reference to fig. 1. The pure states, D2p , are the extreme points of the convex set. The pure state density matrix saturates the inequality (4.4) and are the points on the surface of the ball, i.e. Poincaré sphere S 2 , of rank 1. Only one state is completely unpolarized, i.e. D2u = 12 σ 0 . The points on S 2 are in one-to-one correspondence with the unit rays in C2 . It is worth observing that the mapping from S 3 of unit spinors in C2 to the spheres S 2 in R3 is known as the Hopf vibration (see Appendix A).

Fig. 1. The Poincaré sphere S 2 is the unit sphere surrounding the origin of the Cartesian coordinate orthonormal basis (e1 , e2 ). The normalized Stokes parameters [ σ1 , σ2 , σ3 ]T constitute the components of the Poincaré vector u that represents the state of polarization of an arbitrary pure state for a plane wave (|u| = 1). The longitude 2ψ and latitude 2χ of point M are related to the azimuth and the ellipticity angles of the polarization ellipse of the wave. Each point on S 2 corresponds to a unique state of polarization. The north pole N ([ 0, 0, 1 ]T ) represents left circularly polarized light. The south pole S ([ 0, 0, −1 ]T ) represents right circularly polarized light. Points on the equator (2χ = 0) represent linearly polarized light. Elliptical polarization states lie between the poles and equator. The positive directions of the angle 2ψ and 2χ are defined according to the adopted sign convention.

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The decomposition of a mixed state, point M in fig. 2, as a convex sum of two (2) (1) pure states D2p (point P1 ) and D2p (point P2 ) is given geometrically by drawing (1)

(2)

a straight line through this state; D2p and D2p are its intersections with S 2 . Since convex combinations of two pure states are represented by points on the segment they determine, we see that it admits infinitely many decompositions into pure states. The center of S 3 can be decomposed into an infinite number of pairs of orthogonal pure states corresponding to the many diameters through it. This nonunique decomposability of mixtures is evident. Consider, for simplicity, the section of the sphere S 3 of fig. 1 by a plane through the center, and take a nonpure state M as in fig. 2. After remarking that the convex combinations of two pure states are represented by the points on the segment they determine, we see that M admits infinitely many decompositions into pure states, with weight determined by a sort of barycentric calculus. The two orthogonal (because diametrically opposed) pure states denoted by P1 and P2 in fig. 2 carry one decomposition (the unique decomposition into orthogonal states); the two nonorthogonal pure states Q1 and Q2 carry another decomposition. Observe that by iterated convex combinations we can even decompose M into more than two pure states; for instance the triple P1 , R and S does the trick. More generally we can say that M can be decomposed into

Fig. 2. Schematic illustration of the nonunique decomposability of a mixed state. The point M admits infinitely many decompositions into pure states. Diametrically opposite points on S 2 correspond to mutually orthogonal rays or Hilbert space vectors. It is worth observing that the center of the Poincaré sphere S 3 has a unique status: it can be decomposed into infinitely many pairs of orthogonal states, for there are infinitely many diameters through it. This corresponds to the fact that the center, representing the matrix 12 σ 0 , has a doubly degenerate eigenvalue.

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a family of pure states if it lies inside the polygon, i.e. the convex hull, determined by that family. A mixed state can be uniquely decomposed into a convex sum of a pure state (projection operator D2p , point M in fig. 3) and the completely unpolarized state (point O): D2 = P2 D2p + (1 − P2 )D2u .

(4.5)

The degree of polarization of the wave can be defined naturally as # 3 $1/2

1/2 2 2 P2 = 2 tr (D2 ) − 1 σ j = = |u|.

(4.6)

j =1

The physical relevance of P2 is manifold. First, it has an algebraic significance since it can be presented abstractly as the scalar invariant of D2 . Since the two eigenvalues of D2 can be expressed as λ1,2 = (1 ± P2 )/2, the degree of polariza-

Fig. 3. Schematic illustration of the convexity property of the set of polarization states for a plane wave on the unit ball S 3 . Pure states correspond to surface points (e.g. A) and mixed states to interior points (e.g. M). The partially polarized state described by point M, is represented by the vector OM whose length is the degree of polarization P .

4, § 4]

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Polarization of a plane wave

Fig. 4. Illustration of the definition of the degree of polarization P2 in terms of the eigenvalues of the density matrix D2 . In this approach, we first define what unpolarized means and then measure how far from this state the field is. Two-dimensional unpolarized light is represented by the point U2 .

tion of a plane wave can be also written as13 P2 = λ1 − λ2 = (1 − 4λ1 λ2 )1/2 .

(4.7)

Secondly, it has also a direct geometric interpretation since the set of states whose degree of polarization is P2 is the sphere of radius P2 . Another geometric interpretation of eq. (4.7) has been given by Saastamoinen and Tervo [2004] by considering √the distance between a given mixed state and the unpolarized state, equal to P2 / 2. Figure 4 shows a graph obtained using this procedure that illustrates the variation of this distance with the degree of polarization P2 . As can be seen in fig. 4, any possible mixed state can be identified as a point M in the allowable region of the λ1 –λ2 plane delimited by the triangle OP1 P2 . In this representation, the unpolarized state U2 is located at point [ 12 12 ]T , while the completely polarized states are located in P1 and P2 . Thirdly, an alternative, but equivalent definition of P2 from eq. (4.6) is that it can be interpreted as the length of the Stokes vector. The interrelation of these various approaches arises through the SU(2) symmetry of the polarization states.

13 Observe that the two eigenvalues of D satisfy the relation: λ2 − λ + (1 − tr((D )2 ))/2 = 0. 2 2

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4.2. Some important consequences and concrete applications When a quasimonochromatic plane wave is passed through a linear and deterministic optical device, e.g. a scattering medium or an optical instrument, the input Ei and output Eo Jones vectors are linearly related by virtue of the linearity of Maxwell’s equations. This allows us to write the relation Eo = J2 Ei ,

(4.8)

where J2 defines the 2 × 2 Jones complex-valued matrix. A basic consequence of eq. (4.8) is that the output density matrix is related to the input one by the congruency transformation D2o =

1 J2 D2i J+ 2, g

(4.9)

14 where g = σ 0 o / σ 0 i = tr(J2 D2i J+ 2 ) is the gain of the optical medium. For a passive medium, 0 g 1. By taking the determinant of both sides of eq. (4.9) and making use of eq. (4.6), we get the polarization transfer equations (Brosseau [1996])

and

| det(J2 )|2

1 − (P2o )2 = 1 − (P2i )2 g2

(4.10a)

| det(J2 )|2 2 1− (P2o ) − (P2i ) = 1 − (P2i ) (4.10b) . g2 There are important lessons to be gleaned from the discussion of eqs. (4.10a) and (4.10b). In general, pure input states will evolve into mixed states. However, these equations imply that an incident pure state remains pure after interaction with the optical medium although a mixed state can increase or decrease its degree of polarization. Since the topology of the problem at hand is isomorphic to a sphere, depolarization of a plane wave occurs with no topological changes. The different mechanisms of depolarization are connected to a contraction in the Poincaré space while increasing the purity character is a dilating process, see Brosseau [1998]. The reader can easily show that in the important case where J2 is unitary, then P2i = P2o . Optical retarders which introduce a phase difference between vibrations polarized in orthogonal states of linear polarization and systems exhibiting optical activity both fall into this category. 2

2

14 It is worth noting that in nuclear physics, the scattering of a particle may be described by a density operator which transforms the initial to the final spin state in a similar way as eq. (4.7). Within this dσ = tr(J D J+ ) ≡ g. formalism the differential cross-section for a collision is dΩ 2 2i 2

4, § 4]

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One of the conveniences of the Poincaré sphere is that it provides a geometrically appealing view of the transformation of a polarized light interacting with optical (linear) devices in terms of rotations of states. Note that the SU(2) group is homomorphous to the special orthogonal group in three dimensions SO(3, R) (Biedenharn and Louck [1981], Spanier [1966]), a result that leads to important mappings for the complex space of elliptical polarization states into the real space of the Stokes vector, i.e. stereographic projections (Brosseau [1998]). This result also has implications for the definition of absolute phase in polarization measurements (Penrose and Rindler [1984]). Our viewpoint also carries with it many interesting and important questions about how dynamical features of the polarization states are related to the symmetry of the Stokes parameters. It is worth noting that aspects of spin- 21 have been reported which are operative to study the evolution of polarization states of light propagating in an arbitrary anisotropic dielectric media, e.g. the analogy between the evolution of the Stokes vector and the spin-vector formalism of Bloch developed for nuclear magnetic resonance (Brosseau [1998], Urbantke [1991], Schlichter [1990], Meier and Zakharchenya [1984], Zapasskii and Kozlov [1995]). In fact, spin (pseudospin) decoherence is considered to be a universal phenomenon associated with TLS. To proceed, one considers the Bloch equation for the density matrix that evolves in the standard fashion ih¯ ∂t D2 = [H, D2 ],

(4.11)

where H is the Hamiltonian operator, [A, B] is the notation for a commutator and h¯ is the reduced Planck constant. Equation (4.11) belongs to a class of differential equations known as integrable Hamiltonian systems. Taking H = 3j =0 Hj σ j , and using eq. (4.2), the equation for the three-dimensional Poincaré vector u, i.e. the equation of motion for the TLS, can easily be obtained; it is ∂t u = × u,

(4.12)

where the torque vector is = 2H/h¯ . Consequently, if H is time-independent, eq. (4.12) is the equation of motion of u around H, which is familiar from nuclear magnetic resonance, that is, spin precession around the direction of a constant magnetic field. It is straightforward to show, on taking the dot product of eq. (4.12) with u and , that u · u and · u are both invariants of the motion. The former implies that the motion takes place on the Poincaré sphere; a pure state will remain pure. The latter implies that the motion is confined to a plane. Consequently, the orbit of motion is given by the intersection of the two preceding surfaces and is the circle lying on the Poincaré sphere. Figure 5 displays

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Fig. 5. Schematic illustration of the precession of the Poincaré vector u on the unit sphere. As time evolves, the tip of u traces out a curve on S 2 . The orbit of motion is closed and periodic, starting from an initial state u0 .

a typical orbit of motion. It is worth mentioning that Tratnik and Sipe [1987a, 1987b, 1987c] also considered the polarization dynamics of a pulse propagating through a nonlinear medium, in the limit of small nonlinearities, anisotropies and dispersion, using the SO(3) covariance of the Stokes parameters. The symmetry approach to solving differential equations can be found in several books (Bluman and Cole [1974], Bluman and Kumei [1989], Olver [1986]). In short, it can be stated as follows: a symmetry group of a system of differential equations is a Lie group acting on the space of independent and dependent variables in such a way that solutions are mapped into other solutions. Knowing the symmetry group allows one to determine some special types of solutions invariant under a subgroup of the full symmetry group, and in some cases one can solve the equations completely. As emphasized by Venkatesh and Ram [1976a, 1976b], polarization and dynamics are two aspects, or two alternative modes of description, of an interaction process. Interestingly, it was also realized in recent years that TLS, such as

4, § 4]

Polarization of a plane wave

347

Fig. 6. Plot of the entropy S2 as function of the degree of polarization P2 , i.e. scalar invariant tr(D22 ), of a plane wave.

atoms and spins, can be used as prototypes for the quantum information carrier in quantum computing. The whole set of quantum operations can be accomplished through geometric operations arising from the geometric phase. In the light of these considerations, the polarization entropy is defined as

S2 (P2 ) = − tr(D2 ln D2 ) = − ln s(P2 ) , (4.13)

with s(x) = 12 (1 − x)(1−x)/2 (1 + x)(1+x)/2 . The limiting value are S2 (P2 = 0) = ln 2 and S2 (P2 = 1) = 0, with a monotonic decay from a completely unpolarized wave to a completely polarized wave as displayed in fig. 6. Substitution of eq. (4.10b) into eq. (4.13) yields the following equation for the entropy production which provides a natural description of the “degree of disorder” associated with partially polarized waves propagating in linear nonimage forming optical media s(P2i ) S2 = S2o − S2i = ln s(P2o ) s(P2i ) . = ln (4.14) s( 1 − (1 − (P2i )2 )(| det(J2 )|2 /g 2 )) Brosseau and Bicout [1994] have shown that the wave transmission is reversible if the entropy is conserved and if no absorption occurs; otherwise the wave transmission is irreversible. Typical examples of reversible devices are rotators and com-

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pensators, whereas the transmission of a plane wave normally incident through a plane polarizer is accompanied by a variation of entropy, a feature that is consistent with the fact that the polarization process is irreversible. As defined in this way the entropy is obviously an extensive quantity (Brosseau [1998], Weaver [1978], Gyftopoulos and Çubusku [1997]). Other traditional thermodynamic functions for a partially polarized wave can be defined to characterize the statistical state of the system. The details of the calculations are not shown here in the interest of brevity; the interested reader may wish to consult Brosseau [1998] for the details. A precise definition of the polarization mechanism may be the following: the purification (polarizing) procedure allows us to distill from a set of mixed states a subset of states with a higher degree of purity P2 , i.e. it stochastically increases the purity by filtering out some of the unpolarized part of the partially polarized incident wave. For completeness we indicate that Picozzi [2004] has shown that the definitions of entropy and degree of polarization we have chosen are not relevant for nonlinear optical plane waves, because they would lead to the erroneous conclusion that a genuine process of reversible propagation would be characterized by an irreversible process of entropy production. This arises because a general property of a nonlinear interaction is to couple the evolution of second-order statistical moments to higher-order moments so that the Stokes parameters cannot consistently describe the nonlinear propagation of a partially polarized plane wave (Svirko and Zheludev [2000], Sagdeev, Usikov and Zaslavsky [1988]). One may also introduce the “relative entropy” that measures the closeness of any output state D2 with respect to a corresponding state D′2 as

S2 D2 D′2 = tr(D2 ln D2 ) − tr D2 ln D′2 1 − P2 1 + P2 1 + P2 1 − P2 ln ln + . (4.15) = 2 1 − P1 2 1 + P1

The relative entropy is a monotonic, increasing function of P2 with its maximum occurring when D2 is completely polarized and D′2 is totally unpolarized. If D′2 = D2i and D2 = D2o , then is the relative entropy of the input and output polarization states of a plane wave that has traversed a linear and deterministic optical device characterized by a Jones matrix J2 . Note that if J2 is unitary, then S2 (D2o D2i ) = 0. Any nonunitary Jones matrix, e.g. an anisotropic absorber, will cause the relative entropy to be positive. Theoretical details can be found in Barakat [1996]. It should, perhaps, be emphasized again that in the context of elastic multiple scattering of polarized light by a dense random collection of uncorrelated and noninteracting spherical dielectric particles, we undertake the task of numerically

4, § 5]

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349

establishing the size and polarization state dependences of the characteristic depolarization length (Bicout, Brosseau, Martinez and Schmitt [1994], Brosseau and Bicout [1994]). In a medium containing particles small compared to the wavelength (Rayleigh regime), the characteristic length of depolarization for incident linearly polarized light is found to exceed that for incident circularly polarized light, while the opposite is true in a medium composed of particles large compared to the wavelength (Mie regime). These numerical results are compared with measurements on suspensions of polystyrene latex spheres in water. The striking feature of this problem, where no energy exchange between radiation and scatterer takes place, is that the stationary state corresponds both to the state of minimum production of radiation entropy and to the state of maximum entropy.

§ 5. Polarization of an arbitrary wave Having carefully considered the planar structure of electromagnetic waves, we are now ready to move on and examine what happens in the case of a wave which is not plane, but three-dimensionally structured. It could be thought of as a superposition of plane waves all traveling in different directions without any preferred direction, as in a cavity, for example. The following discussion extends to higher dimensions with larger groups. Because the whole problem is three-dimensional (N = 3) we need an approach that deals with the field vector E = [ Ex Ey Ez ]T itself rather than just with its transverse components. If we consider, as before, monochromatic waves, we represent the Jones vector at a given point r = [ x y z ]T in space by ˜ ˜ + iq(r) exp(−iωt), E(r, t) = p(r) (5.1)

˜ ˜ where p(r) and q(r) are real vectors in three dimensions. Here again we find that at each point of space the Jones vector simply rotates in a plane to describe an ellipse. The only difference is that p˜ and q˜ are three-dimensional vectors defining the plane of the ellipse, whereas p and q were two-dimensional vectors perpendicular to Oz. As in the paraxial case the vectors p˜ and q˜ are conjugate semidiameters of the polarization ellipse. All this refers to a particular point in space in the wavefield (Born and Wolf [1999]). The shapes and planes of the polarization ellipse are, of course, different at different places in space. Nye [1983, 1999] has explored this three-dimensional “continuum” of polarization ellipses being executed in time to identify structurally stable features as landmarks in the pattern. Observe that the ellipse has a sense of rotation but it is not a straightforward matter to assign a handedness to each point of an arbitrary monochromatic wavefield

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[4, § 5

because that would require knowing a direction of propagation, i.e. the analogue of Oz in the paraxial case. We describe this system as a pseudospin-1 (ThLS) and the state of polarization has a SU(3) symmetry (Table 1).

5.1. Density matrix and polarization descriptors 2, 3 In that case H = C3 , so the convex set of states has three strata, ranks a 1, h∗ g with dimensions 4, 7, 8, respectively (Bloore [1976]). Typically D3 = h b f ∗ g∗ f c

with a, b, c real, f , g, h complex, and a trace condition a + b + c = 1. The positivity conditions are now: a 0, b 0, c 0, |f |2 bc, |g|2 ca, |h|2 ab, and det(D3 ) = abc + 2 Re(f gh) − (a|f |2 + b|g|2 + c|h|2 ) 0. For the present purpose, the remarkable result is that the matrices D3 of rank 3 [for which det(D3 ) > 0] form a convex region of R8 . The four-dimensional subregion of pure states of rank 1 sits in the 7-dimensional boundary of mixed states of rank 2. Every point in this region can be written as a convex decomposition of 3 pure states, and of not less than 3. In terms of the eight normalized Stokes parameters (Table 1) the most general density matrix is ⎡ ⎤ √ 1+ O34 / 3+ O33 O31 −i O32 O35 −i O36 1⎣ √ O31 +i O32 1+ O34 / 3− O33 O37 −i O38 ⎦ , D3 = (5.2) √ 3 3 3 O +i O O3 +i O3 1−2 O3 / 3 5

O3j ’s

6

7

8

4

denote the Gell-Mann matrices (Appendix A). It is interesting where the to examine the analogies between eqs. (4.2) and (5.2): O33 and O34 appearing only on the diagonal generalize σ 1 ; the terms in the symmetric, off-diagonal part, O31 , O35 and O37 , generalize σ 2 , and O32 , O36 and O38 , appearing in the antisymmetric, imaginary part, σ 3 . It can be readily seen that the trace of D3 in eq. (5.2) is equal to 1, as required. In Section 3 we saw that the problem in determining DN is to determine what sets of observables will in fact yield N 2 − 1 independent equations for this purpose. For N = 3, we need to find a minimum of eight independent observables. Two different types of definitions are in use; depending on one’s taste one uses the so-called Cartesian tensor operators ui , uij or the spherical tensor operators tkq . If one deals only with vector polarization, the Cartesian notation is appealing. For tensor polarization the spherical tensor operators have the advantage of simpler transformation under rotation of the coordinate system. The procedure we have adopted arises from the fact that the number N 2 is related to the number of independent elements in a sequence of tensors defined in terms of angular momentum

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operators, and referred to elsewhere in theoretical physics as multipoles (Robson [1974], Fano and Racah [1959]). Thus, a multipole tensor operator tkq of rank k, usually called an irreducible spherical tensor operator (Appendix B), has 2k + 1 2 independent components, q = −k to +k, and because N−1 k=0 2k + 1 = N , we assume that any observable in H can be expressed as a linear combination of the N 2 operators included in the set of all tensor operator components tkq , k = 0 to N − 1. Stated differently, the tkq represent a minimal set of noncommuting observables (Robson [1974], Fano and Racah [1959]). Following the Cartesian approach, the density matrix D3 can be expressed in a convenient way, in analogy with a general spin- 12 system [see eq. (4.2)]: 3 1 O30 + u · S + 3uij Sij . D3 = (5.3) 3 2 It is apparent that the specification of the state of polarization requires the knowledge of the three parameters ui and the five parameters uij describe the mixed state of a general spin-1 system, i.e. eq. (B.4) (Dalitz [1952]). Equation (5.3) needs to be properly understood and interpreted. In particular, one needs to introduce two different types of polarization descriptors: a polarization (axial) vector in a similar way as previously, u = tr(SD3 ),

(5.4)

where S denotes the vector of spin matrices Si for a spin-1 in a Cartesian basis (see Appendix B), and a quadratic polarization tensor uij = tr(Sij D3 ),

(5.5)

where Sij is a symmetric tensor defined by the expression Sij = 21 (Si Sj +Sj Si )− 2 3 3 δij O0 . The vector part is also called the orientation and the tensor part the alignment (Dalitz [1952]). Note that the tensor uij may be taken to be symmetric and traceless. While the Stokes vector u specifies the polarization of a plane wave completely, for a nonplane wave it is possible that ui = 0, yet uij = 0 so that the ordering in polarization space is generally more complicated than for the plane wave. Dennis [2004] has remarked that the axial vector u represents an average of the angular momentum in the spin language, i.e. the average sense of rotation of the ellipses in the ensemble of waves. For a general mixed state all eight Stokes parameters (u and uij components) are independent, while for a pure state only four of the eight components are independent. If the polarization vector u is zero and all the components of the polarization tensor uij are zero, then polarization is completely absent and the density matrix D3u ∝ O30 . An example of a completely polarized state has

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u = [ 0 0 1 ]T and u11 = u22 = − 31 , u33 = 32 . A completely aligned pure state has u = 0 and uij uij = 83 . In this case the four independent parameters can be chosen to be u11 , u22 , u33 and u12 . Now D3 can be readily shown to have two scalar invariants: tr((D3 )2 ), and tr((D3 )3 ) or det(D3 ). Unfortunately, the eigenvalues of D3 do not have a simple geometric interpretation in terms of u and uij . In like fashion as one was able to define the degree of polarization of the wave P2 as a function of its scalar invariant, Barakat [1977] suggested to define two “degrees of polarizations” in terms of the above scalar invariants.15 Along the same lines, a different approach was recently given by Setälä, Shevchenko, Kaivola and Friberg [2002], Lindberg, Setälä, Kaivola and Friberg [2004], Setälä, Lindfors, Kaivola, Tervo and Friberg [2004] and Lindfors, Setälä, Kaivola and Friberg [2005]. These authors suggest another definition of the degree of polarization, # 8 $1/2 2 3 1 1/2 3 2 P3 = Oj . = tr(D3 ) − 2 3

(5.6)

j =1

Note that a different coefficient appears in front of their expression owing to the slightly different basis matrices they used (Setälä, Shevchenko, Kaivola and Friberg [2002], Lindberg, Setälä, Kaivola and Friberg [2004], Setälä, Lindfors, Kaivola, Tervo and Friberg [2004], Lindfors, Setälä, Kaivola and Friberg [2005]). This definition was arrived at by considering the two limiting cases of unpolarized wave and totally polarized wave that are idealizations in practical calculations. An unpolarized wave is characterized by a density matrix, i.e. D3 = 31 O30 . Thus tr((D3u )2 ) = 13 and P3 = 0. Recalling that for a pure state of polarization we can write tr((D3p )2 ) = tr(D3p ) = 1, we obtain P3 = 1. Observe that P3 cannot be considered as the length of the vector u, in contrast with the two-dimensional case. The reader may question whether eq. (5.6) is the most appropriate descriptor of the degree of polarization of a nonplane wave. Some physical insight into the concept of three-dimensional polarization can be obtained by referring to the preceding discussion. Multiplying the equality (B.4) by D3 and taking the trace

15 In fact, one of the two “degrees of polarizations” suggested in Barakat [1977, 1983] is the same as eq. (5.6). Two comments are in order. First, it is worth noting that in Barakat [1977, 1983] a formula is suggested as the degree of polarization in the N -dimensional case. For N = 2, it reduces to eq. (4.6), and for N = 3 to eq. (5.6). Second, a similar formula for the degree of polarization had, in fact, been put forward in the geophysics literature already a couple of decades ago, see, i.e. Samson [1980a], Samson and Olson [1980b].

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of both sides of the resulting equality, we can show that k 2 k 2 1

t . tr (D3 )2 = q 3

(5.7)

k=0 q=−k

If the wave is completely polarized, then tr((D3 )2 ) = 1; in a mixed state tr((D3 )2 ) < 1. Therefore we can readily show that 31 2k=0 kq=−k |tqk |2 1. Now using eq. (5.7) we have for pure states of polarization 1 2 1 1 1 2 1 2 2 tq + tq = 1. + 3 3 3 q=−1

(5.8)

q=−2

The reader can readily show (from the results of Appendix B) that for a pure state of polarization, the expectation values of the Lakin operators are: t01 = 3 2 − |E |2 )/I , t 1 = − 3 ( E ∗ E + E ∗ E )/I , t 1 = −(t 1 )∗ , ( |E | x z z x y 1 −1 1 2 2 y 3 3 2 2 2 2 ∗ ∗ t0 = √ (( |Ex | + |Ez | )/I − 2/3), t1 = − 2 ( Ex Ey − Ey Ez )/I , 2 √ 2 = −(t 2 )∗ , t 2 = − 3 E ∗ E /I , t 2 = (t 2 )∗ , where I denotes the total t−1 x z 1 2 −2 2 intensity of the wavefield. Substituting these values into the individual terms of the left-hand side of eq. (5.8), we obtain 1 2 1 |Ex |2 − |Ez |2 2 | Ex∗ Ey + Ey∗ Ez | 1 1 2 tq = + 3 2 I I

(5.9a)

q=−1

and

2 1 2 2 3 |Ex |2 + |Ez |2 2 2 tq = − 3 2 I 3 q=−2

+

| Ex∗ Ey − Ey∗ Ez | 2 I

+2

|Ex |2 |Ez |2 . I2

(5.9b)

All these quantities refer to a particular point in space. Substituting eqs. (5.9a) and (5.9b) into eq. (5.8), it is easy to show that eq. (5.8) is satisfied for a pure state. Our previous discussion indicates that eqs. (5.9a) and (5.9b) can be interpreted as the weights of respectively the vector and tensor polarization in the wave. If eq. (5.9a) vanishes, there is no vector polarization in the wave. This observation leads us to the following conclusion: eqs. (5.9a) and (5.9b) define the maximum possible weights of the vector and tensor polarization in the wave, which are realized only for a pure state.

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The degree of polarization of the wave can readily be found by substituting eq. (5.7) into eq. (5.6): # # 2 k $$1/2 1 k 2 P3 = (5.10) . tq − 1 2 k=0 q=−k

Thus a general definition has been arrived at. It should be noted that the degree of polarization, whatever the dimensionality of the space of polarization states, represents an average local order parameter. It is widely believed that the only order parameter required to describe the average statistical correlation between the electric field vector components at a given point in space is the degree of polarization. Not accepting this common-sense view, we have argued that a new additional order parameter, is necessary for the complete average local description of polarization of an arbitrary wave in three dimensions. Some comments are in order before continuing. First, it is worthwhile asking the following question: What are the key differences between the polarizations of plane and nonplane waves? A special situation is the case for which one of the components ! "of the Jones vector is zero, i.e. Ez = 0, then eq. (5.2) reduces to D2 0 D3 = 0 0 which is analogous to the plane-wave case. Stated in mathematical terms, this is simply a consequence of the fact that SU(2) is a subgroup of the 2 =0 SU(3) algebra. In terms of tqk ’s, the transversality condition requires that t±2 1 2 and t±1 = t±1 so that the density matrix can be written ) ⎡ t01 t1 0⎤ −1 2⎣ ⎦. 3 1 1 D3 = (5.11) 2 − t0 0 3 −t1 0 0 0 Now by definition, the density matrix for a plane wave is given by eq. (4.2). It follows that the Stokes parameters are given by σ 1 = 2 23 t01 − 1 and 1 . This result is consistent with the description in terms of σ 2 ± i σ 3 = ∓ 23 t±1 a triplet of dipole operators (Appendix B) verify. Noting that for an√unpolarized √ 1 3 0 1 k plane wave the only nonzero tq are t0 = 1, t0 = 2 2 and t02 = 3t01 − 2, we observe that such a wave, strictly speaking, is partially polarized from the point of view of the three-dimensional analysis, i.e., P3 = 12 as obtained from eq. (5.10). Thus a plane wave cannot be completely unpolarized in the sense of the definition in eq. (5.10). We recover a result of Setälä and co-workers (Setälä, Shevchenko, Kaivola and Friberg [2002], Setälä, Kaivola and Friberg [2002]), albeit obtained from a different approach. Figure 7 provides a simple physical interpretation of this result. Since the oscillations in the two-dimensional wavefield

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Fig. 7. Symbolic illustration of the difference between the two-dimensional and three-dimensional approaches for treating the case of an unpolarized wave.

are restricted to a single plane, the field is not random in the direction orthogonal to that plane. Thus, such field cannot be unpolarized in the three-dimensional approach. It should be also noted that for a plane wave the degree of polarization in the sense of the definition [eq. (5.10)] cannot be less than 12 . Stated differently, because of the isotropic and disordered nature of the three-dimensional unpolarized state, the anisotropy hidden (along the x-direction in fig. 7) in the twodimensional unpolarized state makes these two states fundamentally different. A notable illustration of this point has been developed by Ellis and Dogariu [2005a]. Figure 8 displays a graph of the decomposition of the density matrix into three types of components, where the axes represent the orthogonal states of polarization, and the lengths of the cylinders denote the energetic contribution of a specific field component (Ellis and Dogariu [2005a]). Based on this realization, and using energy-density arguments, a physical definition of the degree of polarization can be advanced as a measure of the energy density in the polarized state with respect to the total energy of the field, i.e. P3 = (λ1 − λ2 )/(λ1 + λ2 + λ3 ), where the λj ’s denote the eigenvalues of cross-spectral coherency matrix (Ellis, Ponomarenko, Dogariu and Wolf [2005]). It is clear that 0 < P3 < 1, with P3 representing completely unpolarized field and P3 = 1 completely polarized field at a given point in space. It should be noted that when the field is two-dimensional, λ3 = 0, so P3 reduces to the usual expression for the degree of polarization, i.e., eq. (4.7). It is also worth noting that the preceding argument has been “generalized” by Alodzhants and Arakelyan [1998, 2004] in the statement that a state with SU(3) symmetry is polarized to a greater extent than the state with SU(2) symme-

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Fig. 8. Generalized state of polarization of a three-dimensional field. The lengths of the cylinders represent the energy density in the specific, uncorrelated and orthogonal states of polarization. The λj ’s denote the eigenvalues of cross-spectral coherency matrix (adapted with permission from Ellis and Dogariu [2005a]).

try. Furthermore, such statement may yield useful rules of thumb for engineering practice. For further discussion of this question, the reader may wish to consult the articles by Alodzhants and Arakelyan [1998, 2004]. Does this mean that the two-dimensional case can be viewed as a limiting case when the direction of propagation of the wave is known? For this purpose, let us consider the case of a wave with curved wavefronts. The simplest such wave is the type that is emitted by a point source that radiates waves isotropically in all directions such as displayed in fig. 9. At a given point in space, M, the temporal evolution of the electric field can be viewed as the evolution of a transverse component, ET , confined to the plane tangent to the spherical wavefront at M and the concomitant evolution of a component which is normal to that plane, EL . If we view the sphere simply from the perspective of symmetry, the only symmetry operations that leave the sphere invariant are the continuous rotations about the axis along which the longitudinal component is directed. Based on this picture, the evolution of the transverse component can be treated as a plane wave, i.e. since the field is harmonic the end point of the transverse electric field vector traces out a closed curve (ellipse) in this plane with increasing time. In this context, a further point to be mentioned is that the fact that the electric field is confined to a plane does not imply that it should necessarily have a polarized component. Remarkably, the orthogonal component evolution will contribute to the degree of polarization of the wave at point M only in the near field since the amplitude of the longitudinal component decays ∝ 1/R, where R denotes the distance OM (Born and Wolf [1999]). The factor 1/R accounts for the conservation of energy in the outgoing wave. Interestingly, even if the longitudinal and transverse components

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Fig. 9. Geometric configuration of a spherical wave illustrating the difference between three-dimensional and two-dimensional degrees of polarization. ET and EL denote the transverse and longitudinal components of the electric field vector at point M, respectively.

of the electric field vector are assumed to be uncorrelated, the degree of polarization of that wave, P3 , will differ from the degree of polarization, P2 , of the transverse wave. This arises because the latter does not contain any information on the energy density of the longitudinal component. Second, and in analogy with the two-dimensional case, an alternative approach to obtaining an expression for the degree of polarization is to consider the eigenvalues of D3 . This yields16

1/2 P3 = 1 − 3(λ1 λ2 + λ1 λ3 + λ2 λ3 ) . (5.12) In an analysis similar to that of Section 4.1 for the case of two-dimensional fields and originally presented by Setälä, Lindfors, Kaivola, Tervo and Friberg [2004], the degree of polarization, P3 , appears to represent the distance between the un-

polarized state and any mixed state, i.e. equal to 23 P3 , as illustrated in the perspective view of fig. 10. The allowed values of λ1 , λ2 and λ3 are inside and on the regular tetrahedron OP1 P2 P3 . Three of the four vertices P1 , P2 and P3 denote completely polarized states, and the unpolarized state is located at point U3 = [ 13 13 13 ]T . However, the reader is cautioned that there is a remark2 16 The three eigenvalues of D satisfy the relation λ3 − λ2 + (1−(P3 ) ) λ − 2 (tr(D )3 − (P )2 ) = 0. 3 3 3 2 3

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Symmetry properties and polarization for an arbitrary wavefield

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Fig. 10. Perspective view illustrating the definition of the degree of polarization for a three-dimensional field. Three-dimensional unpolarized light is represented by the point U3 .

able difference between the two cases displayed in figs. 4 and 10. In fig. 10, the endpoint of the line defined by the mixed state and the unpolarized state is not necessarily a pure state but is generally located at some point on the line between two pure states (Saastamoinen and Tervo [2004]). This point is consistent with fact that any mixed three-dimensional polarization state cannot be expressed as a linear combination of a pure state and the unpolarized state as is the case for a plane wave. Related to this geometric interpretation of the degree of polarization, we further note that Luis [2005] proposed that this parameter can be expressed as the trace distance between the field correlation matrix (density matrix, in our notation) and the identity matrix associated with unpolarized light.17 As an illustration of the above development, Appendix C considers the situation of a ran17 While it is always possible to associate to any field a degree of polarization with a clear physical meaning, it should not be expected that this simple measure will provide a complete description of the polarization characteristics. As anticipated from the fact that D3 has two scalar invariants, one single parameter is insufficient to describe the local order random electromagnetic field. This is further illustrated in fig. 3 in Ellis and Dogariu [2005a].

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dom evanescent wave with electric field, represented by a statistically stationary ensemble, that is incident on a half-space containing black-body radiation. Third, we would like to note that, as we see from the analysis presented here, a mixed state (8 Stokes parameters) cannot be decomposed into an incoherent sum of a pure state (4 Stokes parameters) and the completely unpolarized state. We would like also to point out that, in analogy with the plane wave, a mixed state can be expressed as the sum of three completely polarized states, as was anticipated above, by writing18 ⎡ ⎤ |a1 |2 a1 a2∗ a1 a3∗ D3 = ⎣ a1∗ a2 |a2 |2 a2 a3∗ ⎦ a1∗ a3 a2∗ a3 |a3 |2 ⎤ ⎡ ⎤ ⎡ 0 0 0 |b1 |2 0 b1 b3∗ +⎣ 0 (5.13) 0 0 ⎦ + ⎣ 0 |c2 |2 0 ⎦ . ∗ b1 b3 0 |b3 |2 0 0 0

Five parameters are associated with the first matrix, two with the second, and one with the third, making a total of eight parameters. Since the decomposition in eq. (5.13) or (5.3) is written as a sum of three density matrices, a decomposition in terms of a single purely polarized part and the unpolarized part such as in eq. (4.5) is, in general, impossible. Fourth, as was previously noted, it is possible to have a polarized wave with 1 = 0 which is a pure tensor-polarized wave. For example, the wave with t01 = t±1

√ 00 0 density matrix 0 1 0 has all the tq1 , tq2 equal to zero except t02 = − 2. Con000

versely, a pure vector-polarized wave, i.e. all tq2 = 0, is only partially polarized. Fourth, an equal mixture of opposite “spin states” leads to a density matrix which is not equal to the density matrix D3u , i.e. ⎤ ⎤ ⎡1 ⎡ ⎡ ⎤ 1 0 0 0 0 0 0 0 3 1 1⎣ (5.14) 0 0 0 ⎦ + ⎣ 0 0 0 ⎦ = ⎣ 0 31 0 ⎦ . 2 2 0 0 0 0 0 1 0 0 13

As discussed in the previous section, plane waves are particularly nice geometries which are directly based on the Poincaré sphere S 2 description of pure states. We have seen in Section 4 how to construct geometrically the convex set of density matrices, i.e. eq. (4.2), with the pure states as the extremal points. A strategy for explicit evaluation of how the four-dimensional subregion of pure states of 18 This was also used in the first demonstration of three-dimensional polarimetry by Ellis and Dogariu [2005b].

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Symmetry properties and polarization for an arbitrary wavefield

[4, § 5

rank 1 sits in the 7-dimensional boundary of mixed states of rank 2 has been reported by Bloore [1976]. From a practical point of view, we must be aware of the limitations of the approach. In particular, regions of space of dimension greater than 4 are hard to visualize. Bloore made the important remark that one can to some extent imagine a four-dimensional region as a family of three-dimensional slices because any plane slice of the region is again convex. However, the topological boundary of the convex domain of positivity of the density matrix D3 is much more complex to describe, even if this boundary has, in fact, quite interesting geometrical properties, involving curved edges and rulings of various dimensions, and a general kind of “obliqueness” [stemming from the fact that the unitary group U(3) of C3 ] is, by its action on density matrices, homomorphically mapped onto a genuine subgroup of the group SO(8, R) of rotations in state space. The interested reader is directed to Bloore [1976]. A variety of theoretical methods have been developed to deal with the polarization and alignment of spin-1 states. For example, the Majorana19 construction should prove useful in “visualizing” the conditions that the density matrix describe a pure state, i.e. the Majorana representation in which a pure state E (spin-1) is modeled as a “star” consisting of the polarization vectors of 2 spin- 12 systems. These two vectors have a complicated expression in terms of the pure state E. The two sets of spherical angles (θi , φi ) giving the directions of these vectors correspond to the 4 parameters identifying a pure state and are determined by the roots of a certain polynomial of degree 2 (Majorana [1932]). We should also mention the work of Mullin, Keller, Hammer and Good [1966] who made a somewhat related use of the previous works of Majorana [1932] and Biedenharn [1958, 1959]. We shall not go through the algebra here, but merely report the results. In Mullin’s approach, an arbitrary pure state is described by two real unit vectors v1 and v2 . Such a state determines a direction u = tr(SD3 ) = v1 × v2 . It was shown that a pure state can be decomposed into two orthogonal parts, one of which is completely polarized (this occurs when v1 is perpendicular to v2 ) and the other of which is completely aligned (which occurs when v1 and v2 are parallel), with the axis of the aligned part along the direction determined by the polarized part. In general a pure state determines an orthogonal triplet of axes, v1 − v2 , v1 + v2 , and v1 × v2 . For mixed states they were able to propose measures of the degree of purity, i.e. ξ = 23 (tr(D3 )2 − 31 ) = P32 (eq. (43) of Mullin, Keller, Hammer and Good [1966]), and of the relative amount of polarization and alignment, i.e. which are analogous to the second and third terms of the left-hand side 19 The Italian physicist Ettore Majorana, who mysteriously disappeared while traveling by ship from Palermo to Naples in 1938 at the age of 32, is best known for his contributions to quantum theory.

4, § 5]

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of eq. (5.8), respectively. Another point of view has been reported by Ramachandran and Mallesh (Ramachandran and Mallesh [1989], Ramachandran, Mallesh and Ravishankar [1984]), who described an alternate geometrical representation for the state of polarization of spin-1 particles.

5.2. Some applications Up to this point, our focus has been on the development and extension of the usual plane-wave method for describing the state of polarization of an arbitrary wave. Having established several quantitative results and matters of principle, this section illustrates several applications. When an optical wave passes through a linear medium or non-image-forming device, its density matrix is in general modified. Consider the generalized 3 × 3 Jones matrix J3 defined by Eo = J3 Ei .

(5.15)

It is a simple matter to show that if an input light with a density matrix D3i is passed through any system described by the Jones matrix J3 , then the output wave is characterized by a density matrix which can be written in the form D3o =

1 J3 D3i J+ 3, g

(5.16)

where g denotes the transmittance of the optical medium. By evaluating eq. (5.16) for incident unpolarized light it is easily shown that the gain is given in that case by gu = 13 tr(J3 J+ 3 ). Now multiplying the equality (5.16) by D3o and taking the trace of both sides of the resulting equality, we obtain

1

+ tr (D3o )2 = 2 tr J+ 3 J3 D3i J3 J3 , g

(5.17)

which cannot be simply related to tr((D3i )2 ), as in the two-dimensional case, i.e. eq. (4.10a). Thus, from eq. (5.17) we learn that, in strong contrast with the planewave case, an incident pure state is not generally in the form of a pure state after interaction with the optical medium. Similarly, the operation of a unitary transformation on polarization states is, like in the two-dimensional case, interpreted as the operation of a phase retarder, and the degree of polarization P3 is unchanged because tr((D3 )2 ) remains invariant. However, as remarked by Dennis [2004], there is a more serious problem with treating D3 completely in analogy with D2 : there is no obvious physical interpretation via optical elements of 3 × 3 unitary

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Symmetry properties and polarization for an arbitrary wavefield

[4, § 5

transformations. This is because in three dimensions the ensemble of waves does not share a common direction of propagation, in general, contrary to the planewave case for which one considers an ensemble of realizations with the same direction but different polarization states. In like fashion, we can define a generalized 8 × 8 Mueller matrix which is not so intuitive physically (Barakat [1996], Luis [2002]). Although the question of the physical meaning of each element of this generalized Mueller matrix is a complex one and lends itself to a variety of rather subjective answers, we mention that Robson [1974] proposed to divide the Mueller matrix elements into four groups, each containing specific information about the polarization transfer between vector and tensor polarizations. An experimental test of this statement would be of considerable interest. It is worthwhile considering the polarization entropy of an arbitrary wavefield

S3 = − tr D3 ln(D3 ) , (5.18)

which is expected to depend on the two scalar invariants tr((D3 )2 ) and tr((D3 )3 ) (Brosseau [1998]). The plot of S3 as a function of these two scalar invariants in fig. 11 shows the monotonic growth of the entropy as the system evolves from a pure to a mixed state. The experimental characterization of the three-dimensional polarization of an optical wavefield needs to be addressed. Characterizing the polarization state in a nonparaxial optical field is not a trivial task, because the properties of polarization-sensitive elements, such as birefringent systems, depend in general on the direction of propagation of the light. Dandliker, Tortora, Vaccaro and Nesci

Fig. 11. Plot of the entropy S3 as function of the two scalar invariants tr((D3 )2 ) and tr((D3 )3 ).

4, § 6]

Postscript

363

[2004] discussed the possibility of the possibility of measuring a pure state by using a heterodyne. Coherent light diffracted by microstructures produces such fields. Alodzhants and Arakelyan [1998, 2004] proposed an interferometric device for measuring the Gell-Mann parameters of an optical field having SU(3) symmetry. Setälä and co-workers (Setälä, Shevchenko, Kaivola and Friberg [2002], Lindberg, Setälä, Kaivola and Friberg [2004], Setälä, Lindfors, Kaivola, Tervo and Friberg [2004], Lindfors, Setälä, Kaivola and Friberg [2005]) have recently proposed that the focal region of a high-numerical-aperture imaging system is a good candidate for experimental studies of the polarization properties of three-dimensional electromagnetic fields, e.g. using scanning-probe techniques (Rhodes, Nugent and Roberts [2002]). The complete determination of the polarimetric characteristics of a threedimensional field requires measurements of the field correlations in three mutually perpendicular directions. This could be realized by using, for instance, three orthogonal dipole-like probes which are overlapped spatially and which are detected simultaneously. In the optical domain however, such an approach seems very difficult to implement. However, very recently this experimental hurdle has been overcome. Ellis and Dogariu [2005b] demonstrated that the full three-dimensional polarimetry of a general state can be obtained by placing in the point of interest a probe which couples all three components of the field and then re-emits the radiation. This probe acts as a secondary source for the radiation which is being sensed by a conventional detector, placed away from the point where measurements are made. By changing the coupling characteristics of the probe, sufficient information is generated for determining all elements of the cross-spectral density matrix.

§ 6. Postscript In closing, we would like to emphasize that the above pseudospin analysis should not be confused with the polarization of the photon, for which noncommutative Stokes operators need to be defined. The photon is ordinarily assumed to be a spin-1 particle whose mass is exactly zero,20 but because of the transversality of the electromagnetic wave in free space, the spin component along the direction 20 Note that if there is any deviation from zero, it must be very small, since Maxwellian electromagnetism has been very well verified (in the classical domain). A nonzero photon mass would give rise to a wavelength dependence of the speed of light in free space, the possibility of longitudinal electromagnetic waves, a leakage of static electric signals into conductive enclosures, and a more rapid (exponential or Yukawa) fall-off of magnetic dipole fields with distance than the usual inverse cube

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Symmetry properties and polarization for an arbitrary wavefield

[4, § 7

of propagation behaves as a two-valued (instead of a three-valued) degree of freedom. In quantum optics, the scalar Stokes parameters have to be replaced by Hermitian Stokes operators but the noncommutability of these operators precludes the simultaneous measurement of the quantities represented by them. The interested reader may consult articles by Jauch and Rohrlich [1955], Case [1988], Luis [2002], Robson [1974] and Viana Ramos [2005] for interesting discussions of the differences between classical and quantum descriptions of the density matrix relevant to polarized light. For example, Luis [2002] reported that a full characterization of polarization by means of higher order than two correlations is required in quantum optics because no quantum state has a definite polarization ellipse for the same reasons that quantum particles do not follow definite classical trajectories. Various kinds of random fields can be globally unpolarized. Ellis and Dogariu [2004, 2005c] have used fourth-order field correlations to discriminate between different types of unpolarized wavefields. Among other applications, quantum effects associated with the polarization of an optical wavefield are of great interest for fundamental research in teleportation (Angel’skii, Ushenko, Arakhelyuk, Ermolenko, Burkovets and Ushenko [2000], Berezin, Kurochkin and Tolkachev [1989]), quantum information processing (Burkalov and Klyshko [1999], Burkalov, Chekhova, Karabutova, Klyshko and Kulik [1999], Karasev and Masalov [1993]), and nonclassical states, e.g. squeezed states of light with suppressed fluctuations of one of the Stokes parameters (Alodzhants and Arakelyan [1998, 2004]) and states of light with “hidden” polarizations (Klyshko [1997]).

§ 7. Summary and prospects Our rationale for these notes is that by viewing the polarization states of a statistically stationary narrow-band electromagnetic field as pseudospin states with SU(N ) symmetry, the geometrical description of the convex set of states faithfully reflects the possibilities of relevant polarization descriptors – we can truly say that we have provided a universal accounting for the mathematical structure of polarization states. Our main emphasis has been to expose the underlying physics as transparently as possible: the convexity property of the density matrix permits us

dependence. Electromagnetism in the presence of nonzero photon mass is described by the Maxwell– Proca equations (see, e.g., Barrow and Burman [1984], Goldhaber and Nieto [1971], Proca [1937]). The possibility of nonzero photon mass has been studied by many authors, e.g. Bass and Schrödinger [1955], de Broglie and Vigier [1972].

4, § 7]

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365

to “see” geometrically how the pure states (completely polarized) and the mixed states (partially polarized) of different ranks are positioned relative to each other in the convex set of states. The SU(N ) symmetry of the density matrix is similar to that of a pseudospin system, i.e. an N-level system in statistical physics. The geometric algebra of the complex Hilbert space H = RN unifies the Stokes parameters, the density (coherency) matrix, the entropy, the Jones and Mueller matrices, etc. in a single formalism and constitutes a descriptor of the polarization states of an arbitrary wavefield. Various theorems have been quoted, and physical arguments presented, to motivate the claim that the seminal concept underlying polarization of an arbitrary (nonplane) electromagnetic wavefield is the concept of symmetry. We note that in this entirely classical picture one is able to work with a self-contained and unified theory. It is important to stress that this has important practical as well as theoretical implications for the collection and interpretation of experimental data. We exhibit the hierarchy of hypersurfaces for a pseudospin- 21 (plane wave) and a pseudospin-1 (nonplane wave). The pseudospin- 12 set is too simple to show the general features whereas the pseudospin-1 has too many dimensions to be “imaginable”. The SU(3) symmetry gives a consistent description of the richness of phenomena in nonparaxial light situations, where in general there is no well-defined propagation direction. A number of theoretical predictions for understanding the polarization properties of electromagnetic fields which fluctuate in three dimensions have been made. For example, by employing the irreducible tensor operators, we are led to specify the polarization completely by defining vector and tensor polarizations. This vector and this tensor are related by several inequalities, and the interpretation is compared to that in which unitary invariants of the density matrix are studied. However, we emphasize that, unlike in the two-dimensional case, the properties of D3 are complicated by the fact that polarization information is contained within both the orientation vector and the alignment tensor. The Poincaré sphere is a particular case, i.e. plane wave, of illustrating the general fact that pure states form the extremal points of the convex set. In closing it should be emphasized that we have been rather abstract here and the problem remains of connecting the theoretical arguments presented here with specific arguments. We also hope that this review will draw attention to the value of geometry in interpreting optical phenomena of contemporary interest. What next? The nearly five decades that have passed since Wolf and others invented the coherency (density) matrix formalism have borne witness to a proliferation of innovations and applications. The field of coherence and polarization optics is at an exciting stage of development. Even though some of the basic concepts regarding optical coherence theory have been around for a while, much remains to be understood and discovered. It is only very recently that measurement

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Symmetry properties and polarization for an arbitrary wavefield

[4, § 7

techniques have matured to the point that experiments can be successfully conducted. There are, of course, a number of points still to be investigated, regarding the fundamental issue of precisely how this geometric-algebra picture jives with other work in polarization optics, i.e. the non-Abelian Jones and Mueller matrix formalisms encountered in the propagation and scattering of polarized waves. One important question in this regard is how this three-dimensional approach can be related to near-field optics and nano-optics problems (Keller [2000], Ohtsu and Kobayashi [2004], Pohl and Courjon [1993], Nieto-Vesperinas and Garcia [1993], van Hulst and Lewis [1998]). For example, related to the emerging nanotechnology, a recent experiment has demonstrated nanoscale control of optical properties on a length scale much less than a light wavelength (Müller, Sönnichen, von Poschingen, von Plessen, Klar and Feldmann [2002]). We are at the threshold of an exciting period in nanoscale optics because we apparently now have available an optical technique which allows us to investigate certain material properties on a length scale much smaller than the wavelength of light. The development of the scanning near-field optical microscope has revolutionized optical imaging and led to applications ranging from biophysics to optical data storage (e.g., Betzig, Trautman, Harris, Weiner and Kostelak [1991]). The full potential of most of the more recent polarization optics developments has yet to be realized. We may not need to wait very long for answers. Another exciting arena for polarization states with SU(3) symmetry is nonlinear optics (Svirko and Zheludev [2000], Menyuk and Wai [1994], Kolokolov and Turitsyn [2004]). For example, Mayer [1991] has shown that such states can be generated during the quantum-parametric decay of a photon of a pump wave into two photons, namely, an idler wave and a signal wave, upon the formation of quantum states of an optical field with SU(3) symmetry in an anisotropically cubic nonlinear medium. Many, ourselves included, expect that optics will have increasingly important things to say about optical biomedical imaging (Bowen, Treps, Schnabel and Lam [2002], Korolkova, Leuchs, Loudon, Ralph and Silberhorn [2002], Tuchin [2000], Yodh and Chance [1995]). This consideration relies on that fact that, of the four basic forces in Nature, only the electromagnetic force is effective on a scale comparable to biological organisms as exemplified by the human visual and nervous systems.21 Diffusing waves are now used to view body and structure. This is made

21 Biology and engineering have a complex, often discordant, relationship. Living systems are intrinsically messy, so most biologists spend a lot of time analyzing untidy subjects such as variation, ecological interactions and the multitudes of tangled genetic, developmental and metabolic pathways that organisms use. In biology, noise is often a kind of signal, and generalizable principles are hard to find. By contrast, engineers can take a more reductionist approach to the world, deducing and testing

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Acknowledgements

367

possible by a spectral window that exists within tissues in the 700–900 nm region, in which wave transport is dominated by scattering rather than absorption. Nearfield imaging of tissue characterization in turbid media using diffusing light fields offers new noninvasive opportunities for clinical diagnosis of human body structure and function. By analyzing how light interacts with tissues, these “optical body scanners” would help doctors diagnose lesions such as tumors, and would determine whether they are malignant or begnin. For that purpose, understanding how light propagates through a highly scattering medium is an important step in resolution of the problem. Light is both absorbed and scattered by biological tissues, resulting in diffuse transmittance. Characteristics such as intensity, coherence, and polarization of the incident light change as it is absorbed and scattered inside the tissue. The extent of these changes depends on the nature of light to be used, the type of tissue through which it propagates, and the tissue thickness. Thus, the light that emerges from the medium has very different characteristics than the incident light. Recent advances in techniques for studying optical fields with subwavelength features for which the longitudinal component of the electric field vector becomes important allow now a more meaningful dialog between experimentalists and theorists. Surely, this subject can keep any of us busy to some day. And, as is always the case in science, the best is yet to come.

Acknowledgements This chapter is a much belated revision of lecture notes we provided for the Workshop on Polarization and Coherence held in May 2004 at CREOL, Orlando (UCF), the hospitality and financial support of which are acknowledged gratefully. It is a pleasure to acknowledge useful discussions with Professor Emil Wolf and Jeremy Ellis. We are grateful to them for sharing their insights with us. The Laboratoire d’Électronique et Systèmes de Télécommunications is Unité Mixte de Recherche CNRS 6165.

the inherent principles and mechanisms by which things fail, work or can be made to work. Despite these differences, the two fields are vitally important to one another. The physical world poses many basic challenges to all living creatures, which in turn have evolved an astonishing array of solutions. Many of them perform so well that we marvel at their superiority to human-made devices. But many of the important biological facts have not been put into a unified framework from which they could be predicted, and many are intrinsically not predictable.

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[4, A

Appendix A: Lie groups in polarization optics This appendix provides a rushed overview of Lie algebra that may be useful for the reader who is interested in going through the calculations in detail. While this material is well known in other areas of theoretical physics, its application to polarization optics is relatively unknown, and hence we have provided a selective review suitable for newcomers to the field. For pedagogical purposes we also define our notation and convention used throughout the paper. Group theory is a branch of mathematics with important applications in physics via the association of algebraic properties of group elements with specific operations, e.g. rotation, on physical systems (Jacobson [1979], Sanches-Mondragon and Wolf [1986], Cornwell [1984], Nash and Sen [1983], Misner and Wheeler [1957]). It also formalizes the concept of symmetry in solutions to differential equations and can be used to extract invariant features in physics. For another angle on the meaning of these groups, it is useful to recall that groups can be realized as groups of matrices in many different ways: such a realization is a representation of the abstract group. The theory of linear representations of finite groups emerged in a series of papers by Frobenius in 1896–1897. This was at first couched in the language of characters but soon evolved into the formulation now considered standard, in which characters give the traces of representing linear transformations. The purpose of this appendix is to briefly review the key features of Lie groups, named after the 19th-century mathematician Sophus Lie who developed much of the theory of such groups. Of special importance to our purpose is the algebraic structure of the special unitary groups, SU(N ), the group of N × N unitary matrices with unit determinant. In the general case, we need N 2 − 1 parameters for a complete characterization. Thus, for the low-order unitary groups under consideration in this review: SU(2) requires three base states and SU(3) requires eight base states.

A.1: SU (2) Here the three basis states are represented by the Pauli matrices:

0 −i 0 1 1 0 σ1 = . , σ3 = , σ2 = i 0 1 0 0 −1

(A.1)

Of the many remarkable properties of the Pauli matrices, we list those that have the greatest bearing on what follows. First, these matrices obey the rules tr(σ i σ j ) = 2δij ,

(A.2a)

4, A]

Appendix A

σ i σ j + σ j σ i = 2δij σ 0 ,

σ i σ j − σ j σ i = 2ieij k σ k ,

369

(A.2b) (A.2c)

where δij and eij k denotes the Dirac delta and Kronecker symbols, respectively. The anticommutation relations, i.e. eq. (A.2b), and commutation relations, i.e. eq. (A.2c), lead to such diverse applications as unitary transformation of the Dirac equation (Weaver [1976]) and derivation of the electron’s magnetic moment (Galindo and Sanchez del Rio [1961]). Second, the general element of SU(2) can be written succinctly using the matrix exponential as U2 = exp(iσ · n), where σ represents the triple of Pauli matrices and n is a real 3-vector defining an axis for rotations in R3 . Observe that if n is real, then σ · n is Hermitian. There are many ways to express a SU(2) element corresponding to an arbitrary rotation, but the most concise form employs the exponential function σ 3φ σ 1θ σ 2φ U2 = exp i exp i exp i 2 2 2)

cos(θ/2) exp(i(ψ + φ)/2) i sin(θ/2) exp(i(φ − ψ)/2) = , i sin(θ/2) exp(i(ψ − φ)/2) cos(θ/2) exp(−i(ψ + φ)/2) (A.3) where ψ, θ and φ are the Euler angles. In topological terms, eq. (A.3) can be interpreted as arising from the Hopf map S 3 → S 2 , which uses the fact that the sphere S 3 can be considered as an S 1 bundle with base S 2 (Hopf fibration), so that (φ, θ ) may be identified with the polar angles on the Poincaré sphere, and ψ is the angle of the S 1 fiber. This statement implies that S 3 is a (nontrivial) product of a two-dimensional sphere S 2 and a circle S 1 . The S 1 fiber corresponds to electromagnetic gauge transformations [S 1 is the group space of the gauge group U(1)], and S 2 is the two-dimensional unit sphere surrounding the origin.22 Since we mentioned earlier that quaternions found applications in polarization optics, we note in passing that quaternions of norm 1 form a group under multiplication which is SU(2). The unit quaternions act via conjugation as rotations of the

22 The conciseness in employing exponential functions to express rotation operators is evident in Euler’s theorem in complex analysis, the Q matrices in Cayley–Klein parameters for finite rotations, and the vector exponentials in Hamilton’s quaternions. The first is contained in the third because complex numbers form a subalgebra of quaternions. The second is also equivalent to the third, because the Pauli matrices in the rotation Q matrix are intimately related to quaternions. This relation is provided by the Clifford algebra (Gurlebeck and Sprossig [1997], Hestenes and Sobcyk [1984], Jancewicz [1988], Lounesto [2001]).

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Smmetry properties for an arbitrary electromagnetic wavefield

[4, A

3D space of “pure imaginary” quaternions. As is well known, this gives a homomorphism from SU(2) onto the 3D rotation group SO(3) and we have the coset space identifications S 2 = SU(2)/U(1) = SO(3)/SO(2). The unit quaternions also act via left and right multiplication as rotations of the 4D space of all quaternions. This gives a homomorphism from SU(2) × SU(2) onto the 4D rotation group SO(4).

A.2: SU (3) We may construct a typical set of eight basis matrices in place of the three Pauli basis matrices considered for SU(2). The simplest spin-1 matrix representation is 3 × 3. A convenient set are the Gell-Mann matrices (Weaver [1976], Gell-Mann and Ne’eman [1964]):

0 −i 0 01 0 1 0 0 G1 = 1 0 0 , G2 = i 0 0 , G3 = 0 −1 0 , 000 0 0 0 0 0 0

00 0 0 0 −i 00 1 G6 = 0 0 1 , G5 = 0 0 0 , G4 = 0 0 0 , 010 i 0 0 100

00 0 1 10 0 G8 = √ 0 1 0 . G7 = 0 0 −i , 3 0 0 −2 0 i 0 Observe that in his discussion of 3 × 3 coherency matrices, Roman [1959a, 1959b] considered a set of 3 × 3 matrices that satisfy a Kemmer algebra. This algebra differs from that of Gell-Mann in two ways: (1) it has different commutation and anticommutation rules, and (2) the coherency matrix is not a multiple of the 3×3 unit matrix, in contradiction with the general result that a density matrix proportional to the unit matrix corresponds to complete ignorance of the state of the system. To put it slightly differently, it should be unaffected by any rotation. Indeed it is a well-known result from the theory of rotation groups (Schur’s lemma) that the only density matrices that commute with all of the rotation matrices are multiples of the unit matrix. Observe that the Gi matrices are not SU matrices themselves, but rather Hermitian generators of the SU group. Thus, the general element of SU(3) can be written in a matrix exponential as U3 = exp(iG · m), where G denotes the set of Gell-Mann matrices and m is an eight-element real vector. The corresponding spin-1 analog of eqs. (A.2) are the algebraic relations tr(Gi Gj ) = 2δij ,

4 Gi Gj + Gj Gi = 2dij k Gk + δij G0 , 3

(A.4a) (A.4b)

4, B]

371

Appendix B

Gi Gj − Gj Gi = 2ifij k Gk ,

(A.4c)

where fij k is real and totally antisymmetric like the Kronecker symbol eij k of eq. (A.2), while dij k is real and totally symmetric. A final point worth mentioning is that the use of group theory in polarization optics was discussed by many authors, including Simon and Mukunda [1998], Byrne [1971], Takenaka [1973], Han, Kim and Noz [1997, 1999] and Cloude [1986].

Appendix B: Madison convention for the density matrix of massive spin-1 particles In this appendix we briefly discuss the density matrix of spin-1 particles. There are various formulations of this problem, one of which is most commonly encountered in the quantum-mechanical literature. The interested reader may wish to consult Robson [1974], Fano and Racah [1959] and Griffiths [1987] for details. Dating back to the pioneering work of Ramachandran and Umerjee [1964] (also called the Madison convention in the nuclear physics literature, Jauch and Rohrlich [1955]), the 3 × 3 Hermitian density matrix D3 for massive spin-1 particles takes the form ⎡ ⎤ √ 1⎢ D3 = ⎢ 3⎣

1+

−

3 1 √1 2 2 t0 + 2 t0 3 1 2 2 (t1 +t1 )

√ 2 3t2

3 1 2 2 (t−1 +t−1 )

√

2 3t−2

⎥

3 1 2 ⎥ 1− 2t02 2 (t−1 −t−1 ) ⎦ , − 32 (t11 −t12 ) 1− 32 t01 + √1 t02 2

(B.1)

in terms of the irreducible spherical tensor parameters tkq of rank k = 1, 2 and q = −k to +k, with the definition that the tqk ’s are the expectation values of the Lakin operators tqk = tkq (Robson [1974], Darden [1971], Lakin [1955]). Note that since the density matrix is normalized in accordance with the trace 1 condition, the zero-rank tensor t00 coincides with the unit matrix. By irreducible tensor operator of rank k we mean an aggregate of quantities tkq which transform on rotations of the coordinate frame according to the (2k + 1)-dimensional irreducible representation of the three-dimensional rotation group. Since the tkq ’s form a complete set, the density operator (of order 2s + 1) corresponding to a system of particles with a given spin s can be represented in the form of a linear superposition of them 2s

D2s+1 =

1 k k + tq tq . 2s + 1 q k=0

(B.2)

372

[4, B

Smmetry properties for an arbitrary electromagnetic wavefield

Hence, to specify D3 completely requires the knowledge of the values of all nine tqk with k = 1, 2 and q = −k to +k. The t1q and t2q are known as the polarization vector and quadratic (rank 2) polarization tensor, respectively. Observe that the tkq ’s are not Hermitian, i.e. they satisfy the condition (tkq )+ = (−1)−q tk−q . The tkq constitute in fact a complete orthogonal set in operator set. Use of this decomposition theorem renders our problem simple while it would otherwise be extremely difficult. The virtue of this procedure is of course that we can characterize a sequence of observables as “monopole”, a triplet of “dipoles”, five “quadrupoles”, seven “octupoles”, and so on. It should, perhaps, be emphasized in this context that in the particular case of spin- 12 particles the density matrix can be expanded in terms of the multipole operators which are the 2 × 2 spin tensors t1q : the identity (a monopole) t00 and a triplet of dipole operators t10 = σ 3 , t1±1 = ∓ √1 (σ 1 ± iσ 2 ) verify coherent 2 through the text. By inserting these parameters into eq. (B.2), one is led to the density matrix # $ 1

1 1 + D2 = (B.3) tq1 t1q σ0 + = (σ 0 + u · σ ), 2 2 q=−1

which is similar to eq. (4.2). Within the language of angular momentum, the only observable is the mean angular momentum vector proportional to the magnetic dipole of the system (Bohn [1991]). When one deals with spin-1 particles, it is conventional for computing the density matrix to use the following set of irreducible tensors of rank 1: t10 = √

3 2 S3 ,

√1 (3S2 − 2O3 ), t2 = − 3 ((S1 + 0 1 3 2 2 3 2 2 iS2 )S3 + S3 (S1 + iS2 )) and t2 = 2 ((S1 + iS2 ) ), where the spin matrices are

10 0 01 0 0 −i 0 S1 = √1 1 0 1 , S2 = √1 i 0 −i , and S3 = 0 0 0 . With this choice, 2 010 2 0 i 0 0 0 −1

t1±1 = ∓

3 2 (S1

√

± iS2 ), and rank 2: t20 = √

one obtains the spin-tensor expansion of the density matrix, from eq. (B.2), in the form # 2 k $ 1 k k + 3 1 3 D3 = (B.4) tq tq O0 + u · S + 3uij Sij , = 3 3 2 k=0 q=−k

which is eq. (5.3). As previously observed, eq. (B.4) leads us to define a polarization vector u = tr(SD3 ) and a quadratic polarization tensor uij = tr(Sij D3 ), where Sij is a symmetric tensor defined by the expression Sij = 1 2 3 2 (Si Sj + Sj Si ) − 3 δij O0 . Within the angular momentum interpretation, a pure state is now identified by its dipole and electric quadrupole moments, with the

4, B]

Appendix B

373

property that a pure state has its dipole moment orthogonal to a symmetry plane of the quadrupole (Bohn [1991]). Because the diagonal elements of the density matrix must be positive definite, this restricts the possible values for t01 and t02 . From eq. (B.1) we find that ) 1 3 1 1 (B.5a) t0 + √ t02 0, 1+ 3 2 2 √ 1

1 − 2t02 0 (B.5b) 3 and ) 1 3 1 1 2 (B.5c) t + √ t0 0. 1− 3 20 2 Noting further that these three inequalities must be satisfied simultaneously, we can readily see that ) ) 3 3 1 t0 − (B.6a) 2 2 and √ 1 − 2 t02 √ . (B.6b) 2 Observe that t01 and t02 do take their maximum values simultaneously as displayed in fig. 12. In like fashion, we can rewrite the other elements of D3 as 1

− √ t11 + t12 = r1 r2 exp(iΔ21 ), 6 1 1 − √ t1 − t12 = r2 r3 exp(iΔ32 ). 6

(B.7a) (B.7b)

Recalling that r12 + r22 + r32 1 and adding eqs. (B.7a) and (B.7b) to obtain t11 , then noting that the maximum value of |t11 | occurs when both phase factors are equal, we can prove that 1√ 0 t11 3. 2

(B.8)

1 |, Equation (B.8) provides us with an explicit condition that is also satisfied by |t−1 2 | and |t 2 |. |t±1 ±2 Parenthetically, we note that these considerations may clearly be generalized to any total spin, and it is noteworthy that for spin-s particles, the density matrix being of order 2s + 1, the polarization state would be defined by specifying a

374

Smmetry properties for an arbitrary electromagnetic wavefield

[4, C

Fig. 12. The allowed values of t01 and t02 are inside and on the triangle ABC. The points A, B and C denote completely polarized states.

polarization tensor of rank 2s. In that case, there are 4s(s + 1) independent Stokes parameters. In closing, we observe that the equations relating the tqk = tkq and uij have been listed by Darden [1967, 1971].

Appendix C: Degree of polarization of a ﬁeld consisting of a superposition of an ensemble of evanescent waves of random amplitude and of black-body radiation It is instructive to find out how the degree of polarization for a three-dimensional field is calculated, because the calculation will tell us something about the relevance of the formalism developed in Section 5.1 in dealing with the characterization of the polarization state for any random, arbitrary statistically stationary electromagnetic field. For that purpose, the case of a random evanescent wave with electric field, represented by a statistically stationary ensemble, that is incident on a half-space z 0 containing black-body radiation is particularly interesting. A solution to this problem was found by Ellis, Ponomarenko, Dogariu and Wolf [2005]. Reference is made to this article for the details of the study. Let us consider that the wave is linearly polarized along the x-direction and decays exponentially in the positive z-direction. Using the superposition principle, Ellis, Ponomarenko, Dogariu and Wolf [2005] were able to show that the coherency matrix for the total field is given by the sum of the coherency matrices

4, C]

Appendix C

375

of the evanescent wave and the black-body radiation. For that case the eigenvalues of the density matrix D3 are given by (in our notation) I (bb) + I (e) exp(−2kz) , 3I (bb) + I (e) exp(−2kz) I (bb) , λ2 = λ3 = (bb) 3I + I (e) exp(−2kz)

λ1 =

(C.1) (C.2)

where I (e) and I (bb) represent the initial intensity of the evanescent field (i.e., the intensity at z = 0) and the intensity of the black-body radiation, respectively, and k denotes the wavevector component along the z-axis. The degree of polarization P3 (z) of the total field can now be evaluated by substituting eqs. (C.1) and (C.2) into eq. (5.12): # $ (e) 3 + 2 I(bb) exp(−2kz) 1/2 I P3 (z) = 1 − 3 . (e) (3 + I(bb) exp(−2kz))2 I

The degree of polarization is plotted as a function of the dimensionless variable kz in fig. 13 for three selected values of

I (e) . I (bb)

It is interesting to observe that these

Fig. 13. The degree of polarization as a function of kz for a field consisting of a superposition of an ensemble of evanescent waves of intensity I (e) at the plane z = 0 and of black-body radiation of intensity I (bb) occupying the half-space z 0. The evanescent wave decays exponentially in amplitude with decreasing z (adapted with permission from Ellis, Ponomarenko, Dogariu and Wolf [2005]).

376

Symmetry properties and polarization for an arbitrary wavefield

[4

graphs are coincident with those given by Ellis, Ponomarenko, Dogariu and Wolf [2005] who used a different definition of the degree of polarization of the field, (bb)

i.e. P3 (z) = 1/(3 I (e) exp(2kz)+1). This point was already made by Luis [2005] I who proved that the definitions (31) and (17) of Ellis, Ponomarenko, Dogariu and Wolf [2005] coincide when λ1 λ2 = λ3 . References Agarwal, G.S., 2005, J. Mod. Opt. 52, 651. Allen, L., Eberly, J.H., 1975, Optical Resonance and Two-Level Atoms, Wiley, New York. Alodzhants, A.P., Arakelyan, S.M., 1998, Zh. Éksp. Teor. Fiz. 113, 1235. [JETP 86 (1998) 672.] Alodzhants, A.P., Arakelyan, S.M., 2004, Opt. Spectrosk. 97, 424. [Opt. Spectrosc. 97 (2004) 453.] Angel’skii, O.V., Ushenko, A.G., Arakhelyuk, A.D., Ermolenko, S.B., Burkovets, D.N., Ushenko, Yu.A., 2000, Opt. Spectrosc. 89, 973. Opt. Spectrosc. 89 (2000) 1050, for an interesting discussion of eqs. (13) and (14). Aravind, P.K., 1992, Opt. Commun. 94, 192. Arvind, K., Mallesh, S., Mukunda, N., 1997, J. Phys. A 30, 2417. Balakrishnan, R.A., Bishop, R., Dadoloff, R., 1990, Phys. Rev. Lett. 64, 2107. Barakat, R., 1963, J. Opt. Soc. Am. 53, 1771. Barakat, R., 1977, Opt. Commun. 23, 147. Barakat, R., 1983, Opt. Acta 30, 1171. Barakat, R., 1996, Opt. Commun. 123, 443. Barakat, R., Brosseau, C., 1993, J. Opt. Soc. Am. A 10, 529. Barnes, W.L., Dereux, A., Ebbesen, T.W., 2003, Nature (London) 424, 824. Barrow, J.D., Burman, R.R., 1984, Nature (London) 307, 14. Bass, L., Schrödinger, E., 1955, Proc. Roy. Soc. London A 232, 1. Baylis, W.E., Bonenfant, J., Derbyshire, J., Huschilt, J., 1993, Am. J. Phys. 61, 534. Berezin, A.V., Kurochkin, Y.A., Tolkachev, E.A., 1989, Quaternions in Relativistic Physics, Nauka i Technika, Minsk. Berry, M.V., 1984, Proc. Roy. Soc. London A 392, 45. Berry, M.V., 1987a, Proc. Roy. Soc. London A 414, 31. Berry, M.V., 1987b, J. Mod. Opt. 34, 1401. Berry, M.V., 2000, in: Singular Optics, SPIE, Alushta, Crimea, pp. 1–12. Berry, M.V., Dennis, M.R., 2001, Proc. Roy. Soc. London A 457, 141. Betzig, E., Trautman, J., Harris, T., Weiner, J., Kostelak, R., 1991, Science 251, 1468. Bhandari, R., 1988, Phys. Lett. A 133, 1. Bicout, D., Brosseau, C., Martinez, A., Schmitt, J.M., 1994, Phys. Rev. E 49, 1767. Biedenharn, L.C., 1958, Ann. Phys. (New York) 4, 104. Biedenharn, L.C., 1959, Ann. Phys. (New York) 6, 399. Biedenharn, L.C., Louck, J.D., 1981, Angular Momentum in Quantum Physics: Theory and Application, Addison–Wesley, Redwood City, CA. Bloore, F.J., 1976, J. Phys. A: Math. Gen. 9, 2059. Blum, K., 1996, Density Matrix, Density Matrix Theory and Applications, 2nd edition, Plenum, New York. Bluman, G.W., Cole, J.D., 1974, Similarity Methods for Differential Equations, Springer-Verlag, New York. Bluman, G.W., Kumei, S., 1989, Symmetries and Differential Equations, Springer-Verlag, New York.

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E. Wolf, Progress in Optics 49 © 2006 Elsevier B.V. All rights reserved

Chapter 5

Quantum cryptography by

Miloslav Dušek Department of Optics, Palacký University, 17. listopadu 50, 77200 Olomouc, Czech Republic e-mail: [email protected]

Norbert Lütkenhaus Institut für Optik, Information und Photonik, Universität Erlangen-Nürnberg, Staudtstr. 7/B3, 91058 Erlangen, Germany e-mail: [email protected]

Martin Hendrych ICFO – Institut de Ciències Fotòniques, Parc Mediterrani de la Ternotogia Avda. Canal Olímpic, s/n 08860 Castelldefels (Barcelona), Spain e-mail: [email protected]

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(06)49005-3 381

Contents

Page § 1. Ciphering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 2. Quantum key distribution . . . . . . . . . . . . . . . . . . . . . . . .

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§ 3. Some other discrete protocols for QKD . . . . . . . . . . . . . . . . .

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§ 4. Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 5. Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 6. Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 7. Supporting procedures . . . . . . . . . . . . . . . . . . . . . . . . . .

425

§ 8. Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 9. Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 1. Ciphering 1.1. Introduction, cryptographic tasks There is no doubt that electronic communications have become one of the main pillars of the modern society and their ongoing boom requires the development of new methods and techniques to secure data transmission and data storage. This is the goal of cryptography. Etymologically derived from the Greek κρυπτ oς, ´ hidden or secret, and γραϕ η, ´ writing, cryptography may generally be defined as the art of writing (encryption) and deciphering (decryption) messages in code in order to ensure their confidentiality, authenticity, integrity and non-repudiation. Cryptography and cryptanalysis, the art of codebreaking, together constitute cryptology (λoγ ´ oς, a word). Nowadays many paper-based communications have already been replaced by electronic means, raising the challenge to find electronic counterparts to stamps, seals and hand-written signatures. The growing variety of applications brings many tasks that must be solved. Let us name a few. The fundamental task of cryptography is to allow two users to render their communications unintelligible to any third party, while remaining intelligible for the two legitimate users. The goal of identification is to verify the identities of the communicating parties. Another cryptographic task is secret sharing: A secret, e.g., a password, is split into several pieces in such a way that when a certain minimal subset of the pieces is put together, the secret is recovered. Other cryptographic applications are, for example, digital signatures, authentication of messages, zero-knowledge proofs, and so on. At all times people have wished to have the possibility to communicate in secrecy so as to allow nobody to overhear their messages. Archeological excavations have revealed that various types of cryptography were already used by ancient civilizations in Mesopotamia, India or China (Kahn [1967]). Four thousand years ago, ancient Egyptians used modified hieroglyphs to conceal their messages. In the Iliad, Homer depicts how Proetus, the king of Argolis, sends Bellerophon to Lycia with “a lethal message, coded symbols inscribed on a folded tablet” (Homer [8th c. B.C.]). 383

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In the 5th century B.C., the Spartans in Greece designed the Skytale cryptodevice, based on transposition of letters (Old Spartan Facts). A stripe of parchment or leather was wound around a wooden baton, across which the message was then written. When the end of the line was reached, the baton was rotated. After the parchment was unwrapped, the letters looked scrambled and only a person who possessed a baton of identical shape could recover the message. Another favorite and easy cipher is the substitution cipher, which substitutes each letter of a message with another letter, a number or a symbol. An example is the Caesar cipher (Stinson [1995]). To communicate between the Roman legions scattered over the Roman republic, Gaius Julius Caesar used a cipher in which each letter of a message was advanced by three letters in the alphabet; A was replaced by D, B was replaced by E, C by F, and so on. A similar substitution cipher is described in Kama Sutra. During the Middle Ages, most cryptosystems were based on transposition, substitution or a combination of both (Leary [1996]). However, neither of these ciphers is secure, because it is possible to break them exploiting various characteristic properties of the language, such as the frequency of individual letters and their clusters. The invention of the telegraph in the 1830s enormously facilitated communications between people. This ancestor of modern communications, however, had a serious drawback from the cryptographic point of view – the content of the transmitted message was known to the telegraph operator. As a consequence, various codebooks were designed by people and companies that wanted to keep their communications private. The codebooks translated significant words and phrases into short, nonsensical words. The codes served two purposes: first, they reduced the size of the message and thus decreased cost because telegrams were charged per transmitted character; and second, if the codebook was kept secret, the codes became a cipher. The two world wars of the 20th century accelerated the development of new cryptographic techniques. Cryptographers tried to design a system where the encryption and decryption algorithms could be publicly known, but the secrecy of the message would be guaranteed by some secret information, the cryptographic key, shared between the users. In 1917, Gilbert S. Vernam proposed an unbreakable cryptosystem, hence called the Vernam cipher or One-Time Pad (Vernam [1926]). Its unconditional security has been proved by Claude E. Shannon (in terms of information theory) in 1949 (Shannon [1949]). The One-Time Pad is a special case of the substitution cipher, where each letter is advanced by a random number of positions in the alphabet. These random numbers then form the cryptographic key that must be shared between the sender and the recipient. Even though

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the Vernam cipher offers unconditional security against adversaries possessing unlimited computational power and technological abilities, it faces the problem of how to securely distribute the key. For this reason it did not become widespread as Vernam had hoped. On the other hand, for many military and diplomatic applications the security of communications outweighs the severe key management problems. The Vernam cipher was used by Russian Intelligence to pass US atomic secrets to Moscow. Che Guevara encrypted his messages to Fidel Castro by means of the One-Time Pad. It was employed in securing the hot line between Washington and Moscow and it is said to be used for communications between nuclear submarines and for some embassy communications. We will return to the Vernam cipher later, as it is very expedient for quantum key distribution. In 1918, Arthur Scherbius invented an ingenious electric cipher machine, called Enigma, which was patented a year later (Deavours and Kruh [1985]). The Enigma consisted of a set of rotating wired wheels, which performed a very sophisticated substitution cipher. After various improvements, it was adopted by the German Navy in 1926, Army in 1928, and Air Force in 1935, and it was used by the Germans and Italians throughout World War II. The military Enigma had an incredible 159 × 1018 possible settings (cryptographic keys). The immense number of potential keys led Alan Turing to construct the first electronic computer, which helped break the Enigma ciphers in the course of the War. Today a personal computer can unscramble an Enigma-encrypted message within minutes. But contemporary processors are even faster than original Pentia.

1.2. Asymmetrical ciphers (public-key cryptography) A new surge of interest in cryptography was triggered by the upswing in electronic communications in the late 1970s. It was essential to enable secure communication between users who have never met before and share no secret cryptographic key. The question was how to distribute the key in a secure way. The solution was found by Whitfield Diffie and Martin E. Hellman, who invented public-key cryptography in 1976 (Diffie and Hellman [1976]). The ease of use of public-key cryptography, in turn, stimulated the boom of electronic commerce during the 1990s. Notice, however, that asymmetric ciphers can provide users who have never met with a secret channel but cannot – without the help of a Trusted Authority – prove the identity of users. Public-key cryptography requires two keys – the public key and the private key, which form a key pair. The recipient generates two keys, makes the public key public and keeps his private key in a secret place to ensure its private possession.

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The algorithm is designed in such a way that anyone can encrypt a message using the public key, however, only the legitimate recipient can decrypt the message using his/her private key. Of course, there is the problem of authenticity of the public key. Therefore public keys are distributed through Trusted Authorities in practice. The security of public-key cryptography rests on various computational problems believed to be intractable. The encryption and decryption algorithms utilize the so-called one-way functions. One-way functions are mathematical functions that are easy to compute in one direction, but whose inversion is very difficult (by “difficult” it is meant that the number of the required elementary operations increases exponentially with the length of the input number). It is, e.g., very easy to multiply two prime numbers, but to factor the product of two large primes is already a difficult task. Other public-key cryptosystems are based, e.g., on the difficulty of the discrete logarithm problem in Abelian groups on elliptic curves or other finite groups. However, it is important to point out that no “one-way function” has been proved to be one-way; they are merely believed to be.1 Public-key cryptography cannot provide unconditional security. We speak of computational security. Today the most widely used public-key system is the RSA cryptosystem. RSA was invented in 1977 by Ronald Rivest, Adi Shamir and Leonard Adleman (Rivest, Shamir and Adleman [1978]), on whose names the acronym is based. RSA exploits the difficulty of factoring large numbers. The receiver picks two large primes p and q and makes their product public. Further, he chooses two large natural numbers d and e [such that (de − 1) is divisible by (p − 1)(q − 1)]. The product pq together with the number e constitutes the public key. Using this key, anyone can encrypt a message P (< pq) employing a simple algorithm: C = P e mod pq, where C is the resulting cipher text. The cipher text can easily be decrypted if the private key d is known: P = C d mod pq. However, inversion of the algorithm without knowing the private key d requires finding the prime factors of the modulus. Although there are several other ways to attack the RSA system, the most promising one still seems to be to attempt to factor the modulus. In 1976 Richard Guy wrote (Guy [1976]): “I shall be surprised if anyone regularly factors numbers of size 1080 without special form during the present century”. The first challenge to break a 425-bit RSA key (equivalent to 129 decimal digits) was published in Scientific American in 1977 (Gardner [1977]). Ronald Rivest calculated that the time needed to factor a 125-digit number, the product 1 This belief is based on the experience that even years of effort of many experts have not proved the opposite.

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of two 63-digit primes, would be least 40 × 1015 years (about one million times the age of the Universe) with the best factoring algorithms then known. However, 17 years later, in 1994, new factoring algorithms had been discovered and computer power had advanced to such a level that it took 1600 computers (and two fax machines!) interconnected over the Internet only 8 months. Today a single PC could do the same job. While breaking 425-bit RSA required a large number of computers, in February 1999 it was only 185 machines that managed to factor a 465-bit RSA modulus in 9 weeks. At that time, 95% of e-commerce on the Internet was protected by 512-bit keys (155-digit number). A 512-bit number was factored in August 1999 by 292 machines. This means that even 512-bit keys do not provide sufficient security for anything more than very short-term security needs. All these challenges have served to estimate the amount of work and the cost of breaking a key of a certain size by public efforts. It is obviously much more difficult to estimate what can be achieved by private and governmental efforts with much larger budgets. A network of computers is not the only way to factor large integers. In 1999 Adi Shamir proposed the TWINKLE device (Shamir [1999]) – a massively parallel optoelectronic factoring device, which is about three orders of magnitude faster than a conventional fast PC and can facilitate the factoring of 512- and 768-bit keys. Today it is already recommended to move to longer key lengths and to use key sizes of 2048 bits for corporate use and 4096 bits for valuable keys. Another menace to the security of public-key cryptography could originate from the construction of a quantum computer. The decryption using a quantum computer would take about the same time as the encryption, thereby making public-key cryptography worthless. Algorithms capable of doing so have already been developed (Shor [1994]) and first experiments with small-scale quantum computers successfully pave the way for more sophisticated devices (Vandersypen, Steffen, Breyta, Yannoni, Sherwood and Chuang [2001]).

1.3. Symmetrical ciphers (secret-key cryptography) In secret-key cryptography users must share a secret key beforehand. The common key is then used for both encryption and decryption.2 Secure key distribution is the main drawback of secret-key cryptosystems. The security of com2 Secret-key cryptography can even provide its users with unconditional security if they share a sufficiently long key (using Vernam cipher). But symmetric algorithms with the key shorter than the message are not unconditionally secure.

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munications is reduced to the security of secret-key distribution. In order to avoid the necessity of personal meetings or courier services for exchanging the secret key, some users use public-key cryptography to distribute the key, which is then used in a secret-key cryptosystem. In such a case, even if the symmetric cipher was unconditionally secure the security of the whole system will be degraded to computational security. These so-called hybrid systems have gained a widespread use, because they combine the speed of secret-key systems with the efficiency of key management of public-key systems. They have been used for electronic purchases, financial transactions, ATM transactions and PIN encryptions, identification and authentication of cellular phone conversations, electronic signatures, and many other applications, whose number is swelling. The most widespread secret-key cryptosystem is the Data Encryption Standard (DES) and its variations. Due to its frequent use in the hybrid systems, it is the most often used cryptosystem ever. DES was developed by IBM and the US government in 1975 and was adopted as a standard two years later. DES is an example of a block cipher – an algorithm that takes a fixed-length string of plain text and transforms it through a series of operations into a ciphertext of the same length. In the case of DES, the block size is 64 bits. The transformation depends on the key. The algorithm consists of a cascade of 16 iterations of substitutions and transpositions and can easily be implemented in hardware, where it can reach very high speeds of encryption. DES has experienced a similar wave of attacks as public-key cryptosystems. The algorithm uses a 56-bit key, which is reused to encrypt the entire message. As a consequence, it is only computationally secure. In 1997, RSA Data Security, Inc. published their first challenge to decrypt a plain-text message scrambled by DES. It took 96 days to break it. The researchers applied “brute force” by searching the entire keyspace of 256 possible keys on a large number of computers (Wiener [1997]). In January 1998, a new prize was offered. The winner of the contest used the idle time of computers connected to the Internet. More than 50,000 CPUs were linked together. The key was found in 41 days (DES Cracker 1). Another group of codebreakers chose a different approach. They built a single machine, which revealed the encrypted message “It’s time for those 128-, 192- and 256-bit keys” after only 56 hours, searching at a rate of 88 billion keys per second (DES Cracker 2). In the challenge in January 1999, the two previous winners combined their efforts to find the key in only 22 hours and 15 minutes, testing 245 billion keys per second. In 1993, Michael Wiener had designed a DES key search machine which, based on 1997’s technology, would break DES in 3.5 hours (Wiener [1997]). The same machine based on 2000’s technology would take only 100 seconds

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(Silverman [2000]). The exhaustive search is not the only possible attack on DES. During the 1990s, other successful attacks were proposed that exploit the internal structure of the cipher (Biham and Knudsen [1998]). Cryptographers attempted to improve the security of DES. Triple DES, DESX and other modifications were developed. In October 2000, a four-year effort to replace the aging DES culminated in the announcement of a new standard, the Advanced Encryption Standard (AES). It uses blocks of 128 bits and key sizes of 128, 192 and 256 bits. This standard was approved in December 2001 and went into effect in May 2002. How long will it last? In summary, the security of conventional techniques relies on the assumption of limited advancement of mathematical algorithms and computational power in the foreseeable future, and also on limited financial resources available to a potential adversary. Computationally secure cryptosystems, no matter whether publicor secret-key, will always be threatened by breakthroughs, which are difficult to predict, and even steady progress of code-breaking allows the adversary to “reach back in time” and break older, earlier captured, communications encrypted with weaker keys. Another common problem of conventional cryptographic methods is the socalled side-channel cryptanalysis (Rosa [2001]). Side channels are undesirable ways through which information related to the activity of the cryptographic device can leak out. The attacks based on side-channel information do not assault the mathematical structure of cryptosystems, but their particular implementations. It is possible to gain information by measuring the amount of time needed to perform some operation, by measuring power consumption, heat radiation or electromagnetic emanation. The problem of side channels will be further discussed in Section 8.7.

1.4. Vernam cipher, key distribution problem Classical cryptography can provide an unbreakable cipher, which resists adversaries with unlimited computational and technological power – the Vernam cipher. The Vernam cipher was invented in 1917 by the AT&T engineer Gilbert S. Vernam (Vernam [1926]), who thought it would become widely used for automatic encryption and decryption of telegraph messages. The Vernam cipher belongs to the symmetric secret-key ciphers, i.e., the same key is used for both encryption and decryption. The principle of the cipher is that if a random key is added to a message, the bits of the resulting string are also random and carry no information about the message. If we use the binary logic,

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unlike Vernam who worked with a 26-letter alphabet, the encryption algorithm E can be written as EK (M) = (M1 + K1 , M2 + K2 , . . . , Mn + Kn ) mod 2,

(1.1)

where M = (M1 , M2 , . . . , Mn ) is the message to be encrypted and K = (K1 , K2 , . . . , Kn ) is the key consisting of random bits. The message and the key are added bitwise modulo 2, or exclusive OR without carries. The decryption D of ciphertext C = EK (M) is identical to encryption, because double modulo-2 addition is the identity, therefore M = DK (C) = (C1 + K1 , C2 + K2 , . . . , Cn + Kn ) mod 2.

(1.2)

For this system to be unconditionally secure, three requirements are imposed on the key: (1) The key must be as long as the message; (2) it must be purely random; (3) it may be used only once.3 This was shown by Claude E. Shannon (Shannon [1949]), who laid the foundations of communication theory from the cryptographic point of view and compared various cryptosystems with respect to their secrecy. Until 1949 when his paper was published, the Vernam cipher was considered unbreakable, but that had not been mathematically proved. If any of the above requirements is not fulfilled, the security of the system is jeopardized. A good example is the revelation of the WW-II atomic spies because of repetitive use of the key incorrectly prepared by the KGB (NSA publications). The main drawback of the Vernam cipher is the necessity to distribute a secret key as long as the message, which prevented it from wider use. The cipher has so far found applications mostly in the military and diplomatic services. It is here that quantum mechanics comes in handy and readily offers a solution. Quantum mechanics gives us the power to detect eavesdropping. Taking into account the problem of authentication, that requires the communication parties to share a certain amount of secret information, quantum cryptography provides a tool for unlimited secret-key growing.

3 If a key K is used twice to encode two different messages M and M ′ into ciphertexts C and C ′ , then one can see that (C1 + C1′ , C2 + C2′ , . . . , Cn + Cn′ ) mod 2 = (M1 + M1′ , M2 + M2′ , . . . , Mn + Mn′ ) mod 2.

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§ 2. Quantum key distribution 2.1. The principle, eavesdropping can be detected As mentioned above, the main problem of secret-key cryptosystems is the secure distribution of keys. While the security of classical cryptographic methods can be undermined by advances in technology and mathematical algorithms, the quantum approach can provide unconditional security. The principle of quantum cryptography consists in the use of non-orthogonal quantum states. Its security is guaranteed by the Heisenberg uncertainty principle, which does not allow us to discriminate non-orthogonal states with certainty and without disturbing the measured system. Within the framework of classical physics, it is impossible to reveal potential eavesdropping, because information encoded into any property of a classical object can be acquired without affecting the state of the object. All classical signals can be monitored passively. In classical communications, one bit of information is encoded into two distinguishable states of billions of photons, electrons, atoms or other carriers. It is always possible to passively listen in by splitting off part of the signal and performing a measurement on it. In quantum cryptosystems the inviolateness of the channel is constantly tested by the use of non-orthogonal quantum states as information carriers. Because information is encoded into states with non-zero overlap, it cannot be read, copied or split without introducing detectable disturbances. It should be noted that quantum mechanics does not avert eavesdropping; it only enables us to detect the presence of an eavesdropper. Since only the cryptographic key is transmitted, no information leak can take place when someone attempts to listen in. When discrepancies are found, the key is simply discarded and the users repeat the procedure to generate a new key.

2.2. Quantum measurement Measurement in quantum physics differs substantially from the measurement in classical physics. According to quantum theory any measurement can distinguish with certainty (i.e., without errors or inconclusive results) only among specific orthogonal state vectors (that form the so-called measurement basis). Non-orthogonal states cannot be distinguished perfectly. Furthermore, quantum measurement disturbs the system in general. If the system is in a state that cannot be expressed as a multiple of one of the measurement-basis vectors but only

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as their linear superposition, then this state is changed after the measurement. The original state is “forgotten” during the measurement process and randomly changed to the state corresponding to one of the basis vectors. Exactly this is the key feature of the quantum world that enables to detect the eavesdropping. Eavesdropping is nothing else than a kind of measurement on the information carrier. If non-orthogonal states are used in transmission, eavesdropping must disturb some of them, i.e., induce errors. With a suitably designed protocol, these errors can later be discovered by the legitimate users of the channel, as will be seen in Section 2.4.

2.3. Quantum states cannot be cloned The linearity of quantum mechanics prohibits the cloning of arbitrary unknown quantum states (Wootters and Zurek [1982]). A device intended to make a copy of, say, a photon with horizontal polarization |H needs to perform the following operation: |copier0 |blank |H → |copier1 |H |H ,

(2.1)

and similarly for orthogonal vertical polarization |V |copier0 |blank |V → |copier2 |V |V ,

(2.2)

where |copier0 is the initial state of the copier, |copier1 and |copier2 are its final states and |blank denotes the initial “empty” state of the ancillary system (photon) to which the information (polarization state) should be copied. However, if we want to copy a linear superposition of states |H and |V , we obtain

|copier0 |blank α|H + β|V

= α|copier0 |blank |H + β|copier0 |blank |V → α|copier1 |H |H + β|copier2 |V |V ,

(2.3)

which is different from the required state

|copier3 α|H + β|V α|H + β|V

= |copier3 α 2 |H |H + αβ|H |V + βα|V |H + β 2 |V |V ,

(2.4)

regardless of whether states |copier1 and |copier2 are identical (and equal to

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|copier3 ) or not. The unitarity of quantum evolution requires that H |V blank|blank copier0 |copier0 = H |V H |V copier1 |copier2 , (2.5) which can be satisfied only when the states to be copied are orthogonal. Thus, the general state of a quantum object cannot be copied precisely. Duplicating can be done only approximately so that any of the resulting states is not exactly equal to the original. An optimal universal machine for approximate cloning of qubits was first designed by Bužek and Hillery [1996].

2.4. Protocol BB84 Quantum key distribution (QKD) was born in 1984 when Charles H. Bennett and Gilles Brassard came up with an idea for securely distributing a random cryptographic key with the help of quantum mechanics (Bennett and Brassard [1984]). Hence, the protocol is called BB84. Drawing upon Stephen Wiesner’s ideas about unforgeable quantum money (Wiesner [1983], original manuscript written circa 1969), Bennett and Brassard presented a protocol that allows users to establish an identical and purely random sequence of bits at two different locations, while allowing to reveal any eavesdropping with a very high probability. The crucial point of the BB84 protocol is the use of two conjugated bases. The sender of the message encodes logical zeros and ones into two orthogonal states of a quantum system. But for each bit she randomly changes this pair of states – i.e., she chooses one of two bases. Each state vector of one basis has equal-length projections onto all vectors of the other basis. That is, if a measurement on a system prepared in one basis is performed in the other basis, its outcome is entirely random and the system “loses all the memory” of its previous state. In fact, the non-orthogonal signal states are used for testing the transmission channel – checking it for eavesdropping. We need not consider any particular quantum system. However, in order to provide an example let us suppose that information is encoded into polarization states of individual quanta of light – photons. One basis can consist, e.g., of horizontal and vertical polarization states of photons, |H and |V , respectively; let us call this basis rectilinear. The other basis, diagonal, would consist of states of linear polarizations at 45◦ (anti-diagonal, |A ) and 135◦ (diagonal, |D ), and 1

|A = √ |H + |V , 2 1

|D = √ |H − |V . (2.6) 2

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These four states satisfy the following relations: H |V = A|D = 0, H |H = V |V = A|A = D|D = 1,

H |A 2 = H |D 2 = V |A 2 = V |D 2 = 1 . 2

(2.7)

Any measurement in the rectilinear (diagonal) basis on photons prepared in the diagonal (rectilinear) basis will yield random outcomes with equal probabilities. On the other hand, measurements performed in the basis identical to the basis of preparation of states will produce deterministic results.4 At the beginning, the two parties that wish to communicate, traditionally called Alice and Bob, agree that, e.g., |H and |A stand for the bit value “0”, and |V and |D stand for a bit value “1”. Now Alice, the sender, generates a sequence of random bits that she wants to transmit, and randomly and independently for each bit she chooses her encoding basis, rectilinear or diagonal. Physically this means that she transmits photons in the four polarization states |H , |V , |A and |D with equally distributed frequencies. Bob, the receiver, randomly and independently of Alice, chooses his measurement bases, either rectilinear or diagonal. Statistically, their bases coincide in 50% of cases, when Bob’s measurements provide deterministic outcomes and agree perfectly with Alice’s bits. In order to know when the outcomes were deterministic, Alice and Bob need an auxiliary public channel to tell each other what basis they had used for each transmitted and detected photon. This classical channel may be tapped, because it transmits only information about the bases used, not about the particular outcomes of the measurements. Whenever their bases coincide, Alice and Bob keep the bit. On the other hand, the bit is discarded when different bases had been chosen, or Bob’s detector failed to register a photon due to imperfect detector efficiency or to the photon having been lost somewhere on the way. Any potential eavesdropper, traditionally called Eve, who listens into this conversation can only learn whether they both set the rectilinear or diagonal basis, but not whether Alice had sent a “0” or “1”. The protocol is outlined in Table 1.

4 We could also consider a third basis consisting of right- and left-circular polarizations whose

vectors satisfy relations analogous to eqs. (2.7). Any two of these three mentioned bases suffice for secure quantum key distribution.

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Quantum key distribution Table 1 BB84 Protocol. 1st line: Alice’s random bits. 2nd line: Alice’s random polarization bases, with “+” and “×” standing for rectilinear and diagonal bases. 3rd line: actual polarization of transmitted photons. 4th line: Bob’s random detection bases. 5th line: polarization of detected photons, with “rand” standing for a random outcome. 6th line: Bob publicly announces his measurement bases. 7th line: Alice publicly replies when Bob set the correct measurement basis. 8th line: the cryptographic key 0 × |A × |A

× OK 0

1 × |D

1 + |V

+ – –

+ OK 1

+ rand

+ |V

0 + |H

0 + |H

1 × |D

0 + |H

1 × |D

– – –

× OK 1

× rand

× rand

+ rand

× – –

× – –

+ – –

+ lost

× |D

2.5. Eavesdropping, intercept–resend attack If Eve is present and wants to eavesdrop on the channel, she cannot passively monitor the transmissions (a single quantum cannot be split and its state cannot be copied without introducing detectable disturbances, as discussed above). What Eve can do is either to intercept the photons sent by Alice, perform measurements on them and resend them to Bob, or to attach some probe to the signal photon, i.e., to let some system in her hands interact with the quantum system carrying information, keep it, and measure it later. To understand the effect of eavesdropping we will first consider only the intercept–resend attack. As Alice alternates her encoding bases at random, Eve does not know what basis to make a measurement in. She must choose her measurement bases at random as well. Half the time she guesses right and she resends correctly polarized photons. In 50% of cases, though, she measures in the wrong basis, which produces errors. For example, let us suppose that Alice sends a “1” in the rectilinear basis, i.e., state |V , Eve measures in the diagonal basis, and Bob measures in the rectilinear basis (otherwise the bit would be discarded). Now, no matter whether Eve detects and resends |A or |D , Bob has a 50% chance to get |H , i.e., a binary “0”, instead of |V . Thus, if we consider a continuous intercept–resend eavesdropping, Bob finds on average errors in 25% of the bits that he successfully detects. If Alice and Bob agree to disclose part of their strings in order to compare them, they can discover these errors. When they set identical bases, their bit strings should be in perfect agreement. When discrepancies are found, Eve is suspected of tampering with the photons, and the cryptographic key is thrown away. Thus, no information leakage

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occurs even in the case of eavesdropping. If their strings are identical, the key is deemed secure and secret,5 and can be used for the above-mentioned Vernam cipher to encrypt communications. Since the bits used to test for eavesdropping are communicated over the open public channel they must always be discarded, and only the remaining bits constitute the key. An intercept–resend attack is not the optimal eavesdropping strategy. However, any interaction with the data carriers that can provide Eve with any information on the key always causes errors in transmission. In order to leave the original states intact, Eve could try to attach a probe and let it interact with the information carrier: |a |E → |a |Ea

and

|b |E → |b |Eb ,

(2.8)

where |a and |b denote two possible states of the information carrier, |E is the initial state of Eve’s probe and |Ea and |Eb are its final states. Any unitary interaction has to conserve the following inner product: a|b E|E = a|b Ea |Eb .

(2.9)

If the states |a and |b are non-orthogonal, a|b = 0, the equality (2.9) can be fulfilled only if Ea |Eb = 1, i.e., when the final states of Eve’s probe are identical. Eve thus cannot gain any information. It is apparent that for Eve to discriminate between two non-orthogonal states she must disturb the state of the measured objects, and thereby inevitably cause errors in transmissions. A more detailed discussion of sophisticated eavesdropping strategies will be provided in Section 8. It should be mentioned that no physical apparatus is perfect and noiseless. Alice and Bob will always find discrepancies, even in the absence of Eve. As they cannot distinguish between errors stemming from eavesdropping and errors from the noise of the apparatus, they conservatively attribute all the errors in transmissions to Eve. From the number of errors, the amount of information that has potentially leaked to Eve can be estimated. Afterwards Alice and Bob reconcile their bit strings using an error-correction technique to arrive at an identical sequence of bits. This sequence is not completely secret. Eve might have partial knowledge about it. To eliminate this knowledge, they run a procedure called privacy amplification. Privacy amplification is a method enabling them to distill a 5 The probability that eavesdropping will not be detected decreases exponentially with increasing number of bits compared.

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secret bit string from their data in such a way that Eve would know even a single bit of the distilled string only with an arbitrarily small probability. Both of these procedures, error correction and privacy amplification, will be described in detail in Section 7.

§ 3. Some other discrete protocols for QKD 3.1. Two-state protocol, B92 Besides BB84, other protocols were designed. In 1992, C.H. Bennett showed (Bennett [1992]) that two non-orthogonal states are already sufficient to implement secure QKD. Let Alice choose two non-orthogonal states and send them to Bob in random order. When Bob performs projections onto subspaces orthogonal to the signal states, he sometimes learns Alice’s bit with certainty and sometimes obtains an inconclusive outcome. After the transmission, Bob tells Alice when he detected a bit. In this case, he does not announce the basis used, because a basis in which he detected a photon uniquely identifies the bit Alice had sent. This protocol is usually called B92. However, such a scheme is secure only in lossless systems or if the losses are very low. In case of higher losses, an eavesdropper could sit in the middle and make measurements on the quantum states. If she has obtained an inconclusive result, she blocks the signal, while if she has detected the sent state, she re-sends a correct copy to Bob, because she knows the state with certainty. To compensate for the blocked photons, she can send a pulse of higher intensity so that Bob cannot observe any decrease in the expected transmission rate.

3.2. B92 protocol with a strong reference pulse One possibility to counteract the above-mentioned eavesdropping strategy against the B92 protocol is to encode bits into a phase difference between a dim pulse (with less than one photon in average) and a classical strong reference pulse (Bennett [1992]). This means the laser pulse is split into strong and weak parts on a highly unbalanced beamsplitter. Both Alice and Bob can introduce a phase shift between these pulses. On Bob’s side both pulses are combined again on an unbalanced beamsplitter where they interfere. Bob can also monitor the presence of all strong pulses. Now, when Eve gets an inconclusive result, she cannot suppress the strong pulse because Bob must receive all pulses. However, when Eve blocks only the

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dim pulse, interference of the bright pulse with vacuum (instead of the dim pulse) will lead to errors. Similarly, if Eve tries to fabricate her own dim or bright pulse (or both of them) and send it (them) to Bob she will inevitably cause detectable errors. Even though the B92 protocol can be unconditionally secure if properly implemented, Eve can now acquire more information on the key for a given disturbance than in the case of the BB84 protocol (Fuchs, Gisin, Griffiths, Niu and Peres [1997]).

3.3. Six-state protocol In the six-state protocol, three non-orthogonal bases are used (Bruss [1998], Bechmann-Pasquinucci and Gisin [1999]) that Alice and Bob randomly alternate. If we denote the two conjugate bases employed in the BB84 protocol as {|0 , |1 } ¯ |1 }, ¯ where and {|0 ,

¯ = √1 |0 + |1 , |0 2

1 ¯ = √ |0 − |1 , |1 2

(3.1)

¯¯ |1 } ¯¯ with then the third basis is {|0 , ¯¯ = √1 |0 + i|1 , |0 2 1 ¯¯ = √ |0 − i|1 . |1 2

(3.2)

The probability that Alice and Bob choose the same basis is now 1/3.6 But this disadvantage against BB84 is outweighed by the fact that eavesdropping causes a higher error rate. For example, a continuous intercept–resend attack induces on average 33% of errors here, compared to 25% in the BB84 protocol. In general, the maximal mutual information between Eve and Alice is smaller than in the BB84 scenario. Besides, the symmetry of the signal states simplifies the security analysis.

6 Factors like 1/3 for the six-state protocol or 1/2 for BB84 are not essential. In fact, the communication can proceed in only one orthogonal basis and the other non-orthogonal states can be sent randomly from time to time just to test the channel for the presence of an eavesdropper. So if the probabilities of bases are “biased” in favor of one of the bases, these factors can asymptotically reach unity (Lo, Chau and Ardehali [2005]).

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3.4. SARG protocol The SARG protocol (labeled after the names of its authors) was proposed to beat the photon-number splitting attack (PNS)7 in QKD schemes based on weak laser pulses. It relies on Eve’s inability to perfectly distinguish between two nonorthogonal states (Scarani, Acín, Ribordy and Gisin [2004], Branciard, Gisin, Kraus and Scarani [2005]). In contrast to BB84, two values of a classical bit are encoded into pairs of non-orthogonal states. However, to implement the SARG protocol one can keep the same hardware as for BB84 and modify only the classical communication between Alice and Bob. Alice prepares four quantum states and Bob makes measurements exactly as in the BB84 protocol. But Alice does not reveal the basis but the pair of non-orthogonal signal states such that one of these states is the one she has sent. Bob correctly guesses the bit if he finds a state orthogonal to one of two announced non-orthogonal states (for details see Scarani, Acín, Ribordy and Gisin [2004]). In comparison with the BB84 protocol, SARG enables to increase the secure QKD radius when the source is not a single-photon source.

3.5. Decoy-state protocols The decoy-state method represents another way to counteract PNS attack on QKD schemes using weak laser pulses (Hwang [2003], Wang [2004a, 2004b], Lo, Ma and Chen [2005], Ma [2004]). It can substantially prolong the distance over which secure communication is possible. If this method is used with the BB84 protocol the secure key rate is proportional to the overall transmittance even if the light source is an attenuated laser (the secure key rate for standard BB84 is linearly dependent on transmittance only in the case of a single-photon source, with weak laser pulses it is proportional to the square of the transmittance). The idea is based on the observation that by adding some decoy states, one can estimate the behavior of vacuum, single-photon and multi-photon states individually. Hence, Alice sometimes sends an additional, decoy, state with a different intensity than the states used for the key transmission (but with the same wavelength, timing, etc.). These decoy states serve only for testing Eve’s presence. Eve does not know when Alice sends the decoy states and she cannot identify them.

7 In the photon-number splitting attack Eve exploits multi-photon states present in weak laser pulses; see Section 8.5.4.

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Changes made on these decoy states by Eve’s PNS attack enable Alice and Bob to detect the PNS eavesdropping. The essence of the decoy-state method is as follows: The conditional probability Yn that Bob detects a signal – providing that Alice’s source has emitted an n-photon state – must be the same for both the signal and the decoy states. When no eavesdropper is present it must be equal to the following value given by the parameters of the apparatus: decoy signal Yn (3.3) = Yn = 1 − (1 − η)n (1 − pdark ) + pdark , = Yn where η is the total transmission efficiency and pdark is the probability of the detector dark count. The PNS attack inevitably changes some Yn . The quantities Yn are not directly measurable. But what Bob can directly determine is the total detection rate for a given mean photon number μ of Alice’s pulses, Qμ = e−μ

∞ n=0

Yn

μn . n!

(3.4)

If Alice and Bob use decoy states with different mean photon numbers they can estimate values of Yn for some photon numbers n and check whether they correspond to the expected values. The security of the decoy-state method with the BB84 protocol under the “paranoid” assumptions (Gottesman, Lo, Lütkenhaus and Preskill [2004]) has been analyzed by Lo, Ma and Chen [2005].

3.6. Entanglement-based protocols Another class of QKD protocols is based on quantum entanglement. The security of the original proposal was ensured by checking the violation of Bell’s inequalities (Ekert [1991]). The simplified version of the protocol works in a very similar way as BB84 (Bennett, Brassard and Mermin [1992]). 3.6.1. Entanglement, Bell’s inequalities Two or more quantum systems are entangled if their global state cannot be expressed as a direct product or a statistical mixture of direct products of any quantum states of the individual systems. Entanglement leads to many interesting effects unknown in classical physics. It is at the basis of quantum teleportation (Bennett, Brassard, Crépeau, Jozsa, Peres and Wootters [1993]) and it is responsible for the effectiveness of quantum computation (Nielsen and Chuang [2000]).

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Asher Peres said that “Entanglement is a trick that quantum magicians use to produce phenomena that cannot be imitated by classical magicians.” (Bruss [2002]). In 1935 Einstein, Podolsky and Rosen (Einstein, Podolsky and Rosen [1935]) formulated a Gedankenexperiment employing two particles prepared in an entangled state to argument against the completeness of quantum theory. They used the fact that the result of any potential measurement on one subsystem of the properly chosen entangled pair can be predicted with certainty after the proper measurement on the other subsystem. Following this fact and a few “natural” assumptions (namely the assumptions of locality and reality) they concluded that there must simultaneously exist “elements of reality” for two complementary observables. However, in 1964 John Bell (Bell [1964]) showed that there is no local realistic theory that would give the same predictions as quantum mechanics. Namely, quantum mechanics predicts different values of certain correlations of measurement results on a bipartite system in a specific entangled state. He derived his famous inequalities that must be satisfied by any local realistic theory but that may be violated by quantum theory. Let us denote A(n1 ) and B(n2 ) random variables, getting discrete values ±1, corresponding to measurement results on two separated but somehow correlated particles, where the settings of respective measurement devices are represented by unit vectors n1 and n2 (note that A depends only on n1 and B depends only on n2 – this reflects the locality condition). The randomness of A and B is supposed to be caused only by some random parameters λ that may be common for both the particles and that we do not know (the premise of reality). The Bell inequality, in the form derived by Clauser, Horne, Shimony and Holt [1969], states that

C(n1 , n2 ) + C n′ , n2 + C n1 , n′ − C n′ , n′ 2, (3.5) 2 1 2 1 where C(n1 , n2 ) is the correlation function C(n1 , n2 ) = A(n1 )B(n2 ) = A(n1 , λ)B(n2 , λ) dρλ .

(3.6)

Now, let us try to describe such a situation in quantum language, assuming two spin-1/2 particles in the following entangled state: 1

|ψ = √ |n, + 1 |n, − 2 − |n, − 1 |n, + 2 , 2

(3.7)

where the state vectors |n, ± correspond to two orthogonal projections of spin to direction n. Then the quantum prediction for the correlation function reads C(n1 , n2 ) = ψ|(n1 · σ 1 )(n2 · σ 2 )|ψ ,

(3.8)

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where σ 1 , σ 2 are vectors of Pauli matrices. If we choose the settings of the measurement apparatuses in such a way that n2 with n1 , n1 with n′2 and n′1 with n2 include angle 45◦ , while n′1 with n′2 include angle 135◦ , we readily find that √

C(n1 , n2 ) + C n′ , n2 + C n1 , n′ − C n′ , n′ = 2 2 > 2. (3.9) 1 2 1 2 3.6.2. Original Ekert protocol and its simplified form

According to Ekert’s protocol (Ekert [1991]), Alice and Bob each obtain one particle from a pair of spin-1/2 particles in the state (3.7). (In fact, it does not matter whether they share two entangled spin-1/2 particles or, e.g., two photons with entangled polarizations.) Alice and Bob perform measurements on their respective particles in three bases defined by three orientations of their measurement devices (e.g., Stern–Gerlach apparatuses). For simplicity let us suppose that they use only directions lying in the plane perpendicular to the trajectory of the particles. Alice’s bases make angles with respect to the vertical 0◦ , 45◦ , 90◦ , and Bob’s bases are making 45◦ , 90◦ , 135◦ . There are nine possible combinations. After the quantum transmission, during which Alice and Bob randomly and independently set their measurement bases, the settings are publicly announced. When identical bases were used, the outcomes of their measurements are correlated and become the cryptographic key. The probability that Alice and Bob use the same basis is 2/9. The outcomes of measurements in the other bases are used to verify the violation of the Clauser–Horne–Shimony–Holt inequality (3.5). An eavesdropper attempting to correlate his probe with the other two particles would disturb the purity of the singlet state (3.7), which would result in a smaller violation of the inequality or no violation at all. A year later Bennett, Brassard and Mermin [1992] proposed a simpler entanglement-based protocol without invoking Bell’s theorem directly. Here, both Alice and Bob choose only from two bases corresponding to two perpendicular orientations of their spin-measurement devices in a way very similar to the BB84 protocol. In fact, the only difference from BB84 is that Alice does not send particles in a chosen spin (or polarization) state but she measures her particle from the entangled pair in one of two conjugated bases. She must select bases randomly and independently from Bob. The rest is the same as in BB84: After the transmission Alice and Bob compare their bases and keep only those results for which they used the same bases. 3.6.3. Passive setup The system for entanglement-based QKD can even be designed in such a way that it can be operated entirely in a passive regime without any externally driven ele-

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ments (e.g., polarization rotators or phase modulators; Rarity, Owens and Tapster [1994]). Each particle from the entangled pair “may freely decide” on a beamsplitter in which basis it will be measured. This means, both the random key bits and random measurement basis are chosen directly by the genuine randomness of the nature.

§ 4. Experiments 4.1. QKD with weak laser pulses Attenuated lasers are often used as sources in practical QKD devices. If the spectral width of the laser pulses is much smaller than their mean frequency, the state of light can well be approximated by a monochromatic coherent state. The photon-number distribution of the coherent state is governed by the Poisson statistics. The multi-photon pulses can cause problems due to the PNS attack. Eve could always split off one photon and perform a measurement on it without introducing an error. This potentially leaked information must be taken into account (see Section 8.5.3 and 8.5.4). The trick that beats this attack appears in the decoystate method (see Section 3.5). 4.1.1. Polarization encoding The very first QKD experiment, that took place in 1989 (Bennett and Brassard [1989], Bennett, Bessette, Brassard, Salvail and Smolin [1992]) was based on polarization encoding for the BB84 protocol. For the description of the protocol we refer the reader to Section 2. A light-emitting diode (LED) generated light pulses that were subsequently attenuated by an interference filter and polarized by a polarizer (see fig. 1). The qubits were encoded in the polarization of photons by means of Pockels cells. The quantum channel was 32 cm of free air. Bob analyzed the polarization states using a Wollaston prism, which was preceded by another Pockels cell for choosing his polarization basis. The output ports of the prism were monitored by photomultipliers. Four years later, Gisin’s group at the University of Geneva replaced the free-air optical path by a 1 km optical fiber (Müller, Bréguet and Gisin [1993], Bréguet, Müller and Gisin [1994]). A semiconductor laser at 800 nm was used to generate light pulses that were detected by silicon avalanche photodiodes. Since the optical fiber deforms the polarization state of light, a manually adjustable polarization controller was employed to compensate for temporal changes of polarization.

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Fig. 1. First QKD experiment (Bennett and Brassard [1989]).

Bends and twists of the optical fiber induce birefringence, which gives rise to different velocities of the orthogonal polarization components of light that result in changes of the polarization state. Since the degree of polarization degrades slowly in fibers, the same stress-induced birefringence can, on the other hand, be used to compensate for this deformation. A fiber spool of a suitable diameter can act as a fractional wave plate. Franson and Ilves [1994] proposed a QKD device with an active polarizationalignment feedback loop. Such a system was demonstrated to work over a distance of 1 km (Franson and Jacobs [1995]). The first experiment with Alice and Bob being placed in different laboratories (in this case even different towns, Geneva and Nyon) was performed by the Geneva group (Müller, Zbinden and Gisin [1995], Müller, Zbinden and Gisin [1996]). Error rates of only 3–4% were achieved between two stations connected by a 23 km fiber deployed under Lake Geneva. In order to reduce fiber losses, a laser at 1.3 µm was used and the photons were detected by liquid-nitrogencooled germanium avalanche photodiodes. Using optical fiber is not the only way to implement QKD at a distance. Another approach is to try to communicate directly through free space. Unlike fibers, the atmosphere is non-birefringent, therefore polarization encoding is very suitable. The feasibility of free-space QKD was shown by Jacobs and Franson [1996], who managed to communicate over 150 m in a fluorescent-tube-illuminated corridor and over 75 m outdoors in daylight. It was the first free-space implementation of QKD after the celebrated 1989 Bennett and Brassard experiment, and there were more to come. The Los Alamos group first exchanged keys at 1 km by night

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bouncing the photons between mirrors (Buttler, Hughes, Kwiat, Luther, Morgan, Nordholt, Peterson and Simmons [1998a], Buttler, Hughes, Kwiat, Lamoreaux, Luther, Morgan, Nordholt, Peterson and Simmons [1998b]), then performed point-to-point communication in daylight over 0.5 km (Hughes, Buttler, Kwiat, Lamoreaux, Morgan, Nordholt and Peterson [2000a]) and eventually over 1.6 km (Buttler, Hughes, Lamoreaux, Morgan, Nordholt and Peterson [2000]). A distance of 1.9 km at night was covered by Gorman, Tapster and Rarity [2001]. Hughes, Nordholt, Derkacs and Peterson [2002] then demonstrated free-space QKD over 10 km. Free-space QKD over the largest distance so far was performed by the Munich group of H. Weinfurter (Kurtsiefer, Zarda, Halder, Gorman, Tapster, Rarity and Weinfurter [2002a], Kurtsiefer, Zarda, Halder, Weinfurter, Gorman, Tapster and Rarity [2002b]). Unlike the other groups, they moved to the high altitudes of the Alps to take advantage of thinner air and less air turbulence. Alice was located on the summit of Zugspitze (2962 m) and Bob was on the 23.4 km-distant Karwendelspitze (2244 m). Demonstration of free-space QKD with a single-photon source based on a nitrogen-vacancy center in diamond (see Section 5.1.3) was done by Beveratos, Brouri, Gacoin, Villing, Poizat and Grangier [2002] (indoor experiment over 50 m) and by Alléaume, Treussart, Messin, Dumeige, Roch, Beveratos, BrouriTualle and Grangier [2004] (this later experiment of the same group was operated outdoors over 30 m at night). 4.1.2. Phase encoding In phase encoding, different polarizations (as used in polarization encoding) are replaced by different phase shifts between two arms of a Mach–Zehnder interferometer. Alice controls the phase shift in one arm of the interferometer, Bob controls the phase shift in the other arm. If Alice’s and Bob’s phase shifts are the same or differ by 180◦ , then the behavior of the photon at Bob’s beamsplitter is deterministic because of constructive interference in one of the outputs and destructive interference in the other one. If the total phase shift between the arms is different from an integer multiple of 180◦ , photons are detected randomly at both detectors. In case of the BB84 protocol, Alice encodes bit values into four non-orthogonal quantum states. She sends weak light pulses to the interferometer and randomly sets phase φA to 0◦ , 90◦ , 180◦ or 270◦ . Bobs randomly (and independently of Alice) sets phase φB to 0◦ or 90◦ . These two values correspond to the measurement

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in “rectilinear” and “diagonal” bases, respectively: φA

+: ×:

0◦ . . . “1”, 90◦ . . . “1”,

180◦ . . . “0” 270◦ . . . “0”

φB

+: ×:

0◦ 90◦

However, in practice it is impossible to keep the same and stable phase conditions in two different arms of the Mach–Zehnder interferometer over long distances. The way to solve this problem was proposed already by Bennett [1992]. Two communicating parties can employ a time multiplex and use only one optical fiber to interconnect their devices (see fig. 2). Now two unbalanced Mach– Zehnder interferometers are used. The path difference between the longer and shorter arm of each interferometer is larger than the width of the laser pulse.8 But the path differences are the same for both interferometers. The case where the photon first goes through the longer (L) arm and then through the shorter (S) one is indistinguishable from the case when it first passes the shorter and then the longer arm. This path indistinguishability results in the interference at the last beamsplitter. Thus for the “central peak” (see right-most panel of fig. 2) the system behaves exactly in the same way as a single balanced Mach–Zehnder interferometer. This peak is selected by the proper timing of detection and the events when the photon passed either through both shorter or through both longer arms are ignored. The first system based on phase encoding was built by Townsend, Rarity and Tapster [1993a] (see also Townsend, Rarity and Tapster [1993b]). The signal was sent through 10 km of fiber in a spool. Later the system was modified so that the polarization in the long arms was rotated by 90◦ in both interferometers and the time multiplex was supplemented by a polarization multiplex. That is, at the output of Alice’s interferometer and at the input of Bob’s interferometer there were polarization beamsplitters. This technique suppresses the lateral non-interfering

Fig. 2. Setup for phase-encoded QKD with a double Mach–Zehnder interferometer.

8 If the pulse width is in the order of nanoseconds then the path difference is usually a few meters.

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peaks (Townsend [1994]). Further the distance was increased to 30 km (Marand and Townsend [1995]). Townsend [1997] also tested a wavelength-division multiplex to execute both QKD and classical communication through the same fiber on different wavelengths. A QKD system with a double Mach–Zehnder interferometer was realized also in Los Alamos National Laboratory (Hughes, Luther, Morgan and Simmons [1996], Hughes, Morgan and Peterson [2000b]); it was tested in an installed optical fiber up to a distance of 48 km. Another fiber-based system (at 830 nm) was realized by Dušek, Haderka, Hendrych and Myška [1999b]. It implemented active stabilization of the interferometers and programmed all supporting procedures for practical QKD. The system was used as a quantum identification system (for mutual identification of the users) at a distance of 500 m. A system with silica-based integrated-optic interferometers was built by Kimura, Nambu, Hatanaka, Tomita, Kosaka and Nakamura [2004] and tested at a distance over 150 km. Toshiba Research Europe developed an automated system at 1550 nm with a new method for active interferometer stabilization (a “stabilization” pulse goes after each signal pulse) and tested it at distances up to 122 km (Gobby, Yuan and Shields [2004], Yuan and Shields [2005]). Systems using either the polarization encoding or double Mach–Zehnder interferometers require an active stabilization to compensate drifts and fluctuations of polarizations and/or phases. Müller, Herzog, Huttner, Tittel, Zbinden and Gisin [1997] proposed an interesting way to implement a QKD device (using phase encoding) where all optical and mechanical fluctuations are automatically passively compensated (the principle of this autocompensation is based on an earlier idea of Martinelli [1989]). Two strong mutually delayed pulses of orthogonal linear polarizations go from Bob to Alice. At Alice’s side they are attenuated (part of them is also used for synchronization purposes), the first pulse is phase-shifted (this is the way Alice encodes the information), and both pulses are reflected on a Faraday mirror. The Farady mirror, which is a Feraday rotator followed by a mirror, exchanges their vertical and horizontal polarization components. Then these two dim pulses return to Bob. Because they go back through the same line but have properly modified polarizations by the Faraday mirror, all the polarization distortions caused by birefrigence experienced by the pulses in their first trip are compensated during the return trip. At the end the sent vertical polarization returns as horizontal one and vice versa. At Bob’s side the first pulse passes the longer arm of an unbalanced Mach–Zehnder interferometer while the second pulse passes its shorter arm (the pulses are separated by a polarization beamsplitter and then their polarizations are made the same). In one of the arms Bob now

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applies his phase shift. Because the original delay between the pulses was created by the same unbalanced interferometer no stabilization of this interferometer is needed. Since no special optical adjustment is necessary to operate this set-up it is usually called “plug&play” system. However, there are also some drawbacks: The fact that pulses must first go from Bob to Alice and then back complicates the timing of the whole process and may effectively decrease the transmission rate. The problem is, especially, with Rayleigh backscattering. To suppress its contribution to the error rate the strong pulses coming from Bob should not meet with the weak pulses propagating in the opposite direction. Further, because the strong pulses must travel the entire path from Bob to Alice before they are attenuated and the information is encoded, Eve has an opportunity to change some of their properties, e.g., their photon statistics. The system is also more sensitive to certain “Trojan horse” attacks (see Section 8.7). The first experimental realization was done by Zbinden, Gautier, Gisin, Huttner, Muller and Tittel [1997]. The key was exchanged over a 23-km-long optical fiber installed under Lake Geneva. Later the fully automated system was tested on the same fiber (Ribordy, Gautier, Gisin, Guinnard and Zbinden [2000]). The implemented protocol was BB84. The system was operated at 1300 nm. A similar autocompensating system operating at 1300 nm was independently developed at IBM (Bethune and Risk [2000]); it was tested on a 10-km-long fiber in a spool. In this set-up the pulses sent by Bob had a reduced intensity to avoid Rayleigh backscattering. Synchronization was provided by classical pulses at 1550 nm using a wavelength-division multiplex. Nielsen, Schori, Sørensen, Salvail, Damgård and Polzik [2001] built a system working at 1310 nm and distributed a key over 20 km in fiber. The group of A. Karlsson demonstrated that the “plug&play” technique can be implemented in fibers also at 1550 nm (Bourennane, Gibson, Karlsson, Hening, Jonsson, Tsegaye, Ljunggren and Sundberg [1999]). Later the operation of an improved Geneva “plug&play” setup at 1550 nm was demonstrated over a 67-km-long optical-fiber link between Geneva and Lausanne (Stucki, Gisin, Guinnard, Ribordy and Zbinden [2002]). The first experimental demonstration of the decoy-state method (see Section 3.5) was done by Zhao, Qi, Ma, Lo and Qian [2005]. Their set-up used a modified commercial QKD “plug&play” system manufactured by id Quantique. The distribution was tested over a distance of 15 km. The protocol was based on the BB84 scheme together with a practical implementation of the decoy-state method with only one decoy state. The average intensities of the signal and decoy states were chosen to be 0.8 and 0.12 photons, respectively. Roughly 88% of signal states and 12% of decoy states were transmitted.

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Gisin, Ribordy, Zbinden, Stucki, Brunner and Scarani [2004] proposed a new technique for practical QKD, based on a specific protocol and tailored for an implementation with weak laser pulses. The key is obtained by a simple measurement of the times of arrival of the pulses incoming to Bob. The presence of an eavesdropper is checked by an interferometer built on an additional monitoring line. Each logical bit is encoded into a sequence of two pulses: either one empty and one non-empty or vice versa. There is a phase coherence between any two non-empty pulses because a mode-locked laser is used as a source. Some pulses are reflected at Bob’s beamsplitter and go to the unbalanced Mach–Zehnder interferometer (monitoring line). Here is where quantum coherence plays a role. If coherence is not broken, only the detector at the particular output of the interferometer may fire at certain instants. This enables to detect eavesdropping. The first experimental realization of this protocol was done by Stucki, Brunner, Gisin, Scarani and Zbinden [2005].

4.2. Entanglement-based protocols The principle of entanglement-based protocols was explained in Section 3.6. In practical realizations only the entangled states of photons are used. However, different kinds of entanglement can be employed: For example, entanglement in polarizations of photons, entanglement in energy and time, entanglement in orbital angular momentum or so-called “time-bin” entanglement which is a special case of energy–time entanglement. Experiments with QKD using photon pairs often utilized set-ups and took up on experiments examining the violation of Bell’s inequalities. Besides QKD, the distribution of entanglement between distant users can be beneficial also for other tasks like quantum teleportation, quantum dense coding, quantum secret sharing, etc. However, there is the problem of coupling between the property used to encode the qubits and the other properties of the carrier electromagnetic field, which rises during the propagation in a dispersive medium. This form of decoherence gradually destroys quantum correlations between the photons.9 For example, polarization-mode dispersion makes two values of polarization-encoded qubit distinguishable also in the temporal domain and so wipes out quantum correlations between polarizations. Similarly, chromatic dispersion degrades energy–time entanglement.

9 This effect also has a positive aspect: It prevents unintentional information leakage in unused degrees of freedom (Mayers and Yao [1998]).

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4.2.1. Polarization entanglement With polarization entanglement, Alice and Bob are each provided by one photon of an entangled pair of one of the following forms: 1

√ |V A |V B ± |H A |H B , 2

1

√ |V A |H B ± |H A |V B , 2

(4.1)

where |V and |H denote single-photon states with vertical and horizontal linear polarizations, respectively. The pairs are prepared by a parametric downconversion process in nonlinear optical crystals. Polarization entanglement is created either by one crystal using phase matching of type II (in a proper geometrical lay-out) or by two crystals with type-I phase matching that are placed closely one by one but with optic axes oriented perpendicularly to each other. Alice and Bob are equipped with polarization analyzers that can rapidly change measurement polarization bases, e.g., electro-optical polarization modulators followed by polarizing beamsplitters (with photon counters behind them). The first two experiments were reported in 2000. Zeilinger’s group (Jennewein, Simon, Weihs, Weinfurter and Zeilinger [2000]) used a BBO10 crystal, cut for type-II phase matching and pumped by an argon-ion laser, to generate photon pairs at 702 nm (both photons had the same wavelength). Their analyzers consisted of fast modulators, polarizing beamsplitters, and silicon avalanche photodiode (APD) detectors. They demonstrated QKD over 360 m in installed singlemode fibers. Kwiat’s group at Los Alamos (Naik, Peterson, White, Berglund and Kwiat [2000]) worked with two BBO crystals of type-I phase matching pumped by an argon-ion laser and also produced photon pairs with degenerate wavelengths at 702 nm. They implemented the original Ekert protocol and demonstrated QKD in free space at a distance of a few meters. In addition, they experimentally simulated different eavesdropping strategies. In a more recent experiment by Poppe, Fedrizzi, Lorünser, Maurhadt, Ursin, Böhm, Peev, Suda, Kurtsiefer, Weinfurter, Jennewein and Zeilinger [2004] in Vienna, the secret key was distributed over 1.45-km-long installed fiber (between a bank and the City Hall). Polarizationentangled pairs at 810 nm were produced by type-II parametric down-conversion in a BBO crystal pumped by a semiconductor laser. The distribution of entanglement over 13 km in free space was demonstrated by Peng, Yang, Bao, Jun-Zhang, Jin, Feng, Yang, Yang, Yin, Zhang, Li, Tian and Pan [2004]. It was used both to prove a space-like separated violation of Bell’s inequality and to realize a QKD-based BB84-like protocol. It utilized type-II parametric down-conversion 10 β-BaB O . 2 4

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in BBO crystal pumped by an argon-ion laser. The wavelengths of entangled photons were 702 nm. 4.2.2. Energy–time entanglement, phase encoding In energy–time entanglement, the employed two-photon entangled states have the approximate form

dω ξ(ω)|ω A |ω0 − ω B ,

(4.2)

where |ω denotes a single-photon state at frequency ω, ω0 is the optical frequency of the pump laser and ξ(ω) expresses the distribution of individual frequency components. The pairs are again produced by parametric downconversion in nonlinear optical crystals. Photons in states close to that given by eq. (4.2) – neglecting vacuum and multi-pair contributions – are generated when the crystal is pumped by a laser with a large coherence time. Alice and Bob obtain one photon each and they let them pass through identically unbalanced Mach– Zehnder interferometers (one interferometer at Bob’s side, one at Alice’s side). The path length difference between the longer and shorter arms of each interferometer must be larger than the coherence length of generated photons but shorter than the coherence length of the pump laser. The path differences must be the same for both interferometers. The instants of detection of two photons from a pair are very tightly correlated (of the order of hundreds of femtoseconds) but the particular times of these coincident detections are uncertain and random. Therefore Alice and Bob cannot distinguish between the situations when both photons went through the longer arms of their interferometers and when both of them went through the shorter arms (this leads to fourth-order interference). Alice and Bob choose their measurement bases by changing the phase shifts between the arms of their interferometers (e.g., they can randomly and independently alternate shifts 0◦ and 90◦ ). When their phase difference is 0◦ , the measurement autcomes are deterministic. When the phase difference is ±90◦ , the results are random. Events when one photon went through a shorter arm and the other through a longer one are ignored. This arrangement was originally devised by Franson [1989] for another purpose. Its use for practical QKD in fibers was proposed by Ekert, Rarity, Tapster and Palma [1992]. The set-up can be further modified to be operated completely in a passive way (Rarity, Owens and Tapster [1994]) – see Section 3.6.3.

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This QKD scheme was first realized by Ribordy, Brendel, Gautier, Gisin and Zbinden [2001] of the University of Geneva. They used a KNbO3 crystal pumped by a doubled Nd-YAG laser to create entangled pairs with asymmetric wavelengths 810 nm and 1550 nm. The wavelength 810 nm was advantageous in that efficient and low-noise Si-APD photon counters could be used at Alice’s side (the distance between the source and Alice’s analyzer was very short). The wavelength 1550 nm of the other photon fitted the low-loss window of optical fibers, so this photon traveled the longer distance between the source and Bob. Bob was connected to the source by an 8.5-km-long optical fiber in a spool (a dispersionshifted fiber was used to limit the decoherence induced by chromatic dispersion). It should be noted that a passive set-up was implemented. Two measurement bases (at each terminal) were passively randomly selected using a polarizing beamsplitter. One physical interferometer behaved like two interferometers with different phase settings for two different polarizations of light. 4.2.3. Time-bin entanglement, phase–time encoding Time bin entanglement (or phase–time encoding) is similar to the phase encoding described above. But now one more unbalanced Mach–Zehnder interferometer is placed in the pump beam, and a pulsed source is used to pump the crystal. The apparatus is shown schematically in fig. 3. The generated pair can be described

Fig. 3. Schematic setup for QKD using time-bin entanglement.

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by the following state: 1

√ eiφ |S A |S B − |L A |L B , 2

(4.3)

with S and L denoting contributions from pump pulses going through the shorter and the longer arm of the interferometer, respectively. The path length differences of all three interferometers should be the same. Now Alice can detect a photon in three different time windows (after each laser pulse): The first corresponds to the situation when both the pump pulse and Alice’s photon went through the shorter arms (SS), the second corresponds to the combination of the shorter and the longer arm or vice versa (SL or LS), and the third corresponds to the situation when both the pump pulse and Alice’s photon went through the longer arms (LL). The same holds for Bob’s detections. To establish the secret key Alice and Bob publicly agree on the events when both of them detected a photon (no matter at which detector) either in the first or in the third time window, but do not reveal in which one, and on the events when they both registered detector clicks in the second time window, without revealing at which detector. In the first case they assign different bit values to the first and third time window (Alice and Bob must have correlated detection times). The second case (both photons detected in the second time window) is formally equivalent to the above described phase-encoding method. This technique was proposed by Brendel, Gisin, Tittel and Zbinden [1999] (who have also built the source of pairs) and the QKD experiment was performed by Tittel, Brendel, Zbinden and Gisin [2000]. The system was tested only in the laboratory. A KNbO3 crystal was pumped by a pulsed semiconductor laser diode. The wavelength of down-converted photons was 1310 nm. Later, the distribution of time-bin entangled qubits was demonstrated over 50 km of optical fiber (Marcikic, Riedmatten, Tittel, Zbinden, Legre and Gisin [2004]).

§ 5. Technology 5.1. Light sources 5.1.1. Attenuated lasers For practical QKD systems, attenuated lasers are still the only reasonable light sources (except systems using entangled pairs). The radiation from a laser can usually be well described by a single-mode coherent state exhibiting Poissonian

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photon-number distribution μn −μ (5.1) e n! with μ the mean photon number. Clearly, a highly attenuated laser pulse with very small μ represents a good approximation of a single-photon Fock state (or rather a superposition of states |0 and |1 ) because the ratio pmulti /p(1) of the probability of more than one photon, pmulti = ∞ n=2 p(n), and a single-photon probability, p(1), goes to 0 as μ → 0. The only problem is the increasing fraction of vacuum . . states (n = 0). For example, if μ = 0.1 then p(0) = 0.905, p(1) = 0.090 and . pmulti = 0.005. Empty pulses decrease the transmission rate. A more important problem arises from detector dark counts. Because detectors must be active for all pulses including empty ones the dark-count rate is constant while the rate of nonempty pulses decreases with decreasing μ. This prevents the use of arbitrarily low mean photon numbers. The mean photon number must be chosen according to several criteria. The existence of detector dark counts and the losses in the system prompt us to use a mean photon number as high as possible. On the other hand the potential leakage of information through multi-photon pulses prompts us to use a mean photon number as low as possible. The optimal mean photon number is the one that maximizes the secure key rate for given conditions. It results from the trade-off between the value of the detection rate and the shortening of the key due to privacy amplification (because of multi-photon contributions, privacy amplification shortens the resulting distilled key substantially if μ is too high, namely if μ η where η is the line transmittance; Lütkenhaus [2000]). A good measure of the quality of imperfect single-photon sources is the secondorder autocorrelation function of the source, g2 = I 2 / I 2 , i.e., the correlation measured in a Hanbury–Brown–Twiss-type experiment (I means optical intensity). It can be approximately calculated as g2 ≈ 2p(2)/[p(1)]2 if p(1) ≫ p(2) ≫ ∞ n=3 p(n). The value g2 = 1 corresponds to the Poissonian case, while g2 < 1 indicates sub-Poissonian distribution. p(n) =

5.1.2. Single-photon sources: parametric down-conversion Another way to prepare quasi-single-photon states is to use photon pairs generated by spontaneous parametric down-conversion (SPDC) (Hong and Mandel [1986]). Here the crucial point is a tight time correlation between photons in the pair. In the ideal case, if one places a photon-number detector into the path of one member of the pair (say, into the idler beam) and detects one photon, then at the same time (i.e., in a very short time window of the order of hundreds of femtoseconds) there must be one photon also in the other (signal) beam.

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In reality, due to losses in the signal beam caused mainly by an inefficient coupling into the fiber and partly by dark counts of the trigger detector, there may be no photon in the signal beam even if the trigger detector has clicked. However, the probability of this event is relatively low – today typically about 30%. Nearly all practically applicable detectors cannot distinguish the number of photons, and their quantum efficiency is substantially lower than 100%. Therefore, there is also non-zero probability of having more than one photon in the signal beam after the trigger detection. (Notice that the number of photons in one mode is thermally distributed and the total number in all modes obeys the Poissonian distribution.) On the other hand, the efficiency of the conversion of a pump photon into a pair of sub-frequency photons is very low, typically about 10−10 , so the probability of generating multi-photon states is also low.11 Besides, there are techniques that allow us to eliminate partly multi-photon states. They are based on the division of the idler beam, used for triggering, into several detectors. Events with more than one detector clicks are discarded. This spatial division can be subˇ stituted by time division using one detector behind a delay loop (Reháˇ cek, Hradil, Haderka, Peˇrina and Hamar [2003]). The important advantage of a SPDC quasi-single-photon source in comparison with an attenuated laser is a substantial reduction of the portion of vacuum contributions, i.e., empty signals. From the technological point of view these sources seem feasible. Diode-laserpumped SPDC sources emitting in the near-infrared region can be made compact and robust (Volz, Kurtsiefer and Weinfurter [2001]). 5.1.3. Single-photon sources: color centers A progressive direction in the research of single-photon sources is represented by color centers in diamond. Color centers are defects in a crystal lattice due to impurities and vacancies. Crystals with such defects can be prepared relatively easily and are stable. The key advantage of sources based on color centers is that they work at room temperature. Particularly, nitrogen-vacancy centers in synthetic diamond were intensively studied (Kurtsiefer, Mayer, Zarda and Weinfurter [2000], Brouri, Beveratos, Poizat and Grangier [2000], Beveratos, Brouri, Gacoin, Poizat and Grangier [2001]). These centers consist of a substitutional nitrogen atom and a vacancy at an adjacent lattice position. The individual nitrogen atom is excited by a focused 11 Take a source that generates 105 pairs per second in average and consider a 1 ns detection window, then this probability is about 10−4 .

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laser beam at 532 nm. Due to fluorescence the atom consequently emits a photon with the spectrum centered around 690 nm. Strong anti-bunching is observed. A weaker point is the broad spectrum of the generated pulses (nearly 100 nm): the optical properties of the transmission medium (absorption, refractive index, etc.) change over such a large interval of wavelengths. However, recently a new kind of crystal defect was found that can emit photons at 802 nm with a spectral width of only about 1 nm (at room temperature). This color center consists of a nickel ion surrounded by four nitrogen atoms in a genuine diamond (Gaebel, Popa, Gruber, Domhan, Jelezko and Wrachtrup [2004]). The main problem of single-photon sources based on color centers is a rather low collection efficiency – currently just about 0.1% for bulk crystals. The situation is slightly better for diamond nanocrystals12 – currently over 2% (Beveratos, Brouri, Gacoin, Villing, Poizat and Grangier [2002]). The way how to increase the collection efficiency is to put the crystal into an optical cavity that suppresses emission to all other spatial modes except the preferred one. There have already been experiments with quantum cryptography using singlephoton sources based on nitrogen-vacancy centers (Beveratos, Brouri, Gacoin, Villing, Poizat and Grangier [2002], Alléaume, Treussart, Messin, Dumeige, Roch, Beveratos, Brouri-Tualle and Grangier [2004]). QKD in free space was demonstrated at a distance of 50 m. 5.1.4. Single-photon sources: quantum dots Quantum dots are semiconductor nanostructures (“artificial atoms”) (Santori, Pelton, Solomon, Dale and Yamamoto [2001], Moreau, Robert, Manin, ThierryMieg, Gérard and Abram [2001], Zwiller, Blom, Jonsson, Panev, Jeppesen, Tsegaye, Goobar, Pistol, Samuelson and Björk [2001], Hours, Varoutsis, Gallart, Bloch, Robert-Philip, Cavanna, Abram, Laruelle and Gérard [2003], Baier, Pelucchi, Kapon, Varoutsis, Gallart, Robert-Philip and Abram [2004]). By a suitable preparation a two- or more-level electronic system can be obtained. Photon emission comes from recombination of an electron–hole pair. Electron–hole pairs can be created either by optical pumping by a pulsed or continuous-wave laser or by an electric current (Yuan, Kardynal, Stevenson, Shields, Lobo, Cooper, Beattie, Ritchie and Pepper [2002]). Various preparation techniques for quantum dots exist. Materials used are, e.g., GaAs, GaAlAs or InP. The wavelength of emitted light is determined mainly by the material used. Sources operating at telecom wavelengths are possible (Takemoto, Sakuma, 12 The subwavelength size of nanocrystals suppresses problems with the high refraction at the sample interface.

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Hirose, Usuki, Yokoyama, Miyazawa, Takatsu and Arakawa [2004]). The spectral width of a generated pulse depends on the number of excited energy levels and the average number of electron–hole pairs created. The main practical drawback of quantum-dot photon sources is the need for cooling to liquid-helium temperature. The latest research promises a shift to temperatures of about 100 K (Mirin [2004]), but the photon-number distribution of such “high-temperature” sources is worse. Another problem is the very low collection efficiency (usually from 10−4 to 10−3 ). This means that the probability of obtaining an empty pulse is high. The efficiency can be increased (up to about 10−1 ) by placing the quantum dot in an integrated solid-state microcavity (Gérard, Sermage, Gayral, Legrand, Costard and Thierry-Mieg [1998]). The first demonstration of QKD using a quantum-dot single-photon gun was done by Waks, Inoue, Santori, Fattal, Vuckovic, Solomon and Yamamoto [2002]. It operated in free space to a symbolic distance of one meter. 5.1.5. Single-photon sources: single atoms and molecules Another way to generate single-photon-like states is to make use of radiative transitions between electronic levels of a single atom (ion) or molecule. Single ions caught in a trap and placed inside (or sent into) an optical cavity where they interact both with the excitation laser beam and the vacuum field of the cavity (Kuhn, Hennrich and Rempe [2002], Keller, Lange, Hayasaka, Lange and Walther [2004]) could represent single-photon sources with good properties (e.g., a narrow spectrum and high collection efficiency owing to the presence of the cavity). However, the practical feasibility of such sources is still low because of their technological complexity (among others, high vacuum is needed). Experiments with single organic-dye molecules are simpler because these are usually caught in a polymer matrix (Brunel, Lounis, Tamarat and Orrit [1999], Fleury, Segura, Zumofen, Hecht and Wild [2000], Treussart, Alléaume, Le Floch, Xiao, Courty and Roch [2002]) or put in a solvent (Kitson, Jonsson, Rarity and Tapster [1998]) and the source is operated in usual environmental conditions and room temperatures. The photon statistics of generated states is reported to be good. The advantage is also a large scope of wavelengths that can be generated. The critical problem here is the limited stability of the molecules. Due to photobleaching even the most stable dyes survive only a few hours of continuous excitation. 5.1.6. Entanglement source: spontaneous parametric down-conversion By spontaneous parametric down-conversion (SPDC) one can prepare photons entangled in energies (wavelengths), momenta (directions), and/or polarizations.

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Any of these features can be used for the purposes of QKD based on Ekert-type protocols (see Section 3.6). In SPDC, one photon from a pump laser is converted, with a certain (small) probability, into two sub-frequency photons, with the total energy and momentum being conserved. Since no couple of possible frequencies and wavevectors of two generated photons is preferred the resulting quantum state is given as a superposition of all allowed cases – it is an entangled state. SPDC occurs in nonlinear optical media, such as crystals of KNbO3 , LiIO3 , LiNbO3 , β-BaB2 O4 , etc. Very promising SPDC sources are periodically poled nonlinear materials, namely waveguides in periodically poled lithium niobate (Tanzilli, De Riedmatten, Tittel, Zbinden, Baldi, De Micheli, Ostrowsky and Gisin [2001]).

5.2. Detectors 5.2.1. Avalanche photodiodes The detectors most widely used in QKD systems with discrete variables are undoubtedly avalanche photodiodes (APD). In APD a single photoelectron generated by an impinging photon is multiplied by collision ionization. This is because APD single-photon detectors are operated in a so-called Geiger mode: On the junction a reverse voltage is applied that exceeds the breakdown voltage. Thus the impinging photon triggers an avalanche of thousands of carriers. To reset the detector the avalanche must be quenched, which can be done in a passive or an active way. In passive quenching, a large resistor is placed in the detector circuit, causing the voltage on the APD to decrease after the avalanche starts. In active quenching, the bias voltage is lowered by an active control circuit. This solution is faster so that higher repetition rates can be reached (up to 10 MHz). Another possibility is to work in a so-called gated mode when the bias voltage is increased above the breakdown voltage only for a short, well-defined period of time. To detect photons at specific wavelengths different materials of detector chips are needed. For the visible and near-infrared region (up to 1.1 µm) a silicon APD can be used. Nowadays they are well elaborated. Compact counting modules with integrated Peltier cooling and active quenching are commercially available, offering low dark-count rates (below 50 per second), high quantum efficiencies (up to about 70%), and maximum count rates reaching 10 MHz. Cooling to temperatures of about −20◦ C is necessary to keep the numbers of dark counts induced by thermal noise in a reasonable range. Note that dark counts, i.e., events when the

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detector sends an impulse even if no photon has entered it, represent an important limiting factor in the operation range of QKD (see Section 6). For telecom wavelengths used in fiber communications, 1300 nm and 1550 nm, the silicon detectors cannot be applied. For 1300 nm germanium and InGaAs/InP detectors can be used. Germanium detectors require cooling to liquid-nitrogen temperatures (77 K). Typical quantum efficiencies are about 15%, dark-count rates about 25 × 103 pulses per second (at 77 K). For 1550 nm even germanium detectors cannot be used any more and currently the only generally available detectors for this wavelength window are based on InGaAs (on InP substrate). These detectors are now in common use for both telecom wavelengths. InGaAs detectors must also be cooled to low temperatures. In practice this can be done either by three-stage Peltier thermoelectric coolers (down to about −60◦ C, i.e., 213 K) or by compact Stirling engines (down to about −100◦ C, i.e., 173 K). Today’s typical performance of InGaAs APD at 1550 nm with a Peltier cooler is as follows: quantum efficiency about 5–10%, dark-count rate (in gated mode) about 104 s−1 , maximal repetition frequency about 100 kHz–1 MHz (i.e., dead time about 1–10 µs). With a Stirling cooler (−100◦ C): quantum efficiency above 10%, dark-count rate hundreds per second (in gated mode), and maximal repetition frequency about 100 kHz–1 MHz. It turns out that the dark-count rate increases with increasing detection efficiency. It is always necessary to find a trade-off between these quantities. As the number of dark counts increases with temperature, better overall performance can be achieved at lower temperatures. Also increasing signal repetition frequency leads to the growth of the number of dark counts because of the increasing probability of afterpulses.13 Let us also mention another effect that can play a negative role in quantum cryptography. When the avalanche is quenched all charge carriers recombine. This brings the diode into an insulating state again, a full photodetection cycle is finished and the diode is ready for the next event. However, some recombinations are radiative – this results in so-called backflashes. These dim light pulses propagate back to the communication channel and they could reveal the information on Bob’s basis setting to an adversary. That is, they represent a serious side channel and must be carefully eliminated (blocked) by proper filters (Kurtsiefer, Zarda, Mayer and Weinfurter [2001]). An interesting possibility to improve the performance of QKD with APD detectors at telecom wavelengths could be the combination of parametric frequency up-conversion with efficient silicon APDs, instead of direct use of InGaAs APDs. 13 After the avalanche is quenched some charge carriers may stay trapped on impurities. Their delayed recombination can lead to so-called afterpulses – unwanted output impulses of the detector.

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The up-conversion in periodically poled lithium niobate can be rather efficient whereas it introduces only relatively low noise. The overall quantum efficiency in combination with a silicon APD detector could then be comparable with the detection efficiency of an InGaAs APD while the dark-cont rate would be lower (Diamanti, Takesue, Honjo, Inoue and Yamamoto [2005]). This fact could increase the operation distance of QKD. 5.2.2. Quantum dot detectors A Quantum Dot Resonant Tunneling Diode is a semiconductor device with a quantum dot layer encased inside a resonant tunneling diode structure (Blakesley, See, Shields, Kardynał, Atkinson, Farrer and Ritchie [2005]). In the diode two n-doped GaAs layers are separated by a double-barrier insulating AlGaAs layer and followed by a InAs self-assembled quantum dot layer. The resonant tunnel current through this double-barrier structure is sensitive to the capture of a hole excited by the photon by one of the quantum dots in the adjacent dot layer. The capture of a hole by the dot can switch the magnitude of the current flowing through the device. The maximum detection efficiency measured with the device at 550 nm was 12%. However, the reasonable dark-count rate of 4000 s−1 was achieved with a detection efficiency of only 5%. The device was cooled to 77 K. The measured sample could detect a new photon every 150 ns, which corresponds to a repetition rate of about 6 MHz (Blakesley, See, Shields, Kardynał, Atkinson, Farrer and Ritchie [2005]); however, it is mainly limited by external electronics, and improvement to about 100 MHz is expected in the near future. Note that a detector manufactured from GaAs cannot be used in the region of telecom wavelengths. Detectors for these wavelengths have to be built from other materials like InP. 5.2.3. Visible light photon counters Visible Light Photon Counters (VLPC) are semiconductor detectors consisting of two main layers, an intrinsic silicon layer and a lightly doped arsenic gain layer (Waks, Inoue, Oliver, Diamanti and Yamamoto [2003], Kim, Takeuchi, Yamamoto and Hogue [1999]). When a single photon is absorbed a single electron–hole pair is created. Due to a small bias voltage applied across the device, the electron is accelerated towards the transparent contact on one side while the hole is accelerated towards the gain region at the opposite side. Donor electrons in this region are effectively frozen out in impurity states because the device is cooled to an operation temperature of about 6 K. However, when a hole is accelerated into the gain region it easily kicks the donor electrons into the conduction

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band by impact ionization. Scattered electrons can create subsequent impact ionization events resulting in avalanche multiplication. When a photon is detected, a dead spot of several microns in diameter is formed on the detector surface, leaving the rest of the detector available for subsequent detection events. If more than one photon is incident on the detector, it will be able to detect all the photons as long as the probability that multiple photons land on the same location is small. Therefore these detectors could perform efficient photon number state detection (photon number count). However, in practice they can well discriminate only between zero, one and more photons because of multiplication noise. The quantum efficiency of a VLPC is about 90% and its dark-count rate is about 2 × 104 s−1 at 543 nm (at 6 K). 5.2.4. Superconducting detectors To detect single photons physical processes in superconductors can also be employed. Several different principles have been proposed that are now experimentally tested. All these detectors require a cryogenic environment. The first kind of detector, usually called Superconducting Single Photon Detector, consists of thin strips of superconducting material, such as niobium nitrate, interconnected to form a meander-shaped “wire” (Verevkin, Zhang, Sobolewski, Lipatov, Okunev, Chulkova, Korneev, Smirnov, Goltsman and Semenov [2002]). In this “wire” the current bias below the critical current of the material is maintained. An impinging photon breaks a Cooper pair and generates a hotspot that forms a resistive potential. The width of the strips is designed in such a way that the current forced around the hotspot exceeds the critical current. This results in the increase of resistance and a voltage signal indicating the detection of the photon. Recent measurements show that at 1300–1550 nm the samples have quantum efficiency up to 10%, dark-count rate about 0.01 s−1 and counting rate over 2 GHz (Verevkin, Pearlmany, Slyszyz, Zhang, Currie, Korneev, Chulkova, Okunev, Kouminov, Smirnov, Voronov, Goltsman and Sobolewski [2004]). The measurements were done at 2.5 K (liquid-helium temperature). Another type of superconducting detector is a Transition Edge Sensor (Miller, Nam, Martinis and Sergienko [2003]). These sensors consist of superconducting thin films electrically biased in the resistive transition. Their sensitivity is a result of the strong dependence of resistance on temperature in the transition and the low specific heat and thermal conductivity of materials at typical operating temperatures near 100 mK. The device produces an electrical signal proportional to the heat produced by the absorption of a photon. These detectors can even determine the number of impinging photons, i.e., they can perform a photon count.

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Observed efficiency at 125 mK is about 20%, dark-count rate about 0.001 s−1 (Miller, Nam, Martinis and Sergienko [2003]). The newest results show an even better performance with a quantum efficiency over 80% at 1550 nm (Rosenberg, Lita, Miller and Nam [2005]). Unfortunately, these detectors are very slow (dead time is about 15 µs) because it is necessary to remove the heat deposited by each photon (Miller, Nam, Martinis and Sergienko [2003]). The next possibility is a Superconducting Tunnel Junction Detector (Fraser, Heslop-Harrison, Schwarzacher, Holland, Verhoeve and Peacock [2003]). It consists of two superconducting electrodes separated by an insulating layer, together forming a Josephson junction. To suppress the tunneling current through the junction, a magnetic field parallel to the electrodes (parallel to the tunnel barrier) is applied. Incident photons break Cooper pairs, which changes the tunneling rate according to the absorbed energy. The operating temperature is on the order of hundreds of millikelvins. These detectors are able to register photons from infrared to ultraviolet.

5.3. Quantum channels 5.3.1. Fibers The most promising channels for terrestrial QKD are undoubtedly single-mode optical fibers. The lowest attenuations of standard telecom fibers are at 1300 nm (about 0.35 dB/km) and at 1550 nm (about 0.2 dB/km). Unfortunately, for these wavelengths standard silicon-based semiconductor photodetectors cannot be used. In principle, it is possible to use special fibers and work around 800 nm, where these efficient detectors are available. But the attenuation of fibers at these wavelengths is relatively high, about 2 dB/km, and such fibers are not used in an existing infrastructure. Therefore, attention is focused on standard telecom fibers and there is an effort to develop low-noise and efficient detectors for wavelengths 1300 nm and 1550 nm. The losses in fibers represent one of the two main factors (see Section 6) limiting the operation range of QKD systems (notice that an attenuation of 0.20 dB/km means 99% loss after 100 km). Other problems are the strong temperature dependence of some optical properties of fibers, the disturbance of polarization states of light in fibers due to the geometrical phase and the birefringence, and the dispersion. The distortion of polarization is a crucial obstacle for the use of any kind of polarization encoding of information. Therefore, fiber-based QKD systems usually employ phase-encoding schemes. However, even in such a case the output

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polarization state must be under control. Fortunately, if the fiber is fixed the polarization properties are relatively stable. Dispersion affects the temporal width of the broad-spectrum light pulses. Therefore, sources generating broad-band signals are not well suited for fiber QKD. Nevertheless, there is still a possibility to work near the wavelength of 1310 nm where silica fibers have zero chromatic dispersion, or to use fibers with a special refractive-index profile, which have zero dispersion shifted near 1550 nm. 5.3.2. Free space Quantum key distribution can also be accomplished through free space. The advantage of this approach is that the atmosphere has very low absorption around the wavelengths of 770 nm and 860 nm where relatively efficient and low-noise silicon semiconductor detectors can be used. Besides, no optical cables have to be installed. Also, the atmosphere is not birefringent at these wavelengths and is only weakly dispersive. The disadvantage is that the free-space communication can be used only at line-of-sight distances, and no obstacle may exist between communicating parties. There are also other drawbacks: the performance is highly dependent on the weather, pollution and other atmospheric conditions. There are huge differences in attenuation for different kinds of weather. For instance, for wavelengths near 860 nm the attenuation of clear air can be below 0.2 dB/km, changes in moderate rain to about 2–10 dB/km, and in heavy mist can exceed 20 dB/km. Further, up to altitudes of about 15–20 km there is considerable atmospheric turbulence. Another problem is spurious influence of the background light, especially the ambient daylight. Finally, there is the problem of beam divergence: due to diffraction the diameter of the beam can be considerably enlarged at large distances. This effect can cause additional loss if only a part of the beam is captured by the receiver.

§ 6. Limitations Two main technological obstacles presently inhibit the wide spread of quantum key distribution: limited operational range and low transmission rates.

6.1. Transmission rate The key factor limiting the raw key rate is the detector’s deadtime (i.e., recovery time of the detector). In the case of avalanche photodiodes (APD), immediately

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after the detection event the detector is not ready for another detection. First of all, the avalanche of charge carriers must be quenched. However, there is also a problem with the so-called afterpulses – clicks of the detector caused by spontaneous transitions from long-living traps (levels in a forbidden band) populated by the preceding avalanche. It is necessary to wait until all carriers have left the detection (depleted) region. Typical APD deadtime is from about hundred nanoseconds to a microsecond. The next factor decreasing transmission rate appears if an attenuated laser is used as a source for QKD. Due to security requirements (suppression of multiphoton pulses) the mean photon number per pulse must be fairly below one, although this leads to a high vacuum fraction of signals. Of course, the crucial decrease of transmission rate is due to losses in the channel. The rate of distilled key is further decreased by error-correction and privacyamplification procedures. The higher the error rate, the shorter is the distilled key that is obtained from the same amount of raw key.

6.2. Limit on the distance The maximal distance over which secure QKD can be established decreases with increasing losses and increasing detector noise. The detector dark-count rate is constant (for a given detector and settings). But the key rate decreases with increasing distance due to cumulative losses. So, the relative number of erroneous bits caused by dark counts continues to grow until it is so high that secure QKD is impossible. Standard amplifiers cannot be used as they would affect the states of photons in a similar manner as eavesdropping. Present-day technology allows secure operation up to about 100 km.

6.3. Quantum repeaters The use of entangled pairs for QKD (see Section 3.6) offers an important advantage. It enables extension of the radius of secure communication to practically arbitrary distance (at least in theory). This can be reached by quantum repeaters (Dür, Briegel, Cirac and Zoller [1999]). These can do “distributed error correction” without revealing any information on the key. The communication channel is divided into shorter segments each containing a source of entangled pairs. At the ends of each segment a distillation of entanglement (Bennett, Brassard, Schumacher, Popescu, Smolin and Wootters [1996]) is performed. It

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produces a smaller number of “repaired” highly entangled pairs from an originally higher number of pairs damaged during transmission. Individual segments are “connected” by means of an entanglement swapping method (Bennett, Brassard, ˙ Zeilinger, Horne and Ekert Crépeau, Jozsa, Peres and Wootters [1993], Zukowski, [1993]). So finally Alice and Bob possess highly entangled pairs. § 7. Supporting procedures 7.1. Estimation of leaked information Real devices like polarizers, fibers, detectors, etc. are never perfect and noiseless. Therefore we always have to tolerate a certain amount of errors. However, we cannot be sure that these errors do not stem from Eve’s activity (Eve could, e.g., replace some noisy part of the system by a better – less noisy – one) so we have to attribute all errors to Eve. Fortunately, from the observed error rate it is possible to estimate the information leaked to Eve and then “shorten” the established key in such a way that Eve’s information on the new, shorter key is arbitrarily small. First, Alice and Bob randomly choose a certain number of transmitted bits and compare them publicly to estimate the error rate. The higher the number of compared bits is, the higher is the probability that the actual error probability does not exceed the estimated value. Assuming the most general attack allowed by the laws of quantum physics one can find the boundary of the amount of information Eve could get on the key, in dependence on the error rate caused by the attack. For the simplest intercept–resend attack described before (assuming non-continuous eavesdropping) Eve gets an average information per bit I = 2ε, where ε is the bit-error rate. Of course, this attack is not optimal. The limiting (“worst”) values of I (ε) depend both on the protocol and implementation. These problems will be discussed in more detail in Section 8. 7.2. Error correction for classical bit strings When Alice and Bob create a sifted key by sorting out signals for which Bob has used the “wrong” bases, their key sequences need not be exactly the same. This may be caused either by eavesdropping or by “technological” noise. Therefore, Alice and Bob must correct or eliminate the erroneous bits. Here we describe a simple error-correction procedure proposed by Bennett, Bessette, Brassard, Salvail and Smolin [1992]. Alice and Bob first agree on a random permutation of the bit positions in their strings to randomize the location of errors. Then they partition the permuted

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strings into blocks of size k such that single blocks are believed to be unlikely to contain more than one error (block size is a function of the expected bit-error rate). For each block, Alice and Bob compare the block’s parity. Blocks with matching parities are tentatively accepted as correct. If parities do not agree, the block is subjected to a bisective search, disclosing further parities of sub-blocks, until the error is found and corrected. To remove errors that have remained undetected (e.g., because they occurred in blocks or sub-blocks with an even number of errors), the random permutation and block parity disclosure is repeated several more times, with increasing block sizes. Once Alice and Bob estimate that at most a few errors remain in the data as a whole, they change the strategy (at this point, the block parity disclosure approach becomes much less efficient because it forces Alice and Bob to reveal at least one parity bit in each block). Now they publicly choose random subsets of the bit positions in their entire respective data strings and compare the parities. If disagreement is found, the bisective search is undertaken, similar to that described above. The procedure is repeated several times, each time with a new independent random subset of bit positions, until no error is left. Alice and Bob are now in possession of a string that is almost certainly shared but only partly secret. The revealed parity bits represent an additional information leaked to Eve that must be taken into account. In order to avoid this leakage of information during the reconciliation process either the exchanged parity bits must be One-Time-Pad encrypted or the information that is additionally made available to the eavesdropper must be taken into account during the privacy amplification step. Other error-correcting (or reconciliation) procedures are described by Brassard and Salvail [1993] (among others the procedure that leaks the minimum amount of information during reconciliation) and by Sugimoto and Yamazaki [2000]. Note that the error correction shortens the bit string at least to a fraction 1 − h(ε), where ε is the error rate and h(p) = −p log2 (p) − (1 − p) log2 (1 − p) is the Shannon entropy. This is the so-called Shannon limit. Practical error-correcting procedures are less efficient and shorten the bit string even more.

7.3. Privacy amplification for classical bit strings Let us suppose that both Bob and Eve have already made measurements and they have some classical information on the key bits sent by Alice.14 If Bob has higher 14 If Eve has attacked the transmission using quantum probes she can wait with measurements on her probes until Alice and Bob carry out all necessary supporting procedures and she can then modify her

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information on the key sent by Alice than Eve [I (B; A) > I (E; A)],15 then Alice and Bob can establish a new secret key, such that Eve has negligible information on it, using only one-way communication. First, Alice and Bob have to carry out an error-correction procedure in order to have the exactly same bit sequences. At that point, Alice and Bob possess identical strings, but those strings are not completely private. Next, they proceed with the following algorithm, called privacy amplification (Bennett, Brassard and Robert [1988], Bennett, Bessette, Brassard, Salvail and Smolin [1992], Bennett, Brassard, Crépeau and Maurer [1995]). Alice, at random, picks N bits, [X1 , X2 , . . . , XN ], from the sifted key and performs an exclusive OR logic operation on them (XOR, denoted here by ⊕), which finds their sum modulo 2 (in fact she calculates a parity bit): [X1 ⊕X2 ⊕· · ·⊕XN ]. She tells Bob which bits she did the operation on, but does not share the result. Bob then carries out the same operation with his bits on the same positions: [Y1 ⊕ Y2 ⊕ · · · ⊕ YN ] and keeps the result. As we have supposed that Alice’s and Bob’s bit strings are exactly the same (Xi = Yi ), Bob’s result must also be the same as that of Alice. Bob and Alice next replace each N-tuple of key bits with the calculated XOR value (these values represent a new key). Meanwhile, if Eve, who has many errors in her key, tries the same operation, it only compounds her mistakes, thus her information decreases. For example, if Eve knows the correct value of each bit with a probability p = 12 (1 + δ) then she will know the parity bit with the probability p ′ = 21 (1 + δ N ) < p when δ < 1. To put it in a more formal way, Alice and Bob share an n-bit string S, and we suppose that Eve knows at most k bits of S. Alice and Bob wish to compute an r-bit key K, where r < n, such that Eve’s expected information about K is below some specified bound. To do so, they must choose a compression function g : {0, 1}n → {0, 1}r and compute K = g(S). The procedure described above is an example of a good compression function. It has been shown by Bennett, Brassard, Crépeau and Maurer [1995] that if Eve knows k deterministic bits of S, and Alice and Bob choose their compression function g at random from the socalled universal class of hash functions, g : {0, 1}n → {0, 1}r where r = n − k − s for some safety parameter s ∈ (0, n − k), then Eve’s expected information about K = g(S) is less than or equal to 2−s /ln 2 bits.

measurements. The procedures described below are useful even in such a case. More about security issues in Section 8. 15 I (X; Y ) = H (X) + H (Y ) − H (X, Y ) with H the Shannon entropy; see Section 8.2.

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It is worth noting that if even a single discrepancy is left between Alice’s and Bob’s data after the error correction procedure, then after privacy amplification their final bit strings will be nearly completely uncorrelated.

7.4. Advantage distillation for classical bit strings Even if the mutual information on the key of Bob and Alice is lower than the mutual information of Eve and Alice [I (B; A) I (E; A)] it may still be possible to establish a secret shared key by means of a two-way classical communication16 (assuming a noiseless and authenticated classical public channel; Maurer [1993]). Alice takes an N -bit block, [X1 , X2 , . . . , XN ], of the sifted-key bits, generates a random bit C and makes the following encoding (where ⊕ means XOR again; note that all bits of the block are XORed with the same bit C): [X1 ⊕ C, X2 ⊕ C, . . . , XN ⊕C]. Finally she sends this encoded block to Bob. Bob then computes [(X1 ⊕ C) ⊕ Y1 , (X2 ⊕ C) ⊕ Y2 , . . . , (XN ⊕ C) ⊕ YN ], where [Y1 , Y2 , . . . , YN ] is his block of the sifted-key bits corresponding to Alice’s block. Bob accepts only if the result consists of the equal bits, i.e. either [0, 0, . . . , 0] or [1, 1, . . . , 1]. In this case he sets either C ′ = 0 or C ′ = 1, respectively, as an element of his new key [note that if Xi = Yi then (Xi ⊕ C) ⊕ Yi = C]. If Bob’s calculation results in different bits Bob rejects the block. This procedure is repeated with the other blocks of the sifted key and other random bits C. In other words, Alice and Bob make use of a repeat code of length N with only two codewords [0, 0, . . . , 0] or [1, 1, . . . , 1]. The sequence of random bits C sent by Alice and accepted by Bob represents a new key generated by Alice, and the sequence of bits C ′ accepted by Bob represents a new key received by Bob. In this way, the probability that Bob accepts erroneously bit C sent by Alice goes down with increasing N as ε N , where ε is the bit-error rate in the original sifted key. Eve, on her side, has to use a majority vote to guess the bit C. Hence, Bob’s information on C may be larger than Eve’s information even if Bob’s information on Alice’s bits [X1 , X2 , . . . , XN ] is lower than Eve’s one. On the new key the error correction and privacy amplification may be applied subsequently.

7.5. Authentication of public discussion In practice, the “auxiliary” information transmitted through the open channel during QKD could be modified, as it is difficult to create a physically unjammable 16 Two-way communication is necessary anyway for basis announcement in BB84.

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classical channel. For example, Eve can cut both the quantum and classical channels and pretend to be Bob in front of Alice. Therefore the authentication of the messages sent over the open channel is necessary (the recipient must be able to check that the message has come from the “proper” sender and that it has not been modified). This procedure requires additional “key” material to be stored and transmitted. For quantum cryptography to provide unconditional security, the procedure used for authentication of public discussion must also be unconditionally secure. Such authentication algorithms exist (Wegman and Carter [1981], Stinson [1995]). They are based, e.g., on the so-called orthogonal arrays. The length of the authentication password must always be greater than the length of the authenticated message, but the authentication tag (the additional information sent together with the message to verify its origin and integrity) is relatively short. This authentication tag itself is One-Time-Pad encrypted to avoid leaking information on the authentication password to Eve. A small random sequence of the same length as the authentication tag, used for its encryption, needs to be renewed after each QKD transmission (it may be “refilled” from the established keystring). For example, if the cardinality of the set of authenticated messages is (p d − 1)/(p − 1), where p is a prime and d 2 an integer, an authentication code can be created with p d keys and p authentication tags. The deception probability is then 1/p (Stinson [1995], Dušek, Haderka, Hendrych and Myška [1999b]). Clearly, the authentication requires Alice and Bob to meet each other at the beginning in order to exchange an authentication password and primary One-TimePad key for encrypting the authentication tag. After each transmission, this key is replaced by a new one, obtained from the transmitted sequence. Therefore, the QKD cryptosystem works rather as an “expander” of shared secret information: Some initial shared secret string is needed but later it can be arbitrarily expanded.

§ 8. Security It is the goal of QKD to deliver secret keys to the users. It differs from classical key distribution schemes in that, in QKD, we can actually prove the security of the final key under a very limited number of natural assumptions. These include, for example, that an eavesdropper cannot have access to the data inside the devices of Alice and Bob. In an experimental implementation one cannot directly demonstrate secure quantum key distribution: security cannot be measured as such. Security is a theoretical statement referring to specific protocols to generate a secret key from the data we obtain in an experiment. These protocols depend on observable parameters, such as the error rate, the mean photon number of the source and the loss rate

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of the signals. So in an experiment, one verifies the model assumptions of the theoretical security analysis and demonstrates that one can operate the device such that the observed parameters allow the generation of a secret key following the protocol. It is important that the awareness of this point increases. Let us have a closer look at the problem of real-life implementations of QKD schemes (see Section 4). All devices we are using will be imperfect to some degree. Moreover, all quantum channels show imperfections, for example in the form of a polarization mode dispersion, dephasing in interferometric schemes, and, dominantly, loss (Gisin, Ribordy, Tittel and Zbinden [2002]). Basic QKD protocols test for the presence of an eavesdropper by looking for changes in the quantum-mechanical signals. As a result of imperfections we have to face the situation that Alice and Bob end up with data that deviate from the ideal ones. Therefore they would have to abort QKD in an idealized simple protocol that only tests for the presence of an eavesdropper: we have to assume the worst-case scenario that the degradation of the data is not due to the channel imperfections, but might come from an active eavesdropper. The eavesdropper could be correlated with the data of Alice and Bob, thus having some information about them. Moreover, in general Alice and Bob do not even share an error-free bit-string. It turns out that there are ways to create a secret key despite these imperfections. For this, Alice and Bob apply some postprocessing procedures by publicly communicating over a classical, authenticated channel. Typically, these procedures include error correction and privacy amplification (see Section 7). It is important to know what key rate can be achieved from the data without compromising security. The parameters for the public discussion protocols come from the security proofs. In this section we will give some background on security proofs and report on the present status for different protocols.

8.1. Attacks on ideal protocols Before we start to analyze the security of QKD in more detail, let us have a look at how Eve could actually perform her eavesdropping activity. From the theory of quantum-mechanical measurements we know that any eavesdropping can be thought of as an interaction between a probe and the signals. Eve can then measure the probe to obtain information about the signals. We distinguish three main types of eavesdropping attacks: Individual attack: In the individual attack Eve lets each signal interact with a separate probe. Eve performs then a measurement on each probe separately

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after the interaction. This type of attack is easy to analyze since it does not introduce correlations between the signals. Collective attack: The collective attack starts as does the individual attack, as each signal interacts with its own independent probe. At the measurement stage, however, Eve can perform measurements that act on all probes coherently. We know from quantum estimation theory that such measurements can in some cases give more information about the signals than individual measurements. For the analysis it is convenient that this attack also does not introduce correlations between the signals. Coherent attack: This is the most general attack an eavesdropper can launch on the quantum signals exchanged between Alice and Bob. Actually, one can assume the worst-case scenario that Eve has access to all signals at the same time. Then the sequence of signals is described by one high-dimensional quantum state, on which Eve can perform a measurement via a single probe. This type of interaction can introduce any type of correlations, also between subsequent signals, as seen by Alice and Bob. Further variations of these attacks can be obtained by distinguishing whether Eve has to measure her probes before Alice and Bob continue their protocol, e.g., by exchanging basis information in the BB84 protocol, or whether she can delay her measurement until the very end of the protocol executed by Alice and Bob. Note that Eve does not necessarily have to measure the probe to extract information about the key. The secret key will be used to encrypt a secret, or be used in a different cryptographic application, which might also use quantum tools. So Eve might use her probes from the QKD protocol to attack the subsequent cryptographic application. The problem whether we can separate the security analysis of the different steps is known as composability. This has been addressed recently by Ben-Or, Horodecki, Leung, Mayers and Oppenheim [2004] showing that also in the quantum case the generation of a secret key via QKD can be separated from the use of this key later on. This is especially important since part of this secret key will be used to authenticate the public channel of subsequent QKD exchanges. Another question is that of the assumptions to which extent an eavesdropper can exploit imperfections of Alice’s and Bob’s devices. As an example, consider single-photon detectors: they are affected by dark counts and have a non-ideal detection efficiency (see Section 5.2). In a paranoid picture, we assume that Eve can exploit even these imperfections. She might reduce or eliminate dark counts by a suitable pulse sequence inserted into the optical fiber leading to Bob’s detectors. By a change of wavelength, she might increase the detection efficiency. Clearly, a precaution against each individual known attempt can be taken, though it will be hardly possible to exhaustively list all possible attacks. In a paranoid

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picture, we are on the safe side even if Eve could really do all those things. Actually, it turns out that this paranoid picture is extremely helpful in providing actual security proofs. On the other hand, we can hope to protect against eavesdropping activities that manipulate Bob’s detectors. In that case, the secure key rate will increase clearly. However, it turns out that it is technically harder to provide unconditional security proofs in this scenario. In the history of QKD, the individual attack played a crucial role (Fuchs, Gisin, Griffiths, Niu and Peres [1997], Ekert, Huttner, Palma and Peres [1994], Lütkenhaus [1996], Slutsky, Rao, Sun and Fainman [1998]) since it was easy to analyze in conjunction with the generalized privacy amplification method. However, presently the individual attack scenario is losing its relevance as methods have been developed to prove unconditional security, that is, security against coherent attacks. Actually, it is widely believed that for typical protocols one needs only to consider collective attacks, though only recently steps have been made to prove this (Renner [2005]).

8.2. Secure key rates from classical three-party correlations A typical, practical QKD protocol consists of two phases: Phase I. A physical setup generates quantum-mechanical signals. These are distributed and subsequently measured. As a result, Alice and Bob hold classical data describing their knowledge about the prepared signals and the obtained measurement results. Phase II. Alice and Bob use their authenticated classical channel to talk about their data, for example by sifting their data, performing error correction and privacy amplification. The important question is, how exactly to convert the data obtained in Phase I into a secret key in Phase II. To understand this process and its limitations, let us have a look into the classical world. Unconditional security is also being discussed in classical information theory. There one starts from identically and independently distributed random variables with a probability distribution P (A, B, E) for data of Alice, Bob and Eve. Once one assumes correlations of a given type, described by P (A, B, E), one can investigate whether public discussion protocols can turn these data into a secret key. There are two main results in this context. The first is about a lower bound on the rate achievable. This has been given by Csiszár and Körner [1978]. Remember that the Shannon entropy H (A) of a random variable A, which takes

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values a with probability p(a), is defined as H (A) = − a∈A p(a) log2 p(a), and the Shannon entropy of a joint probability distribution is analogously defined as H (A, B) = − (a∈A) p(a, b) log2 p(a, b) (Cover and Thomas [1991]). Then b∈B

the Shannon mutual information between two parties holding the random variables A and B, respectively, with a joint probability distribution p(a, b) is given by I (A; B) = H (A) + H (B) − H (A, B),

(8.1)

and the lower bound for the maximal secure key rate, R, is given by (Csiszár and Körner [1978])

R max I (A; B) − I (A; E), I (A; B) − I (B; E) . (8.2)

This lower bound can be achieved, if positive, in the following way: Alice and Bob perform error correction (see Section 7.2) via a one-way method, either by Alice giving error-correction information to Bob or vice versa, depending on whether the first or second expression in eq. (8.2) is bigger. If we encode the errorcorrection information with a One-Time Pad to avoid leakage of additional correlations to Eve, then this reduces the effective key rate by the fraction 1 − I (A; B) of the original data. In the second step, Alice and Bob perform privacy amplification, shortening their key by the fraction I (A; E) or I (B; E), depending on the chosen communication direction. In total we find the key rate given on the right-hand side of eq. (8.2).17 Surprisingly often we find that this classical lower bound is also cited and used in a QKD scenario, where an optimization over individual attacks is performed to give bounds on Eve’s information about Alice’s or Bob’s data. Note that the use of the Csiszár–Körner formula is restricted to the classical case of independently and identically distributed random variables. This can only be justified if we restrict Eve to individual attacks, which are not necessarily optimal compared to coherent or collective attack. Additionally, we have to assume that Eve attacks all signals in precisely the same fashion, and that she measures the probes of each signal immediately. It is clear that the key rates predicted from this procedure can give a rough indication of what to expect from a more detailed security analysis, but cannot replace it. The second important result in the classical three-party situation is due to Maurer (Maurer [1993], Maurer and Wolf [1999]). This result gives an upper 17 Alternatively, one can send the error-correction information unencoded; then the final key is shortened in privacy amplification giving the same effective secret-key rate (Cachin and Maurer [1997], Lütkenhaus [1999]).

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bound on the extractable secret-key rate for given P (A, B, E). It can be expressed in terms of the conditional mutual information I (A; B|E), which is defined as18 I (A; B|E) = H (A|E) + H (B|E) − H (A, B|E). The formal definition of the upper bound, the intrinsic information, is + H (B|E) − H (A, B|E) , I (A; B ↓ E) = min H (A|E) E→E

(8.3)

(8.4)

where we minimize over all possible mappings from the random variable E to the [i.e., over all possible random distributions P (A, B, E) conrandom variable E sistent with P (A, B)]. The intrinsic information measures how much Bob learns about Alice’s data by looking at his own data after Eve announces her data (or a function of her data). The bound is then given by R I (A; B ↓ E).

(8.5)

If Bob’s data depend only on Eve’s announcement, but no longer on Alice’s data, then the intrinsic information vanishes and we find that no secret key can be generated. Note that this statement is true for all possible public discussion protocols Alice and Bob might come up with (Maurer and Wolf [1999]). By evaluating the lower and upper bounds one finds a wide gap between them. Actually, no known protocols achieve the rate of the upper bound. The method of advantage distillation (see Section 7.4) taps into the gap (Maurer [1993]). There are cases where the lower bound is initially zero, but after the application of an advantage distillation step the lower bound for the new, conditional, correlations is positive.

8.3. Bounds on quantum key distribution So far we have been talking about the classical scenario. There we had to assume a specific form of the joint probability distribution P (A, B, E). In quantum mechanics we can infer from the observations on Alice’s and Bob’s side something about the ways Eve might be correlated to their data, so we are in a stronger position. At the same time, we have some added complications: Eve is free to maintain her probes in a quantum-mechanical state. We cannot force her to measure her probe, thus reducing her probe to classical data. So we cannot directly use quantum mechanics to consider the class of joint probability distributions P (A, B, E) 18 The conditional Shannon entropy is defined as H (X|Y ) = − x∈X,y∈Y p(y)p(x|y) log2 p(x|y)

with p(x|y) being a conditional probability.

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that are compatible with the observations to apply the Csiszár–Körner result. Here we have to find new lines of argumentation to provide the security statements, including new lower bounds. However, in one point the classical statements can be applied directly: the result of Maurer on upper bounds on the key rates is valid for QKD. Any individual attack compatible with the observations and quantum mechanics allows us to derive a valid upper bound (Moroder, Curty and Lütkenhaus [2005]). We obtain this upper bound by choosing a measurement on the individual probes. This results in a classical probability distribution P (A, B, E) and subsequently we obtain an upper bound on the key rate in the quantum case according to inequality (8.5). Other bounds are given e.g. by the regularized relative entropy of entanglement (Horodecki, Horodecki, Horodecki and Oppenheim [2003a], Christandl and Renner [2004]). This idea allows us to address a question that is important for experimental quantum key distribution: which types of correlated data generated by a set-up of Phase I can lead at all to a secret key via a suitable designed protocol in Phase II? More specifically, given a set of signals for Alice and a choice of measurement devices for Bob, and given that one finds some joint probability distribution P (A, B) for the signals and measurement results using some quantum channel under Eve’s control: can we at all generate a secret key from these data? What would be an upper bound for the data rate we can obtain? As a (partial) answer it turns out that a necessary condition for generating a secret key from these data is that they cannot be explained as coming from an entanglement-breaking channel (Curty, Lewenstein and Lütkenhaus [2004]). Such a channel breaks the entanglement of an entangled input state by acting on that sub-system of a bi-partite state which passes through it. It has been shown by Horodecki, Shor and Ruskai [2003b] that each entanglement-breaking channel can be represented by a so-called intercept–resend attack (see Section 2.5). In this attack Eve performs some measurement on Alice’s incoming signal, transmits the measurement result over a classical channel, and then feeds a new quantum state into Bob’s measurement device which depends only on Eve’s measurement result. If the data cannot be explained in this way, we say that the data contain quantum correlations. In this situation it has been shown that the intrinsic information does not vanish (Acín and Gisin [2005]). It is easy to see that we cannot generate a secret key from data that can be explained as coming from an entanglement-breaking channel. Just have a look at the joint probability distribution of Alice, Bob and Eve, regarding Alice’s signals and Bob’s and Eve’s measurement results. This class of channels assures that the joint probability distribution for Alice and Bob conditioned on Eve factors as P (A, B|E) = P (A|E)P (B|E). One can insert this into the definition of the

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= E) and finds quickly that the intrinsic inforintrinsic information (using E mation vanishes, using H (A, B|E) = H (A|E) + H (B|E). This means that the upper bound on the key rate vanishes and no secret key can be generated. This principle allows us to narrow down the parameter regimes in which QKD can be successfully performed at all for specific setups. For specific protocols, e.g. choice of signals and measurement devices, one can convert the question whether a given set of data can be explained by an entanglement-breaking channel into the problem of proving the existence of entanglement of a virtual bi-partite quantum state (Curty, Lewenstein and Lütkenhaus [2004], Curty, Gühne, Lewenstein and Lütkenhaus [2005]). This can be done e.g. using the idea of entanglement witnesses (Horodecki, Horodecki and Horodecki [1996]). Since general security proofs can be quite complicated, it makes sense for newly proposed QKD protocols to check first for which parameter regime of the channel the upper bound does not vanish. Note that once we verify the presence of quantum correlations we only satisfy a necessary condition for secure QKD, but we still need to provide a protocol of Phase II together with a security proof to achieve QKD. It is not clear whether one can always generate a secret key once one has quantum correlations.

8.4. Security proofs It is now time to present ideas on how to construct protocols in Phase II which turn the observed correlated data into a secret key. The key requirement in quantum key distribution is that at the end of such a protocol, the quantum system in Eve’s hand should be uncorrelated with the output of the protocol: the secret key. There are several ideas on how one can achieve this goal. Consider a quantum channel which transmits faithfully two non-orthogonal states. One can show that in this case Eve cannot have interacted with the signals; more precisely, starting with a general interaction with a probe and adding the constraint that the interaction leaves two non-orthogonal signal states invariant, one can show that the output of this action is a tensor product between the probe and the signal states. This guarantees that the probe cannot be correlated with the signals or Bob’s measurement results: the state of the probe is independent of these classical data of Alice and Bob. Clearly, in a realistic noisy channel, we cannot expect to be able to use this principle directly. However, there is an analogy in classical information transfer. As we learned from Shannon, one can use noisy classical channels to transmit classical messages perfectly. The trick is to use classical error-correction codes

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that encode the original message as so-called codewords. The encoded message is sent through the noisy channel. The effect of the noise on the codewords can be detected and the errors can be corrected. This mechanism works asymptotically perfect. Something similar can be done by using Quantum Error-Correction Codes (QECC); Calderbank and Shor [1996], Steane [1996]. Again, the basic idea is to take the non-orthogonal signal states from the source, to encode them into a longer sequence of signals that are transmitted through the channel, and then to decode the original states asymptotically error-free. This can be done in principle, though in this form it would require Alice and Bob to perform encoding and decoding operations on several signals, which is beyond our present experimental capability. Based on this idea, and using earlier results by Mayers (Mayers [1996, 2001], Shor and Preskill [2000] showed that one can adapt the basic idea of quantum error-correction codes so that the quantum protocol becomes equivalent to the standard BB84 protocol in which Alice sends a random sequence of signals and Bob measures them in a randomly selected basis. In that case, the decoding operation of the QECC turns into classical error correction and privacy amplification and no quantum manipulation capabilities are required. Let us have a look at this method in more detail. A QECC can correct errors which are introduced by the channel. The Shor–Preskill security proof is based on the Calderbank–Shor–Steane QECC (Calderbank and Shor [1996], Steane [1996]) which divides the errors into bit and phase errors. That is, without loss of generality, the channel applies to each signal qubit either an error operator, the σx or the σz , or it applies the identity operator. One encodes the signal qubits into quantum codewords, e.g., into a larger number of qubits, which are then sent over the channel. As long as the number of qubits affected by error operators is sufficiently low, the action of the channel can be reverted, thanks to the additional structure that is provided by the codewords. The reversion of the σx corresponds to the classical bit-error correction. The errors coming from σz will not be corrected, as we are interested only in the bit values of the original quantum signals. Instead, we choose the QECC structure such that, in principle, we could have corrected the errors in the quantum domain. This happens by including redundancy in the signals. Taking out this redundancy is exactly what happens in the privacy amplification procedure. We note that one essential step is to estimate the number of phase and bit errors, since the security hinges on the fact that one could in principle correct these errors. Therefore, in fact, it is an essential task to estimate the number of errors from the observable data. From this estimation, we can then determine the parameters characterizing the classical bit-error correction and privacy amplification. It is im-

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portant to reduce this estimation problem from the quantum level to the level of classical estimation theory. In the case of the BB84 protocol and the Shor–Preskill proof this is straightforward, due to the symmetry. For other protocols more advanced methods have been developed (Tamaki, Koashi and Imoto [2003b], Koashi [2004]). Let us proceed to the next principle for security proofs. The principle exploiting the QECC method uses effectively only one-way communication. This idea can be extended to two-way communication, which turns out to tolerate higher noise levels in the channel. So far, we have been using the idea that it is sufficient to create an effective perfect channel between Alice and Bob to guarantee that Eve decouples from Alice and Bob. Another way to achieve this goal is to establish maximally entangled states between Alice and Bob. Once Alice and Bob verify this property; they can be assured that Eve is decoupled from their bi-partite states. This is what is commonly referred to as monogamy of entanglement. Clearly, once we have effective perfect channels via QECC, we can achieve the distribution of maximally entangled states. For this, Alice prepares these states locally and sends one subsystem of each state to Bob via the effective, perfect, channel. This method can be generalized in the way that Alice sends the subsystems via the noisy channel to Bob. The important idea is that Alice and Bob then perform entanglement distillation to regain a reduced number of maximally entangled states (Bennett, DiVincenzo, Smolin and Wootters [1996]). This assures that Eve is decoupled from their states. Actually, the use of one-way QECC is one method to do this, though there are two-way protocols that can tolerate a higher error threshold. In practical QKD it is important to find those entanglement distillation protocols that can be translated again in classical post-processing of data. An example of this is the protocol and security proof based on the BB84 protocol by Gottesman and Lo [2003] and Chau [2002]. For quite a while it seemed that the security of QKD can always be expressed as an underlying entanglement purification protocol. However, recently it has been shown by the Horodecki family and Oppenheim (Horodecki, Horodecki, Horodecki and Oppenheim [2003a]) that one can go even further. They showed that one can create secret keys also from states that are bound entangled, that is, from states that cannot be distilled to maximally entangled states. The important idea behind their protocols is that there are certain global unitary operations acting on their systems only, which cannot actually be performed by Alice and Bob due to their spacial separation, but which would turn the bound entangled states into products of maximally entangled states and some remaining systems. Again, Eve is then decoupled from the maximally entangled system. Alice and Bob obtain their secret key by measuring the maximally entangled state in a predefined

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basis. The discussed global unitary operations now have the property that they leave these measurement results invariant. So the key data will be the same with or without applying the unitary operation. Since the key is secure after application of the global unitary operation of Alice and Bob, it is also secure without performing this operation. The security is therefore not based directly only on the distillability of maximally entangled states.

8.5. Specific attacks Before we turn to the security results for given protocols, we list a few specific attacks, especially those applicable to realistic implementations of QKD going beyond the simple qubit picture. 8.5.1. Intercept–resend attack We understand by intercept–resend attack any attack where Eve performs a complete measurement on the signals sent out by Alice. A special version has already been introduced in Section 2.5. Eve then transmits the classical measurement result and prepares a new quantum state close to Bob’s detection device. In this way, she cuts out all channel imperfections. As we have seen before, the resulting correlations will not allow Alice and Bob to create a secret key. The simplest example is an intercept–resend attack in the BB84 protocol: Eve performs a measurement of the BB84 signals in one of the signal bases and prepares a state which corresponds to her measurement result. For example, if she measures in the horizontal/vertical polarization basis and obtains a vertically polarized photon, she prepares such a vertical polarized photon for Bob. Actually, in the sifted key, that is for those signals where Alice’s and Bob’s polarization bases agree, this leads to an error rate of 25%. This error rate is composed of an error rate of 0% whenever Eve uses the same basis as Alice and Bob, and 50% whenever her basis differs from theirs. It follows that for data with more than 25% average error rate QKD cannot be successfully completed. 8.5.2. Unambiguous state discrimination attack Let us turn to an attack that is a special case of an intercept–resend attack. It applies whenever the signal states sent by Alice are linearly independent. In this case, Eve can measure the signals with an unambiguous state discrimination (USD) measurement so that with some probability she learns, without error, the exact signal, while in the remaining cases she is left without any information about the signal states (Dušek, Jahma and Lütkenhaus [2000]). She can now

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selectively continue her attack. For example, she might forward a new signal to Bob only in those cases where she knows the signal for certain, while she might send no signal at all (corresponding to sending the vacuum state) in the remaining cases. With this strategy she is able to mimic a lossy channel. As a result, the data obtained by Alice and Bob show no obvious trace of eavesdropping whenever Bob obtains a signal. Despite this absence of visible disturbance of the signal degree of freedom, no secure key can be created. A typical protocol for which this problem arises is the variation of the B92 protocol (Bennett [1992]) which uses single photons in non-orthogonal polarization states together with single-photon detection (see Section 3.1). This protocol becomes insecure once the transmissivity of the channel sinks below a threshold which depends on the non-orthogonality of the signal states. The threshold is defined as the transmissivity where the probability of success of the USD measurement equals the detection probability for Bob via the lossy channel. In our example, the success probability of the USD measuresucc = 1 − | ϕ |ϕ | and Bob obtains the fraction η of signals, ment is given as PUSD 0 1 where η is the transmissivity of the channel. Then we find for the threshold of the transmissivity the expression (Tamaki, Koashi and Imoto [2003a]) ηthresh = 1 − ϕ0 |ϕ1 .

(8.6)

8.5.3. Beam-splitting attack

The beam-splitting attack is a very natural attack for any optical implementation of QKD. The reason is that a lossy optical transmission line is very well described by a model consisting of an ideal line in which a beamsplitter is inserted which mimics the loss of the original line. Now Eve gets hold of the signal emerging from the second output of the beamsplitter, while Bob obtains the transmitted part. In some protocols, Eve can in these cases learn a fraction of the signal deterministically (Bennett, Bessette, Brassard, Salvail and Smolin [1992], Dušek, Jahma and Lütkenhaus [2000]). This is the case, for example, in implementations of the BB84 protocol with weak laser pulses instead of single photons. Alice here prepares weak laser pulses in the BB84 polarizations such that the signals also contain multi-photon pulses. The beamsplitter in Eve’s attack gives, for some of the signals, some or even all photons of a signal pulse to Eve. She waits until Alice and Bob publicly communicate the polarization bases of the signals and measurement results. Then she measures her photons in the correct basis and obtains Alice’s signals deterministically. If also Bob received at least one photon, then Eve knows deterministically also a bit of the sifted key (Inamori, Lütkenhaus and Mayers [2001]). One can show that the secret-key rate is therefore bounded

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by R pexp − psplit ,

(8.7)

where pexp is the probability that a signal enters the sifted key, and psplit is the joint probability that Eve obtains at least one photon of the signal and that this signal enters the sifted key. In the case of weak laser pulses with mean photon number μ, we find

R 1 − e−μη 1 − e−μ(1−η) . (8.8) Actually, this upper bound is positive for all values of the average photon number μ and of the total transmissivity η. It is clear that this attack cannot be excluded by Alice and Bob by any additional test of the channel since it represents the physical model of the channel. 8.5.4. Photon-number splitting attack In the beam-splitting attack the photons of the incoming signal states are distributed statistically to Eve and Bob. In principle, Eve could implement a more effective method (Dušek, Haderka and Hendrych [1999a], Lütkenhaus [2000], Brassard, Lütkenhaus, Mor and Sanders [2000]). We have seen that Eve learns an element of the sifted key whenever she and Bob obtain at least one photon. The beamsplitter, however, sometimes sends all photons of multi-photon pulses either to Eve or Bob. The improved eavesdropping attack, called photon-number splitting attack, starts with Eve performing a quantum non-demolition measurement of the total photon number of the signals. Whenever Eve finds a multi-photon signal, she deterministically splits one photon off, sending the other photons to Bob. Additionally, whenever she finds a single photon, she either blocks the signal or she performs a standard eavesdropping method on it and sends it on to Bob. As we see, errors in the polarization of the signal arise only by the eavesdropping on the single-photon signals. Ignoring this effect for the moment, we again find an upper bound on the possible secret-key rate in analogy to the formula for the beam-splitting attack (Brassard, Lütkenhaus, Mor and Sanders [2000]): R pexp − pmulti ,

(8.9)

where now pmulti is the joint probability that Alice sent a multi-photon signal and the signal enters the sifted key, while pexp is the total probability that a signal enters the sifted key. We can evaluate this bound for a Poissonian photon number distribution with average photon number μ and a single-photon transmissivity η

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for the channel. In this case we find R (1 + μ)e−μ − e−μη ,

(8.10)

which is positive only for certain combinations of μ and η. Generally, for given μ there is a cut-off transmissivity below which no secure key rate can be generated. Note that for a realization of this attack it is important that Eve can suppress signals at will (here some single-photon signals) without paying any penalty in form of an error rate (see Sections 3.4 and 3.5).

8.6. Results So far we have discussed the principles of security proofs and specific attacks. Now we will summarize results of complete security analysis as they are known so far. The results are typically given only in the limit of a large number of signals, so that all statistical effects of finite sequences of signals can be neglected. 8.6.1. Bennett 92 protocol with single photons The Bennett protocol of 1992 (B92 protocol) uses only two non-orthogonal signal states. As discussed before, this protocol is prone to the USD attack. Nevertheless, it is possible to achieve unconditional secure key distribution over lossy channels by adapting the overlap of the input signal states. This protocol has been analyzed for lossless channels (Tamaki, Koashi and Imoto [2003b]) and for lossy channels (Tamaki and Lütkenhaus [2004]). There is no explicit closed formula for the key rate; for a detailed discussion see the original publications. 8.6.2. BB84 protocol with single photons The security of the BB84 protocol is well studied (Mayers [1996, 2001], Shor and Preskill [2000]). Mayers’ proof did not make use of random permutations of the signals and resulted in a secure key rate given by R = 1 − h(ε) − h(2ε),

(8.11)

where ε is the observed error rate and h(x) is the binary entropy function given by h(x) = −x log2 x − (1 − x) log2 (1 − x). The secure rate given by Shor and Preskill is higher, as they include a random permutation of the signals, so that they obtain R = 1 − 2h(ε).

(8.12)

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The cut-off error rate in this scenario is about 11%. However, we know that one can verify quantum correlations up to 25%. Gottesman and Lo [2003] proposed a two-way communication protocol in the public-discussion part of the protocol (Phase II) which can come closer to this upper bound. It has been improved by Chau [2002] to tolerate 20%. This is at present the highest known error-rate threshold for the BB84 protocol. For this protocol, any loss in the channel reduces the rates only by a prefactor corresponding to the single-photon transmissivity. The key rates are given here without the prefactor 1/2 which would be expected since only in half of the cases the signal bases of Alice and Bob match. As Lo, Chau and Ardehali [2005] pointed out, Alice and Bob can choose the probabilities for the two signal bases asymmetrically. In the limit, they use basically only one basis, and test only a small number of signals in the other basis. Though this requires a larger sampling size, we can nevertheless get rid of the factor 1/2 in the rate formulas. 8.6.3. The six-state protocol The six-state protocol can be analyzed in similar fashion to the BB84 protocol. This has been done by Lo [2001] who found the key rate 3 3 3 1 R = 1 + 1 − ε log2 1 − ε + ε log2 ε . (8.13) 2 2 2 2 Again, we made use of the idea that one can use the three bases of the protocol asymmetrically so as to avoid a prefactor 1/3. Also for this protocol there are improved two-way protocols. The best error threshold found so far is given by Chau [2002] as 27.6%. 8.6.4. BB84 protocol with weak laser pulses For practical realizations the BB84 using weak laser pulses has special importance. The security of this protocol has been investigated by Inamori, Lütkenhaus and Mayers [2001]. For this case we do not only have the key rate for long sequences, but also the complete analysis for finite key sizes. It extends the Mayers’ results for single-photon BB84, and therefore does not use the random permutation of signals. This random permutation has been introduced by Gottesman, Lo, Lütkenhaus and Preskill [2004], so that the final key rate in the long-key limit is given by ε , R = (1 − Δ) − h(ε) − (1 − Δ)h (8.14) 1−Δ

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[5, § 8

where Δ is the fraction of signals received by Bob which might have leaked all its signal information to Eve via a multi-photon process. This fraction is given via the multi-photon probability of the source, pmulti , and the total signal detection probability for Bob, pexp , as Δ=

pmulti . pexp

(8.15)

This result holds against the most general attack of Eve, the coherent attack where Eve may delay her measurements. Moreover, it allows to give reasonable secret key rates already in the paranoid picture where all of Bob’s detection imperfections (dark counts, detection efficiency) are ascribed to Eve. Clearly one can optimize the parameters of the experimental set-up. By varying the mean photon number μ of the signals we find that one should choose approximately μ ≈ η so that the key rate scales as R ∼ η2 , where η is the total transmissivity. 8.6.5. BB84 with weak laser pulses and decoy states The BB84 protocol with weak laser pulses gives a rate of R ∼ η2 which is mainly given by the photon-number splitting attack. One possibility to avoid this attack is to use the so-called decoy-states (Hwang [2003], Lo, Ma and Chen [2005], Wang [2004a, 2004b]). Here Alice tests the channel not only with signals having one average mean photon number. Instead, she randomly varies the mean photon number; this she might do with two, three, or many intensity settings. The idea is that Eve can now no longer complete the full PNS attack. Of course, she can still split one photon from each multi-photon pulse, but she can no longer block the correct number of single-photon signals for each subset of signals with the same average photon number. Effectively, this forces Eve back to use the beam-splitting attack only. This basic idea is supported by the full security analysis (Lo, Ma and Chen [2005]), and one finds that the final key rate scales as R ∼ η, which is a clear improvement of the performance of these schemes. Indeed, distances of more than 100 km are now possible without giving up a conservative, paranoid security notion. 8.6.6. B92 with a strong phase reference pulses Another approach to improve the rate of QKD protocols is the use of coherent states with phase reference. The idea here is, again, to make it impossible for Eve to suppress signals without paying a penalty. The ability to do just that is what makes the USD attack and the PNS attack so powerful. This scheme has been

5, § 8]

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analyzed by Koashi [2004], who confirmed that in this case the secure key rate scales again as R ∼ η. 8.7. Side channels and other imperfections So far we have discussed the security assuming that the signals are prepared exactly as described in the protocol. However, in physical realizations there might be many imperfections. For example, the preparation of different signal polarizations might also affect other degrees of freedom of the signals, for example the timing or the spectrum of the signals. Therefore, by monitoring other than the intended degrees of freedom Eve might obtain information about the signal which is not captured in the typical security analysis. This situation applies also to classical cryptography where measurable quantities such as power consumption might help to break classical ciphers. Other imperfections come into play. Consider the detection process: typically, we assume that the choices of signals happen at random. What if Eve can have some information about the basis or signal choice beforehand, if the detectors show some dependence of the chosen signal basis, or if Eve could manipulate the detectors to some degree? One example is Eve’s strategy to apply a simple intercept–resend attack mimicking Bob’s measurement strategy. Then Eve forwards not only a single photon, but a strong light pulse in the polarization that corresponds to the measurement result. If Bob’s and Eve’s measurement bases agree, Bob just recovers the signal without error. When the bases disagree, with almost certainty Bob will find that both of his single-photon detectors fire. If Bob discards these events, this opens a loophole for Eve to manipulate Bob. For this reason, Bob has to keep those events, effectively increasing the error rate since he has to assign a random outcome. Further, the setting of Bob’s measurement basis could be betrayed by detector backflashes (see Section 5.2.1). Eve could also try to flash to Alice’s device and hope to get her setting by measuring the reflected light. All similar possibilities must be carefully considered and eliminated. These questions are currently under investigation. One often finds the term “Trojan horse attack”, as coined by Lo [2001], for any attack which exploits the circumstance that Alice’s and Bob’s devices do not operate only on the degree of freedom specified in the ideal protocol. It turns out that many imperfections, once one has a quantitative bound on them, can be dealt with (Gottesman, Lo, Lütkenhaus and Preskill [2004]). As long as they are small, the influence on the resulting key rates is small.

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§ 9. Prospects It is apparent that quantum cryptography is now ready to offer efficient and userfriendly systems providing an unprecedented level of security. While classical methods are still safe enough for short-lifetime encryption, quantum cryptography may prove valuable when thinking with longer prospects. The progress in the development of quantum computers can play a significant role in speeding up the increase of the need for QKD in the IT market. Quantum key distribution can also be well combined with existing infrastructure. Even QKD with very low bit rate (hundreds of bits per second) can significantly improve security of contemporary cryptosystems. It enables, e.g., to change the secret key for symmetric ciphers like AES several times per second. The widespread use of QKD is now restrained mainly due to the limited operational range (up to about 100 km). There are three main technological challenges in improving this situation: Substantial reduction of noise of detectors working at wavelengths suitable for fiber communications (1550 nm), the development of ultra-low-attenuation fibers (based, e.g., on photonic crystals), and the development of quantum repeaters. A challenging opportunity for future global secure networks is long-distance quantum communication between Earth and a satellite, between two satellites or between satellite and plane (Aspelmeyer, Jennewein, Zeilinger, Pfennigbauer and Leeb [2003]). The disturbing influence of the atmosphere constraints terrestrial free-space quantum cryptography to short-range communications. On the other hand in outer space and higher levels of the atmosphere (above 10 km) only losses due to beam geometry are important.

Acknowledgements This work was partially supported by the SECOQC project of the EC (IST-2002506813), by the Deutsche Forschungsgemeinschaft via the Emmy-Noether Programme, by the project MSM6198959213 of the Ministry of Education of the Czech Republic, and by project 202/05/0486 of Czech Science Foundation. The authors would like to thank Marcos Curty, Hauke Häseler, and Miroslav Gavenda for their feedback on the manuscript.

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E. Wolf, Progress in Optics 49 © 2006 Elsevier B.V. All rights reserved

Chapter 6

Optical quantum cloning by

Nicolas J. Cerf Centre for Quantum Information and Communication, Ecole Polytechnique, Université Libre de Bruxelles, 1050 Bruxelles, Belgium

Jaromír Fiurášek Department of Optics, Palacký University 77200 Olomouc, Czech Republic

DOI: 10.1016/S0079-6638(06)49006-5

ISSN: 0079-6638 455

Contents

Page § 1. Introduction and history . . . . . . . . . . . . . . . . . . . . . . . . .

457

§ 2. Overview of quantum cloning machines . . . . . . . . . . . . . . . .

466

§ 3. One-to-two quantum cloning as a CP map . . . . . . . . . . . . . . .

480

§ 4. N-to-M universal quantum cloning . . . . . . . . . . . . . . . . . . .

495

§ 5. Universal cloning of photons . . . . . . . . . . . . . . . . . . . . . . .

504

§ 6. Phase-covariant cloning of photons . . . . . . . . . . . . . . . . . . .

525

§ 7. Cloning of optical continuous variables . . . . . . . . . . . . . . . . .

533

§ 8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

541

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

542

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

542

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§ 1. Introduction and history 1.1. The no-cloning theorem The history of quantum cloning can be traced back to the controversial story of a paper by Herbert [1982] entitled “FLASH – A superluminal communicator based upon a new kind of measurement”. In this paper, submitted in early 1981 to Foundations of Physics, Herbert was discussing an idealized laser gain tube which would produce, via stimulated emission, macroscopically distinguishable states of light from an incoming single photon in any polarization state. The claim was that the noise in this process would, in principle, not prevent perfectly identifying the polarization state of the photon. This process would supposedly open the way to faster-than-light communication, a possibility with which any physicist feels uncomfortable since it violates causality. Today, more than twenty years later, it is publicly known that GianCarlo Ghirardi and Asher Peres were requested to review this paper. The first of them recommended its rejection, based on the argument that the linear nature of quantum mechanics must prevent such a process to exist, see van der Merwe [2002]. The second referee wrote, see Peres [2002], that he had realized the paper was wrong, but nevertheless recommended its publication because he expected that finding the error would raise a considerable interest! Herbert’s paper was then published, and, funnily enough, the prediction of Peres happened to be true. Soon afterwards, Wootters and Zurek [1982] published a paper in Nature, entitled “A single quantum cannot be cloned”, which arrived at essentially the same conclusions as those drawn by Ghirardi in his anonymous referee report dated April 1981, which itself was turned into a paper two years later, see Ghirardi and Weber [1983]. Wootters and Zurek realized that, if one can build a “cloning machine” that produces several clones of the horizontal- and vertical-polarization states of an incoming photon, then circularly polarized states cannot yield circularly polarized clones. Instead, due to the linearity of quantum mechanics, one gets a linear superposition of vertically polarized clones and horizontally polarized clones. Indeed, if the cloning machine is such that |H |C → |H, H |CH ,

|V |C → |V , V |CV , 457

(1.1)

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[6, § 1

where |H and |V are horizontal- and vertical-polarization states of the original photon, |C is the initial state of the cloning machine, and |CH and |CV are the (arbitrary) final states of the cloning machine, then the left- and right-circularly polarized states, |L = 2−1/2 (|H + i|V ) and |R = 2−1/2 (|H − i|V ), are transformed as

|L |C → 2−1/2 |H, H |CH + i|V , V |CV = |L, L |CL ,

|R |C → 2−1/2 |H, H |CH − i|V , V |CV = |R, R |CR , (1.2) |CL and |CR being (arbitrary) final states of the cloning machine. As a consequence, the cloning of circularly polarized states fails, even in the special case where |CH = |CV . This is the simplest explanation of what is known today as the quantum no-cloning theorem. Independently of this story, a related paper by Dieks [1982] was published almost simultaneously in Physics Letters, also showing that the “FLASH” proposal by Herbert was flawed. Here, the proof relies on the existence of EPR states, see Einstein, Podolsky and Rosen [1935], which give rise to quantum correlations between spatially separated systems. If two photons are prepared in the EPR state

|EPR = 2−1/2 |H, V − |V , H , (1.3)

it is well known that measuring the linear polarization of one of them in the horizontal–vertical basis allows one to immediately predict the outcome of a measurement of the linear polarization of the second one in the same basis, even if the measurement events are separated by a space-like interval. For example, if the first photon is found to be in the |H state, then the second photon will necessarily be observed in the |V state. This property holds for any measurement basis. It had been realized since the early times of quantum mechanics that this property, called quantum entanglement, does not permit superluminal communication. Indeed, the statistics of any measurement performed on one of the twin photons remains unchanged irrespectively of the measurement (or, more generally, the operation) applied on the second one. Dieks noticed, however, that if it was possible to perfectly clone one of the twin photons when the other had been measured, then superluminal communication would become possible; hence, cloning must be impossible. Assume that Alice measures the first photon either in the horizontal–vertical linear polarization basis or in the left–right circular polarization basis depending on whether she wants to transmit a 0 or a 1 to Bob. In the former case, the second photon will be found by Bob to be in a balanced mixture of the |H and |V states, while in the second case it will be in a balanced mixture of the |L and |R states.

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These two mixtures are indistinguishable (they are characterized by the same density operator, proportional to the identity I ), which is why quantum mechanics is said to “coexist peacefully” with special relativity. However, if the second photon could be cloned perfectly, Bob would then get either a balanced mixture of |H, H and |V , V , or a balanced mixture of |L, L and |R, R . These mixtures being distinguishable, Bob would have a way to infer Alice’s bit instantaneously (with some error, which however can be made arbitrarily small as the number of clones increases). Dieks concluded from this paradox that such a cloning transformation cannot be consistent with quantum mechanics. It appears that the quantum no-cloning theorem is thus one of those scientific results that have been rediscovered several times, at least by Dieks, Ghirardi, Wootters and Zurek. Actually, it can be argued that it was already implicitly used by Stephen Wiesner in his famous paper entitled “Conjugate coding” written in the 1970s but published only in the 1980s (Wiesner [1983]) which is sometimes considered to be the founding paper of quantum information theory. In some sense, the no-cloning theorem was already intrinsically contained in the roots of quantum mechanics and is thus trivial; on the other hand, its discovery has contributed to revisiting quantum mechanics in an information-theoretic language, which has had a decisive influence on the dramatic development of quantum information science over the past decade.

1.2. Beyond the no-cloning theorem Soon after the publication of the quantum no-cloning theorem, another paper appeared in Nature, written by Mandel [1983]. In this paper, entitled “Is a photon amplifier always polarization dependent?”, Mandel drew attention to the physical origin of the impossibility of making a perfect amplifying apparatus for light, namely spontaneous emission. He showed that, if the amplifier is a single twolevel atom with a dipole moment μ, then the amplification of an incoming photon with polarization vector ε depends on the scalar product between μ and ε. If the polarization vector ε of the incoming photon is parallel to the dipole moment μ, then the state |1 ε will, after some interaction time, evolve into a state containing the desired two-photon state |2 ε due to stimulated emission. On the contrary, if ε is orthogonal to μ, then the two-photon component of the resulting state corresponds to |1 ε |1 ε¯ , where ε¯ is a polarization vector orthogonal to ε. This is due to spontaneous emission, which spoils the amplification since one of the two photons has the wrong polarization ε¯ . In other words, with such a simple one-atom amplifier, the final state depends on the polarization of the incoming photon.

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Interestingly, Mandel noticed that if we consider a more elaborate amplifier made of two such atoms with orthogonal dipole moments (μ1 and μ2 ), it does become possible to amplify the photon independently of its polarization, although this process suffers from the unavoidable noise originating from spontaneous emission. Assuming that the two atoms interact similarly with the incoming photon, one understands intuitively that if one atom amplifies the photon “well” (when ε is close to μ1 ), then the second atom amplifies it “poorly” (because ε is then approximately orthogonal to μ2 ). The balance between these two effects results in an amplification that does not depend on ε. By filtering out the resulting two-photon component, one gets 2 1 (1.4) |2 ε 2| ⊗ |0 ε¯ 0| + |1 ε 1| ⊗ |1 ε¯ 1| 3 3 irrespective of ε. In some sense, the perfect cloning of polarization via stimulated emission works with probability 2/3, while spontaneous emission blurs the polarization with probability 1/3. Mandel’s paper remained mostly unnoticed and, remarkably, one had to wait more than ten years before the notion of quantum cloning machine, which was implicitly contained in this paper, became popular. In a seminal paper, Bužek and Hillery [1996] realized that, although perfect quantum cloning is ruled out by the no-cloning principle, some imperfect cloning may be possible. They found out that a qubit (two-level quantum system) that is in an unknown state can be approximately duplicated, resulting in two pretty good clones of the original state. This result holds in full generality, regardless of the physical variable carrying the qubit, so it goes much beyond the polarization-independent amplification of a single photon considered before. This paper had a considerable impact at the time because quantum information was born, and it had been realized how fruitful it is to investigate quantum mechanics using an information language. Consider a qubit in the state |ψ = α|0 + β|1 , where |0 and |1 form an orthonormal basis of the Hilbert space, while α and β are arbitrary complex numbers satisfying |α|2 + |β|2 = 1. Bužek and Hillery [1996] addressed the following formal problem: find a transformation acting on an original qubit in state |ψ together with an auxiliary system (commonly viewed as the cloning machine itself) that produces two clones with the same fidelity and is state-independent, or universal. If the cloning machine is initially put in state |C , then |1 ε |0 ε¯ →

|0 |C → |Σ0 ,

|1 |C → |Σ1 ,

(1.5)

with the final states |Σ0 and |Σ1 belonging to the product Hilbert space HA ⊗ HB ⊗ HC , where HA and HB denote the spaces of the two clones (called A and B)

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Fig. 1. Quantum cloning machine. The input of the transformation consists of the state |ψ to be cloned and the initial state of the cloning machine |C . The two clones are contained in outputs A and B, while C refers to an ancilla, commonly viewed as the cloning machine itself.

and HC denotes the space of the cloning machine C, see fig. 1. By linearity, an arbitrary qubit state |ψ is cloned as |ψ |C → α|Σ0 + β|Σ1 ≡ |Σ .

(1.6)

The fidelity of the clones, which measures the overlap between the input state and each clone, is given by fA (ψ) = ψ| TrBC (Σ)|ψ ,

fB (ψ) = ψ| TrAC (Σ)|ψ ,

(1.7)

where Tr denotes the trace and Σ ≡ |Σ Σ| is a short-hand notation for the density operator of a pure state. Bužek and Hillery [1996] showed that, under the constraint that fA (ψ) = fB (ψ) is independent of ψ, quantum mechanics permits the existence of a cloning transformation which achieves a fidelity as high as 5 (1.8) ≃ 0.833. 6 This transformation, which is called a quantum cloning machine, is given by ) ) 2 1 + |0 |C → |Σ0 ≡ |00 AB |0 C + Ψ AB |1 C , 3 3 ) ) 2 1 + |11 AB |1 C + Ψ AB |0 C , |1 |C → |Σ1 ≡ (1.9) 3 3 f univ =

where |Ψ + = 2−1/2 (|01 + |10 ) is one of the Bell states, while |0 C and |1 C denote two orthogonal states of the cloning machine. It is easy to check, by tracing over the cloning machine, that the two clones of an input state |0 are left in the joint state ρAB = TrC (Σ) =

2 |00 00| + 3

1 + + Ψ Ψ 3

(1.10)

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[6, § 1

which is equivalent to eq. (1.4) given the bosonic statistics of photons. More generally, if the input state is |ψ , the first term on the right-hand side of eq. (1.10) becomes a projector onto |ψ ⊗2 , while the second term is some (ψ-depending) maximally-entangled state. Therefore, by tracing over one of the clones, the resulting state of the other clone is 2 1 |ψ ψ| + I, 3 6 2 1 ρB = TrAC (Σ) = |ψ ψ| + I, (1.11) 3 6 where I denotes the identity operator, confirming that the two clones are left in the same state. They can be viewed each as emerging from a quantum depolarizing channel: they are found in the right state |ψ with probability 2/3, while they are replaced by a random qubit I /2 with probability 1/3. Soon after the publication of this paper, it was proved that this machine is actually the optimal universal cloning machine, that is, the highest fidelity of cloning permitted by quantum mechanics is indeed 5/6, see Bruss, DiVincenzo, Ekert, Fuchs, Macchiavello and Smolin [1998]. This discovery by Bužek and Hillery [1996] triggered an immense interest and initiated an entire subfield of quantum information science devoted to quantum cloning. In particular, further studies addressed cloning in dimensions larger than 2, state-dependent cloning (considering a restricted set of input states), the so-called N -to-M cloning (where one produces M identical clones out of N identical replicas of the original), asymmetric cloning (where the clones have unequal fidelities), the cloning of orthogonal qubit states, the cloning of continuousvariable states (such as coherent states), economical cloning (where no ancillary space is necessary), probabilistic cloning (which is not deterministic, i.e., it does not succeed with probability 100%), and even the cloning of quantum entanglement (instead of quantum states). These numerous results will be reviewed in Section 2. Aside from its utmost importance for the foundations of quantum mechanics, the study of quantum cloning has drawn a lot of interest probably also because it is closely connected to quantum key distribution (QKD), see, e.g. the review by Dusek, Lutkenhaus and Hendrych [2006]. Indeed, in many cases, the cloning machine is known to be the most powerful eavesdropping strategy against QKD protocols: the eavesdropper duplicates the quantum state and sends one clone to the authorized party, while keeping the second clone for later measurement. The characterization of cloning machines is therefore crucial for assessing the security of these QKD protocols (this particular connection is outside the scope of the present review, and will not be discussed any further). ρA = TrBC (Σ) =

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1.3. Quantum cloning without signaling Before entering the detailed study of quantum cloning, it is interesting to backtrack for a moment and further discuss the proof of the quantum no-cloning theorem based on a pair of entangled photons due to Dieks [1982]. As explained earlier, if Alice measures her photon either in the horizontal–vertical linear polarization basis or in the left–right circular polarization basis, and if Bob is able to clone his photon perfectly, then he obtains two distinguishable two-photon mixtures, which apparently makes superluminal signaling possible. A natural idea, due to Gisin [1998], is to assume that Bob’s cloning machine must necessarily introduce some intrinsic noise, and determine the minimum amount of noise that must be added so that causality ceases to be violated. Remarkably, it so happens that the minimum noise needed to comply with causality exactly coincides with that of the optimal universal cloning machine. In other words, the upper bound on quantum cloning can be derived from simple principles. As shown earlier, the two clones of the universal machine emerge each from a quantum depolarizing channel, see Bužek and Hillery [1996]. This channel can be interpreted as giving rise to a shrinking of the vector representing the qubit state in the Bloch sphere. Using the Bloch representation ρ=

I +m·σ , 2

(1.12)

where m is a vector isomorphic to state ρ and σ ≡ (σx , σy , σz ) is the vector of Pauli matrices, we see from eq. (1.11) that a state associated with m yields two clones which are in a state associated to 2m/3, independently of the orientation of m. Therefore, this universal cloning machine is sometimes also said to be isotropic. Following Gisin [1998], consider that the (pure) state of the original qubit is associated with the (unit-norm) vector m, and let us restrict our search to cloning machines that are symmetric and isotropic, that is, the clones are in the states ρA (m) = ρB (m) =

1 + ηm · σ , 2

(1.13)

where η is an unknown “shrinking factor” (0 η 1). It is easy to check that η is related to the fidelity by fA (m) = fB (m) =

1+η . 2

(1.14)

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Using eq. (1.13), the Hilbert–Schmidt decomposition of the joint state of the two clones can be written as I + ηm · σ ⊗ I + I ⊗ ηm · σ + j,k tj,k σj ⊗ σk , ρAB (m) = (1.15) 4 where the matrix tj,k measures the quantum correlations between the clones. Gisin [1998] went on to derive constraints on tj,k that result from covariance and causality. The covariance property (which will be explained in detail later on) means, physically, that rotating the original qubit around, say, the z-axis before cloning must be equivalent to cloning the original qubit and then rotating each of the two clones by the same amount around the z-axis. Following Dieks’ argument, the causality condition is taken into account by imposing that ρAB (m1 ) + ρAB (−m1 ) = ρAB (m2 ) + ρAB (−m2 )

(1.16)

which expresses the fact that the two-clone states corresponding to two indistinguishable mixtures of input states, {m1 , −m1 } and {m2 , −m2 }, are themselves indistinguishable. Putting all these conditions on tj,k together, one can show that the maximum value of η that preserves the positivity of the two-clone state, ρAB (m) 0, is η = 2/3; hence fA = fB = 5/6. This provides an alternate proof of the optimality of the qubit universal cloner of Bužek and Hillery [1996]. For completeness, let us mention that such a use of the no-signaling condition has been criticized in Bruss, D’Ariano, Macchiavello and Sacchi [2000], the argument being that the linearity and trace-preservation properties of the cloning map (which, combined, imply the no-signaling condition) are not sufficient, strictly speaking, and need to be supplemented with the complete positivity condition in order to bound the cloning fidelity. This simple technique, however, has proved to be successful to recover conditions on probabilistic cloning, see Hardy and Song [1999], on asymmetric universal cloning, see Ghosh, Kar and Roy [1999], or even to find a new class of real cloning machines, see Navez and Cerf [2003].

1.4. Content of this review The rest of this review will be devoted to the study of quantum cloning machines, as well as their optical realization. Let us sketch the content of the following sections. Section 2 provides an overview of the main papers that have been written in this context, focusing on the results but skipping the derivations. The numerous classes of quantum cloning machines will be presented (universal cloners, Pauli

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or Heisenberg cloners, phase-covariant cloners, Fourier-covariant cloners, groupcovariant cloners, real cloners, entanglement cloners, continuous-variable cloners, probabilistic cloners, or economical cloners). In Section 3 we will consider the issue of quantum cloning from a formal point of view, based on the description of the associated completely positive (CP) map and the notion of covariance. This study will be restricted to 1-to-2 cloning, and will focus on the isomorphism between CP maps and operators. It will be shown that finding the optimal cloning map reduces to a semidefinite programming problem, which can be solved efficiently by numerical methods. It will also be shown that the unitary realization of a cloning map based on the “double-Bell” ansatz provides a simple and efficient tool to investigate cloning analytically. Some examples of d-dimensional 1-to-2 cloners will be provided. This formal study will be extended in Section 4 to N -to-M cloning machines in d dimensions, but will be restricted to the case of universal cloning. The derivation of the optimal cloning transformation as well as the optimality proof will be detailed. In addition, the extension to asymmetric cloning machines and the notion of universal-NOT gate will be discussed. The reader who is mainly interested in the optical realization of cloning machines and not so much in their theoretical derivation may skip Sections 3 and 4, and proceed immediately to the following sections. In Section 5 the optical implementation of the universal quantum cloning machines will be analyzed in details. Cloning experiments relying on stimulated parametric down-conversion will be described first, followed by those relying on the symmetrization that can be obtained with a Hong–Ou–Mandel interferometer. Next, the optical realization of (universal) asymmetric cloning machines will be discussed, as well as the (universal) cloning of a pair of orthogonal qubits. In Section 6 the phase-covariant cloning machines will be developed for qubits as well as d-dimensional systems, in a 1-to-2 or N-to-M configuration. The experimental realization of phase-covariant cloning for photonic qubits will be described. In Section 7 the generalization of quantum cloning to states belonging to an infinite-dimensional Hilbert space will be considered. In particular, the cloning of coherent states of light by phase-insensitive amplification will be explained, as well as the experimental realization of continuous-variable cloning using linear optics, measurement, and feed-forward. The cloning of a finite-width distribution of coherent states will be analyzed, as well as the cloning of a pair of conjugate coherent states. Finally, Section 8 concludes.

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§ 2. Overview of quantum cloning machines 2.1. Universal cloning machines This section will be devoted to a summary of the various cloning machines that have been introduced in the literature, following the chronology as well as possible. Soon after the universal quantum cloning machine was discovered by Bužek and Hillery [1996], the question arose whether this machine was optimal. As already mentioned, this cloning machine is required to be symmetric, that is, the two clones must have equal fidelities fA (ψ) = fB (ψ) ∀ψ. In addition, it must be universal (or state-independent), which means that all states are cloned with the same fidelity, independent of ψ. It was proven by Bruss, DiVincenzo, Ekert, Fuchs, Macchiavello and Smolin [1998] that it is indeed the optimal symmetric universal duplicator for qubits, so that f = 65 is indeed the highest fidelity allowed by quantum mechanics in this case. In the same paper, the concept of optimal state-dependent cloning machines was also introduced, that is, transformations that optimally duplicate only a particular subset of the input states. Almost simultaneously, Gisin and Massar [1997] introduced the concept of N-to-M quantum cloning machines, which transform N identical replicas of an arbitrary state, |ψ ⊗N , into M > N identical clones. They were able to prove for low N that the optimal universal N-to-M cloning of qubits is characterized by the fidelity univ = fN→M

M(N + 1) + N . M(N + 2)

(2.1)

Incidentally, this confirms the optimality of the 1-to-2 universal cloning machine univ = 5 . The quantum network that realizes this 1-to-2 universal with fidelity f1→2 6 cloning of qubits was described by Bužek, Braunstein, Hillery and Bruss [1997], and was extended to 1-to-M universal cloning in Bužek and Hillery [1998b]. Note also that when the number of clones M increases for fixed N, the cloning fidelity decreases. This can simply be interpreted as a spreading of quantum information over more clones. In the limit M → ∞, the cloning transformation tends to a measurement, which confirms that the optimal (state-independent) estimation of the state |ψ ⊗N of N identical qubits has a fidelity

N +1 (2.2) N +2 as originally derived in Massar and Popescu [1995]. Then, in early 1998, the extension of quantum cloning machines to higherdimensional spaces was considered independently by Bužek and Hillery [1998a], Cerf [1998] and Werner [1998]. The form of the optimal universal 1-to-2 cloner in univ = fN→∞

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dimension d was conjectured by Bužek and Hillery [1998a], Cerf [1998], while the derivation and full optimality proof of the universal d-dimensional N-to-M cloner was given by Werner [1998], Keyl and Werner [1999]. The optimal fidelity of the universal 1-to-2 cloner of d-dimensional states (or qudits) was shown to be univ (d) = f1→2

d +3 2(d + 1)

(2.3)

while that for arbitrary N and M > N is univ (d) = fN→M

M(N + 1) + (d − 1)N . M(N + d)

(2.4)

In Cerf [1998], the cloning of d-dimensional systems was actually investigated in a more general setting: a large class of symmetric or asymmetric, universal or state-dependent, 1-to-2 cloning machines was introduced in arbitrary dimension d. The optimality of this class of cloners was only conjectured, but, in the special case of a symmetric and universal cloner, eq. (2.3) was also derived. For the set of asymmetric universal 1-to-2 cloning machines, the balance between the fidelity of the two clones, fAuniv (d) = ηA +

1 − ηA , d

fBuniv (d) = ηB +

1 − ηB , d

(2.5)

was characterized by the simple relations ηA = 1 − α 2 ,

ηB = 1 − β 2 ,

α2 +

2αβ + β 2 = 1, d

(2.6)

where ηA and ηB are the “shrinking” factors associated with the clones (η is the probability that the input state emerges unchanged at the output of the quantum depolarizing channel). Here, α and β are positive real variables. It is instructive to notice that in the limit d → ∞, the cloning of quantum information resembles the distribution of a resource that can strictly not be shared: the probability that |ψ is found in one clone is complementary to the probability that it is found in the second clone, that is, ηA + ηB = 1. Finally, even more general quantum cloning machines were obtained in the special case of qubits (d = 2) in an independent work by Niu and Griffiths [1998]. There, the 1-to-2 asymmetric and state-dependent cloning of a qubit was investigated in full generality, and, in particular, formulas (2.6) were recovered for d = 2 without any assumption. Note also that the universal cloning of mixed states in a symmetric subspace was studied by Fan [2003], while entanglement properties of cloning transformations were investigated by Bruss and Macchiavello [2003].

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2.2. Pauli and Heisenberg cloning machines The results of Cerf [1998] were later expanded for the case of qubits (Cerf [2000a]), and for the case of d-dimensional systems (Cerf [2000b]). The specificity of the approach to quantum cloning underlying these papers is that one considers the cloning of a system that is initially maximally entangled with another system instead of the cloning of a pure state. This second system acts as a “reference” by keeping a memory of the original state after the cloning has been achieved. The final state of the “reference”, the two clones, and the cloning machine then fully characterizes the cloning transformation, as a consequence of the isomorphism between completely positive (CP) maps and operators (this will be explained in detail in Section 3.1). By choosing an appropriate form for this final state, one generates a large class of quantum cloning machines. For qubits (d = 2), this class corresponds to the so-called Pauli cloning machines, whose clones emerge from two – possibly distinct – Pauli channels. In a Pauli channel, the input qubit undergoes one of the three Pauli rotations {σx , σy , σz } or the identity I with respective probabilities {px , py , pz , 1 − px − py − pz }. For example, it was shown that the whole class of symmetric Pauli cloning machines corresponds to Pauli channels with probabilities px = x 2 , py = y 2 and pz = z2 , with x, y, z satisfying the condition 1 . (2.7) 2 The action of these Pauli cloners is easy to understand √ knowing that, if the original qubit is in an eigenstate of σx , namely (|0 ±|1 )/ 2, then it is rotated by an angle π around the y-axis (z-axis) under σy (σz ) while it is left unchanged (up to a sign) by σx . Therefore, the cloning fidelity of the eigenstates √ of σx is 1 − py − pz . Similarly, the eigenstates of σy , namely (|0 ± i|1 )/ 2, are cloned with fidelity 1−px −pz , while the eigenstates of σz , namely |0 and |1 , are cloned with fidelity 1−px −py . The universal 1-to-2 symmetric cloning machine simply corresponds 1 . Note that these Pauli cloning machines appear to be to px = py = pz = 12 a special case of the state-dependent cloning transformations considered in Niu and Griffiths [1998]. The quantum circuit for the asymmetric universal cloning of qubits was described in Bužek, Hillery and Knight [1998]. These considerations can be extended to d dimensions in order to obtain the set of so-called Heisenberg cloning machines, whose clones emerge from two – possibly distinct – Heisenberg channels. In a Heisenberg channel, the d-dimensional input state undergoes, according to some probability distribution, one of the d 2 error operators Em,n (with 0 m, n d − 1) that form the discrete Weyl– Heisenberg group. It can be shown that the probability distribution of the Em,n x 2 + y 2 + z2 + xy + xz + yz =

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errors for the first clone is dual, under a Fourier transform, to that of the second clone. This corroborates the fact that if one clone is close-to-perfect (its associated error distribution is peaked), then the second clone is very noisy (its associated error distribution is flat). More precisely, this fidelity balance between the two clones can be shown to result from a no-cloning uncertainty principle, akin to the Heisenberg principle, see also Cerf [1999]. The quantum circuit realizing these Heisenberg cloning machines was described by Braunstein, Bužek and Hillery [2001]. Recently, the optimality of this entire class of (Pauli or Heisenberg) quantum cloning machines has been proven rigorously by Chiribella, D’Ariano, Perinotti and Cerf [2005] in the following sense: under some general invariance conditions, the cloners of this class coincide with all the extremal cloners. Therefore, for a given (invariant) figure of merit, it is sufficient to search the optimal cloner within this class to be guaranteed that the solution thus found is the global optimal cloner.

2.3. Phase- and Fourier-covariant cloning machines In 2000 an important class of state-dependent qubit cloning machines, named phase-covariant cloning machines, was introduced by Bruss, Cinchetti, D’Ariano and Macchiavello [2000]. It is defined as a transformation that clones all the balanced superpositions of basis states with the same (and highest) fidelity. These states |ψ =

|0 + eiφ |1 , √ 2

(2.8)

with φ being an arbitrary phase, are located on the equator of the Bloch sphere. The optimal cloner also fulfills the covariance condition with respect to the rotation of φ, that is, cloning the rotated original qubit is equivalent to cloning the original qubit followed by a rotation of each of the clones. The optimal phasecovariant symmetric 1-to-2 cloner was found to have a fidelity pc

f1→2 (2) =

1 1 + √ ≃ 0.854, 2 8

(2.9)

univ (2) = 5/6. which is higher than that of the corresponding universal cloner, f1→2 In contrast, the resulting fidelity for the states |0 and |1 , corresponding to the univ (2). In some poles of the Bloch sphere, is equal to 3/4, which is lower than f1→2 sense, it is possible to clone some restricted set of states (the equator) better at the expense of cloning some other states (near the poles) worse.

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Interestingly, this phase-covariant cloner can be viewed simply as a special case √ Cerf, Durt and Gisin [2002]. If we take x = y = 1/ of the Pauli cloners, see 8 √ and z = 12 − 1/ 8, which satisfies eq. (2.7), we indeed recover the√same cloner: the eigenstates of σx are cloned with fidelity 1 − py − pz = 21√+ 1/ 8, while the eigenstates of σy are cloned with fidelity 1−px −pz = 21 +1/ 8. It was observed by Cerf, Durt and Gisin [2002] that imposing these 4 states lying symmetrically on the equator to be cloned with the same fidelity results in the phase-covariant cloner, which actually gives the same fidelity for all states on the equator (the deep reason for this equivalence was found by Chiribella, D’Ariano, Perinotti and Cerf [2005]). Finally, we verify that this Pauli cloner clones the eigenstates of σz with a lower fidelity 1 − px − py = 34 . One can summarize the results on qubit cloning machines by noting that the eigenstates of the three Pauli matrices play the role of three mutually unbiased (MU) bases for qubits (MU bases are such that the modulus of the scalar √ product of any two states taken from distinct bases is 1/ d, with d the dimension). One can thus define three generic classes of qubit cloning machines, namely, the universal cloner (which can be obtained by imposing the states of 3 MU bases to be cloned with the same and highest fidelity), the phasecovariant cloner (if the states of only 2 MU bases are cloned equally), and some particular Pauli cloner (if the states of all 3 MU bases are cloned with unequal fidelities). The cloning of qubits having been essentially covered, it became natural to turn to the state-dependent cloning of qutrits (d = 3). Cerf, Durt and Gisin [2002] defined four kinds of Heisenberg cloning machines for qutrits, depending on whether four, three, two or none of the MU bases are cloned with the same fidelity. If none of the MU bases are cloned with equal fidelities, one has a particular Heisenberg cloning machine. On the contrary, if all four MU bases are cloned with the same fidelity, one recovers eq. (2.3) for d = 3 in the case of symmetric cloning, that is, the qutrit universal cloner with fidelity univ f1→2 (3) =

3 . 4

(2.10)

If three MU bases are requested to be cloned with the same fidelity, one gets the so-called double-phase-covariant qutrit cloner, with fidelity pc f1→2 (3)

√ 5 + 17 = ≃ 0.760, 12

(2.11)

univ (3). This cloner, which was independently derived by slightly higher than f1→2 D’Ariano and Lo Presti [2001], has the property that it clones with the same and

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highest fidelity all the balanced superpositions 1

|ψ = √ |0 + eiφ1 |1 + eiφ2 |2 (2.12) 3 for arbitrary phases φ1 and φ2 (it is also covariant with respect to both φ1 - and φ2 -rotations). This can be understood by noting that if we complete the computational basis with any triplet of bases in order to make 4 MU bases, these 3 bases only consist of balanced superposition states. In analogy with the qubit case, it then appears that imposing these 3 bases to be cloned with the same (and highest) fidelity results in a cloning machine that clones all states (2.12) equally well, that is, the double-phase-covariant cloner. Finally, we may impose that two MU bases that are dual under a Fourier transform are cloned with the same (and highest) fidelity, the other two being also cloned with an equal (albeit lower) fidelity. For example, the computational basis {|0 , |1 , |2 } and the dual basis j

= 3−1/2 2k=0 γ j k |k with j = 0, 1, 2 and γ = e2πi/3 form such a pair of MU bases. We then get the so-called Fouriercovariant cloner for qutrits, see Cerf, Durt and Gisin [2002], with fidelity Fourier f1→2 (3) =

1 1 + √ ≃ 0.789 2 12

(2.13)

pc

which is even higher than f1→2 (3) as expected since the considered set of input states is smaller than for the double-phase-covariant cloner. This cloner is covariant with respect to a Fourier transform, hence it clones two Fourier-conjugate bases with the same fidelity. Note that, except in dimension 2, one cannot always map any two MU bases onto any other two MU bases, so that the Fourier-covariant cloner is not the unique transformation that clones equally well two MU bases. Indeed, Durt and Nagler [2003] showed that, in dimension 4, the cloner for two MU bases conjugate under a Fourier transform differs from the cloner for two MU bases conjugate under a double Hadamard transform. In the special case of qubits (d = 2), however, all pairs of MU bases are unitarily equivalent, so that the Fourier-covariant and Fourier (2) = f pc (2). phase-covariant cloners coincide, f1→2 1→2 2.4. Group-covariant cloning machines In D’Ariano and Lo Presti [2001], a general method for optimizing the groupcovariant cloners was derived. More specifically, they considered the optimal cloning transformations that are covariant under a proper subgroup Ω of the universal unitary group U (d). For example, the universal qubit cloner is covariant

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with respect to U (2), while the phase-covariant qubit cloner is covariant with respect to U (1). They used this technique to derive the symmetric double-phasecovariant cloner for qutrits corresponding to eq. (2.11), as well as the 1-to-3 symmetric phase-covariant cloner for qubits, associated with the fidelity pc

f1→3 (2) =

5 ≃ 0.833. 6

(2.14)

Owing to the complexity of the group-theoretical parametrization of CP maps underlying this technique, its applicability seems rather limited. Nevertheless, using another method, Fan, Matsumoto, Wang and Wadati [2001] were able to derive the optimal 1-to-M symmetric phase-covariant cloning of qubits, yielding the fidelity , √ M(M+2) 1 + , M even, pc f1→M (2) = 12 M+14M (2.15) + , M odd. 2 4M More recently, D’Ariano and Macchiavello [2003] succeeded in applying the theory of group-covariant cloning in order to confirm eq. (2.15), as well as to find a pc general expression for fN →M (2) and the associated N-to-M cloner. This expression, which was partly conjectured in Fan, Matsumoto, Wang and Wadati [2001], is quite complex, and depends on whether N and M have the same parity. It was noticed that if the parities do not match then the cloner that optimizes the fidelity of each of the clones does not coincide with the optimal cloner with respect to the global fidelity (measuring how well the joint state of the clones approximates |ψ ⊗M , if |ψ is the state of the original). In the case of qutrits (d = 3), D’Ariano and Macchiavello [2003] also found the optimal 1-to-M symmetric double-phasepc covariant cloner. The expression for its fidelity f1→M (3) is rather complex, and depends on M modulo 3.

2.5. High-d state-dependent cloning machines In parallel with this series of results on group-covariant cloning involving several originals and clones but in low dimensions, both the phase-covariant and Fouriercovariant 1-to-2 cloning machines were extended to arbitrary dimensions d. Cerf, Bourennane, Karlsson and Gisin [2002] derived the d-dimensional symmetric Fourier-covariant cloner, and showed it to be characterized by the fidelity Fourier (d) = f1→2

1 1 +√ . 2 4d

(2.16)

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It clones equally well two MU bases that are conjugate under a Fourier transform, such as the computational basis {|0 , . . . , |d − 1 } and the dual basis d−1 j k γ |k with j = 0, . . . , d − 1 and γ = e2πi/d . The asymmetj

= d −1/2 k=0 ric Fourier-covariant cloners were also characterized in the same paper. Then, Fan, Imai, Matsumoto and Wang [2003] derived the d-dimensional symmetric multi-phase-covariant cloner, giving the fidelity √ 1 d − 2 + d 2 + 4d − 4 pc . f1→2 (d) = + (2.17) d 4d It clones with the same (and highest) fidelity all balanced superpositions of the states of the computational basis, with arbitrary phases. This result was independently derived by Lamoureux and Cerf [2005], Rezakhani, Siadatnejad and Ghaderi [2005], who also extended it to asymmetric cloners. Note also that the role of multi-phase-covariant cloners in the context of entanglement-based QKD protocols was first studied by Durt, Cerf, Gisin and Zukowski [2003] for qutrits, then by Durt, Kaszlikowski, Chen and Kwek [2004] for d-dimensional systems.

2.6. Cloning a pair of orthogonal qubits Another possible variant of the problem of cloning was studied by Fiurášek, Iblisdir, Massar and Cerf [2002], who introduced universal cloning machines that transform 2 qubits that are in an antiparallel joint state |ψ |ψ ⊥ into M clones of |ψ , with ψ|ψ ⊥ = 0. It was proven that for sufficiently large M such a cloner outperforms the standard 2-to-M cloner. One has the fidelity √ (M + 2)/(3M) 1 univ f1,1→M (2) = + (2.18) 2 2 univ (2) for M > 6. In some sense, it is better to replace which is greater than f2→M one of the two original states |ψ by its orthogonal state |ψ ⊥ if the goal is to produce M > 6 clones. This effect can be understood at the limit M → ∞, univ (2) = f1,1→∞

1 1 + √ ≃ 0.789, 2 2 3

(2.19)

that is, for the optimal measurement of a pair of antiparallel qubits. Indeed, it had been noticed earlier by Gisin and Popescu [1999] that measuring |ψ |ψ ⊥ yields univ (2) = 3 . An interpretation more information than measuring |ψ ⊗2 , with f2→∞ 4 of this property lies in the dimension of the Hilbert space spanned by |ψ |ψ ⊥ , which is 4, while |ψ ⊗2 only spans the three-dimensional symmetric subspace of 2 qubits, H+ .

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2.7. Entanglement cloning machines Another problem, related to quantum cloning, has been investigated by Lamoureux, Navez, Fiurášek and Cerf [2004]. They showed that the amount of entanglement contained in a two-qubit state cannot be cloned exactly, in analogy with the impossibility of cloning the state itself. If a cloning machine is devised that produces maximally entangled clones for maximally entangled qubit pairs at the input, then it cannot yield unentangled clones for all product states at the input. Nevertheless, the approximate cloning of entanglement is very well possible. Lamoureux, Navez, Fiurášek and Cerf [2004] defined a class of 1-to-2 entanglement cloning machines which are universal over the set of maximally entangled two-qubit states. The symmetric cloner of this class provides two clones of all maximally entangled two-qubit states with optimal fidelity √ 5 + 13 entang ≃ 0.717 f1→2 (2 × 2) = (2.20) 12 corresponding to an entanglement of formation 0.285 e-bits. In contrast, all product states are transformed into unentangled clones. This was recently extended to the cloning of entanglement for (d × d)-dimensional systems by Karpov, Navez and Cerf [2005]. The fidelity of the optimal symmetric entanglement cloner that is universal over the set of maximally entangled (d × d)-dimensional states is 4 d2 − 2 2 1 d2 + 1 entang + 1+ 2 . f1→2 (d × d) = (2.21) 4 d2 − 1 d d2 − 1 Note also that the broadcasting of entanglement via local cloning was investigated by Bužek, Vedral, Plenio, Knight and Hillery [1997].

2.8. Real cloning machines Still another class of d-dimensional 1-to-2 cloners was introduced by Navez and Cerf [2003], and named real cloning machines. It is defined as a transformation that clones all real superpositions of the computational basis states with the same (and highest) fidelity. The optimal 1-to-2 symmetric real cloner in dimension d was shown to have fidelity √ 1 2 − d + d 2 + 4d + 20 real . f1→2 (d) = + (2.22) 2 4(d + 2) Note that in dimension d = 2, the set of real states forms a circle in the Bloch real (2) = sphere which is unitarily equivalent to the equator, so that we have f1→2

6, § 2]

Overview of quantum cloning machines

pc

Fourier (2) = f1→2 (2) = f1→2

1 2

475

√ + 1/ 8. For any dimension d > 2, one has

pc

univ real Fourier (d) < f1→2 (d) < f1→2 (d) < f1→2 (d). f1→2

(2.23) entang

real (4) = f Note that for d = 4 we have the identity f1→2 1→2 (2 × 2). This comes from the fact that the set of maximally-entangled two-qubit states is isomorphic to the set of four-dimensional real states.

2.9. Highly-asymmetric cloning machines Iblisdir, Acín, Cerf, Filip, Fiurášek and Gisin [2005] introduced the concept of multipartite asymmetric N-to-M cloning machines (with M > 2). These machines are highly asymmetric in the sense that they produce M clones of unequal fidelities. A very general group-theoretical approach to the construction of the multipartite asymmetric cloning machines for qubits was then presented by Iblisdir, Acín and Gisin [2005]. It was applied to several particular examples such as the asymmetric 1 → N + 1 cloning machine, which produces two kinds of clones, one clone with fidelity f A and N clones with fidelity f B . The optimal fidelities read 2 A f1→N+1 = 1 − x2, 3

1 1 2 B = + x + x 1 − x 2 N (N + 2) , f1→N+1 (2.24) 2 3N where x ∈ (0, 1) parametrizes the class of optimal 1 → N +1 asymmetric cloners. Note that eq. (2.24) holds only for N > 1. It also was conjectured, based on exact analytical calculations for low N, that the optimal N → N +1 asymmetric cloner, which produces, from N replicas of a qubit, N clones with fidelity f A and a single clone with fidelity f B , achieves 2 x2, N(N + 2) ) 2 N 1 =1− x − 1 − x2 . 2 N +2

A fN→N+1 =1− B fN→N+1

(2.25)

The extension to d-dimensional systems was considered by Fiurášek, Filip and Cerf [2005] who investigated the universal asymmetric quantum triplicator, which produces, from a single replica of a qudit, three clones with three different fidelities f A , f B and f C . A simple parametric description of the class of optimal universal highly-asymmetric triplicators was provided, extending eqs. (2.5) and (2.6).

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It was proved that the optimal fidelities can be expressed as

2βγ d −1 2 2 A β +γ + , f1→1+1+1 = 1 − d d +1

d −1 2 2αγ B f1→1+1+1 =1− α + γ2 + , d d +1

2αβ d −1 2 C f1→1+1+1 =1− α + β2 + , d d +1

[6, § 2

(2.26)

where the positive real parameters α, β, γ satisfy the normalization condition α2 + β 2 + γ 2 +

2 (αβ + αγ + βγ ) = 1. d

(2.27)

2.10. Continuous-variable cloning machines Another interesting extension of quantum cloning, often referred to as continuousvariable quantum cloning, concerns the case of quantum systems lying in an infinite-dimensional Hilbert space. Cerf, Ipe and Rottenberg [2000] investigated 2 † the cloning of the set of coherent states |α = e−|α| /2 eαa |0 , with |0 denoting √ the vacuum state, a † being the bosonic creation operator, and α = (x+ip)/ 2 being a c-number which defines the position (x, p) of |α in phase space. Here x and p are the so-called quadrature components. A set of 1-to-2 (symmetric or asymmetric) cloning machines that are covariant with respect to the Weyl group of displacements in phase space was derived. The symmetric 1-to-2 Gaussian cloner was found to have fidelity 2 (2.28) ≃ 0.667 3 and was conjectured to be optimal. It causes an independent Gaussian noise on x and p, with a variance equal to one shot-noise unit. Thus, the two clones are left in a thermal state (containing on average 12 thermal photon) which is displaced by α. Let us also mention the independent derivation of this 1-to-2 Gaussian cloner as well as its extension to multiple clones (M > 2) by Lindblad [2000]. Cerf and Iblisdir [2000] later derived an upper bound on the fidelity of the symmetric N -to-M Gaussian cloners, based on a link with state estimation theory. Since it coincided with eq. (2.28) for N = 1 and M = 2, this proved that the above cloner is indeed the optimal cloner by means of a Gaussian operation. Cerf and Iblisdir [2001c] then showed that this 1-to-2 Gaussian cloner can be realized simply by use of an optical parametric amplifier of gain 2 followed by a balanced CV f1→2 =

6, § 2]

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Overview of quantum cloning machines

beamsplitter. The cloning noise then originates from the vacuum fluctuations of the ancillary modes that are coupled to the input mode. Cochrane, Ralph and Dolinska [2004] showed that if the ensemble of input coherent states has a finite width, the 1-to-2 Gaussian cloning can be achieved with a higher fidelity. Clearly, if the task is to clone a coherent state drawn from a distribution that is peaked around the origin of phase space, the vacuum state is a very good approximation of the original state, so cloning with a fidelity close to CV corresponds to the opposite situation of one is possible. The above fidelity f1→2 an infinitely wide input distribution, that is, an arbitrary input coherent state. For an input coherent state distributed according to a Gaussian distribution of zero mean and given variance, Cochrane, Ralph and Dolinska [2004] gave a closed formula for this fidelity as a function of the variance. The optimal N-to-M Gaussian cloning transformation that achieves the abovementioned upper bound was obtained by Braunstein, Cerf, Iblisdir, van Loock and Massar [2001] and Fiurášek [2001a], yielding CV fN→M =

MN . MN + M − N

(2.29)

N . N +1

(2.30)

As for discrete-dimensional states, these cloners tend, at the limit M → ∞, to the optimal measurement of |α ⊗N , with fidelity CV = fN→∞

The optical realization of these symmetric N-to-M cloners was also described there, while it was generalized to asymmetric 1-to-2 cloners in Fiurášek [2001a]. In the latter case, the balance between the fidelities of the two clones follows fACV =

1 , 1 + σA2

fBCV =

1 , 1 + σB2

σA σB =

1 , 2

(2.31)

which corresponds to the no-cloning uncertainty relation derived in Cerf, Ipe and Rottenberg [2000]. Here, σA2 and σB2 are the variances of the added noise on clone A and B, while one shot-noise unit is taken as 1/2. Finally, Cerf and Iblisdir [2001a] characterized a more general class of Gaussian cloners, which transform N replicas of an arbitrary coherent state |α and N ′ replicas of its phase-conjugate |α ∗ into M clones of |α and M ′ clones of |α ∗ , with N − N ′ = M − M ′ . For well-chosen ratios N ′ /N , this cloner was shown to perform better than the (N + N ′ )-to-M cloner. In addition, the special case of the balanced Gaussian cloner, with N = N ′ and M = M ′ , was shown to be optimal among all cloners in this class in the sense that it yields the highest

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fidelity for fixed N + N ′ and M + M ′ , namely CV fN,N →M,M =

4M 2 N . 4M 2 N + (M − N)2

(2.32)

4N , 4N + 1

(2.33)

Interestingly, in the limit M → ∞ we have CV fN,N →∞,∞ =

which means that the optimal measurement of |α ⊗N |α ∗ ⊗N gives the same fidelity as the optimal measurement of |α ⊗4N , instead of |α ⊗2N as a simple counting of states seems to imply. This advantage of phase conjugation was first noted by Cerf and Iblisdir [2001b]. Returning to the question of the symmetric N-to-M cloning of coherent states, Cerf, Krüeger, Navez, Werner and Wolf [2005] have recently investigated the question of whether the above Gaussian cloners really provide the absolute highest fidelity or, instead, transformations outside the realm of Gaussian operations need to be considered. Against all intuition it was shown that, provided M is finite, the cloning transformation that optimizes the single-clone fidelity is slightly non-Gaussian. For example, the optimal symmetric 1-to-2 non-Gaussian cloner of coherent states was shown to have fidelity CV,NG f1→2 = 0.683

(2.34)

CV f1→2

= 2/3 ≃ 0.667. In contrast, the optimal cloners of costrictly larger than herent states with respect to the global fidelity remain Gaussian. This discrepancy between optimal cloners with respect to single-clone or global fidelities is reminiscent of the situation for phase-covariant cloners in finite-dimensional spaces. For a review on continuous-variable quantum cloning, see Cerf [2003] and Braunstein and van Loock [2005].

2.11. Probabilistic cloning machines All cloning machines listed above are deterministic, i.e., they always produce (imperfect) clones. However, one can also consider probabilistic cloning machines, which sometimes fail to generate clones but, if they succeed, generate clones exhibiting higher fidelities than those achieved by the best deterministic cloners. The concept of probabilistic cloning was introduced by Duan and Guo [1998a, 1998b], Chefles and Barnett [1998, 1999] who investigated the cloning of a discrete finite set of pure states. They showed that a set of linearly independent states can be copied perfectly with some probability p. In particular, an exact 1-to-2

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cloning of two generally nonorthogonal pure states |ψ1 and |ψ2 is possible with probability p1→2 =

1 . 1 + | ψ1 |ψ2 |

(2.35)

The probabilistic cloning was then extended to infinite continuous sets of input states by Fiurášek [2004]. It was shown that the optimal universal cloning cannot be improved by using a probabilistic cloning strategy, due to a very high underlying symmetry of the problem. Nevertheless, if one considers cloning of some restricted set of states, then probabilistic cloning may become useful. A particular example is the optimal N-to-M phase-covariant cloning of qubits, where the optimal probabilistic cloner achieves the single-clone fidelity pc,prob

fN→M (2) =

N 1 , 1 M k + 2 (M − N ) 2M k=0

pc,prob

where [x] denotes the integer part of x. For N > 1 the fidelity fN→M (2) is larger pc than the fidelity fN→M (2) of the optimal deterministic phase-covariant cloning.

2.12. Economical cloning machines The 1-to-2 cloning transformation for d-dimensional systems (qudits) can typically be expressed as a unitary operation on the Hilbert space of three qudits – the input, a blank copy, and an ancilla. The presence of an ancilla significantly affects the experimental implementation of the cloning operation, which becomes more complicated and sensitive to decoherence. These problems, which might drastically reduce the achieved cloning fidelity, may significantly be suppressed if an “economical” approach is followed, which avoids the ancilla. The 1-to-2 cloning is then realized as a unitary operation on two qudits only: the input and the blank copy. This is obviously much simpler to implement because it requires less qudits and two-qudit gates, and it requires to control the entanglement of a pair of qudits only. It is thus likely to be much less sensitive to noise and decoherence than its three-qudit counterpart. To date, the only 1-to-2 cloning machine for which an economical realization is known is the phase-covariant qubit cloner due to Niu and Griffiths [1999], which optimally clones all states on the equator of the Bloch sphere, |ψ A = 2−1/2 (|0 A + eiφ |1 A ). The qubit to be cloned is coupled to another qubit which becomes the second copy and is initially prepared in state |0 B . The unitary two-

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qubit transformation reads 1

|1 A |0 B → √ |0 A |1 B + |1 A |0 B , 2 for a symmetric phase-covariant cloner. It can easily be extended to an asymmetric setting as we will show in eq. (6.4). The possibility of economically realizing various 1-to-2 cloning machines for qudits has been analyzed in detail by Durt, Fiurášek and Cerf [2005]. They showed that the economical universal cloning is not possible for any d. It was also argued that the optimal 1-to-2 phase-covariant cloning of qudits does not admit economical implementation for any d > 2, and this assertion was rigorously proved for d 7. A suboptimal economical phase-covariant cloner was nevertheless constructed, which does not require an ancilla and achieves the fidelity √ 2

1 pc,econ f1→2 (d) = 2 d − 1 + d − 1 + 2 , (2.36) 2d which is only slightly below that of the optimal cloner. Similarly, it was argued that the 1-to-2 Fourier-covariant cloning cannot be realized economically, albeit in dimension d = 2 (in which case it is unitarily equivalent to the phase-covariant cloner). The concept of economical cloning can be extended to N-to-M machines. As shown by Fan, Matsumoto, Wang and Wadati [2001], the optimal N-to-M phasecovariant cloning of qubits (d = 2), which maximizes the single-clone fidelity, admits an economical implementation for any N and M > N . Moreover, the economical phase-covariant cloning of d-dimensional systems (qudits) is also possible provided that M = kd + N, where k is an integer, see Buscemi, D’Ariano and Macchiavello [2005]. |0 A |0 B → |0 A |0 B ,

§ 3. One-to-two quantum cloning as a CP map 3.1. Isomorphism between CP maps and operators A very useful characterization of cloning relies on the isomorphism between completely positive (CP) maps S : Hin → Hout and positive semidefinite operators S 0 acting on Hin ⊗ Hout , where Hin and Hout denote, respectively, the input and output Hilbert spaces of S, see Jamiolkowski [1972] and Choi [1975]. To ⊗2 construct this isomorphism, consider a maximally entangled state on Hin , d−1 + Φ = √1 |j |j , d j =0

(3.1)

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where d = dim(Hin ). If the map S is applied to the second subsystem of |Φ + while the first one is left unchanged, then the resulting (generally mixed) quantum state is isomorphic to S and reads

S = [I ⊗ S] dΦ + ,

(3.2)

where Φ + ≡ |Φ + Φ + | and I stands for the identity map, while the prefactor d is introduced for normalization purposes. The map S can be characterized in terms of the state S as follows: T ρout = S(ρin ) = Trin ρin ⊗ Iout S ,

(3.3)

where “in” labels the input space, I is the identity operator, and “T ” denotes the transposition in the computational basis. If the map S is trace-preserving then S satisfies the condition Trout [S] = Iin ,

(3.4)

while the complete positivity condition on S translates into S 0. In the following, we shall make this description specific to the 1-to-2 quantum cloning machines, which produce two copies of a single d-dimensional system (qudit), see Fiurášek [2001b]. The output Hilbert space is endowed with a tensor product structure, Hout = HA ⊗ HB , where the subscripts A and B label the two clones. For each particular input state |ψ , the joint state of the clones is T ⊗ IAB S , S(ψ) = Trin ψin

(3.5)

T is a rank-one where ψ ≡ |ψ ψ|. It will be useful in the following to note that ψin ∗ , where “*” denotes the complex conjugation in the projector onto the state |ψin computational basis. Using eq. (3.5), the fidelity of the clones A and B is given by

T ⊗ ψ A ⊗ IB S , FA (S, ψ) = Tr (ψA ⊗ IB )S(ψ) = Tr ψin T ⊗ IA ⊗ ψ B S . FB (S, ψ) = Tr (IA ⊗ ψB )S(ψ) = Tr ψin

(3.6)

The symmetric cloning machines are defined as the maps S verifying FA (S, ψ) = FB (S, ψ) ∀ψ. Otherwise, the cloning machines are called asymmetric. When considering a universal cloning machine, we require that both FA (S, ψ) and FB (S, ψ) are independent of ψ, for all states ψ in Hin . Other cloning machines, such as the phase-covariant, Fourier-covariant, or real cloning machines will correspond to a constant fidelity over a restricted set R of input states ψ.

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3.2. Covariance condition In what follows, we will always assume that the set R of input states ψ is invariant under the action of the group G(Ω) of unitaries {Uω | ω ∈ Ω}, that is, Uω RUω† = R

∀ω ∈ Ω.

(3.7)

The universal cloning machine is the special case Uω ∈ SU(d). A useful figure of merit to measure the quality of cloning is the global fidelity, T ⊗ ψA ⊗ ψB S , F (S, ψ) = Tr (ψA ⊗ ψB )S(ψ) = Tr ψin (3.8)

which measures how well the joint state of the two clones approximates ψ ⊗2 . When looking for a cloning machine that optimally clones all the states of set R, one generally defines the cloning fidelity of map S as the infimum of the global fidelity over all input states ψ, F (S) = inf F (S, ψ).

(3.9)

ψ∈R

It has been shown by Werner [1998] that, by using the so-called twirling operation, there is no loss of generality in assuming the optimal cloning machine to be covariant with respect to the group G(Ω), hence the cloning fidelity to be state-independent within the set R. The twirling operation consists in randomly applying a unitary Uω to the input state and then undoing this by applying the reverse unitary Uω† to each of the two clones with the probability density dω equal to the Haar measure on the group G(Ω). This results in the twirled map Sω (ψ) dω, Stwirl (ψ) = (3.10) Ω

with the rotated map Sω being defined as

Sω (ψ) = Uω†⊗2 S Uω ψUω† Uω⊗2 .

(3.11)

The core of the argument is that

F (S) = inf F (S, ψ) inf F S, Uω φUω† dω ψ∈R φ∈R Ω F (Sω , φ) dω = inf F (Stwirl , φ) = F (Stwirl ), = inf φ∈R Ω

φ∈R

(3.12)

where we have used the invariance of R under the unitaries Uω , the invariance of the trace function under Uω , and the linearity of the fidelity in S. As a result, the operation of twirling can only increase the cloning fidelity, so that the twirled map Stwirl is at least as good as each of its constituent maps Sω . Finally, as mentioned

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earlier, we note that Stwirl is covariant with respect to the group G(Ω), that is,

∀ω ∈ Ω, ∀ψ ∈ R. Stwirl Uω ψUω† = Uω⊗2 Stwirl (ψ)Uω†⊗2 (3.13)

Physically, this covariance property means that rotating the original state is exactly equivalent to rotating the two clones by the same amount. This also implies that F (Stwirl , ψ) does not depend on ψ, within the set R. In summary, we have shown that when looking for an optimal quantum cloning machine (i.e., a machine that maximizes the worst-case global fidelity), it is sufficient to consider cloning maps that are covariant with respect to the group under which the set of input states R is invariant. The cloning fidelity is therefore stateindependent within the set R. Keyl and Werner [1999] proved that this reasoning also applies more generally to the quantum cloning machines that maximize the single-clone fidelities (FA and FB ) instead of the global fidelity, provided that universal cloning machines are considered. This, however, does not hold for all quantum cloners (see, e.g., the case of phase-covariant or continuous-variable cloners we will be considering later).

3.3. Cloning as a semidefinite programming problem Returning to the characterization of the map S via its associated operator S, we can now use the fact that the optimal cloning machine must have a stateindependent fidelity over the set of input states considered. We can then turn to the average performance of the cloning machines, which is measured by the mean fidelities FA (S, ψ) dψ, FB (S) = FB (S, ψ) dψ, FA (S) = (3.14) ψ

ψ

where the measure dψ determines the kind of cloning machines we are dealing with. In particular, universal cloning machines correspond to choosing dψ to be the invariant measure on the factor space SU(d)/SU(d − 1) induced by the Haar measure on the group SU(d). The fidelities (3.14) can be expressed as linear functions of the operator S, FA = Tr[SRA ],

FB = Tr[SRB ],

(3.15)

where we have defined the positive semidefinite operators T T RA = ψin ⊗ ψA ⊗ IB dψ, RB = ψin ⊗ IA ⊗ ψB dψ. ψ

ψ

(3.16)

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We will see in the following sections how these operators can be calculated for different kinds of cloning machines. Note that for a symmetric cloning machine, one should simply maximize the average of the mean fidelities, F (S) =

1 FA (S) + FB (S) = Tr[SR], 2

(3.17)

with R = (RA + RB )/2. This can be justified by an argument similar to that used for the twirling operation. By averaging over the permutation between the two clones, one obtains a map whose mean fidelity can only be better than that of the original map. Therefore, we can restrict ourselves to cloning transformations that are covariant with respect to the interchange of the clones, hence satisfying FA (S) = FB (S). From this argument based on twirling and permutation, we conclude that maximizing the mean fidelity, averaged over the two clones (with equal weights), should yield a cloning map which has a state-independent and clone-independent fidelity. The asymmetric cloners can also be obtained with the same maximization but by putting different weights in front of FA and FB . An interesting point to note is that finding the optimal cloning map S reduces to a semidefinite programming problem, namely finding the operator S verifying S 0 and TrAB [S] = Iin that maximizes Tr[SR], with R depending on the considered cloning machine (Audenaert and De Moor [2002]). Very efficient numerical methods are available for solving semidefinite programs, see, e.g., Vandenberghe and Boyd [1996]. Even more importantly, it can be shown with the help of Lagrange duality lemma that the optimal cloning trace-preserving CP map, which maximizes Tr[SR], must satisfy (R − λin ⊗ IAB )S = 0, λin ⊗ IAB − R 0,

(3.18) (3.19)

where λ 0 is a positive semidefinite operator whose matrix elements represent the Lagrange multipliers accounting for the trace-preservation constraint TrAB [S] = Iin . Note that λ can be expressed in terms of the optimal CP map, λ = TrAB [SR]. If both eqs. (3.18) and (3.19) are satisfied, then S is the optimal CP map maximizing Tr[SR], a property which is useful to prove and check the optimality of a given map S that is conjectured to be optimal. The proof that eqs. (3.18) and (3.19) imply optimality is rather simple and we briefly sketch it here. Suppose that (3.19) is satisfied, then it holds for any tracepreserving CP map that Tr[S(λ ⊗ I − R)] 0 and Tr[λ ⊗ I S] = Tr λ, due to the trace-preservation condition. It follows that the fidelity is upper bounded by the

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trace of the Lagrange multiplier, Tr[RS] Tr[λ], and that the optimal map which satisfies (3.18) saturates this bound. Very often a simpler method is sufficient to prove the optimality, namely, the fidelity can be bounded by the maximum eigenvalue rmax of R. Since R rmax I , we immediately have F (S) drmax ,

(3.20)

where d = dim(Hin ). If there exists a CP map S which saturates (3.20), then this transformation is optimal. Note, however, that in certain cases such as the cloning of a pair of orthogonal qubits, the bound on fidelity (3.20) is not tight and cannot be saturated.

3.4. Double-Bell ansatz Let us now consider the unitary realization of the cloning map S. We know that any CP map can be realized physically by supplementing the input system with an ancilla (hence, extending the Hilbert space) and acting with a unitary operator in this extended space. Here, the ancilla can be viewed as the cloning machine itself, and it must be traced over after applying the unitary operator. The resulting map can be written as T ⊗ IABC Σ , S(ψin ) = Trin,C ψin (3.21)

where C denotes the cloning machine and Σ is the operator that is isomorphic to this extended map Hin → HA ⊗ HB ⊗ HC . Since this extended map is some unitary operation UABC in the extended space HA ⊗ HB ⊗ HC , the operator Σ must be some (unnormalized) rank-one projector or pure state in the joint space Hin ⊗ HA ⊗ HB ⊗ HC . We thus have √ Σ = |σ σ |, with |σ in,ABC = d(Iin ⊗ UABC )Φ + in,A |0 B,C , (3.22)

where |0 B,C is the (arbitrary) initial state of the blank copy √ B and cloning + machine C, and |Φ is defined as in eq. (3.1). The prefactor d, with d = dim(Hin ), is introduced for normalization purposes, and the cloning map corresponds to eq. (3.5) with S = TrC Σ. Physically, the state |σ has a very simple interpretation, see fig. 2. If we start with two qudits prepared in a maximally entangled state |Φ + and process one of them in the quantum cloning machine while the other one is left unchanged (kept as a reference), then |σ is the joint state of this “reference” qudit (denoted as “in” since it keeps a memory of the input state), the clones A and B, as well

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Fig. 2. Preparation of the state |σ fully characterizing the cloning transformation. The input of the cloning machine is maximally entangled with a reference qudit labeled “in”. The two clones are contained in the outputs A and B, while C refers to an ancilla or the cloning machine itself.

as the cloning machine C. Remember that if we project the reference qudit onto the state |ψ ∗ , then, in the absence of cloning, qudit A is found in state |ψ . By causality, it is irrelevant whether this projection onto |ψ ∗ is done before or after the cloning machine has been applied on qudit A. Therefore, projecting the reference qudit of state |σ onto |ψ ∗ yields the joint state of A, B and C that would have been obtained by cloning the state |ψ , namely |ψ → |ψout ABC = in ψ ∗ σ in,ABC . (3.23)

We can say that |σ fully encodes the information about the cloning of any state. It was suggested by Cerf [1998, 2000a, 2000b] that a generic form for state |σ involving a superposition of double-Bell states may encompass most of the interesting quantum cloning machines, including the universal or state-dependent – symmetric as well as asymmetric – cloners. This so-called double-Bell ansatz √ corresponds to taking |σ = d|A , with |A in,A;B,C =

d−1

m,n=0

∗ am,n |Φm,n in,A Φm,n , B,C

(3.24)

where it is assumed that it is sufficient to use a Hilbert space for the cloning machine C which has the same dimension d as the input or the clones. Note that the Schmidt decomposition of |A for the partition “in” vs. ABC implies that dim(HC ) d. In eq. (3.24), the am,n ’s are complex amplitudes which satisfy the d−1 |am,n |2 = 1, while |Φm,n denote Bell states in normalization condition m,n=0 d dimensions. As the latter states play an important role in what follows, we will first discuss them in detail, as well as some useful related properties. √ Note also that |A is a quantum state of norm one, while |σ has a norm equal to d.

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3.4.1. Useful properties of d-dimensional Bell states A standard generalization of the Bell states in d dimensions is d−1 1 nj γ |j 1 |j + m 2 , |Φm,n = √ d j =0

(3.25)

where 1 and 2 denote two d-dimensional systems. These states form a set of d 2 maximally entangled states of systems 1 and 2, where m, n ∈ {0, 1, . . . , d − 1} and γ = e2πi/d stands for the dth root of unity. Note that, in what follows, the “bra” and “ket” labels are always taken modulo d. In the case of qubits (d = 2), we recover the standard Bell states |00 + |11 |00 − |11 |Φ0,0 = Φ + = , |Φ0,1 = Φ − = , √ √ 2 2 |01 − |10 |01 + |10 , |Φ1,1 = Ψ − = . (3.26) |Φ1,0 = Ψ + = √ √ 2 2

Taking the partial trace of any state Φm,n ≡ |Φm,n Φm,n | over one of the two systems (1 or 2) results in the maximally mixed state,

I (3.27) ∀m, n, d so that the states |Φm,n are indeed maximally entangled. It is easy to check that the states |Φm,n form a complete orthonormal basis in the d 2 -dimensional Hilbert space considered here. The resolution of identity reads Tr1 (Φm,n ) = Tr2 (Φm,n ) =

d−1

m,n=0

1 n(j −j ′ ) ′ γ |j j ⊗ |j + m j ′ + m d m,n ′ j,j δj,j ′ |j j ′ ⊗ |j + m j ′ + m = I12 , =

Φm,n =

(3.28)

m j,j ′

d−1 nj where we have used the identity d1 n=0 γ = δj,0 . Let us focus on the Bell state with m = n = 0, that is, d−1 1 |Φ0,0 = √ |j |j d j =0

(3.29)

which is another notation for the state |Φ + as defined in eq. (3.1). This state is particularly useful because it satisfies the relation (U ∗ ⊗ U )|Φ0,0 = |Φ0,0

(3.30)

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for any unitary transformation U , as can readily be checked by using the unitarity condition U U † = I and the completeness relation j |j j | = I . Note that the symbol “*” denotes the complex conjugation operation in the computational basis {|j }; thus |j ∗ = |j . The identity (3.30), or equivalently

(I ⊗ U )|Φ0,0 = U T ⊗ I |Φ0,0 (3.31)

corresponds to the following useful property: if the joint system 12 is prepared in the state |Φ0,0 and system 1 is projected onto |ψ ∗ , then the resulting state of system 2 is |ψ . Indeed, taking |ψ = U |0 , we have ψ ∗ | = 0|U T , so that

∗ ∗

ψ ψ ⊗ I |Φ0,0 = ψ ∗ 0| ⊗ I U T ⊗ I |Φ0,0

= ψ ∗ 0| ⊗ I (I ⊗ U )|Φ0,0

= d −1/2 ψ ∗ U |0 = d −1/2 ψ ∗ |ψ . (3.32)

Interestingly, it makes no difference whether 1 prepares and sends the state |ψ to 2, or 1 projects its part of a shared entangled state |Φ0,0 onto |ψ ∗ so to create |ψ at a distance on 2. In what follows, we will also need the discrete group of Weyl–Heisenberg operators (also called error operators), namely Em,n =

d−1 j =0

γ j n |j + m j |

(3.33)

with m, n ∈ {0, 1, . . . , d − 1}, generalizing the Pauli matrices for more than two dimensions. For qubits (d = 2), we have E0,0 = I,

E0,1 = σz ,

E1,0 = σx ,

E1,1 = σx σz = −iσy .

(3.34)

In arbitrary dimension, the error operator Em,n shifts the state by m units (modulo d) in the computational basis and multiplies it by a phase so as to shift its Fourier transform by n units (modulo d). Indeed, in the computational basis

} we have Em,0 |j = |j + m , while in the dual basis {j

= √ {|j d−1 j k γ |k } we have E0,n |j

= j + n

. The error operators ful(1/ d ) k=0 fill the following properties: ∗ Em,n = Em,−n ,

† Em,n

=γ

mn

E−m,−n ,

T = γ −mn E−m,n , Em,n

Em,n Eμ,ν = γ

nμ

Em+μ,n+ν .

(3.35) (3.36)

Interestingly, the Bell states can be transformed into each other by applying an error operator locally (on one of the two systems, leaving the other one unchanged),

T |Φm,n = (I ⊗ Em,n )|Φ0,0 = Em,n (3.37) ⊗ I |Φ0,0 .

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This also implies that the Bell states are invariant (up to a phase) under correlated ∗ ⊗E error operators (Eμ,ν μ,ν ). We can check this by calculating

∗

∗ Eμ,ν ⊗ Eμ,ν |Φm,n = (I ⊗ Eμ,ν Em,n ) Eμ,ν ⊗ I |Φ0,0

† |Φ0,0 = (I ⊗ Eμ,ν Em,n ) I ⊗ Eμ,ν = γ mν−nμ (I ⊗ Em,n )|Φ0,0

= γ mν−nμ |Φm,n ,

(3.38)

where we have used property (3.37) as well as † = γ μν Eμ,ν Em,n E−μ,−ν = γ μν−nμ Eμ,ν Em−μ,n−ν Eμ,ν Em,n Eμ,ν

= γ μν−nμ+ν(m−μ) Em,n = γ mν−nμ Em,n .

(3.39)

3.5. Heisenberg cloning machines Returning to the double-Bell ansatz (3.24), the quantum cloning machine is thus completely characterized by the d × d matrix a = {am,n }. The form (3.24) is particularly interesting because, when tracing over B and C, the systems “in” and A are left in a mixed state that is diagonal in the Bell basis, ρin,A =

d−1

m,n=0

|am,n |2 Φm,n

(3.40)

with Φm,n ≡ |Φm,n Φm,n |. Since the original system is maximally entangled with the reference system “in” (the initial state being |Φ0,0 ), this implies that clone A undergoes the error Em,n with probability |am,n |2 . It emerges from a Heisenberg channel characterized by the probability distribution |am,n |2 . An important property of state |A is that, when interchanging clones A and B, it can be re-expressed as a superposition of double-Bell states albeit with different amplitudes, |A in,B;A,C =

m,n=0

with bm,n

d−1

∗ bm,n |Φm,n in,B Φm,n A,C

d−1 1 nx−my = γ ax,y . d x,y=0

(3.41)

(3.42)

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Again, when tracing over A and C, systems “in” and B are left in a Bell-diagonal mixed state, ρin,B =

d−1

m,n=0

|bm,n |2 Φm,n

(3.43)

implying that clone B undergoes the error Em,n with probability |bm,n |2 (it emerges from another Heisenberg channel). Remarkably, eq. (3.42) implies that the matrix b = {bm,n } is related to a = {am,n } by a (bivariate and d-dimensional) discrete Fourier transform, b = F [a]. So, the cloning map can be characterized equivalently by the matrix a (characterizing the noise of clone A) or its Fourier transform b (characterizing the noise of clone B), and we see that the complementarity between these two clones simply originates from a Fourier transform: the more noisy clone A is, the less noisy is clone B. This leads to a no-cloning uncertainty relation, see Cerf [1999, 2000b]. Finally, we can use the ansatz (3.24) to express the map associated with an arbitrary Heisenberg cloner in the simple form |ψ → |ψout = =

d−1

m,n=0 d−1

m,n=0

T

T am,n Trin ψin ⊗ IABC (Φm,n )in,A ⊗ Φm,n BC ∗ am,n Em,n |ψ A ⊗ Φm,n . BC

(3.44)

Incidentally, we note here that by measuring the clone B together with the cloning machine C in the Bell basis, we get a pair of indices (m, n) which can be used to † . This process, which bears undo the noise on clone A simply by applying Em,n some analogy with quantum teleportation, will be exploited in Section 5.3 in order to convert a symmetric cloner into an asymmetric cloner. 3.5.1. Covariance with respect to the Weyl–Heisenberg group It can be proven that the Heisenberg cloning machines are covariant with respect to the discrete Weyl–Heisenberg group of error operators {Em,n }. Recall that Eμ,0 corresponds to a cyclic relabeling of the computational basis states, while E0,ν corresponds to a cyclic relabeling of the dual basis states; Eμ,ν = Eμ,0 E0,ν simply corresponds to a sequence of these cyclic permutations. Thus, Heisenberg cloners are covariant with respect to cyclic permutations of the basis states in the computational and dual basis.

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Using eq. (3.5), it can easily be shown that the covariance condition of the cloning map S with respect to the unitary operator U , namely

S U ψU † = U ⊗2 S(ψ)U †⊗2 ∀ψ ∈ R, (3.45) translates into the condition

∗ U ⊗ U ⊗2 S U T ⊗ U †⊗2 = S

⇐⇒

S, U ∗ ⊗ U ⊗2 = 0

(3.46)

on the operator S that is isomorphic to S. We may also impose that when the original is transformed according to the unitary U , the cloning machine is transformed according to the unitary U ∗ . This condition, named strong covariance, can be expressed as a constraint on state |σ or |A , namely

∗ U ⊗ U ⊗2 ⊗ U ∗ |A in,A,B,C = |A in,A,B,C . (3.47)

It was shown recently that, provided that the set of input states is invariant with respect to the Weyl–Heisenberg group, the class of strongly covariant cloning maps is equivalent to the class of extremal covariant maps, see Chiribella, D’Ariano, Perinotti and Cerf [2005]. Thus, substituting covariance with strong covariance greatly simplifies the search for optimal cloners since, given that the covariant cloners form a convex set, it is sufficient to search among extremal cloners. The strong covariance of the Heisenberg cloners can be checked by using condition (3.47) with U = Eμ,ν for all μ and ν. This equation indeed holds for each component of A, namely ∗

∗ ∗ |Φm,n Φm,n Eμ,ν ⊗ Eμ,ν ⊗ Eμ,ν ⊗ Eμ,ν ∗ = γ mν−nμ |Φm,n γ −(mν−nμ) Φm,n = |Φm,n Φ ∗ , (3.48) m,n

where we have used eq. (3.38). Thus, the Heisenberg cloning machines defined by the ansatz state |A for an arbitrary matrix a have the nice property that they keep the same form when making a cyclic permutation of the basis states (in both the computational and dual bases). This covariance property also implies that the reduced cloning maps are unital. It is trivial to prove that applying an error operator Em,n chosen at random (uniformly among the d 2 possibilities) on an arbitrary state ρ always gives a maximally disordered state, d−1 I 1 † = . Em,n ρEm,n d d2 m,n=0

(3.49)

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Consider an arbitrary input state of the cloner ψ. The covariance and the linearity of the reduced cloning map SA or SB imply that # $ d−1 d−1 1 1 † † Em,n ψEm,n = 2 Em,n SA,B (ψ)Em,n SA,B 2 (3.50) d d m,n=0

m,n=0

so that, using eq. (3.49), we verify that the reduced cloning maps SA,B are indeed unital: I I SA,B (3.51) = . d d 3.6. Three special cases of Heisenberg cloners 3.6.1. Universal cloners Let us now discuss several interesting special cases of Heisenberg cloning machines. The first example is the universal (or isotropic) cloning machine, where the channel underlying each output is a quantum-depolarizing channel. This implies that all of the probabilities pm,n = |am,n |2 except p0,0 must be equal. The same holds for the probabilities qm,n = |bm,n |2 associated with the second clone. These conditions put very strong constraints on the matrix a, whose elements can be thus parametrized by two real coefficients v and x, am,n = (v − x)δn,0 δm,0 + x ⎞ ⎛ v x ··· x ⎜x x ··· x ⎟ ⎟ =⎜ ⎝ ... ... . . . ... ⎠ . x

x

···

(3.52)

x

The Fourier transform yields the matrix elements of b, namely

bm,n = v ′ − x ′ δn,0 δm,0 + x ′ ,

(3.53)

with

v + (d 2 − 1)x v−x , v′ = . (3.54) d d The cloning is a trace-preserving operation so the condition Tr[ρin,A ] = 1 must be satisfied, which provides the normalization constraint

v 2 + d 2 − 1 x 2 = 1. (3.55) x′ =

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If the input state is |0 , the error operators E0,n ∀n, leave it unchanged up to a phase, while all the other Em,n ’s produce a state that is orthogonal to it. Therefore, the fidelities of the two clones for any input state can be expressed as FB = v ′2 + (d − 1)x ′2 .

FA = v 2 + (d − 1)x 2 ,

(3.56)

Note that only a single free parameter x controls the asymmetry of the cloner. We can also characterize the cloner by the fidelity FA of the first clone, namely, x2 =

1 − FA , d(d − 1)

v2 =

(d + 1)FA − 1 . d

(3.57)

The symmetric cloner is obtained by putting x = x ′ , which results in x2 =

1 , 2d(d + 1)

v2 =

d +1 2d

(3.58)

and is associated with the fidelity given in eq. (2.3). Note that, as rigorously proved recently for any d by Fiurášek, Filip and Cerf [2005], this isotropic Heisenberg cloner represents the optimal asymmetric cloning machine which, for a fixed fidelity FA of the first clone, maximizes the fidelity of the second clone FB . Note also that the optimality of the Heisenberg cloners, based on the double-Bell ansatz, was explained by Chiribella, D’Ariano, Perinotti and Cerf [2005] as a consequence of the extremality of these cloners. It is worth stressing that by exploiting this double-Bell ansatz, these machines can be derived almost without any effort as they follow immediately from the general isotropy- and trace-preservation conditions. 3.6.2. Fourier-covariant cloners As a second example, we shall consider the Fourier-covariant machine, which clones equally well two mutually unbiased bases, the computational basis {|k } and the dual basis d−1 1 2πi(kl/d) e |k . l

= √ d k=0

(3.59)

The cloner copies equally well the states of both bases if the matrix a has the form am,n = (v − 2x + y)δm,0 δn,0 + (x − y)(δm,0 + δn,0 ) + y ⎛v x ··· x ⎞ ⎜x y ··· y ⎟ ⎟ =⎜ ⎝ .. .. . . . .. ⎠ , . . . x

y

···

y

(3.60)

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where the parameters satisfy the trace-preservation condition v 2 + 2(d − 1)x 2 + (d − 1)2 y 2 = 1.

(3.61)

The matrix b then has a similar form with v, x and y being replaced by 1 v + 2(d − 1)x + (d − 1)2 y , d 1 x ′ = v + (d − 2)x + (1 − d)y , d 1 ′ y = [v − 2x + y]. (3.62) d The fidelities of the two clones are again given by eq. (3.56) but now we have two free parameters, say x and y. To eliminate one of them, one has to maximize Bob’s fidelity FB for a given value of Alice’s fidelity FA (using the normalization relation), which is a simple constrained optimization problem. The resulting optimal asymmetric cloner is characterized by ) FA (1 − FA ) 1 − FA v = FA , (3.63) x= , y= , d −1 d −1 v′ =

which depends on the single parameter FA . The symmetric Fourier-covariant cloner can again be obtained by setting x = x ′ and y = y ′ , which gives eq. (2.16) for the fidelity. 3.6.3. Phase-covariant cloners As a third example, consider the phase-covariant machine, which optimally clones all balanced superpositions of the form |ψ = d −1/2 |0 + eiφ1 |1 + · · · + eiφd−1 |d − 1 , (3.64)

where the φi ’s are arbitrary phases. Here, it can be easily shown that the matrix a must take the form am,n = (v − y)δm,0 δn,0 + (y − x)δm,0 + x. ⎞ ⎛ v y ··· y ⎜x x ··· x ⎟ ⎟ =⎜ ⎝ ... ... . . . ... ⎠

(3.65)

v 2 + (d − 1)y 2 + d(d − 1)x 2 = 1.

(3.66)

x

x

···

x

while the trace preservation condition is

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The matrix b has the same form, albeit with v, x and y being replaced by 1 v + d(d − 1)x + (d − 1)y , d 1 x ′ = [v − y], d 1 ′ y = v − dx + (d − 1)y . (3.67) d If the input state is 0

(all the states of the dual basis are balanced superpositions), the error operators Em,0 ∀m, leave it unchanged up to a phase, while all the other Em,n ’s produce a state that is orthogonal to it. Therefore, the fidelities of the two clones are again given by eq. (3.56), and we have two free parameters, say x and y. We can eliminate one of them by maximizing FB for a given FA , which yields the optimal phase-covariant cloner. In the special case of a symmetric cloner, we have x = x ′ and y = y ′ , resulting in the fidelity given by eq. (2.17). v′ =

§ 4. N-to-M universal quantum cloning 4.1. Optimal cloning transformation In this section we will focus on universal (state-independent) cloning. An ideal universal N → M quantum cloning machine would be a device that prepares M exact clones of an arbitrary state ψ ∈ H from N copies of ψ. The input Hilbert ⊗N space of the cloning transformation is the symmetric subspace H+ of N qudits, and d = dim H denotes the dimension of the Hilbert space of the input states. As already explained above, exact deterministic quantum cloning is forbidden by the linearity of quantum mechanics, and only approximate copying with fidelity less than unity is possible. As noted before, two different kinds of cloning fidelities are considered in the literature. The global fidelity compares the global state of M clones with the ideal output ψ ⊗M . Let S denote the cloning CP map. Then the global fidelity of cloning univ,G the state ψ can be expressed as FN→M (S, ψ) = Tr[ψ ⊗M S(ψ ⊗N )]. Generally, the fidelity of the cloning can depend on ψ and one may define the cloning fidelity univ,G as the infimum of FN→M (S, ψ) over all input states ψ,

univ,G FN→M (4.1) (S) = inf Tr ψ ⊗M S ψ N . ψ

The single-clone fidelity quantifies how well each clone resembles the desired ′ ⊗N )], output ψ. For the kth clone we can write FNuniv,SC →M (S, ψ, k) = Tr[ψ Trk S(ψ

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where Tr′k denotes the trace over all M qudits except for the kth qudit. When judging the performance of the cloning machine, we should take the infimum of univ,SC (S, ψ, k) over all input states and all M clones and define FN→M

univ,SC (S) = inf inf Tr ψ Tr′k S ψ N , FN→M (4.2) k

ψ

where k ∈ {1, . . . , M}. The universal cloning machine should clone all quantum states equally well, so the fidelity should not depend on ψ. Any transformation S can be converted into a universal cloning transformation whose fidelity is state independent by a twirling operation that consists of applying randomly a unitary U (ω) to each input ψ and then undoing this by applying a unitary U † (ω) to each clone, with the probability density dω equal to the Haar measure on SU(2), see also Section 3.2. The effective map

⊗N ⊗M Stwirl (ψ) = (4.3) U U †⊗M (ω)S U (ω)ψU † (ω) (ω) dω Ω

is covariant, i.e., Stwirl [(U ψU † )⊗N ] = U ⊗M Stwirl (ψ ⊗N )U †⊗M and, conseuniv,G quently, FN→M does not depend on ψ. To guarantee the independence of the single-clone fidelity on the clone index k, it is also necessary to randomly permute the M clones after the twirling. The important feature of the twirling operation and the permutations is that they do not modify the mean fidelity calculated univ,G univ,SC 1 M as the average of FN→M (S, ψ) or M k=1 FN→M (S, ψ, k) over all input states |ψ = U (ω)|ψ0 with the measure dω. Universal cloning has been studied extensively by many authors (Bužek and Hillery [1996, 1998a], Gisin and Massar [1997], Hillery and Bužek [1997], Bužek, Hillery and Knight [1998], Cerf [1998, 1999, 2000a, 2000b], Werner [1998], Niu and Griffiths [1998], Keyl and Werner [1999]). The task of cloning can be rephrased as diluting the quantum information carried by the N input qudits into M output qudits. Universal cloning should not prefer any direction in Hilbert space and should be isotropic. As shown by Werner [1998], the optimal universal cloning operation Sopt can be expressed as follows:

+ D(N, d) + ⊗N Sopt ψ ⊗N = ψ ⊗ I ⊗(M−N ) ΠM,d , Π D(M, d) M,d

(4.4)

+ where ΠM,d is the projector onto the fully symmetric (Bose) subspace of M qudits and d +M −1 D(M, d) = (4.5) M

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is the dimension of this subspace. We can see from eq. (4.4) that the optimal cloning formally consists in attaching M − N blank copies prepared in the maximally mixed state I /d to the input state ψ ⊗N and then projecting the whole state ⊗M . With proper norof M qudits onto the symmetric subspace of M qudits, H+ malization as given in eq. (4.4), Sopt is a trace-preserving completely positive map and can be therefore realized deterministically. The maximal global cloning fidelity achieved by the optimal cloner (4.4) reads D(N, d) (d + N − 1)!M! = . (4.6) D(M, d) (d + M − 1)!N! The density matrix of each output clone is a convex mixture of the input state ψ and the maximally mixed state I /d, univ,G = FN→M

1 (1 − η)I. (4.7) d This expression reveals the high isotropy of universal symmetric quantum cloning which is fully characterized by a single parameter, namely the shrinking factor η(N, M), ρ = ηψ +

N M +d . (4.8) N +d M The single-clone fidelity can be determined immediately from eq. (4.7) and we confirm eq. (2.4), that is, η(N, M) =

univ,SC FN→M =

MN + M + N (d − 1) . M(N + d)

(4.9)

4.1.1. Connection with quantum state estimation There is a close relationship between optimal quantum cloning and optimal quantum state estimation. As shown by Bruss, Ekert and Macchiavello [1998] and Bruss [1999], in the limit of an infinite number of clones, M → ∞, the singleuniv,SC becomes equal to the fidelity of the optimal estimation of clone fidelity FN→M the state ψ from N copies (Massar and Popescu [1995], Bruss and Macchiavello [1999], Hayashi, Hashimoto and Horibe [2004]): N +1 . (4.10) N +d Consequently, in the limit M → ∞ the optimal cloning becomes equivalent to the optimal state estimation from N copies of ψ followed by the preparation of infinitely many copies of the estimated state. This relationship between optimal universal cloning and optimal state estimation can be explored to prove the optimality of the cloning transformation (4.4). It follows from the symmetry, isotropy FN→∞ =

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[6, § 4

and linearity of universal quantum cloning that the single-qudit outputs must have the form (4.7), and that for concatenated universal cloners the shrinking factors multiply. Since the concatenation of the optimal N → M and M → L cloners cannot be better than the optimal N → L cloner, we get η(N, L) η(N, M)η(M, L).

(4.11)

Taking the limit L → ∞, and taking into account that the shrinking factor corresponding to the fidelity (4.10) reads η(N, ∞) = N/(N + d), we get from the inequality (4.11) an upper bound on η(N, M), η(N, M)

N M +d η(N, ∞) = , η(M, ∞) N +d M

(4.12)

which is saturated by the optimal universal cloning transformation (4.4). 4.1.2. Unitary realization and quantum circuit So far the optimal cloning transformation was presented in the form of the rather abstract CP map (4.4). It holds that every trace-preserving CP map admits a unitary realization with the use of an ancilla system. The unitary realization of cloning requires 2(M − N ) ancilla qudits: M − N blank copies and M − N additional ancillas. For the sake of presentation simplicity we will consider here the N → M cloning of qubits (Gisin and Massar [1997]). The unitary cloning transformation can be expressed in a covariant form: U |N ψ in |R anc =

M j =0

αj (M − j )ψ, j ψ ⊥ clones (M − N − j )ψ ⊥ , j ψ anc .

(4.13)

Here |kψ, (N − k)ψ ⊥ denotes a symmetric state of N qubits with k qubits in state |ψ and N − k qubits in an orthogonal state |ψ ⊥ , ψ|ψ ⊥ = 0, |R anc denotes the initial state of the ancilla qubits, and −1/2 1/2 M +1 M −j . αj = (−1)j (4.14) M −N N The generalization of the formula (4.13) to qudits with arbitrary d was obtained by Fan, Matsumoto and Wadati [2001]. In Section 5.1 we shall show that the transformation (4.13) arises naturally in stimulated amplification of light when the qubits are represented by the polarization states of single photons. Quantum information theory teaches us that an arbitrary unitary operation U can be implemented as a sequence of single-qubit rotations and two-qubit

6, § 4]

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Fig. 3. Network for optimal universal 1 → M cloning of qubits. The cloning requires M − 1 blank copies a1 , . . . , aM−1 and M − 1 ancilla qubits b1 , . . . , bM−1 . The cloning consists of the preparation of an entangled state |Φ ab of blank copies and ancillas followed by a sequence of 2(M − 1) C-NOT gates between the input qubit a0 and the blank copies and ancillas.

controlled-NOT gates, UCNOT = |j c |k t = |j c |k ⊕ j t , where ⊕ denotes addition modulo 2, c is the control qubit and t is the target qubit. The quantum network for the optimal universal 1 → M cloning of qubits (Bužek, Braunstein, Hillery and Bruss [1997], Bužek and Hillery [1998b]) is depicted in fig. 3. First, the 2(M − 1) ancilla qubits a1 , . . . , aM−1 and b1 , . . . , bM−1 are prepared in an entangled state |Φ ab

1 = M ×

)

2 M +1

M−1 k=0

ek |M − 1, k a + fk |M − 1, k − 1 a |M − 1, k b , (4.15)

√ with ek = M − k and fk = k(M − k), where |M − 1, k denotes a symmetric state of M − 1 qubits with k qubits in state |1 and M − 1 − k qubits in state |0 . The state (4.15) can be generated by a sequence of single-qubit rotations and C-NOT gates starting from any initial pure state of the ancilla. The cloning itself consists of a sequence of 2(M − 1) C-NOT gates where the qubit a0 that contains the state |ψ to be copied serves as a control qubit and the ancillas are target qubits. This is followed by another sequence of 2(M − 1) C-NOT gates where now the qubit a0 is target and the ancilla qubits are controls. The M clones are stored in the qubits a0 , . . . , aM−1 while the qubits b1 , . . . , bM−1 represent the ancillas.

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4.2. Optimality proof for 1 → M cloning of qubits The optimality of the 1 → 2 symmetric cloning machine for qubits was first proved by Bruss, DiVincenzo, Ekert, Fuchs, Macchiavello and Smolin [1998]. The optimality of the cloning transformation (4.4) for arbitrary number of inputs N, outputs M and dimension d was proved by Werner [1998] for the global fidelity and later by Keyl and Werner [1999] for the single-clone fidelity using powerful group-theoretical techniques. Here we shall present a simple optimality proof for the class of 1 → M universal symmetric cloning machines for qubits. This proof follows the general concept outlined in Section 3.3 where it was shown that the fidelity is upper bounded by the maximum eigenvalue of a certain positive semidefinite operator. This optimality proof with single-clone fidelity being used as a figure of merit is similar to that of Gisin and Massar [1997]; it has been extended to global fidelity (Fiurášek [2001b]). The advantage of this approach is that it can easily be generalized to asymmetric cloning, as will be discussed in the next section. Consider the maximization of the single-clone fidelity and let us assume that the output Hilbert space of the cloning map S is the symmetric subspace, since ⊗M the desired outputs |ψ ⊗M ∈ H+ . Then all the clones have the same fidelity by construction and we can express the operator S that is isomorphic to the CP map S as follows: S=

M 1

i,j =0 k,l=0

Sik,j l |i in j | ⊗ |M, k out M, l|.

(4.16)

The mean single-clone fidelity can be calculated by averaging over the surface of the Poincaré sphere,

univ,SC FN→M (4.17) = Tr ψ T ⊗ ψ Tr′out (S) dψ, ψ

where

Tr′out

denotes tracing over all output qubits except for the first one, and

1 dψ ≡ 4π ψ

|ψ = cos

0

2π π

sin ϑ dϑ dφ,

0

ϑ ϑ |0 + eiφ sin |1 . 2 2

(4.18)

After the tracing and integration, we find that the mean single-clone fidelity is a linear function of S, FNuniv,SC →M = Tr[SRSC ], where the positive semidefinite opera-

6, § 4]

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tor RSC reads RSC =

M 1 (2M − k)|0 0| + (M + k)|1 1| ⊗ |M, k M, k| 6M k=0

+

M−1 1 (M − k)(k + 1)|1 0| ⊗ |M, k + 1 M, k| + h.c. 6M k=0

univ,SC is upper bounded by the maximum According to eq. (3.20) the fidelity FN→M univ,SC eigenvalue of RSC , FN→M 2rSC,max . The matrix RSC has a block diagonal structure and it is easy to show that all eigenstates of RSC have the form α|0 |M, k + β|1 |M, k + 1 . The calculation of the eigenvalues of RSC thus reduces to finding roots of quadratic polynomials, and one finds that RSC has only three different eigenvalues, r1 = (2M + 1)/(6M), r2 = 31 and r3 = 61 . This provides an upper bound FNuniv,SC →M (2M + 1)/(3M) which is saturated by the cloning machine (4.4). This proves that the machine (4.4) is optimal. A similar chain of arguments can be used to demonstrate the optimality of the machine (4.4) when global fidelity is the figure of merit. The mean global fidelity univ,G can be written as FN→M = Tr[SRG ], where T ⊗M RG = (4.19) dψ. ψin ⊗ ψout ψ

With the help of Schur’s lemma this integral can easily be evaluated, and one +,T1 + 1 ΠM+1 , where ΠM+1 is a projector onto the symmetric subobtains RG = M+2 space of M + 1 qubits, and T1 denotes partial transposition with respect to the first qubit. Again, the matrix RG is block diagonal and its eigenvalues can easily be determined analytically. One finds that rG,max = 1/(M + 1) which implies univ,G F1→M 2/(M + 1), and this bound is achieved by the cloner (4.4). 4.3. Universal asymmetric quantum cloning Quantum cloning machines serve as universal distributors of quantum information among several parties. The symmetric cloner divides the information equally between all M copies but it is also possible to distribute the information unequally. A lot of attention has been devoted to universal asymmetric 1 → 2 cloning machines for qubits (Cerf [1998], Bužek, Hillery and Bednik [1998], Niu and Griffiths [1998], Cerf [1999, 2000a]) and qudits (Cerf [2000b], Cerf, Bourennane, Karlsson and Gisin [2002]), which produce two clones A and B with different fidelities FA and FB . The optimal asymmetric cloner can be defined as a machine

502

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that for a given fixed fidelity FA of the first clone maximizes the fidelity FB of the second clone. Such machines can find application, e.g., in eavesdropping on quantum key distribution protocols, where they allow one to investigate the trade-off between the information gained by the eavesdropper and the disturbance observed at the receiver’s station; see, e.g., Dusek, Lutkenhaus and Hendrych [2006]. In terms of the cloning CP map S, the mean fidelities of the two clones can be expressed as FA = Tr[SRA ] and FB = Tr[SRB ], where the positive semidefinite operators Rj are given by T T ψin ⊗ ψA ⊗ IB dψ, RB = ψin ⊗ IA ⊗ ψB dψ. RA = (4.20) ψ

ψ

The optimal asymmetric cloning machine should maximize a convex mixture of the mean fidelities FA and FB (Fiurášek [2003], Lamoureux, Navez, Fiurášek and Cerf [2004], Fiurášek, Filip and Cerf [2005]), F = pFA + (1 − p)FB = Tr[SR],

(4.21)

where R = pRA + (1 − p)RB , and p is a parameter that controls the asymmetry of the cloner. The maximization of F for a given value of p can be equivalently rephrased as a maximization of FB for a fixed value of FA . After some algebra, we find 1 + + ⊗ IB + d(1 − p)Φin,B ⊗ IA . Iin,AB + dpΦin,A R= (4.22) d(d + 1) The maximum eigenvalue of R is d-fold degenerate, with corresponding eigenvector |rmax ; k = α Φ + AR |k B + β Φ + BR |k A , (4.23) where the coefficients α, β 0 are some functions of d and p. By properly normalizing the eigenstates (4.23) we get

2αβ (4.24) = 1. d The operator S isomorphic to the optimal cloning CP map S is proportional to the projector onto the subspace spanned by the eigenstates (4.23). The unitary realization of this map requires a single ancilla qudit C and can be written in a covariant way: |ψ → α|ψ A Φ + BC + β|ψ B Φ + AC . (4.25) α2 + β 2 +

From this expression we can evaluate the fidelities of the two clones, FA = 1 −

d −1 2 β , d

FB = 1 −

d −1 2 α . d

(4.26)

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Note that the parameters α 2 and β 2 are the so-called depolarizing fractions as discussed by Cerf [1998, 2000a, 2000b]. The one-parametric class of optimal universal asymmetric 1 → 2 cloning machines is characterized by eqs. (4.25) and (4.26) together with the normalization condition (4.24).

4.4. Universal-NOT gate The process of optimal quantum cloning is closely connected to another impossible operation in quantum mechanics, the so-called universal-NOT gate for qubits. The hypothetical universal-NOT gate would perfectly reverse any spin- 21 state. This device should thus produce from the input qubit |ψ an orthogonal state |ψ⊥ . However, this is impossible, because the transformation |ψ → |ψ⊥ is anti-unitary. More generally one can consider an extended scenario where N copies of the state |ψ are available and the task is to prepare a single copy of the flipped spin |ψ⊥ . The best approximation to this forbidden operation was found by Gisin and Popescu [1999], Bužek, Hillery and Werner [1999, 2000]. The optimal universal NOT gate SUNOT can be made covariant by twirling so that the fidelity F = ψ⊥ |SUNOT (ψ ⊗N )|ψ⊥ does not depend on |ψ and can be written as FUNOT = Tr[SUNOT RUNOT ], where ⊗N T ψ ⊗ ψ⊥ dψ RUNOT = ψ

+ †⊗N 1 ⊗N (4.27) ⊗ I ΠN+1 U ⊗I , U N +2 see Fiurášek [2001b]. The unitary operation U = iσy provides the link between the states |ψ ∗ and |ψ⊥ , |ψ ∗ = U |ψ⊥ , U |0 = −|1 , U |1 = |0 . The fidelity of the U-NOT gate is bounded by the maximum eigenvalue of RUNOT . Since this operator is proportional to a projector, we immediately find rUNOT,max = ⊗N is d = N + 1, and we 1/(N + 2). The dimension of the input Hilbert space H+ obtain =

N +1 . (4.28) N +2 Remarkably, this fidelity coincides with the optimal fidelity of the estimation of the state |ψ from N copies. If we possess an estimate of |ψ then we can also produce an estimate of |ψ⊥ with the same fidelity, simply by flipping the estimated spin. This implies that the optimal U-NOT gate can be realized by performing the optimal estimation of the state |ψ followed by the preparation of the flipped estimated state. In this way we can generate arbitrarily many approximate FUNOT =

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copies of |ψ⊥ , all with the same fidelity (4.28). Remarkably, the optimal cloning transformation (4.13) simultaneously also implements the optimal approximate U-NOT gate. This machine produces M − N approximate anti-clones, which are stored in the ancillas, see Bužek, Hillery and Werner [1999], De Martini, Bužek, Sciarrino and Sias [2002]. It is possible to generalize the concept of U-NOT gate to qudits, by noting that the state |ψ⊥ is unitarily equivalent to the state |ψ ∗ . The complex conjugation is well defined for any dimension d, and one can look for the transformation that optimally approximates the (generalized) transposition map ψ ⊗N → ψ ∗ ≡ ψ T . Using similar reasoning as before, one can prove that the maximal fidelity of the approximate transposition is equal to the fidelity (4.10) of the optimal state estimation from N copies, see Fiurášek [2004].

§ 5. Universal cloning of photons 5.1. Amplification of light In quantum optics, single photons are very often used as carriers of quantum information. Photons represent ideal flying qubits; they can be transmitted over long distances via low-loss optical fibers and their interaction with the environment is very weak so they do not suffer from a significant decoherence. Quantum bits can be encoded into single photons in various ways. One natural option is to exploit the polarization degrees of freedom and to represent a qubit as a superposition of vertically (|V ) and horizontally (|H ) polarized photons, |ψ = α|H + β|V . Another possibility is to use the so-called time-bin encoding where the photon can be located in one of d different time slots (Marcikic, de Riedmatten, Tittel, Scarani, Zbinden and Gisin [2002], de Riedmatten, Marcikic, Tittel, Zbinden, Collins and Gisin [2004]). Such encoding has been used advantageously for longdistance quantum key distribution. It is not restricted to qubits, and the photon can thus represent a d-dimensional system with arbitrary d (de Riedmatten, Marcikic, Scarani, Tittel, Zbinden and Gisin [2004]). Arbitrary time-bin qubits can be prepared using an unbalanced Mach–Zehnder interferometer. The cloning of the quantum states of single photons requires that the number of output photons be higher than the number of input photons. This simple fact immediately leads to the insight that the optimal copying of photons can be performed by means of amplification of light (De Martini, Mussi and Bovino [2000], Simon, Weihs and Zeilinger [2000a, 2000b]). This is very natural because the goal of quantum cloning is to “amplify” the quantum information carried by the

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photons. Several physical mechanisms can be used for cloning, such as parametric down-conversion or amplification of light in atomic media. In all cases, the cloning is achieved due to the process of stimulated emission, which means that the medium emits preferably photons in the same quantum state as that of the input photons injected into the medium. Most of the quantum cloning experiments based on stimulated amplification of light were carried out using the process of stimulated parametric downconversion. Consider a nonlinear crystal with second-order nonlinearity χ (2) . In such a crystal, a single “blue” pump photon with frequency ωP can be converted into two “red” photons with frequencies ωS and ωI such that ωS + ωI = ωP , which expresses energy conservation. The two down-converted photons are referred to as signal (S) and idler (I), respectively, for historical reasons. An efficient down-conversion requires the conservation of momentum, which translates into the phase-matching condition kS + kI = kP , where kj stands for the wavevector of the j th photon and |kj | = nj ωj /c, where nj is the refraction index at frequency ωj . Efficient phase matching in the nonlinear crystal can be achieved by exploiting the birefringence and using different polarizations for the pump, signal and idler beams. We can distinguish two different kinds of phase matching. In Type-I matching the pump beam is, say, vertically polarized, and both signal and idler are horizontally polarized. On the other hand, in Type-II matching, the signal and idler photons are orthogonally polarized. Besides their polarization states, the signal and idler beams can also be distinguished spatially. So, in nondegenerate Type-II down-conversion we deal with modes AH and AV for the signal beam and BH and BV for the idler beam. It is possible to arrange the configuration of the pump beam and nonlinear crystal and to select only certain directions in the output beams in such a way that the effective Hamiltonian describing this process reads

† † † − aH bV + h.c. H = iκ aV† bH

(5.1)

This Hamiltonian is obtained in the limit of strong coherent pumping, and the coupling constant κ is proportional to the pump-beam amplitude αP and to the second-order nonlinearity χ (2) , while aj† is the creation operator for the j th mode. An essential feature of the Hamiltonian (5.1) is that it is invariant with respect to the simultaneous identical transformation of the polarization basis of signal and idler photons. Mathematically, we have (U ⊗ U )H (U † ⊗ U † ) = H , where U aV U † = uV V aV + uV H aH , U aH U † = uH V aV + uH H aH , the matrix uij is unitary, and identical transformation rules hold for bV and bH . This covariance property guarantees that the cloning process is universal and the cloning fidelity

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Fig. 4. Optimal universal cloning of photons by means of stimulated parametric down-conversion. BS: auxiliary beamsplitter; C1 , C2 – nonlinear crystals; WP – wave plates for input-state preparation; M – mirror; PD – triggering single-photon detector for conditional preparation of N -photon Fock state.

is the same for all input states. It therefore suffices to consider only one particular input state. The cloning of polarization states of photons via stimulated down-conversion is sketched in fig. 4. The signal mode is initially prepared in the N-photon state |ψ ⊗N a . This can be achieved in practice, e.g., by means of spontaneous parametric down-conversion in crystal C1 and conditioning on observing N photons in the output idler mode with photodetector PD. After the passage through crystal C2 , M − N photon pairs can be generated with a certain probability. If this happens, then M clones are present in mode A while mode B contains M − N anti-clones. Note that the cloning is only probabilistic and we cannot predict a priori the number of clones that will be generated. The particular N → M cloning events can be selected only a posteriori by accepting events with M photons detected in mode A or M − N photons in mode B. Let us start with a simple example of 1 → 2 cloning to illustrate all the main features. In this case, the input state is given by a single photon in mode A and a vacuum in mode B. As already explained, without loss of generality we can assume that the photon is vertically polarized and we have |ψin = |1 aV |0 aH |0 bV |0 bH . The generation of the second clone requires that a single photon pair is emitted in the nonlinear crystal. In the first-order perturbation theory, the output state is given by H |1 aV |0 aH |0 bV |0 bH √ ∝ 2|2 aV |0 aH |0 bV |1 bH − |1 aV |1 aH |1 bV |0 bH . (5.2) √ Notice the prefactor 2 which arises because the emission of the second vertically polarized photon in mode A is stimulated by the presence of a vertically

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polarized photon in this spatial mode. This cloning is optimal since it yields the maximum fidelity. We can immediately see that the global fidelity is 23 . To determine the single-clone fidelity we note that with probability 32 both photons in mode A are vertically polarized and with probability 13 only one photon is vertically polarized. So, the probability that one randomly chosen photon in spatial mode A is vertically polarized is 32 × 1 + 13 × 12 = 65 , which is the maximal single-clone fidelity for 1 → 2 cloning of qubits. Several experiments on cloning via parametric down-conversion have been reported (De Martini, Mussi and Bovino [2000], Lamas-Linares, Simon, Howell and Bouwmeester [2002], Pelliccia, Schettini, Sciarrino, Sias and De Martini [2003], Sias, Sciarrino and De Martini [2003], De Martini, Pelliccia and Sciarrino [2004]). The experimental set-up used by Lamas-Linares, Simon, Howell and Bouwmeester [2002] is shown in fig. 5. A nonlinear BBO crystal is pumped by a second harmonic of a Ti:sapphire laser which emits 120-fs-long pulses. A tiny part of the coherent master laser beam is split on the first beamsplitter BS and used as a seed for the down-conversion. With probability p ≪ 1, the beam contains exactly one photon. This beam is fed to the BBO crystal and the output is analyzed us-

Fig. 5. Experimental set-up for optimal universal cloning by means of stimulated parametric down-conversion. The input single-photon state to be cloned is obtained from a weak coherent laser beam. (After Lamas-Linares, Simon, Howell and Bouwmeester [2002].)

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ing a sequence of wave plates, polarizing beamsplitters (PBS) and single-photon photodetectors. The probability of pair generation in the crystal p2 ≪ p which guarantees that the dominant event leading to two photons in mode a and one photon in mode b is when a single photon was in the weak coherent beam a and a single pair was emitted in the crystal. The conditioning on observing a click of the trigger detector D1 is important since it eliminates events with two photons in mode a and no pair generated in the crystal. In the experiment, one measures the number of coincidence clicks of the photodetectors D2 and D3 as a function of the time delay between the input photon beam in mode a and the pump beam. If those two beams do not overlap in the BBO crystal, then there is no stimulated down-conversion and the polarization of the second photon emitted in mode a is fully random. If the two beams overlap, then stimulated amplification sets on and the second photon is emitted preferably with the same polarization as the input photon. Optimal cloning is achieved when the overlap is perfect. A detector setting with a PBS was used to measure the number of orthogonally polarized photon pairs N (1, 1). To detect the number of pairs with the same polarization N (2, 0), the PBS was replaced by a polarizer followed by an ordinary beamsplitter. The observed coincidence rates as a function of the time delay are shown in fig. 6 for three different polarizations. We see that N(1, 1) does not depend on the delay as expected, while N (2, 0) decreases with increasing delay. The average experimental cloning fidelity determined from these data reads F ≈ 0.81 which is very close to the theoretical maximum 56 ≈ 0.833. An improved experimental set-up involving double passage of the pump beam through the nonlinear crystal was developed by Pelliccia, Schettini, Sciarrino, Sias and De Martini [2003] and De Martini, Pelliccia and Sciarrino [2004]), see fig. 7. In this set-up, the photon to be cloned is generated during the first passage of the pump pulse through the crystal. Since the signal and idler beams are entangled, projecting the idler beam onto state |ψ prepares the signal in state |ψ⊥ . The click of the trigger detector DT heralds the preparation of a single photon in the mode labeled −k1 in fig. 7. This photon is then cloned by sending it again through the nonlinear BBO crystal. The delay between the pump and signal is controlled by moving the mirror MP . In this experiment, the states of both clones and the anti-clone were analyzed simultaneously and it was demonstrated that this device accomplishes jointly the optimal 1 → 2 cloning and also the optimal universal NOT gate for qubits. The attained fidelities were FCLON = 0.81 and FUNOT = 0.62. The stimulated down-conversion can be used to probabilistically implement any N → M cloning of qubits (Simon, Weihs and Zeilinger [2000a]) and even qudits (Kempe, Simon and Weihs [2000], Fan, Weihs, Matsumoto and Imai [2002]).

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Fig. 6. Observed coincidence rates as functions of the position of the movable mirror of two clones in identical polarization states (panels A, B, C) and in orthogonal polarization (panels D, E, F) for three different input polarizations. (After Lamas-Linares, Simon, Howell and Bouwmeester [2002].)

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Fig. 7. Simultaneous contextual realization of the optimal cloning machine and the universal NOT gate. Q – quarter-wave plates; WP – half-wave plates; PBS – polarizing beamsplitters; BS – beamsplitters, BBO – nonlinear crystal; D – single-photon detectors; M – movable mirror. (After De Martini, Pelliccia and Sciarrino [2004].)

The unitary transformation induced by the Hamiltonian H can be written in a factorized form as follows: n /2+1 −λ(a b −a b ) † †

† † V H H V , e−iH t = eλ(aV bH −aH bV ) 1 − λ2 tot (5.3) e

† where λ = tanh(κt), t is an effective interaction time and ntot = aV† aV + aH aH + † bH is the total number of photons in spatial modes a and b. Since bV† bV + bH the Hamiltonian H is covariant it is enough to consider the input state |ψin = |N aV |0 aH |0 bV |0 bH . With the help of the factorization (5.3) we find that the corresponding output state reads

e

−iH t

∞

2 N/2+1 λM−N |ΨM , |ψin = 1 − λ

(5.4)

M=N

where

|ΨM =

M−N k=0

(−1)

k

M −k N

× |M − k aV |k aH |k bV |M − N − k bH .

(5.5)

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We can see that the output state (5.4) is a weighted superposition of states |ΨM with different numbers of clones M. The state |ΨM and hence the fidelity of N → M cloning is independent of the coupling strength λ, and only the probability of generating exactly M clones depends on λ. One can also immediately see that the state (5.5) coincides (up to an irrelevant overall normalization factor) with the outcome of the optimal cloning transformation (4.13), hence the universal cloning via parametric down-conversion is optimal. We now extend the concept of cloning via amplification to qudits represented by a single photon in d different spatial modes or time bins. The use of timebin encoding seems to be particularly advantageous since only a single nonlinear Type-I-matched crystal is required, and the pump beam should consist of a sequence of d pulses. We associate creation operators aj† and bj† with the j th time bins of signal and idler beams, respectively. The Hamiltonian governing the evolution of this system can be expressed as Hd = iκ

d

j =1

aj† bj† − aj bj .

(5.6)

This Hamiltonian is invariant with respect to simultaneous unitary transformations of the signal and idler modes, (U ⊗ U ∗ )H (U † ⊗ U T ) = H , where U ∈ SU(d). This covariance property guarantees that the cloning is universal and the cloning fidelity does not depend on the input state, so it suffices to consider the input state |ψin,d = |N a1 |0 a2 · · · |0 ad |0 b1 |0 b2 · · · |0 bd . The unitary operation exp(−iH t) can again be factorized, similarly as in eq. (5.3), and we get M−N |Ψ e−iHd t |ψin,d = (1 − λ2 )N/2+d ∞ M,d , where the state containing M=N λ M clones reads N + m1 |N + m1 a1 |m2 a2 · · · |md ad |ΨM,d = N m

× |m1 b1 |m2 b2 · · · |md bd . (5.7) In this formula, m indicates summation over all vectors m = (m1 , . . . , md ) satisfying dj =1 mj = M − N. The optimality of this cloning transformation can be proved by explicit evaluation of the fidelity. It can be shown that there are

M−N+d−2−m different terms in eq. (5.7) with N + m photons in mode a1 , each d−2

N+m with weight m . The average single-clone fidelity can be thus expressed as M−N 1 N +m M −N +d −2−m N +m F = , N M N d −2 m=0

(5.8)

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where the normalization factor is given by N ≡

M−N m=0

N +m M −N +d −2−m M +d −1 = . N d −2 N +d −1

(5.9)

The summation in eq. (5.8) can be performed with the help of the identity given in eq. (5.9), and one recovers the optimal fidelity (4.9). Instead of parametric down-conversion it is also possible to amplify the light by sending it through an inverted atomic medium (Simon, Weihs and Zeilinger [2000a], Kempe, Simon and Weihs [2000], Fan, Weihs, Matsumoto and Imai [2002]). The atoms should possess d different ground states |gj and an excited state |e . We assume that each atomic transition |e → |gj is strongly coupled to a single optical mode aj and the qudits are represented by single photons in those d modes. The universality of the cloning requires that the coupling strength κ must be the same for all d transitions |e → |gj . In the interaction picture and in the rotating-wave approximation, the interaction of light with atoms is governed by the Jaynes–Cummings Hamiltonian, HJC = κ

d L k=1 j =1

aj† |gj k ek | + h.c.,

(5.10)

where L is the number of atoms and |gj k stands for the ground state |gj of the kth atom. This Hamiltonian satisfies the covariance property U ⊗ U ∗ H U † ⊗ U T = H , where U aj† U † = uj k ak† , U |ek = |ek and U |gj k = dl=1 uj l |glk . Suppose that all L = M − N atoms are initially prepared in the excited state and that all N input photons are in mode a1 . The joint atoms–photons input state reads |ψin,LA = |N a1 |0 a2 · · · |0 ad |e1 · · · |eL . If each atom emits a photon during the passage of the light through the atoms then M clones are generated and all atoms end up in ground states. In the weak-coupling regime we can express the output state conditional on all atoms being in some ground state using the Lth-order perturbation theory: $L # d † † aj bj c |ψin,LA , |ψout ∝ (5.11) j =1

where the operator bj† c is defined as bj† c = L k=1 |gj k ek |. Note that with this notation, the Hamiltonian (5.10) becomes similar to the down-conversion Hamiltonian (5.6). Since the atoms are supposed to be identical, the photons emitted by them do not carry any information about which atom emitted which photon. Consequently, if all atoms emit photons, then the atoms relax to symmetric ground

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state. Suppose that mj photons were emitted to mode aj , with j = 1, . . . , d. The corresponding symmetrized ground atomic state reads −1 |gm = CLm |g1k1 · · · |g1km1 |g2km1 +1 π(k)

· · · |g2km1 +m2 · · · |gdkL−md +1 · · · |gdkL ,

(5.12)

where π(k) denotes summation over all L! values of the subscripts kl , l = 1, . . . , L, which can be obtained as permutations of {1, . . . , L}. The normalization coefficient 2 CLm = m1 !m2 ! · · · md !L!

(5.13)

is chosen such that gm |gm = 1. After some algebra, one finds that the output state (5.11) can be expressed as CLm L! |ψout ∝ a1†m1 a2†m2 · · · ad†md |N a1 |0 a2 · · · |0 ad |gm m !m ! · · · m ! 1 2 d m N + m1 ∝ |N + m1 a1 |m2 a2 · · · |md ad |gm . N m

Since this state is fully equivalent to the state (5.7), the cloning is optimal. Although this result was obtained within the framework of perturbation theory, a detailed analysis reveals that it holds for any interaction strength. It can also be shown that if only M ′ − N < L atoms emit photons and the rest of the atoms remain in the excited state, then M ′ optimal photonic clones are generated (Fan, Weihs, Matsumoto and Imai [2002]). A proof-of principle experiment on cloning via stimulated emission was reported by Fasel, Gisin, Ribordy, Scarani and Zbinden [2002] utilizing a commercially available polarization-insensitive erbium-doped fiber amplifier. The amplifier was injected with a weak vertically polarized coherent signal with mean photon number n¯ in . After the amplification, the output mean numbers n¯ V and n¯ H of vertically and horizontally polarized photons were measured. The fidelity of the amplification process can be simply defined as F = n¯ V /(n¯ V + n¯ H ). The output mean intensities depend linearly on the input intensity (Shimoda, Takahasi and Townes [1957]), n¯ V = Gn¯ in +

1 (G − 1), Q

n¯ H =

1 (G − 1). Q

(5.14)

Here G is the gain of the amplifier and Q is a factor depending on the properties of the amplification process. The term Gn¯ in represents the amplified injected input signal while (G − 1)/Q represents the noise arising due to spontaneous

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emission. For quantum-noise-limited amplification, Q = 1. From eqs. (5.14) we can express G in terms of n¯ in , n¯ out = n¯ V + n¯ H and Q, and we find G = (Qn¯ out + 2)/(Qn¯ in + 2). On inserting this expression into the formula for the fidelity, we obtain Qn¯ out n¯ in + n¯ out + n¯ in . (5.15) Qn¯ out n¯ in + 2n¯ out If we formally replace n¯ in with N (the number of input copies), and n¯ out with M (the number of output clones), then for Q = 1 the formula (5.15) becomes the optimal fidelity of N → M cloning of qubits. Experimentally, G = 1.3 and Q = 0.8 was observed, quite close to the optimal value Q = 1. For instance, the fidelity of 1 → 2 cloning for Q = 0.8 inferred from eq. (5.15) reads F = 0.821 which is only slightly lower than the optimal fidelity F = 65 ≈ 0.833. F =

5.2. Symmetrization We have seen in Section 4.1 that the optimal universal N → M quantum cloning can be accomplished by symmetrizing the state of N input copies and M −N maximally mixed states. Since photons are bosons, the projection onto the symmetric subspace can be easily carried out with the use of linear optics, namely by mixing the M photons on an array of M − 1 beamsplitters and selecting only the events when all photons are collected in a single spatial mode. Let us first illustrate this method on the example of 1 → 2 cloning of polarization states of photons (Ricci, Sciarrino, Sias and De Martini [2004], Irvine, Lamas-Linares, de Dood and Bouwmeester [2004], Sciarrino, Sias, Ricci and De Martini [2004b]). The set-up is schematically illustrated in fig. 8a. The photon in mode A whose state is to be cloned is combined on a balanced beamsplitter BS1 with a blank copy photon prepared in a maximally mixed state. Only the cases when both photons leave the beamsplitter in the left output mode are post-selected, and the two clones are spatially separated by an auxiliary balanced beamsplitter BS2 . At the heart of cloning via symmetrization is the Hong–Ou– Mandel effect (Hong, Ou and Mandel [1987]). If two photons with identical polarization state interfere on a balanced beamsplitter, then they both end up in the same spatial mode and one does not observe any coincidences of one photon in mode A and one in mode B. So, for the input |ψ A |ψ B , there is probability 12 of having two photons in the left output port and probability 12 of splitting them in the two output modes A′ and B ′ . Altogether, the conditional transformation reads |ψ A |ψ B →

1 |ψ A′ |ψ B ′ . 2

(5.16)

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(a)

(b) Fig. 8. Optimal cloning of polarization states of photons via projection onto the symmetric subspace. (a) Optimal 1 → 2 universal cloning based on the interference of two photons on a balanced beamsplitter BS1 . (b) Extension to optimal N → M cloning. The N input states and M − N blank copies in maximally mixed states are combined on an array of M − 1 beamsplitters BSj , and the clones are then separated on another array of M − 1 beamsplitters BS′j .

On the other hand, if the two photons are initially in orthogonal polarization states, |ψ A |ψ⊥ B , then they are distinguishable and do not interfere on BS1 . With probability 14 , the photon in mode A is reflected and the photon in mode B is transmitted and they are both in the left output. Again, there is probability 21 that the two photons will be divided on a balanced beamsplitter BS2 . Since the photon in state |ψ can be either reflected or transmitted on BS2 , the final state of photons in modes A′ and B ′ is a balanced superposition of these two possibilities, namely a symmetric state, 1

(5.17) |ψ A′ |ψ⊥ B ′ + |ψ⊥ A′ |ψ B ′ . 4 Since the projector onto the symmetric subspace acts as Π+ |ψ |ψ = |ψ |ψ and Π+ |ψ |ψ⊥ = 12 (|ψ |ψ⊥ + |ψ⊥ |ψ ), it immediately follows from eqs. (5.16) and (5.17) that the set-up shown in fig. 8(a) implements with probability 41 the projection onto the symmetric subspace followed by a spatial separation of the two photons. The maximally mixed polarization state in mode B can be obtained for instance by preparing the blank copy photon in state |V or |H with probability 21 each. Another, more intriguing option is to send into port B one part of the maximally |ψ A |ψ⊥ B →

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entangled two-photon singlet state |Ψ − = √1 (|ψ B |ψ⊥ C − |ψ⊥ B ψ C ). In 2 this case, if the symmetrization succeeds, then we obtain in the spatial mode C the optimal anti-clone of |ψ , i.e. a state that has a fidelity 32 with |ψ⊥ . The optimal 1 → 2 cloning based on symmetrization has been experimentally demonstrated by two groups (Ricci, Sciarrino, Sias and De Martini [2004], Irvine, Lamas-Linares, de Dood and Bouwmeester [2004]). In both experiments, the input photon whose state was cloned was obtained from a weak coherent beam, and it was combined on a balanced beamsplitter with one photon from a maximally entangled singlet state generated in a nonlinear crystal by means of spontaneous parametric down-conversion. The triple-coincidence events were selected where there were two clones and one anticlone present, and the intensity of the weak coherent beam was adjusted such that the dominant contribution to the triplecoincidence events originated from the cases when there was a single photon in the coherent beam and a single entangled photon pair was generated in the nonlinear crystal. The observed mean cloning fidelities in these two experiments were F = 0.82 (Ricci, Sciarrino, Sias and De Martini [2004]) and F = 0.81 (Irvine, Lamas-Linares, de Dood and Bouwmeester [2004]), respectively. The simpler setup depicted in fig. 8(a), involving only two photons, was also implemented experimentally (Sciarrino, Sias, Ricci and De Martini [2004a]). A single photon pair was generated in a nonlinear crystal. One photon representing the input was prepared in the state |ψ using wave plates while the other photon was randomly prepared in the state |V or |H . This experiment is much simpler than the previous one, because only two-photon coincidence events were observed instead of tree-photon coincidences. This resulted in a much higher rate of cloning, and also in better visibility and mean cloning fidelity F = 0.826 very close to the theoretical maximum F = 0.833. An extension of the symmetrization procedure to M photons is illustrated in fig. 8(b). The photons are combined on an array of M − 1 beamsplitters BSj , and the symmetrization succeeds if all M photons are bunched in the same spatial mode (Sciarrino, Sias, Ricci and De Martini [2004b]). To confirm this we can split the output signal into M different spatial modes using another array of M − 1 beamsplitters BS′j and post-select only events with each of M photodetectors PD registering one photon. We now demonstrate that the array of beamsplitters accomplishes the desired projection onto the symmetric subspace. The symmetric two-mode L-photon states |L, k with L − k photons polarized vertically and k photons polarized ⊗L . We prove our claim by horizontally form a basis in the symmetric space H+ induction. Consider the Lth beamsplitter BSL in the scheme of fig. 8(b). The state impinging from the left is a symmetric L-photon state while a single pho-

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ton impinges on BSL from the bottom. The beamsplitter BSL does not need to be balanced but its transmittance t and reflectance r should be independent of the polarization. In the Heisenberg picture, the mixing of the modes on the beamsplitter is described by linear input output canonical transformations of the creation operators, aV† ,out = raV† ,in + tbV† ,in ,

bV† ,out = rbV† ,in − taV† ,in ,

(5.18)

and similar formulas hold for horizontal polarization. The state transformation on a beamsplitter can be most easily determined by expressing all states in terms of the creation operators acting on the vacuum, 1 a †k a †L−k |vac , |L, k = √ k!(L − k)! H,in V ,in |V = bV† ,in |vac ,

† |H = bH,in |vac .

(5.19)

From eqs. (5.18) we express the “in” operators as linear combinations of the “out” operators and substitute into the formulas (5.19). Using this technique it is easy to show that if all L + 1 photons bunch in the right output mode then the following conditional transformation takes place: √ |L, k |V → tLL rL L + 1 − k|L + 1, k , √ |L, k |H → tLL rL k + 1|L + 1, k + 1 . (5.20) Consider now the projection of the states |L, k |V and |L, k |H onto the symmetric subspace of L + 1 photonic qubits. One finds that ) L+1−k Π+,L+1 |L, k |V = |L + 1, k , L+1 ) k+1 Π+,L+1 |L, k |H = (5.21) |L + 1, k + 1 . L+1 The transformations (5.20) and (5.21) are equivalent up to a state-independent √ prefactor L + 1tLL rL , which proves that the array of M − 1 beamsplitters BS in fig. 8(b) projects the input states onto the symmetric subspace of M qubits. The probability of success of the projection can be determined by comparing the coefficients in eqs. (5.20) and (5.21), and we find P = PS M!

M−1 3 j =1

j

Tj (1 − Tj ),

(5.22)

+ ρin ] is the overlap of the input M-photon state ρin where Tj = tj2 and PS = Tr[ΠM with projector onto the symmetric subspace. The optimal transmittance Tj of the

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[6, § 5

Optical quantum cloning j

j th beamsplitter leading to maximal P can be obtained by maximizing Tj (1−Tj ), which yields Tj,opt = j/(j + 1). Note that Tj,opt does not depend on the input N-photon state. On inserting the optimal Tj into eq. (5.22) we get M! (5.23) . MM Recently, the optimal universal 1 → 3 and 2 → 3 cloning of polarization states of photons via symmetrization was demonstrated experimentally by Masullo, Ricci and De Martini [2004]. The three photons used in the experiment consisted of a pair of photons generated in the process of spontaneous parametric downconversion and a single photon in a very weak coherent beam. These three photons were combined on two beamsplitters, and only the events where all photons bunched in a single spatial mode were chosen by post-selection. Wave plates, a polarizing beamsplitter, an array of beamsplitters and photodetectors were employed to analyze the clones. The experimentally observed fidelity of the 1 → 3 cloning exp th = 79 ≈ 0.778. was F1→3 = 0.758, very close to the theoretical maximum F1→3 exp The observed fidelity of 2 → 3 cloning, F2→3 = 0.894, was also close to the th = 11 optimum value F2→3 12 ≈ 0.917. The symmetrization on a beamsplitter can be naturally extended to qudits. Symmetrization of two photonic qudits represented by a state of a photon in d different spatial modes would require an array of d balanced beamsplitters, each mixing the j th mode of the first and second qudits. It may be more advantageous to work with time-bin qudits, where the symmetrization would require only one balanced beamsplitter where the two photons would interfere. Similarly as before, only the events when the two photons bunch and leave the beamsplitter in the same spatial port have to be post-selected. Note finally that the cloning of the quantum state of a single photon using linear optics was also demonstrated by Huang, Li, Li, Zhang, Jiang and Guo [2001] using a different approach. In their scheme, both clones were represented by the quantum state of just a single photon in several modes, so that the two clones could not be physically separated. Popt = PS

5.3. Universal asymmetric cloning of photons So far, we have presented various optical implementations of symmetric cloning machines. In this section we will consider the optimal 1 → 2 asymmetric cloning of qubits. We will describe two methods, both based on the interference of photons on unbalanced beamsplitters. The first approach, introduced in Section 3.5, is to start from the output of the optimal symmetric cloner and convert it into an output

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519

of the optimal asymmetric cloner, which is given by |Ψ =

1

2 − 2p + 2p 2 × |ψ A |ψ B |ψ⊥ C

− p|ψ A |ψ⊥ B |ψ C − (1 − p)|ψ⊥ A |ψ B |ψ C .

(5.24)

Here p ∈ [0, 1] is an asymmetry parameter and the fidelities of the clones in qubits A and B read (1 − p)2 , 2(1 − p + p 2 ) p2 FB = 1 − . 2(1 − p + p 2 )

FA = 1 −

The symmetric cloner is recovered when p =

(5.25) 1 2

and we have

1 |Ψ sym = √ 2|ψ A |ψ B |ψ⊥ C − |ψ A |ψ⊥ B |ψ C 6 − |ψ⊥ A |ψ B |ψ C .

(5.26)

Suppose first that the second clone (qubit B) and the anti-clone (qubit C) are projected on the singlet state |Ψ − . We obtain 1√ − IA ⊗ ΠBC (5.27) 3|ψ A Ψ − BC , |Ψ sym = 2

− where ΠBC = |Ψ − Ψ − |. The original input state |ψ is perfectly recovered in qubit A. The projection on a singlet forms a part of the Bell measurement, i.e. a measurement in the basis of four maximally entangled Bell states. There is an interesting analogy between eq. (5.27) and the process of quantum teleportation (Bennett, Brassard, Crepeau, Jozsa, Peres and Wootters [1993], Bouwmeester, Pan, Mattle, Eibl, Weinfurter and Zeilinger [1997], Boschi, Branca, De Martini, Hardy and Popescu [1998], Marcikic, de Riedmatten, Tittel, Zbinden and Gisin [2003]). Indeed, as implied by eq. (3.44), the cloning can be deterministically reversed by performing a Bell measurement on one of the clones and the anticlone and applying an appropriate correcting unitary to the first clone (Bruss, Calsamiglia and Lütkenhaus [2001]). In the Bell measurement, the singlet is detected with probability 43 while each of the triplet Bell states is detected with 1 probability 12 , independently of the input state. This full reversal of cloning can be generalized to a partial reversal which converts the symmetric cloner to asymmetric one (Filip [2004a]). The idea is to apply − + to qubits B and C a filter ΠBC + aΠBC , where a ∈ [0, 1] controls the asymmetry.

520

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If a = 0 we get projection on singlet and full reversal, while for a = 1 the two qubits are multiplied by the identity and nothing happens. Let us now consider arbitrary a. The state after filtering,

− + + aΠBC |Ψsym , |Ψproj = IA ⊗ ΠBC (5.28) can be expressed, after normalization, as follows, |Ψproj =

1

6(a 2 + 3) × (3 + a)|ψ A |ψ B |ψ⊥ C

− (3 − a)|ψ A |ψ⊥ B |ψ C − 2a|ψ⊥ A |ψ B |ψ C .

(5.29)

We can immediately see that this state coincides with the outcome of the optimal asymmetric cloner (5.24) and p = (3 − a)/(3 + a). For optical polarization qubits, the filtration (5.28) can be implemented by letting the two photons interfere on an unbalanced beamsplitter and post-selecting only the events when a single photon is detected in each output port. There are two ways for the photons to exit the beamsplitter in different spatial modes: either both photons are reflected or both are transmitted. Unitarity dictates that these two alternatives acquire a mutual phase shift π. If the two photons are in the same state |ψ , then these two alternatives interfere destructively, while if the photons are in orthogonal polarization states there is no interference. The resulting conditional transformation reads |ψψ BC → (R − T )|ψψ BC ,

|ψψ⊥ → R|ψψ⊥ − T |ψ⊥ ψ .

(5.30)

It follows that the unbalanced beamsplitter applies the filter Π − + aΠ + with a = R − T . A schematic set-up of the proposed asymmetric cloning experiment is shown in fig. 9. Optimal symmetric cloning is accomplished by stimulated parametric down-conversion as discussed in detail in Section 5.1. At the output, the two clones are separated on an auxiliary balanced beamsplitter, and one of the clones is combined with the anti-clone on an unbalanced beamsplitter. Successful asymmetric cloning is heralded by a coincident observation of a single photon in each of the modes A, B and C. The second scheme for optimal asymmetric cloning (Filip [2004b]) very closely resembles the scheme for teleportation of polarization states of photons, see fig. 10. The only difference is that the balanced beamsplitter used in teleportation to perform a Bell analysis is replaced by an unbalanced beamsplitter that conditionally applies the filter Π− + aΠ+ . The cloning succeeds if a single photon is detected in each of the modes A, B and C. The initial state in the scheme

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Fig. 9. Optimal universal asymmetric cloning from symmetric cloning. The input photon to be cloned is obtained from a weak coherent beam. BS – auxiliary beamsplitter; SHG – second-harmonic generation; C1 – nonlinear crystal; 50:50 – balanced beamsplitter; T:R – unbalanced beamsplitter; WP – wave plates for input-state preparation; M – mirrors.

Fig. 10. Partial quantum state teleportation as an optimal asymmetric cloning. The input photon to be cloned is obtained from a weak coherent beam. BS – auxiliary beamsplitter; SHG – second-harmonic generation; C1 – nonlinear crystal; T:R – unbalanced beamsplitter; WP – wave plates for input-state preparation; M – mirrors.

shown in fig. 10 is |ψ B |Ψ − AC , and after the interference on a beamsplitter and post-selection we get

Ψ proj = IA ⊗ aΠ + + Π − |ψ B Ψ − . (5.31) BC BC AC

After some algebra we arrive at 1 Ψ proj ∝ 2(1 + 3a 2 ) × (1 + a)|ψ A |ψ B |ψ⊥ C

− (1 − a)|ψ A |ψ⊥ B |ψ C − 2a|ψ⊥ A |ψ B |ψ C .

(5.32)

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[6, § 5

This is again the output state of the optimal asymmetric cloning machine with p = (1 − a)/(1 + a), so the asymmetric cloning can be implemented by means of a partial teleportation. An interesting feature of this scheme is that one of the clones is teleported from Alice to Bob so we can speak about cloning at a distance. The universal asymmetric cloning of polarization states of single photons has been demonstrated experimentally by Zhao, Zhang, Zhou, Chen, Lu, Karlsson and Pan [2005] following the scheme illustrated in fig. 10. In that experiment, a Mach–Zehnder interferometer acted as an effective unbalanced beamsplitter whose transmittance could be controlled by changing the relative path difference between the two arms of the interferometer. In this way it was possible to demonstrate the whole class of asymmetric 1 → 2 cloning machines. 5.4. Cloning of orthogonally polarized photons It was shown in Section 4.1 that the optimal universal quantum cloning and optimal quantum state estimation are closely related, and that in the limit of an infinite number of clones the fidelity of cloning is equal to the fidelity of optimal state estimation. In this context, a very interesting and surprising observation was made by Gisin and Popescu [1999], who found that the state of a single qubit can be estimated better from the state |ψ |ψ⊥ than from the state |ψ |ψ . Picturing the qubits as spin- 21 particles, we can say that the information about the direction is encoded better in two anti-parallel spins than in two parallel ones. The fidelity of the estimation of |ψ from a single copy of the two-qubit state |ψ |ψ⊥ reads (Gisin and Popescu [1999], Massar [2000]), 1 1 ≈ 0.789, 1+ √ F⊥ = (5.33) 2 3 which is slightly higher than the fidelity of optimal estimation from |ψ |ψ , F = 43 . Motivated by this observation we may expect that this advantage of two anti-parallel spins over two parallel ones extends also to cloning. This is indeed the case, provided that the number of clones is large enough. We shall now describe the optimal universal cloning transformation which produces M approximate clones of the state |ψ from a single replica of |ψ |ψ⊥ and maximizes the single-clone fidelity. Making the natural assumption that the output Hilbert space is the symmetric subspace of M qubits, the optimal cloning CP map S can be determined analytically for any M (Fiurášek, Iblisdir, Massar and Cerf [2002]). The mean singleclone fidelity can be expressed as F = Tr[SR], where R has a rather complicated form and can be found in (Fiurášek, Iblisdir, Massar and Cerf [2003]). In contrast

6, § 5]

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523

to the universal cloning with input state |ψ ⊗N , the maximum fidelity cannot be determined from the maximum eigenvalue of R, and one has to solve the extremal equations (3.18) and prove the optimality by checking that the inequality (3.19) is satisfied. Since the input state of the cloner can be obtained as an orbit of the group SU(2), |ψ |ψ⊥ = U ⊗ U |0 |1 , the optimal cloner is covariant and can be expressed as follows, |ψ, ψ⊥ →

M j =0

αj,M (M − j )ψ, j ψ⊥ ⊗ (M − j )ψ⊥ , j ψ ,

where the coefficients αj,M are given by √

3(M − 2j ) 1 j αj,M = (−1) √ +√ . 2(M + 1) 2M(M + 1)(M + 2)

(5.34)

(5.35)

The cloning machine (5.34) is symmetric with respect to the interchange of |ψ and |ψ⊥ . The cloner requires an ancilla whose size is the same as the size of the output Hilbert space, i.e., the ancilla Hilbert space is also a symmetric subspace of M qubits. The ancilla contains M approximate copies of the state |ψ⊥ , and the fidelity of these anti-clones is the same as the fidelity of the clones. The single2 , clone fidelity can be calculated as weighted average of the coefficients αj,M F⊥ (M) =

M M −j 2 αj,M , M

(5.36)

j =0

and after a simple algebra we arrive at ) M +2 1 F⊥ (M) = 1+ . 2 3M

(5.37)

The fidelity monotonically decreases with increasing number of clones M, and in the limit M → ∞ we recover the fidelity (5.33) of the optimal state estimation from |ψ |ψ⊥ . Upon comparing the fidelity F⊥ (M) with the fidelity of the optimal cloner for a pair of identical qubits, F (M) = (3M + 2)/(4M), we see that F (M) F⊥ (M) for M 6, while F⊥ (M) > F (M) for M > 6 and the cloner (5.34) outperforms the standard universal cloner. We have seen in Section 5.1 that the optimal universal cloning of polarization states of photons can be realized by means of stimulated parametric downconversion. It turns out that the optimal cloning with a pair of orthogonal qubits as the input can be performed in the same way, if the photons in states |ψ and |ψ⊥ are fed to the input signal and idler ports of the amplifier, respectively, as

524

Optical quantum cloning

[6, § 5

Fig. 11. Cloning of a pair of orthogonal qubits by means of stimulated parametric down-conversion. BS – auxiliary beamsplitters; Cj – nonlinear crystals; WP – wave plates for input-state preparation; M – mirrors; PD – triggering single-photon detectors for conditional preparation of single-photon states.

schematically illustrated in fig. 11. We assume that the parametric amplification in the nonlinear crystal C3 is governed by the singlet-type Hamiltonian (5.1), which is invariant under the simultaneous rotation of the signal and idler qubits, (U ⊗ U )H (U † ⊗ U † ) = H . Assuming the input state |1 aV |0 aH |0 bV |1 bH , the output state after the amplification in the crystal C3 reads |Ψout =

∞

M=0

λM−1 1 − λ2 |Ψ⊥,M ,

(5.38)

where the state with M clones and M anti-clones is given by |Ψ⊥,M =

M

(−1)j (M − j ) 1 − λ2 − λ2 j =0

(5.39)

× |M − j aV |j aH |j bV |M − j bH ,

with λ = tanh(κt) and t the effective interaction time. In contrast to universal cloning with N identical photons at the input, the state |Ψ⊥,M depends on the strength of the parametric amplification λ. The cloner that produces M copies is obtained by post-selecting only the events with exactly M photons detected in signal and idler spatial modes, which corresponds to the selection of the state |Ψ⊥,M from the superposition (5.38). The fidelity of the cloner depends on λ, F⊥ (M, y) =

3y 2 − 2y(2M + 1) + 32 M(M + 1) , 6y 2 − 6My + M(2M + 1)

(5.40)

6, § 6]

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where y = λ2 /(1 − λ2 ) = sinh2 (κt). The optimal parametric gain which max(M,y) imizes the fidelity (5.40) can be found by solving the equation ∂F⊥∂y = 0, which yields ) M 1 M(M + 2) − . yopt = (5.41) 2 2 3 On inserting the optimal y into eq. (5.40) we recover the fidelity (5.37), hence the optimal cloning of a pair of orthogonal qubits can be achieved by means of stimulated parametric down-conversion with properly chosen gain. § 6. Phase-covariant cloning of photons In Sections 4 and 5 we have focused on the implementation of universal cloning machines that clone all states equally well. In many situations, however, one deals only with a subset of states. An archetypal example is the Bennett–Brassard 1984 (BB84) protocol for quantum key distribution (Bennett and Brassard [1984]), which utilizes four non-orthogonal states |0 , |1 , |0 + |1 and |0 − |1 . If we restrict the range of admissible input states of the cloning machine, then we can expect that the machine will exhibit better performance than the universal cloner and will reach higher fidelity. In this section we shall study phase-covariant cloning machines which optimally clone all states that are balanced superpositions of the computational basis states, d−1 1 iφj |ψ = √ e |j , d j =0

(6.1)

where the phases φj can be arbitrary, see Section 3.6.3. 6.1. Phase-covariant cloning of qubits The simplest and perhaps most important example is the 1 → 2 phase-covariant cloning machine which can be used as the optimal individual eavesdropping attack on the BB84 protocol (Fuchs, Gisin, Griffiths, Niu and Peres [1997]). In contrast to universal cloners, the optimal cloning transformation here depends on whether single-clone fidelity or global fidelity is taken as the figure of merit to be maximized. In the context of eavesdropping on the quantum key distribution protocol, it is natural to consider the single-clone fidelities, since they quantify the amount of information transmitted to the receiver and gained by the eavesdropper. The optimal symmetric 1 → 2 cloning transformation for qubits that maximizes the single-clone fidelity has the following form (Bruss, Cinchetti,

526

[6, § 6

Optical quantum cloning

D’Ariano and Macchiavello [2000], Bruss and Macchiavello [2001], Fan, Weihs, Matsumoto and Imai [2002], Cerf, Durt and Gisin [2002], Karimipour and Rezakhani [2002]): 1 1

|0 |Ain → √ |0 A |0 B |0 C + |0 A |1 B + |1 A |0 B |1 C , 2 2 1 1

|1 |Ain → √ |1 A |1 B |1 C + |0 A |1 B + |1 A |0 B |0 C . 2 2

(6.2)

It is a special case of the Pauli cloner, see Section 2.2. The two clones are √ stored pc in qubits A and B, and the fidelity of each clone reads F1→2 = (1 + 1/ 2 )/2 ≈ 0.855, which is indeed slightly higher than the fidelity of the optimal universal univ = 5 ≈ 0.833. Note that besides a blank copy 1 → 2 cloner for qubits, F1→2 6 qubit, the transformation also requires another ancilla qubit C. However, in contrast to universal cloning, this ancilla is not necessary and one can design a simplified cloning transformation which achieves the same fidelity and requires only two qubits: the input and a blank copy (Niu and Griffiths [1999], Durt and Du [2004]). This is very important from the experimental point of view since it is much easier to realize a two-qubit transformation than a three-qubit transformation. The economic phase-covariant cloner can be obtained by projecting the ancilla C on the basis state |0 (or |1 ). If we project on |0 , then we get |0 A |0 B → |0 A |0 B , 1

|1 A |0 B → √ |0 A |1 B + |1 A |0 B . 2

(6.3)

An alternative economic cloning transformation can be obtained from eq. (6.3) by exchanging 0 and 1. Interestingly, the cloning machine (6.3) is optimal not only for the states on the equator of the Bloch sphere but also for all the states on the northern hemisphere, i.e., all states cos(θ/2)|0 + eiφ sin(θ/2)|1 with θ π/2 (Fiurášek [2003]). The optimal asymmetric cloning machine which produces two clones with different fidelities FA and FB is obtained by breaking the symmetry in the output superposition of |10 and |01 , |0 A |0 B → |0 A |0 B ,

|1 A |0 B → cos ϑ|0 A |1 B + sin ϑ|1 A |0 B ,

(6.4)

and the two fidelities can be expressed as follows: FA =

1 (1 + sin ϑ), 2

FB =

1 (1 + cos ϑ), 2

1 ϑ ∈ 0, π . 2

(6.5)

The phase-covariant cloning machine that maximizes the global two-qubit fi-

6, § 6]

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527

delity has a structure that is qualitatively similar to the cloner (6.2),

1 |0 |Ain → √ |0 A |0 B |0 C + |0 A |1 B + |1 A |0 B |1 C , 3

1 |1 |Ain → √ |1 A |1 B |1 C + |0 A |1 B + |1 A |0 B |0 C , 3

(6.6)

pc,G

and it reaches fidelity F1→2 = 43 which is again higher than the global fidelity of univ,G = 32 . the universal cloner, F1→2 The phase-covariant cloning can be extended to the case when we possess N copies of the state and would like to prepare M clones, M > N. The optimal 1 → M phase-covariant cloning machine was determined by Fan, Matsumoto, Wang and Wadati [2001], who considered the single-clone fidelity as the figure of merit. The structure of the cloning transformation depends of the parity of M. If M is even, there exist two independent cloning transformations, |0 → |M, M/2 − 1 ,

|1 → |M, M/2

(6.7)

and |0 → |M, M/2 ,

|1 → |M, M/2 + 1 ,

(6.8)

where |M, k is a symmetric state of M qubits with k qubits in state |1 and M − k qubits in state |0 . Note that there are in fact infinitely many cloning transformations since any convex mixture of the operations (6.7) and (6.8) is also optimal. On the other hand, if M is odd then we get only one optimal transformation: |1 → M, (M + 1)/2 . |0 → M, (M − 1)/2 , (6.9)

The resulting fidelity is , √ M(2+M) 1 + , F = 21 M+14M 2 + 4M ,

M even, M odd.

(6.10)

The optimality of the cloning transformations (6.7)–(6.9) can be proved using the method that was employed in Section 4.2 to prove the optimality of the 1 → M universal cloning machine for qubits. In particular, the single-clone fidelity can be expressed as F = Tr[SR], where S is the operator isomorphic to the cloning CP map and 1

1 R = I ⊗ Π+,M + |0 1| + |1 0| 4 4 M−1

DM,k |M, k + 1 M, k| + |M, k M, k + 1| , ⊗ k=0

(6.11)

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Optical quantum cloning

[6, § 6

√ where DM,k = (M − k)(k + 1)/M. The fidelity is upper bounded by the maximum eigenvalue rmax of R, F 2rmax , and this bound is saturated by the above phase-covariant cloners. Fan, Matsumoto, Wang and Wadati [2001] also conjectured the structure of the general optimal N → M phase-covariant cloning transformation for qubits. The proposed generalization is straightforward, namely, every input symmetric N-qubit state |N, k is transformed to an M-qubit symmetric state |M, k + j with the constant j adjusted such that the fidelity is maximized. If N and M have the same parity, M = N + 2L, then the suggested cloning map is |N, j → |M, j + L , and the corresponding fidelity is N−1 1 1 N N pc FN→M = + 2 M2N j j +1 j =0 × (N + L − j )(L + j + 1). (6.12)

When M and N have different parities, M = N + 2L + 1, then the two possible cloning transformations are either |N, j → |M, j + L or |N, j → |M, j + L + 1 , and the corresponding fidelity is N −1 1 1 N N pc FN→M = + 2 M2N +1 j j +1 j =0 × (N + L − j + 1)(L + j + 1) + (L + j + 2)(N + L − j ) . (6.13)

The optimality of the fidelity (6.12) was proved by D’Ariano and Macchiavello [2003] exploiting the generic theory of covariant cloning machines, see D’Ariano and Lo Presti [2001]. In contrast, if N and M have different parities, the optimal phase-covariant cloning transformation found by D’Ariano and Macchiavello [2003] differs from eq. (6.13).

6.2. Phase-covariant cloning of qudits Going beyond the cloning of qubits, the 1 → 2 phase-covariant cloning of qudits (6.1) was investigated by Fan, Imai, Matsumoto and Wang [2003], Lamoureux and Cerf [2005] and Rezakhani, Siadatnejad and Ghaderi [2005]. It can be shown that the optimal cloning transformation for qudits (6.1) has the structure d−1

β |j → α|jj AB |j C + √ |j l AB + |lj AB |l C , 2(d − 1) l=j

(6.14)

6, § 6]

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529

where α 2 + β 2 = 1. The two clones are contained in qudits A and B while the qudit C serves as an ancilla. Note that eq. (6.14) is a direct extension of the cloning transformation for qubits (6.2). The coefficients α and β have to be optimized such that the cloning fidelity is maximized. After some algebra one arrives at 1/2 1 d −2 α= , − √ 2 2 d 2 + 4d − 4 1/2 d −2 1 , β= (6.15) + √ 2 2 d 2 + 4d − 4

and the fidelity reads

√ 1 d 2 + 4d − 4 1 F = + (6.16) + . 4 2d 4d In contrast to the phase-covariant cloning of qubits, we cannot get rid of the ancilla C because if we project the ancilla on the computational basis state |k then the conditional map is not unitary. So, for d > 2 it seems impossible to implement the optimal phase-covariant 1 → 2 cloning in an economic way, without ancilla, see Durt, Fiurášek and Cerf [2005].

6.3. Optical phase-covariant cloning In contrast to universal cloning, the optical experimental implementation of phase-covariant cloning machines has received much less attention. This may come as a surprise in view of the apparent simplicity of the optimal cloning transformation (6.3). However, the phase-covariant cloning exhibits much less symmetry than the universal copying, and methods such as stimulated amplification or symmetrization cannot readily be extended to implement the 1 → 2 phasecovariant cloning machine. It is nevertheless possible to conditionally realize the 1 → 2 phase-covariant cloning of photonic qubits with linear optics (Fiurášek [2003]). As usual, the qubits are encoded into polarization states of single photons, and the state to be cloned is a balanced √ superposition of vertical and horizontal polarization, |ψ = (|V + eiφ |H )/ 2. Besides the input state, the cloning requires also a second photon, the blank copy which we assume to be initially prepared in the state |V . Written in the basis of polarization states, the cloning transformation (6.3) becomes |V V → |V V ,

1

|H V → √ |H V + |V H . 2

(6.17)

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Optical quantum cloning

[6, § 6

The cloning machine is shown schematically in fig. 12(a). The input photon and the blank copy are combined on an unbalanced beamsplitter whose transmittance tj and reflectance rj for the vertical (j = V ) and horizontal (j = H ) polarizations are different. Only the events when the two photons leave the beamsplitter in different output ports are post-selected. The principle of operation of the cloner is easy to grasp. If the input |ψ is in state |V , the two photons at the output must be in state |V V since the blank copy is initially in the state |V . On the other hand, if the input to be cloned would be in the state |H then the beamsplitter would produce a superposition of |H V and |V H . By properly choosing rj this superposition can be made balanced and the conditional map becomes exactly the unitary (6.17).

(a)

(b) Fig. 12. Phase-covariant cloning using interference of two photons on an unbalanced beamsplitter. (a) Scheme with a single beamsplitter BS that differently reflects vertical and horizontal polarizations. (b) Alternative set-up involving a polarizing beamsplitter PBS and an unbalanced beamsplitter BS′ whose reflectance does not depend on the polarization.

6, § 6]

Phase-covariant cloning of photons

531

The mixing of the modes on a beamsplitter is governed by the linear canonical transformations † aj,out = rj aj† + tj bj† ,

† bj,out = rj bj† − tj aj† ,

(6.18)

with j = V , H and rj2 + tj2 = 1. The conditional transformation corresponding to selecting only the events with one photon in the left output arm (mode A) and one photon in the right output arm (mode B) reads

|V V → rV2 − tV2 |V V , |H V → rH rV |H V − tH tV |V H .

(6.19)

This transformation becomes fully equivalent to eq. (6.17) if the following conditions are satisfied: √ √ rV2 − tV2 = 2rH rV = − 2tH tV . (6.20) √ These constraints imply that rH = tV , tH = −rV and (rV2 − tV2 ) = 2rV tV . On combining this equation with the normalization rV2 +tV2 = 1 we can determine rV . After simple algebra we obtain 1 1 rV2 = (6.21) 1+ √ . 2 3 The probability of successfully realizing the phase-covariant cloner is given by

2 1 P = rV2 − tV2 = . (6.22) 3 The required beamsplitter with different transmittances for vertical and horizontal polarizations can be simulated by a Mach–Zehnder interferometer with polarization-dependent phase shifters in its arms, such as Soleil–Babinet compensators, so that the phase shift and, consequently, the splitting ratio could be controlled independently for vertical and horizontal polarizations. The set-up could also be modified to work with a beamsplitter whose reflectance is the same for both vertical and horizontal polarizations. This alternative configuration is depicted in fig. 12(b). The signal and blank copy photons are first combined on a polarizing beamsplitter PBS that reflects vertically polarized photons and transmits horizontally polarized photons. The two beams are then recombined on a beamsplitter with reflectance r. If the signal photon is initially vertically polarized, then a vertically polarized photon enters each input port of BS. If the signal photon is polarized horizontally, then it is switched to the right arm and two photons in orthogonal polarization states impinge on the right input port of BS. The polarizing beamsplitter ensures that the role of the transmittance and reflectance

532

Optical quantum cloning

[6, § 6

for the horizontally polarized photon is interchanged with respect to the scheme shown in fig. 12(a). It can easily be shown that this set-up leads to the cloning transformation √(6.17) provided that the reflectance of the beamsplitter is equal to r 2 = (1 + 1/ 3 )/2. In the experiment, it may not be easy to precisely control the transmittance. It is therefore important to investigate how the performance of the set-up shown in fig. 12(b) depends on the reflectance r of the beamsplitter. The cloning transformation remains phase covariant and the cloning fidelity F is the same for both clones and does not depend on φ. However, F becomes a function of r. After some algebra one arrives at the formula for the fidelity of cloning of equatorial qubits,

1 2r(2r 2 − 1) (1 − r 2 ) F = (6.23) 1+ . 2 2r 4 − 2r 2 + 1 It turns out that the cloning is rather robust with respect to the variations of the reflectance of the beamsplitter, and a cloning fidelity F > 0.8 can be achieved for a broad range of beamsplitter reflectances 0.7 r 2 0.9. In the experiment, the required pair of photons can be produced in spontaneous Type-I parametric down-conversion and the desired initial states of the photons can be prepared with the use of wave plates. After cloning, the states of the two clones can be analyzed by a sequence of wave plates, polarizing beamsplitters, and single-photon detectors, similarly as in the experiments on universal cloning.

6.4. Experimental 1-to-3 phase-covariant cloning Remarkably, while the optimal 1 → 2 phase-covariant cloning transformation (6.2) or (6.3) has not yet been implemented for optical qubits, the optimal 1 → 3 phase-covariant cloning of the polarization state of a single photon has been demonstrated experimentally by Sciarrino and De Martini [2004]. The set of cloned states included all linear polarization states cos θ|V + sin θ|H . The first step in the copying process consisted of the optimal 1 → 2 universal cloner described in Section 5.1 which produced two clones and one anti-clone. In the next step, the anti-clone was converted into a clone by applying a unitary transformation σy with the help of a half-wave plate. The final step was to symmetrize the state of the three clones by combining the two clones and the anti-clone on a balanced beamsplitter and selecting only the events where all three photons ended up in the same output spatial mode. In this way three copies of equal fidelity were produced.

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pc

The experimentally observed fidelities were F1→3 (|+ ) = 0.76 for the state pc |+ = 2−1/2 (|V + |H ) and F1→3 (|H ) = 0.80. This should be compared with pc the theoretical maximum F1→3 = 65 ≈ 0.833. It is also instructive to make a univ = 7 ≈ comparison with the fidelity of the optimal universal 1 → 3 cloner, F1→3 9 0.778. One can conclude that the experimental phase-covariant cloning machine operates very close to its theoretical limit, and for certain inputs it achieves better fidelity than what would be possible with universal cloning machine.

§ 7. Cloning of optical continuous variables In Sections 4–6 we have considered the cloning of quantum states in finitedimensional Hilbert spaces. During recent years, however, quantum information processing in systems with infinite-dimensional Hilbert space, such as modes of the electromagnetic field, has attracted a great deal of attention (see, e.g., Braunstein and van Loock [2005]). In this approach, the quantum information is usually encoded into two noncommuting quadrature operators x and p which satisfy canonical commutation relations [x, p] = i. Since these operators have continuous spectra, one speaks of quantum information processing with continuous variables. The universal cloning machine for states belonging to infinite-dimensional Hilbert space can be formally obtained as a limit of the universal cloning machine for qudits when d → ∞. One finds that the single-clone fidelity of the universal 1 → 2 cloner is 21 , which means that the optimal cloning can be achieved by a very simple strategy where the input state is sent with probability 21 to the first or second output, while the other output is prepared in maximally mixed state. Besides being rather trivial, this universal cloner is not of great practical interest because most of the quantum information protocols with continuous variables involve only the so-called Gaussian states. These states have a Gaussian Wigner function and their great advantage is that they can be generated and manipulated relatively easily in the laboratory with the help of linear optical interferometers and optical parametric amplifiers which produce squeezed and entangled Gaussian states.

7.1. Cloning of coherent states Among the Gaussian states, the coherent state is perhaps the best known example. The coherent state |α can be defined as a displaced vacuum state D(α)|0 , where D(α) = exp(αa † − α ∗ a) is the displacement operator. The coherent state is the

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eigenstate of the annihilation operator, a|α = α|α and it is also a minimum uncertainty state. The variance of all rotated quadratures xθ = x cos θ + p sin θ is the same and equal to 21 . The Glauber P -distribution of a coherent state is a Dirac delta function, so that coherent states are not usually considered as nonclassical states in the quantum-optical sense. Still, they are pure quantum states and they carry quantum noise. This makes these states suitable for applications such as quantum key distribution. It has been shown theoretically (Grosshans and Grangier [2002], Grosshans, Cerf, Wenger, Tualle-Brouri and Grangier [2003], Grosshans and Cerf [2004], Iblisdir, Van Assche and Cerf [2004]) and demonstrated experimentally (Grosshans, Van Assche, Wenger, Tualle-Brouri, Cerf and Grangier [2003]) that secure key distribution can be achieved with coherent states and balanced homodyne detection. Let us first consider the optimal Gaussian cloning of coherent states introduced by Cerf, Ipe and Rottenberg [2000] and Lindblad [2000]. In this scenario, the class of admissible cloning transformations is restricted to Gaussian operations, which preserve the Gaussian shape of the Wigner function. Intuitively, one could expect that the Gaussian cloning should be optimal. This is indeed true if the figure of merit is the global M-clone fidelity or if the quality of the clones is quantified in terms of the noise added to the two quadratures x and p (Cerf and Iblisdir [2000]). However, it has been realized recently that, remarkably, the single-clone fidelity of the 1 → 2 cloning of coherent states is maximized by a non-Gaussian cloner (Cerf, Krüeger, Navez, Werner and Wolf [2005]). We begin with the Gaussian 1 → 2 cloning. We require that the mean values of the quadratures of the two clones A and B are equal to the mean values of the quadratures xin and pin of the input coherent state α. This guarantees that the cloning transformation is invariant with respect to the displacements and the cloning fidelity does not depend on the amplitude α. In the Heisenberg picture, the most general Gaussian cloning transformation can be written in the form xA = xin + x˜A , pA = pin + p˜ A ,

xB = xin + x˜B , pB = pin + p˜ B .

(7.1) (7.2)

The operators x˜A , p˜ A , x˜B and p˜ B represent the noise that is added to the two copies during the cloning process, and they all commute with xin and pin . The quadrature operators xA , pA , xB and pB must satisfy the canonical commutation relations, which implies [x˜A , p˜ B ] = −i,

[x˜B , p˜ A ] = −i.

(7.3)

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The Heisenberg uncertainty relation gives a lower bound on the products of the variances of the noise operators, 1 1 (x˜B )2 (p˜ A )2 . (x˜A )2 (p˜ B )2 , (7.4) 4 4 as shown in Cerf, Ipe and Rottenberg [2000] and Grosshans and Grangier [2001]. The cloning should add noise isotropically, that is, the variance of the x and p quadratures of each clone should be the same. Since the noise operators are not correlated with xin and pin , the variance of the quadratures (7.2) of the two clones is the sum of two variances, and the isotropy condition is satisfied if (x˜A )2 = (p˜ A )2 = n¯ A , (7.5) (x˜B )2 = (p˜ B )2 = n¯ B . The two uncertainty relations (7.4) boil down to a single constraint

1 . (7.6) 4 The state of each clone is a mixed Gaussian state, namely a coherent state with added thermal noise with mean number of thermal photons equal to n¯ j , j = A, B. The fidelity of cloning can be most easily calculated from the Husimi Q-function, which is defined as the overlap of the density matrix with the coherent state, Q(β) = π−1 β|ρ|β . The Q-function of the j th clone reads (Fiurášek [2001a]),

1 |β − α|2 Qj (β) = (7.7) exp − . π(1 + n¯ j ) 1 + n¯ j n¯ A n¯ B

The fidelity can be calculated as Fj (α) = πQj (α) = 1/(1 + n¯ j ). The best tradeoff between the fidelities of the two clones is obtained when the equality holds in eq. (7.6), and we get 2 2 , FB = , (7.8) 2 + e2γ 2 + e−2γ where γ is a parameter which controls the asymmetry of the cloning. The fidelity of the optimal 1 → 2 symmetric (γ = 0) Gaussian cloner is F = 2/3. FA =

7.2. Cloning by phase-insensitive amplification If the coherent states are carried by optical modes, then the cloning can be realized with the use of a phase-insensitive amplification of light (Cerf and Iblisdir [2001c], Braunstein, Cerf, Iblisdir, van Loock and Massar [2001], Fiurášek [2001a], Cerf, Iblisdir and Van Assche [2002]). This is a natural and intuitive result, because the idealized perfect cloning amounts to noiseless amplification of

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(a)

(b) Fig. 13. Cloning of coherent states in a nondegenerate optical parametric amplifier. (a) Asymmetric 1 → 2 cloner consisting of a Mach–Zehnder interferometer with a non-degenerate parametric amplifier (NOPA) in one of its arms. The amplification gain and the splitting ratios determine the asymmetry of the cloner. (b) Simplified scheme of a symmetric cloner.

√ the coherent state, |α → | 2α . The optimal amplification that adds the minimum amount of noise can be performed, e.g., in a nondegenerate optical parametric amplifier (NOPA), which transforms the input annihilation operator a as √ √ † aout = Gain + G − 1cin , with c the annihilation operator of the idler mode in the NOPA. The set-up for asymmetric cloning of coherent states is shown in fig. 13(a). It consists of a Mach–Zehnder interferometer with an amplifier in one of its arms. The signal is initially divided into two beams and one beam is amplified such that the total mean intensity is twice the input intensity. The two clones are obtained by recombining the two beams on the second beamsplitter. The splitting ratios of the unbalanced beamsplitters BS1 and BS2 and the intensity gain G of the amplifier can be expressed in terms of the asymmetry parameter γ as follows: √ 2 sinh γ r1 = − , 1 + 2 sinh2 γ G = 1 + cosh(2γ ), e2γ r2 = √ (7.9) . 1 + e4γ The set-up becomes particularly simple for a symmetric cloner. In this case the first beamsplitter disappears and the whole input signal is amplified with

6, § 7]

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537

gain G = 2 and then divided into two modes on a balanced beamsplitter, see fig. 13(b). The procedure for symmetric cloning can be readily extended to the optimal symmetric N → M Gaussian cloning of coherent states. The cloning consists of three steps. First, the whole signal is collected in a single mode using √ an array of ⊗N → | N α . Next, N −1 beamsplitters with properly chosen transmittances, |α the collected signal is amplified with a gain G = M/N. Finally, the amplified signal is distributed among the M modes with the help of another array of M − 1 unbalanced beamsplitters such that the mean complex amplitude in all modes is the same and equal to α. The fidelity of this cloner does not depend on α, and each clone is in a coherent state with thermal noise described by the Husimi function (7.7). The total mean number of thermal photons in all modes is G − 1 = M/N − 1, and the noise is equally divided into M modes, hence the thermal noise in each clone is n¯ = 1/N − 1/M. On inserting this into the expression for the fidelity, F = 1/(1 + n), ¯ we obtain F =

MN . MN + M − N

(7.10)

In the limit of an infinite number of copies, M → ∞, we get F = N/(N + 1) which is the fidelity of optimal estimation of a coherent state from N copies. Similarly as in the case of universal cloning of qubits (Bruss, Ekert and Macchiavello [1998]), the connection between optimal cloning and optimal state estimation can be exploited to prove that eq. (7.10) is the maximal fidelity of the Gaussian N → M cloning of coherent states (Cerf and Iblisdir [2000]). As shown by van Loock and Braunstein [2001], it is also possible to clone coherent states via an extended continuous-variable teleportation. The telecloning requires a specific multimode entangled Gaussian state that can be generated by mixing single-mode squeezed vacuum states on an array of unbalanced beamsplitters (van Loock and Braunstein [2000]). As noted before, the Gaussian machine depicted in fig. 13 is not the optimal one if the single-clone fidelity is taken as the figure of merit (Cerf, Krüeger, Navez, Werner and Wolf [2005]). The optimal non-Gaussian cloner can achieve a fidelity Fmax = 0.6826, which is slightly higher than the maximum fidelity achievable by Gaussian transformations, F = 32 ≈ 0.6667. Interestingly, the optimal non-Gaussian cloner can be obtained from the set-up shown in fig. 13(b) if the input ports of the idler mode of the amplifier and the auxiliary mode of the beamsplitter are fed with a specific non-Gaussian entangled state |ψ BC = ∞ n=0 cn |2n B |2n C . The coefficients cn can be optimized in order to maximize the cloning fidelity which results in the above value Fmax = 0.6826. It should be

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stressed that while the non-Gaussian cloner maximizes the fidelity, the variance of the quadratures of the clones is higher than for the optimal Gaussian cloner. In applications such as quantum key distribution with coherent states and balanced homodyning, where the quantum channel between Alice and Bob is characterized in terms of the first and second moments of the transmitted quadratures, the aim of the eavesdropper is to minimize the quadrature variance instead of the fidelity, and the Gaussian cloning (or its variant entangling Gaussian cloner in case of reverse reconciliation protocol) can be the most dangerous individual eavesdropping attack.

7.3. Experimental cloning of coherent states The first proposal for an experimental continuous-variable cloning machine, due to D’Ariano, De Martini and Sacchi [2001], seemed quite involved as it required a network of parametric amplifiers. Recently, the optimal Gaussian 1 → 2 cloning of coherent states was experimentally demonstrated by Andersen, Josse and Leuchs [2005]. The distinct feature of this experiment is that it does not require an amplifier, the latter being replaced by a clever combination of measurement and feedback. A simplified scheme of the experimental set-up is shown in fig. 14. Mode ain contains the coherent state to be cloned. The beam is split into two parts on a balanced beamsplitter whose auxiliary input port ν1 is in vacuum state. The output annihilation operators thus read 1 a ′ = √ (ain + ν1,in ), 2

1 ν1′ = √ (ain − ν1,in ). 2

The output beam ν1′ is sent to an eight-port homodyne detector, which consists of a balanced beamsplitter followed by two balanced homodyne detectors. This detector effectively measures the operator λ = ν1′ + ν2† , where ν2† is the creation operator of an auxiliary vacuum mode. After the measurement, the mode a ′ is displaced by the amount λ which is in practice √ achieved by mixing this beam with a strong coherent beam with amplitude λ/ 1 − T on a highly unbalanced beamsplitter with transmittance T ≈ 99%. The resulting displaced beam is effectively the amplified input, √ adisp = 2ain + ν2† . The cloning is finished by dividing the amplified beam into two parts with the help of another balanced beamsplitter, thereby preparing the two clones of the input coherent state. The fidelity observed in the experiment was about 65%, very close to the optimal value 2/3 ≈ 0.667.

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Fig. 14. Cloning of coherent states using only linear optics and homodyne detection. ν1 , ν2 and ν3 denote vacuum auxiliary input modes, D is a displacement proportional to the measurement λ. (After Andersen, Josse and Leuchs [2005].)

7.4. Gaussian distribution with finite width Up to now, we have assumed that the distribution of the coherent states that should be cloned is uniform over the entire phase space. However, this is clearly an idealization, since the mean energy of the input state would be infinite. A more realistic scenario, considered by Cochrane, Ralph and Dolinska [2004], is that the coherent states are drawn from a Gaussian distribution with width σ and centered on vacuum, so that the a priori probability that the cloned state is |α is given by 1 |α|2 P (α) = (7.11) exp − 2 . 2πσ 2 2σ This occurs for instance in quantum key distribution with coherent states (Grosshans, Van Assche, Wenger, Tualle-Brouri, Cerf and Grangier [2003]). If the width σ of the distribution (7.11) is finite, then we possess some information that can be explored in order to increase the average cloning fidelity. Also, the probability (7.11) is not invariant with respect to the displacements, so there is no reason to search for a covariant cloner. The fidelity of the cloner may depend on the input state, and the figure of merit that should be maximized is the average fidelity, F = P (α)F (α) d2 α. (7.12)

It turns out that the optimal finite-width symmetric 1 → 2 Gaussian cloning transformation is still amplification followed by beamsplitting on a balanced beamsplitter. However, the gain G depends on σ . After the amplification and

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Optical quantum cloning

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√ beamsplitting, the coherent amplitude in each mode is α G/2 and the mean number of chaotic photons in each mode is n¯ = (G − 1)/2. The fidelity of cloning a particular coherent state |α reads √

2 2(1 − G/2 )2 2 F (α) = (7.13) exp − |α| . G+1 G+1

After averaging over the Gaussian distribution (7.11) we arrive at the expression for the mean fidelity, F=

2

. √ 2G )

G + 1 + 2σ 2 (2 + G − 2

(7.14)

We have to find the maximum of F under the constraint G 1. It turns out that there √ are two different solutions depending on the value of σ . If σ 2 > σth2 = (1 + 2 )/2, then it is optimal to amplify the signal and the optimal gain is G=

8σ 4 . (1 + 2σ 2 )2

(7.15)

On the other hand, if σ 2 < σth2 then it is optimal to simply divide the input signal into two beams without any amplification, and G = 1. The resulting cloning fidelity is , 4σ 2 +2 , σ 2 σth2 , 2 F = 6σ +11 (7.16) √ , σ 2 < σth2 . 2 1+(3−2 2)σ

The average fidelity increases monotonically with decreasing width of the distribution (7.11), and in the limit σ → 0 we get F = 1, as expected. 7.5. Cloning of conjugate coherent states In Section 5.4 we discussed a cloning machine for a pair of orthogonal qubits. This device possesses a natural and very interesting continuous-variable analogue, namely, one can consider a cloning machine for coherent states |α whose input consists of N copies of the state |α and N ′ copies of the complex conjugate coherent state |α ∗ . This problem was analyzed in detail in an even more general setting by Cerf and Iblisdir [2001a]. Without any loss of generality, we can assume that a pair of arrays of beamsplitters is used to collect all into two modes, and the input state of the √ signal √ cloning machine thus reads | N α A | N ′ α ∗ B . The goal of cloning is to produce M copies of |α with minimum added noise. This could again be accomplished

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with the help of a non-degenerate parametric amplifier. While mode A represents the signal input similarly as before, mode B is sent to the idler input port of the amplifier. Assuming amplification with intensity√gain G, the output annihilation √ † . Note that operator of the signal mode is given by aout = Gain + G − 1bin † both terms ain and b in the above formula contribute to the total coherent signal in aout . If the cloning should be performed with unity gain, then G must satisfy √ √ √ √ √ M = G N + G − 1 N ′, (7.17) and we can easily determine G by solving the above quadratic equation. A careful analysis reveals that for certain values of N, N ′ and M the cloning with conjugate inputs could be more efficient than the standard cloning of coherent states. To be fair, we should compare the cloning fidelities for inputs consisting either of N +N ′ copies of |α or of N copies of |α and N ′ copies of |α ∗ . The advantage of dealing with complex conjugate inputs could be most easily illustrated in the limit of an infinite number of clones, M → ∞, where the optimal cloning becomes equivalent with optimal state estimation. It has been shown by Cerf and Iblisdir [2001b] that when possessing a single copy of |α |α ∗ we can estimate |α with fidelity Fc.c. = 54 , which is strictly higher than the estimation fidelity F = 32 corresponding to the input state |α ⊗2 . In the former case the optimal detection strategy is the nonlocal continuous-variable Bell measurement where the quadratures xA + xB and pA − pB are measured simultaneously. § 8. Conclusions The quantum no-cloning theorem is a crucial aspect of modern quantum mechanics and one of the cornerstones of quantum information theory. Besides its fundamental interest for the foundations of quantum physics, the impossibility of exactly copying an unknown quantum state is crucial for the security of quantum key distribution protocols. Going beyond the no-cloning theorem, it is possible to design approximate quantum cloning machines, which enable the copying of quantum information in an optimal – albeit imperfect – way, an issue which has attracted considerable attention over the last decade. This review aims at providing an exhaustive overview of the various quantum cloning machines that have been introduced since the concept was put forward by Bužek and Hillery [1996]. The mathematical description of quantum cloning machines based on the isomorphism between maps and operators is developed in detail. Special attention is also devoted to the experimental optical implementations of these machines. The cloning of single photons has now been accomplished by

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several groups, and these experiments represent a very valuable contribution to the toolbox of available optical methods for quantum information processing. In the course of years, quantum cloning has grown into a genuine subfield of quantum information sciences, which is still currently very active on both the theoretical and experimental side. The advanced methods of preparation, manipulation, and measurement of quantum states of light, whose development has been stimulated to a large extent by the perspectives of quantum information processing, have recently enabled the demonstration of even more complex cloning machines. In the years to come, we anticipate many new achievements and breakthroughs in quantum information sciences, and there is no doubt that quantum cloning will play an important role in these future developments.

Acknowledgements We acknowledge financial support from the EU under projects COVAQIAL (FP6511004) and QAP (FP6-IP-015848), from the Communauté Française de Belgique under grant ARC 00/05-251, and from the IUAP programme of the Belgian government under grant V-18. JF also acknowledges support under the Research projects Measurement and Information in Optics MSM 6198959213 and Center of Modern Optics LC06007, and from grant 202/05/0498 of the Grant Agency of Czech Republic.

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Author index for Volume 49 Arecchi, F.T., 136, 143 Arentoft, J., 259 Ariga, M., 269 Arimondo, E., 100, 105 Arjavalingam, G., 206 Arlt, M., 254 Arriaga, J., 186, 208, 276, 280, 281 Arvind, K., 330 Asakawa, K., 241, 252, 255, 259, 262 Asakura, T., 4, 54 Asano, T., 251, 257, 263–265, 269 Ashcroft, N.W., 186, 188 Aspar, B., 252 Aspelmeyer, M., 446 Asquini, M.L., 135, 136, 163 Astratov, V.N., 222, 224, 231 Atatüre, M., 251 Atkin, D.M., 208, 265, 266, 272 Atkinson, P., 420 Atwater, H.A., 226 Audenaert, K., 484 Augustin, M., 246 Avizonis, P.V., 92 Azizi, K., 257

A Abeyta, A., 202 Abraham, N.B., 163 Abram, I., 416 Abrams, D.S., 284 Acín, A., 399, 435, 475 Adleman, L.M., 386 Agarwal, G.S., 136, 139, 331 Agio, M., 231, 234, 236, 238, 254, 259–261 Agrawal, G.P., 90, 280 Ahopelto, J., 218 Airy, G.B., 11 Akahane, Y., 251, 263–265 Akimoto, O., 162 Akulshin, A.M., 101 Al-Saidi, I.A., 163 Albert, J.P., 210, 252, 286 Alerhand, O.L., 201, 211, 230, 260 Algul, H.G., 210 Allan, D.C., 258, 276 Alléaume, R., 405, 416, 417 Alleman, A., 241 Allen, L., 104, 319, 338 Alodzhants, A.P., 355, 356, 363, 364 Alperovich, V.I., 215 Alsing, P.M., 101 Ammouche, A., 228 Anand, S., 240, 244–246, 254, 259, 260, 262, 263 Andersen, U.L., 538, 539 Anderson, C.M., 215 Anderson, D., 26 Andre, A., 101, 150, 171 Andreani, L.C., 231, 234, 259, 260 Angel’skii, O.V., 364 Arakawa, Y., 231, 249, 251, 416, 417 Arakelyan, S.M., 355, 356, 363, 364 Arakhelyuk, A.D., 364 Aravind, P.K., 330 Ardehali, M., 398, 443

B Baba, T., 193, 229, 241, 249, 251, 252, 255, 267, 269, 270 Badolato, A., 251 Baek, J.-H., 255 Baets, R., 184, 213, 241, 258, 269, 286 Baier, M.H., 416 Bajoni, D., 259 Balakirev, V.G., 217 Balakrishnan, R.A., 330 Baldi, P., 418 Balic, V., 102, 134 Bao, X.-H., 410 Bao, Z., 252 547

548

Author index for Volume 49

Barakat, R., 327, 329, 334, 335, 348, 352, 362 Bardinal, V., 243, 249, 266 Barnes, W.L., 194, 227, 270, 318 Barnett, S.M., 478 Barra, A., 229 Barreiro, S., 101 Barrow, J.D., 364 Bartolo, R.E., 254 Bass, L., 364 Battacharya, P., 220 Baughman, R.H., 223 Baumberg, J.J., 231, 245, 270 Bayindir, M., 260, 262 Baylis, W.E., 319 Beattie, N.S., 416 Becher, C., 250 Bechmann-Pasquinucci, H., 398 Beckx, S., 241 Bednik, R., 501 Behroozi, C.H., 100, 113, 156 Bell, J.S., 401 Bell, P.M., 211 Belland, P., 4, 10 Bellessa, J., 227 Belotti, M., 259 Ben-Or, M., 431 Benisty, H., 181, 192–194, 199, 209, 212–214, 235–237, 243–246, 248, 249, 251, 252, 254, 255, 257, 261–266, 268–270 Bennett, C.H., 393, 397, 400, 402–404, 406, 424, 425, 427, 438, 440, 519, 525 Bennink, R.S., 102 Benyattou, T., 221, 245, 265 Béraud, A., 237, 244, 251 Berenger, J.P., 210 Berezin, A.V., 328, 364 Berger, V., 181, 194, 199, 220, 255, 284 Berggren, M., 252 Berglund, A., 410 Bergman, J., 318 Bergmann, K., 106 Berry, M.V., 330, 331 Bert, N.A., 222 Bertho, D., 201, 210, 229 Bessette, F., 403, 425, 427, 440 Bethe, H.A., 227 Bethune, D., 408 Betzig, E., 366

Beveratos, A., 405, 415, 416 Bewley, W.W., 254 Bhandari, R., 319, 330 Bhat, R., 269 Bi, W.G., 252 Bicout, D., 329, 347, 349 Biedenharn, L.C., 345, 360 Bienstman, P., 184, 213, 241, 258, 259, 269, 286 Biham, E., 389 Birks, T.A., 186, 208, 209, 265, 266, 272, 276 Birner, A., 238 Bishop, R., 330 Biswas, R., 202, 204, 219, 260, 262 Bjarklev, A., 236 Björk, G., 193, 194, 270, 416 Blakesley, J.C., 420 Blanco, A., 215, 218 Bloch, J., 193, 416 Bloemer, M.J., 184, 252 Blom, H., 416 Blondeau, R., 221 Blondelle, J., 269 Bloom, D.M., 219 Bloore, F.J., 335, 336, 338, 350, 360 Blum, K., 333, 334 Bluman, G.W., 346 Blumenthal, D.J., 257 Bo, X.-Z., 218 Boca, A., 101, 171 Bockstaele, R., 184, 213, 269 Bogaerts, W., 241, 286, 287 Bogomolov, V.N., 215, 217, 222 Böhm, H.R., 410 Bohn, J., 372, 373 Boland, B.F., 284 Boller, K.-J., 99 Boltasseva, A., 259 Bonenfant, J., 319 Bonifacio, R., 137 Bonnand, C., 227 Boozer, A.D., 101, 171 Borel, P.I., 261, 262, 287 Borghs, G., 269 Born, M., 5, 15, 35, 194, 333, 338, 349, 356 Boroditsky, M., 269 Boschi, D., 519 Boscolo, S., 227, 258, 261 Bostak, J., 219

Author index for Volume 49 Botten, L.C., 225 Böttger, G., 246 Bouadma, N., 255, 259 Bourennane, M., 408, 472, 501 Bouwmeester, D., 507, 509, 514, 516, 519 Bovino, F., 504, 507 Bowden, C.M., 184, 252 Bowen, W.P., 101, 171, 366 Boyd, R.W., 100, 102, 118, 126 Boyd, S., 484 Bozhevolnyi, S.I., 227, 287 Braje, D.A., 102, 134 Brambilla, M., 135, 136, 163 Branca, S., 519 Branciard, C., 399 Brand, S., 242 Brassard, G., 393, 400, 402–404, 424–427, 440, 441, 519, 525 Braunstein, S.L., 466, 469, 477, 478, 499, 533, 535, 537 Bréguet, J., 403 Bremond, G., 245 Brendel, J., 412, 413 Breyta, G., 387 Briegel, H.-J., 424 Brillat, T., 228 Brillouin, L., 180 Bristow, A.D., 284 Broeng, J., 236, 276 Brommer, K.D., 201, 202, 206, 211 Broner, F., 149–152 Brorson, S.D., 191, 193 Brosseau, C., 319, 321, 327, 329, 333–335, 339, 344, 345, 347–349, 362 Brouri, R., 405, 415, 416 Brouri-Tualle, R., 405, 416 Brown, A., 101, 102, 136, 140–143, 153–155, 171 Brown, E.R., 216 Bru-Chevallier, C., 245 Brunel, C., 417 Brunner, N., 408, 409 Bruss, D., 398, 401, 462, 464, 466, 467, 469, 497, 499, 500, 519, 525, 526, 537 Buck, A.L., 3, 12 Budker, D., 100, 122 Bur, J., 219 Burger, M., 229, 283, 284 Burkalov, A.V., 364 Burkett, W.H., 100, 102, 130, 132, 133, 170

549

Burkovets, D.N., 364 Burman, R.R., 364 Burstein, E., 193 Burt, M.G., 243, 244 Buscemi, F., 480 Busch, K., 181, 193, 199, 215, 218, 221, 238, 239, 283 Butko, V.Y., 215 Buttler, W.T., 405 Bužek, V., 393, 460–464, 466–469, 474, 496, 499, 501, 503, 504, 541 Byrne, J., 371 C Cachin, C., 433 Cada, M., 225 Cai, M., 195 Calderbank, A.R., 437 Caldwell, R.L., 321 Calsamiglia, J., 519 Campbell, C., 93 Campbell, J.P., 3 Campbell, M., 215, 220 Campillo, A.J., 195 Caneau, C., 269 Cao, Q., 214, 287 Carcenac, F., 238 Cardimona, D.A., 101 Cardoso, G.C., 101 Carlin, J.-F., 269 Carlin, J.F., 269 Carlsson, N., 241 Carmichael, H.J., 101, 136, 163 Carminati, R., 228 Carozzi, T., 318 Cartan, E., 338 Carter, J.L., 429 Carter, W.H., 3 Case, K.M., 364 Cassagne, D., 201, 210, 229, 235–237, 243, 244, 246, 251, 252, 266, 286 Cassan, E., 268 Cavanna, A., 416 Cecchi, S., 136, 143 Cerf, N.J., 464, 466–478, 480, 486, 490, 491, 493, 496, 501–503, 522, 526, 528, 529, 534, 535, 537, 539–541 Chai, Y.J., 263 Chan, C.H., 182

550

Author index for Volume 49

Chan, C.T., 186, 189, 201, 202, 208, 210, 215, 225 Chan, T., 215, 221 Chan, Y.S., 215 Chance, B., 366 Chanclou, P., 10 Chandouineau, J.-P., 245, 254 Chang, H., 102, 123, 148, 156 Chang, R.K., 195 Charlton, M.D.B., 231, 245, 270 Chau, H.F., 398, 438, 443 Chefles, A., 478 Chekhova, M.V., 364 Chelnokov, A., 220, 221, 228, 229, 238, 270 Chen, C., 199, 268 Chen, H., 214, 250 Chen, H.X., 100 Chen, J.-L., 473 Chen, J.C., 236, 240, 260 Chen, K., 399, 400, 444 Chen, Y., 220, 228, 259 Chen, Y.-A., 522 Chen, Y.L., 220, 221 Cheng, C.C., 252 Cheroske, A.G., 321 Chigrin, D.N., 199 Chilwell, J.T., 143 Chin, M., 191 Chiou, T.-H., 321 Chiribella, G., 469, 470, 491, 493 Choi, M.-D., 480 Chomski, E., 215, 218 Chong, H.M.H., 261 Chou, C.W., 101, 171 Chow, E., 241, 260, 261 Chowdury, A., 284 Christ, A., 227 Christandl, M., 435 Chu, D.Y., 191, 252 Chuang, I.L., 338, 387, 400 Chulkova, G., 421 Chung, C.S., 3, 15 Chutinan, A., 219, 223, 249, 252, 256, 261, 265 Cinchetti, M., 469, 525, 526 Cirac, J.I., 171, 424 Clauser, J.F., 401 Cloude, S., 371 Coccioli, R., 269 Cochrane, P.T., 477, 539

Cohen-Tannoudji, C., 105 Coldren, L.A., 184, 195 Cole, J.D., 346 Collins, D., 504 Colson, V., 254 Conti, C., 227, 258, 261 Cooper, K., 416 Corbalan, R., 101 Cormack, I.G., 280 Cornwell, J.F., 368 Corzine, S.W., 184, 195 Costard, E., 194, 195, 220, 251, 285, 417 Courjon, D., 366 Courty, J.-M., 417 Courty, J.M., 163, 165, 167 Coutaz, J.L., 221 Cover, T.C., 433 Cowan, A.R., 284 Cox, S., 245 Craford, M.G., 269 Cregan, R.F., 276 Crenn, J.P., 4, 10 Crépeau (Crepeau), C., 400, 425, 427, 519 Crewett, J., 227 Cronin, T.W., 321 Cronin-Golomb, M., 102, 130 Crozat, P., 229 Csiszár, I., 432, 433 Ctyroky, J., 237 Çubusku, E., 335, 348 Cuisin, C., 220, 252, 254, 257, 258, 265 Culshaw, I.S., 213, 231, 233 Currie, M., 421 Curty, M., 435, 436 D Dadoloff, R., 330 Dagenais, M., 147, 148 Dahmen, C., 227 Daido, H., 162 Dale, Y., 416 Dalitz, R.H., 351 Damgård, I., 408 Dandliker, R., 362, 363 Danglot, J., 228 Dantec, N.L., 221 Dapkus, P.D., 254 Darden, S.E., 371, 374 D’Ariano, G.M., 464, 469–472, 480, 491, 493, 525, 526, 528, 538

Author index for Volume 49 Datta, S., 186, 189, 208 Davanço, M., 257 David, A., 212, 236, 269, 270 David, K., 184 David, S., 270 de Bougrenet de la Tocnaye, J.-L., 10 de Broglie, L., 364 de Dood, M., 214 de Dood, M.J.A., 514, 516 De La Rue, R.M., 182, 231, 235, 242–244, 246, 249, 255, 257, 264, 265, 269 de Lustrac, A., 228 De Martini, F., 504, 507, 508, 510, 514, 516, 518, 519, 532, 538 De Micheli, M., 418 De Moor, B., 484 De Neve, H., 192, 268, 269 de Nicola, S., 26 De Riedmatten, H., 418 de Riedmatten, H., 504, 519 Deavours, C.A., 385 Decanini, D., 220 Delbeke, D., 213, 269 Demeester, P., 269 Dementev, A.S., 3 DenBaars, S.P., 236, 269, 270 Denison, D.R., 215 Denning, R.G., 215, 220 Dennis, M.R., 318, 331, 351, 361 Deppe, D., 250 Deppe, D.G., 184, 227, 249 Derbyshire, J., 319 Dereux, A., 318 Derkacs, D., 405 Derouin, E., 252, 257, 265 Deschamps, G., 338 Deserno, R., 136, 143 DeShazer, L.G., 3 Désières, Y., 265 Deutsch, M., 171, 215, 222, 223 Devaux, E., 227 Devenyi, A., 230, 240, 260 Diamanti, E., 420 Diana, F.S., 236, 270 Dickson, L.D., 3 Dieks, D., 458, 463 Diffie, W., 385 Dintinger, J., 227 Dionne, J.A., 226 DiVincenzo, D.P., 438, 462, 466, 500

551

Djoudi, A., 245 Dodabalapur, A., 252, 262 Dogariu, A., 100, 318, 331, 355, 356, 358, 359, 363, 364, 374–376 Döhler, G.H., 269 Dolinska, A., 477, 539 Doll, T., 258 Domarkene, D.P., 3 Domhan, M., 416 Donoghue, J., 102, 130 Dowling, J.P., 184, 252 Dreiser, J., 251 Drexhage, K.H., 227 Driel, H.M.V., 217 Drisse, O., 252, 254, 257, 258, 265 Du, J., 526 Du, Y., 102, 123, 156 Duan, G.-H., 236, 243, 245, 252, 254, 257, 260, 265 Duan, G.H., 258 Duan, L.-M., 101, 171 Duan, L.M., 478 Duan, X.M., 221 Duguay, M.A., 191, 277 Dumeige, Y., 405, 416 Dunbar, L.A., 258, 267 Dunn, M.H., 102, 122 Dür, W., 424 Durrant, A.V., 100 D’Urso, B., 236, 242 Durt, T., 470, 471, 473, 480, 526, 529 Dušek (Dusek), M., 407, 429, 439–441, 462, 502 Dutta, B., 269 Dutton, R.W., 286 Dutton, Z., 100, 113, 156 Duvillaret, L., 221 E Ebbesen, T.W., 227, 318 Eberly, J.H., 104, 319, 338 Echenique, P.M., 215 Economou, E.N., 186, 210 Egen, M., 218 Eggleston, H.G., 335, 336, 338 Eibl, M., 519 Eich, M., 246 Einstein, A., 401, 458 Eisaman, M.D., 101, 171

552

Author index for Volume 49

Ekert, A., 400, 402, 425, 432, 462, 466, 497, 500, 537 Ekert, A.K., 411 El Melhaoui, L., 268 El-Kady, I., 228 Ell, C., 249 Ellis, J., 318, 331, 355, 356, 358, 359, 363, 364, 374–376 Engelen, J.P., 287 Englund, D., 251 Enoch, S., 213, 214, 267 Eradat, N., 223 Erchak, A.A., 270 Eriksson, N., 270 Erland, J., 227, 236 Ermolenko, S.B., 364 Erni, D., 262, 268 Etrich, C., 246 F Fabre, C., 165 Fabre, S., 245 Fainman, Y., 317, 432 Fan, H., 467, 472, 473, 480, 498, 508, 512, 513, 526–528 Fan, S., 181, 199, 204, 219, 231, 236, 240, 241, 252, 263, 265, 269, 270, 284, 286 Fan, W.H., 284 Fano, L., 326 Fano, U., 104, 326, 334, 351, 371 Farrer, I., 420 Fasel, S., 513 Fattal, D., 251, 417 Fedorov, F.I., 327 Fedrizzi, A., 410 Fehrembach, A.-L., 213 Feldmann, J., 227, 366 Felix, C.L., 254 Felson, L.B., 90 Feng, F.-Y., 410 Feng, X.-P., 231 Ferrand, P., 218 Ferrera, J., 240 Ferrini, R., 236, 237, 243–246, 254, 258, 260, 261, 267 Feynman, R.P., 338 Fideli, J.M., 268 Field, J.E., 99, 100, 106, 116, 117 Filip, R., 475, 493, 502, 519, 520

Fink, Y., 199 Firth, W.J., 163 Fiurášek, J., 473–475, 477, 479–481, 493, 500, 502–504, 522, 526, 529, 535 Fleischhauer, M., 100–102 Fleming, J.G., 219, 228 Fleury, L., 417 Fokin, A.V., 215 Föll, H., 181, 193, 199, 215, 221, 238, 239, 283 Forchel, A., 221, 244–246, 249, 251, 254, 259–263 Ford, G.W., 194, 227 Foresi, J.S., 240 Forrest, S.R., 259 Fourier, J., 323 Fox, A.M., 284 Frandsen, L., 259, 261, 262, 287 Frandsen, L.H., 261 Franson, J.D., 404, 411 Fraser, G.W., 422 Friberg, A.T., 90, 318, 331, 352, 354, 357, 363 Fry, E.S., 100, 112, 122 Fuchs, C., 398, 432 Fuchs, C.A., 462, 466, 500, 525 Fuchs, H.J., 246 Fujii, T., 269 Fujita, M., 269 Fukaya, N., 241, 249, 255 Fukumitsu, O., 90 Fulton, D.J., 102, 122 Fushman, I., 251 G Gaborit, F., 254 Gacoin, T., 405, 415, 416 Gadot, F., 228, 229 Gaebel, T., 416 Gailhanou, M., 184 Galindo, A., 369 Galisteo-Lopez, J.F., 225 Gallart, M., 416 Gallet, J.-F., 268 Galli, M., 259 Gaponenko, S.V., 199, 215 Garcia, N., 214, 228, 257, 318, 366 Garcia de Abajo, F.J., 215 García-Santamaría, F., 217 Gardner, M., 386

Author index for Volume 49 Garoche, P., 221 Gaskill, J., 4, 10, 72 Gates, B., 217, 218 Gauthier, D.J., 100 Gauthier-Lafaye, O., 220 Gautier, J.-D., 408 Gautier, J.D., 412 Gavrielides, A., 99 Gaylord, T.K., 213 Gayral, B., 194, 195, 251, 285, 417 Gea-Banacloche, J., 99, 100, 102, 107–115, 125, 128, 163, 170 Gell-Mann, M., 370 Genack, A.Z., 194, 214 Gendry, M., 245 Gerace, D., 231, 259 Gérard, J.-M., 181, 194, 195, 199, 229, 237, 251, 255, 285, 417 Gérard, J.M., 193, 194, 229, 270, 416 Gerardot, B.D., 250 Geremia, J.M., 250 Gersen, H., 287 Ghaderi, A.H., 473, 528 Ghaemi, H.F., 227 Ghirardi, G.C., 457 Ghosh, S., 464 Giacobino, E., 143, 163, 165, 167 Giapis, K.P., 215 Gibbs, H.M., 135, 136, 138, 142, 146, 148, 149, 249 Gibson, F., 408 Giessen, H., 227 Gippius, A., 227 Gippius, N.A., 213, 231, 233 Gisin, N., 398, 399, 403, 404, 407–409, 412, 413, 418, 430, 432, 435, 463, 464, 466, 470–473, 475, 496, 498, 500, 501, 503, 504, 513, 519, 522, 525, 526 Giusfredi, G., 136, 143 Gmitter, T.J., 202, 206, 269 Gobby, C., 407 Goda, S., 134 Gogna, P., 254 Goldhaber, A.S., 364 Goldsztein, G., 150 Goldsztein, G.H., 149–152 Golka, S., 265 Goltsman, G.N., 421 Goobar, E., 416 Good Jr., R.H., 360

553

Goodman, J., 5, 15 Goorskey, D., 102, 118, 119, 121, 122, 140, 142, 153, 155–159, 163, 170, 171 Goorskey, D.J., 100, 102, 103, 118, 121, 123, 136, 140, 141, 163, 165, 167, 170, 171 Gorman, P.M., 405 Gösele, U., 231, 238 Gossard, A.C., 146, 148 Gottesman, D., 400, 438, 443, 445 Grabtchak, S., 215, 218 Gralak, B., 214, 267 Grangier, P., 405, 415, 416, 534, 535, 539 Grant, R.S., 243, 244 Gray, G., 149–152 Gray, H.R., 105 Greffet, J.-J., 228 Griesebock, B., 218 Griffiths, D., 371 Griffiths, R.B., 398, 432, 467, 468, 479, 496, 501, 525, 526 Grillet, C., 242, 256, 258, 265 Gripp, J., 136 Grosshans, F., 534, 535, 539 Gruber, A., 416 Grüning, U., 221, 238 Grynberg, G., 127 Gu, B.-Y., 215 Gühne, O., 436 Guillot, G., 221, 245 Guinnard, O., 408 Guizzetti, G., 259 Guo, G.C., 478, 518 Gurlebeck, K., 369 Guy, R.K., 386 Gyftopoulos, E.P., 335, 348 H Haake, F., 150 Haderka, O., 407, 415, 429, 441 Hafner, C., 262, 268 Hagberg, M., 270 Hagelstein, P.L., 215 Hager, J., 146 Hahn, K.H., 99, 100 Haidinger, W., 321 Hajnal, J.V., 330, 331 Hakuta, K., 99, 101 Halder, M., 405 Halevi, P., 186, 206, 208 Ham, B.S., 161

554

Author index for Volume 49

Hamano, T., 193 Hamar, M., 415 Hamilton, M.W., 143 Hammer, C.L., 360 Han, D., 371 Han, I.-Y., 255 Happ, T., 254, 260 Happ, T.D., 251, 254, 259 Hardy, L., 464, 519 Harman, P.M., 323 Haroche, S., 193, 194 Harpoth, A., 261, 262 Harris, A., 231 Harris, S.E., 99, 100, 102, 106, 108, 113, 116, 117, 126, 132, 134, 153, 156, 159–161, 170, 171 Harris, T., 366 Harrison, R.G., 163 Harrisson, M.T., 215, 220 Harshawardhan, H., 136, 139 Hashimoto, T., 497 Hatanaka, T., 407 Hau, L.V., 100, 113, 156 Hauchi, K., 26 Haus, H.A., 236, 241, 265 Haus, J.W., 201–203 Hayasaka, K., 417 Hayashi, A., 497 He, S., 210 Hecht, B., 417 Hegarty, J., 244 Heidrich, H., 265 Heitmann, H., 193 Hellman, M.E., 385 Hellworth, R.W., 338 Hemmer, P.R., 102, 130, 161 Hendrickson, J., 249 Hendrych, M., 407, 429, 441, 462, 502 Hening, A., 408 Hennessy, K., 251 Hennrich, M., 417 Henry, C., 252 Henry, C.H., 184 Hensel, H.J., 265 Herbert, N., 457 Heremans, P., 269 Herloski, R., 87 Herman, R.M., 88 Hermann, C., 231 Hernandez, G., 102, 132, 133, 135

Herzog, T., 407 Heslop-Harrison, J.S., 422 Hess, O., 231 Hestenes, D., 328, 369 Hetherington, D.L., 219 Heyman, E., 90 Hietala, V., 260 Hillery, M., 393, 460–464, 466–469, 474, 496, 499, 501, 503, 504, 541 Hiloco, L., 165 Hirose, S., 416, 417 Ho, K.M., 186, 189, 201, 202, 204, 208, 210, 219, 225, 260, 262 Ho, S.T., 191, 252 Hofmann, C., 249 Hogue, H., 420 Hohl, A., 150 Holden, A.J., 228 Holland, A.D., 422 Hollberg, L., 100, 101, 112, 122 Holleitner, A.W., 227 Hollinger, G., 245 Holmes, D.A., 92 Holt, R.A., 401 Homer, 383 Hönerlage, B., 224 Hong, C.-S., 189 Hong, C.K., 414, 514 Hong, T., 126 Honjo, T., 420 Hopkins, H.H., 3, 15 Hopkins, S.A., 100 Horibe, M., 497 Horne, M.A., 401, 425 Horng, J.S., 18, 25 Horodecki, K., 435, 438 Horodecki, M., 431, 435, 436, 438 Horodecki, P., 435, 436, 438 Horodecki, R., 436 Hou, H., 241 Houdré (Houdre), R., 181, 184, 205, 209, 235–237, 243–246, 248, 249, 254, 255, 257, 258, 260–267, 269, 270 Hours, J., 193, 416 Howell, J.C., 507, 509 Hradil, Z., 415 Hsü, H.-Y., 328 Hu, E., 236, 250, 251, 270 Hu, E.L., 257 Huang, H., 214, 224

Author index for Volume 49 Huang, J., 223 Huang, Y.F., 518 Huang, Z., 227 Hubert, S., 245 Hübner, U., 246 Hughes, R., 407 Hughes, R.J., 405 Hughes, S., 194, 209, 210, 260, 286 Hugonin, J.-P., 214, 237, 251, 258, 263, 287 Huh, J., 251, 255 Husain, A., 254 Huschilt, J., 319 Huttner, B., 407, 408, 432 Hvam, J.M., 227 Hwang, J.-K., 251, 255 Hwang, W.-Y., 399, 444 I Ibanescu, M., 215, 259 Ibisate, M., 215, 218 Iblisdir, S., 473, 475–478, 522, 534, 535, 537, 540, 541 Iga, K., 193, 269 Ikeda, K., 162 Ikeda, N., 241, 252, 255, 259, 262 Ilegems, M., 184, 205, 249 Iliew, R., 246 Ilves, H., 404 Imada, M., 219, 249, 252, 265 Imai, H., 473, 508, 512, 513, 526, 528 Imamoglu, A., 99–101, 106, 116, 117, 123, 125, 126, 156, 159, 160, 170, 171, 250, 251 Imhof, A., 217, 222 Imoto, N., 101, 438, 440, 442 Inamori, H., 440, 443 Inguva, R., 201 Inoshita, K., 251, 252, 269, 270 Inoue, K., 241, 252, 262, 417, 420 Ipe, A., 476, 477, 534, 535 Ippen, E.P., 193, 219, 240, 270 Irman, A., 215, 224, 225 Irvine, W.T.M., 514, 516 Ishida, K., 252, 255, 259, 262 Ishihara, T., 213, 231, 233 Ishikawa, A., 249 Ishino, N., 220 Iwai, T., 241 Izraël, A., 229

555

J Jaakkola, T., 90 Jack, M.W., 126 Jacobs, B.C., 404 Jacobson, N., 334, 368 Jacquinot, P., 3 Jahma, M., 439, 440 Jain, M., 102 Jalaguier, E., 252 Jamiolkowski, A., 480 Jamois, C., 231 Jancewicz, B., 328, 369 Jang, D.H., 255 Janiak, K., 265 Jaskorzynska, B., 237, 244, 245, 254, 257, 261–263 Jauch, J.M., 364, 371 Jelezko, F., 416 Jennewein, T., 410, 446 Jensen, J.S., 261, 262 Jensen, S.M., 153 Jeppesen, S., 416 Jewell, J.L., 146, 148 Jiang, D.Y., 4 Jiang, W., 284 Jiang, Y.K., 518 Jin, S., 111 Jin, S.-Z., 99, 100, 102, 107–115, 125, 128, 163, 170 Jin, X.-M., 410 Joannopoulos, J.D., 181, 199, 202, 204, 206, 208, 215, 219, 230, 231, 236, 240, 241, 250, 252, 259–262, 265, 269, 270, 284, 286 John, S., 181, 201, 214, 215, 218, 221, 224, 238 Johnson, N.P., 215 Johnson, P.M., 225 Johnson, S.G., 208, 215, 219, 231, 241, 250, 259, 261 Johnston Jr., T.F., 90 Jones, R.C., 327 Jonsson, P., 194, 270, 408, 416, 417 Joshi, A., 101–103, 136, 140–155, 164–169, 171 Josse, V., 538, 539 Jouanin, C., 201, 210, 229, 236, 237, 243, 244, 246, 266 Joulain, K., 228

556

Author index for Volume 49

Jozsa, R., 400, 425, 519 Ju, Y.-G., 255 Jun-Zhang, A., 410 Jung, P., 149–152 K Kadyshevitch, A., 214 Kafesaki, M., 236, 254, 257, 261 Kahn, D., 383 Kaivola, M., 318, 331, 352, 354, 357, 363 Kaliteevski, M.A.J.M., 286 Kalosha, I.I., 215 Kamp, M., 221, 244, 246, 251, 254, 259–263 Kaneko, K., 221 Kang, H., 102, 124, 125, 132, 133, 135, 170 Kanzaki, O., 25 Kapenga, J., 210 Kaplyanskii, A.A., 222 Kapon, E., 416 Kar, G., 464 Karabutova, O.A., 364 Karalis, A., 250 Karasev, P., 364 Kardynał (Kardynal), B.E., 416, 420 Karimipour, V., 526 Karimov, O.Z., 222 Karle, T., 260 Karle, T.J., 263, 287 Karlsson, A., 254, 257, 261, 262, 408, 472, 501, 522 Karlsson, R., 318 Karpov, E., 474 Kash, K., 230, 260 Kash, M.M., 100, 112, 122 Kastler, A., 268 Kaszlikowski, D., 473 Kathuria, Y.P., 92 Katz, A., 328 Katz, D.P., 102, 130 Katz, J., 91 Kawai, N., 241 Kawakami, S., 220, 267 Kawakami, T., 193 Kawashima, T., 267 Kawata, S., 221 Kazarinov, R., 184 Keldysh, L.V., 249 Keller, J.M., 360 Keller, M., 417

Keller, O., 366 Kelly, P.J., 335, 336, 338 Kemhadjian, H., 245 Kempe, J., 508, 512 Kessler, M., 246 Keyl, M., 467, 483, 496, 500 Khanin, Y.I., 99 Khazhinsky, M., 210 Khitrova, G., 249 Kiesel, P., 269 Kim, I., 254 Kim, J., 420 Kim, J.-S., 251, 255 Kim, K., 327 Kim, S.-B., 255 Kim, S.-H., 251, 255 Kim, Y.S., 371 Kimball, D.F., 100, 122 Kimble, H.J., 101, 135, 136, 162, 171 Kimmerling, L.C., 240 Kimura, T., 407 Kippenberg, T.J., 195 Kira, G., 265, 285 Kitaev, V., 215 Kitano, M., 136 Kitson, S.C., 227, 417 Kittel, C., 186, 188 Kitzke, B., 242 Kivshar, Y.S., 284 Klar, T., 227 Klar, T.A., 366 Kleppner, D., 192, 257 Kley, E.-B., 246 Klopf, F., 221 Klyshko, D.N., 364 Knight, J.C., 271, 276, 280, 281 Knight, P.L., 468, 474, 496 Knobloch, A., 269 Knudsen, L.R., 389 Koashi, M., 438, 440, 442, 445 Kobayashi, H., 219 Kobayashi, K., 366 Kobayashi, T., 318 Koch, T.L., 191, 277 Kocharovskaya, O.A., 99 Kochman, B., 220 Koenderink, A.F., 215, 217, 218, 223, 225 Koenig, U., 330 Kofman, A.G., 214 Kogelnik, H., 88, 181, 184

Author index for Volume 49 Kokubun, Y., 191, 277 Kolodziejski, L.A., 270 Kolokolov, I.V., 366 Korka, J.E., 92 Korneev, A., 421 Körner, J., 432, 433 Korolkova, N., 366 Korterik, J.P., 287 Kosaka, H., 267, 407 Kosaka, H.T.K., 267 Kostelak, R., 366 Kotlyar, M.V., 260 Kouminov, P., 421 Koyama, F., 193, 269 Kozlov, G.C., 319, 345 Kraus, B., 399 Krauss, T., 262, 263 Krauss, T.F., 182, 209, 231, 235–237, 242–244, 246, 248, 249, 251, 254, 255, 257, 260, 261, 263–266, 268–270, 284, 287 Kreibig, U., 318 Kristensen, M., 236, 259, 261, 262, 287 Krokhin, A.A., 186, 208 Kronig, R. de L., 185 Krüeger, O., 478, 534, 537 Kruh, L., 385 Kuchinsky, S., 258 Kuech, T.F., 284 Kuhl, J., 227 Kuhn, A., 417 Kuhn, S., 249 Kuipers, L., 287 Kulakovskii, V.D., 249, 251 Kulik, S.P., 364 Kumar, P., 102, 130 Kumei, S., 346 Kumzerov, Y.A., 217 Kunz, K.S., 210 Kuramochi, E., 265, 285 Kurizki, G., 194, 214 Kurland, I., 260 Kurmann, J., 136 Kurochkin, Y.A., 328, 364 Kuroki, Y., 251, 270 Kurtsiefer, C., 405, 410, 415, 419 Kuttner, P., 93 Kutz, S.R., 219 Kuzmiak, V., 225 Kuzmich, A., 100, 101, 171

557

Kwek, L.C., 473 Kwiat, P., 410 Kwiat, P.G., 405 Kwon, S.H., 251, 255 L Labilloy, D., 209, 235, 237, 243, 244, 246, 248, 249, 266 Ladan, F.R., 229 Lagendijk, A., 193, 215, 218, 222, 223, 225 Lakin, W., 371 Lalanne, P., 212–214, 236, 237, 240, 243, 251, 258, 259, 263, 287 Lam, P.K., 366 Lamas-Linares, A., 507, 509, 514, 516 Lambrecht, A., 163, 165, 167 Lamoreaux, S.K., 405 Lamoureux, L.-P., 473, 474, 502, 528 Lange, B., 417 Lange, W., 136, 143, 417 Larsson, A., 270 Laruelle, F., 416 Laval, S., 268 Lavrinenko, A.V., 199 Lawandy, N.M., 224 Lax, M., 90 Le Floch, V., 417 Le Gouezigou, L., 255, 259 Le Vassor d’Yerville, M., 252 Leary, T.P., 384 Lecamp, G., 251 Leclercq, J.L., 221 Lederer, F., 246 Lee, H., 254 Lee, P.T., 254 Lee, R., 252 Lee, R.K., 193, 242, 254, 258, 262 Lee, Y.-H., 251, 255 Lee, Y.H., 251, 255 Lee, Y.J., 251 Leeb, W., 446 Legouézigou, E., 252, 257 Legouézigou (Legouezigou), L., 245, 252, 254, 257, 265 Legouézigou (Legouezigou), O., 254, 265 Legrand, B., 194, 195, 251, 285, 417 Legre, M., 413 Lehmann, V., 221, 238 Leighton, R.B., 338 Lelarge, F., 254, 258

558

Author index for Volume 49

Lemaitre, A., 193 Lemarchand, F., 213 Lenz, G., 88 Leonard, S.W., 215, 218, 238 Leosson, K., 227 Letartre, X., 242, 252, 256, 258, 265 Leuchs, G., 366, 538, 539 Leuenberger, D., 244, 246, 254 Leung, D.W., 431 Leung, K.M., 199, 202, 210 Levenson, A., 284, 285 Lewenstein, M., 435, 436 Lewis, A., 366 Lezama, A., 101 Lezec, H.J., 227 Li, A.-P., 238 Li, C.F., 518 Li, F.L., 214, 224 Li, G.X., 214, 224 Li, L., 212, 213 Li, N., 410 Li, T., 88 Li, W.L., 518 Li, Y., 3, 15, 18, 23, 25, 26, 91, 92, 111 Li, Y.-Q., 99–102, 106–116, 125, 127–129, 163, 170 Li, Z.-Y., 215, 218 Lidorikis, E., 210, 219, 238, 259 Liguda, C., 246 Lin, C.C., 227 Lin, J., 126, 127 Lin, S.Y., 219, 228, 241, 260, 261 Lindberg, J., 318, 331, 352, 363 Lindblad, G., 476, 534 Linden, S., 227 Lindfors, K., 318, 331, 352, 357, 363 Lindle, J.R., 254 Ling, H.Y., 101 Lipatov, A., 421 Lippens, D., 228 Lisak, M., 26 Lita, A.E., 422 Liu, Y.F., 199, 202 Liu, Z.Y., 215 Ljunggren, D., 408 Lloyd-Lucas, F.D., 186, 208, 266 Lo, H.-K., 398–400, 408, 438, 443–445 Lo Presti, P., 470, 471, 528 Lobo, C.J., 416 Lodahl, P., 215, 224, 225

Löffler, A., 249 Lölkes, S., 181, 193, 199, 215, 221, 239, 283 Lombardet, B., 236, 243, 245, 254, 258, 260, 267 Loncar, M., 227, 251, 254, 258, 270 López (Lopez), C., 215, 217, 218 López-Tejeira, F., 217 Lorünser, T., 410 Louck, J.D., 345 Loudon, R., 366 Louisell, W.H., 90 Lounesto, P., 369 Lounis, B., 417 Lourtioz, J.-M., 181, 194, 199, 220, 221, 228, 229, 238, 255, 270 Lourtioz, J.M., 238 Lowenthal, D.D., 3, 4, 34, 66 Lu, B., 102, 130, 132, 133, 170 Lu, C.-Y., 522 Lu, X.H., 214, 224 Luebbers, R.J., 210 Lugiato, L.A., 135–138, 142, 149, 163 Luis, A., 358, 362, 364, 376 Lukin, M.D., 100–102, 112, 122, 146, 171 Luo, C., 215, 284 Luo, Z., 108 Lupton, J., 227 Lupu, A., 268 Lustrac, A.D., 229 Luterova, K., 224 Luther, G.G., 405, 407 Lütkenhaus (Lutkenhaus), N., 400, 414, 432, 433, 435, 436, 439–443, 445, 462, 502, 519 Luyssaert, B., 241, 258 Lyan, P., 268 M Ma, X., 399, 400, 408, 444 Mabuchi, H., 250, 251, 254 Macchiavello, C., 462, 464, 466, 467, 469, 472, 480, 497, 500, 525, 526, 528, 537 Macke, B., 163 Maddox, J., 202 Mahajan, V.N., 3–5, 10, 15, 18, 21, 25, 31, 34, 35, 40, 43, 47, 50, 54, 66, 70, 72, 83, 87, 92 Mahnkopf, S., 254, 260 Maier, M., 194 Maier, S.A., 226

Author index for Volume 49 Mair, A., 146 Majorana, E., 360 Makimoto, T., 88 Mallesh, K.S., 361 Mallesh, S., 330 Mandel, L., 327, 333, 414, 459, 514 Mandel, P., 149–152 Mangan, B.J., 276 Manin, L., 416 Manolatou, C., 241 Manuszak, D., 132 Maradudin, A.A., 201, 225, 226, 228 Marand, C., 407 Marangos, J.P., 100 Marcikic, I., 413, 504, 519 Marcuse, D., 255, 271 Maréchal, A., 35 Marino, A.M., 102 Markard, A., 254 Marmet, L., 99, 101 Marquier, F., 228 Marshall, J., 321 Marti, J., 258 Martin, L., 211 Martinelli, M., 407 Martinelli, R.U., 254 Martinez, A., 329, 349 Martinis, J.M., 421, 422 Martorell, J., 224 Martrou, D., 193 Maruyama, T., 262 März, R., 254 Marzin, J.Y., 229 Masalov, A.V., 364 Massar, S., 466, 473, 477, 496–498, 500, 522, 535 Masullo, L., 518 Matsko, A.B., 102 Matsumoto, K., 472, 473, 480, 498, 508, 512, 513, 526–528 Matsumoto, T., 267 Matsunami, A., 269 Matsuzaki, T., 229 Mattle, K., 519 Maurer, U., 428, 433, 434 Maurer, U.M., 427, 433 Maurhadt, O., 410 Maxwell, J.C., 323, 328 Mayer, A.A., 366 Mayer, S., 415, 419

559

Mayers, D., 409, 431, 437, 440, 442, 443 Maystre, D., 181, 194, 199, 214, 255 Mazilu, M., 268 McCaughan, L., 284 McGurn, A.R., 210, 228 McKnight, W.B., 90 McMahon, O.B., 216 McNab, S., 255 McPhedran, R.C., 225 Meade, R., 240 Meade, R.D., 181, 199, 201, 202, 204, 206, 211, 219, 230, 260 Megens, M., 193, 215, 217, 222, 223, 225 Meier, C., 236, 270 Meier, F., 319, 345 Meier, M., 252 Mekis, A., 252, 259, 260, 262 Menon, V.M., 259 Menyuk, C.R., 366 Mermin, N.D., 186, 188, 400, 402 Merriam, A.J., 102, 132 Meseguer, F., 215, 217, 218 Messin, G., 405, 416 Meyer, H.G., 246 Meyer, J.R., 254 Michel, E., 219 Michel, J., 199 Midrio, M., 227, 258, 261 Mielke, S.L., 136 Miguez, H., 215, 217, 218 Miller, A.J., 421, 422 Mingaleev, S.F., 284 Mirin, R.P., 417 Misner, C.W., 368 Mitschke, F., 136, 143 Mitsugi, S., 265, 285 Mitsunaga, M., 101 Miura, A., 220 Miyamoto, T., 269 Miyashita, T., 262 Miyazaki, H.T., 217 Miyazawa, T., 416, 417 Mizokami, H., 26 Mlynek, J., 136, 143 Mochizuki, M., 252 Modinos, A., 262 Moharam, M.G., 213 Moll, N., 255 Mompart, J., 101 Monat, C., 245, 252

560

Author index for Volume 49

Mondia, J.P., 215, 218, 284 Moosburger, J., 237, 243–246, 254, 261, 263 Mor, T., 441 Morales, J.A., 332, 340 Moreau, E., 416 Morgan, G., 407 Morgan, G.L., 405, 407 Morgan, M.C., 263 Morgenroth, W., 246 Moroder, T., 435 Moroz, A., 212, 225 Morris, G.M., 212, 213 Morthier, G., 184 Moseley, R.R., 102, 122 Motegi, A., 241 Mueller, H., 327 Mugnier, J., 227 Mukai, T., 126 Mukunda, N., 330, 371 Mulet, J.-P., 228 Muljarov, E.A., 213, 231, 233 Müller (Muller), A., 403, 404, 407, 408 Müller, F., 238 Müller, J., 366 Müller, M., 218 Mullin, C.J., 360 Mulot, M., 240, 244–246, 254, 259, 262, 263 Murad, S.K., 243, 244 Murakowski, J., 268 Murata, M., 252 Murray, W.A., 227 Musikhin, Y.G., 222 Mussi, V., 504, 507 Myška, M., 407, 429 N Nagler, B., 471 Naik, D., 410 Nakamura, H., 262 Nakamura, K., 407 Nakamura, M., 267 Nakamura, S., 236, 269, 270 Nakamura, Y., 252 Nakoaka, T., 251 Nalamasu, O., 252 Nam, S.W., 421, 422 Nambu, Y., 193, 407 Narayanan, T., 222 Narducci, L.M., 135, 136, 163

Nash, C., 368 Nau, D., 227 Navarro, E., 332, 340 Navez, P., 464, 474, 478, 502, 534, 537 Nayyar, V.P., 92 Nedeljkovic, D., 258 Ne’eman, Y., 370 Nemoto, S., 88, 91 Nesci, A., 362, 363 Netti, M.C., 231, 245, 270 Nevière, M., 212–214, 231 Nicorovici, N.A., 225 Nielsen, M.A., 400 Nielsen, M.N., 338 Nielsen, P.M., 408 Nieto, M.M., 364 Nieto-Vesperinas, M., 228, 318, 366 Nijboer, B.R.A., 35 Nikolaev, I.S., 215, 224, 225 Nikonov, D.E., 101 Nishioka, M., 249 Nityananda, R., 330 Niu, C.-S., 398, 432, 479, 525, 526 Niu, C.S., 467, 468, 496, 501 Noda, S., 219, 223, 249, 251, 252, 256, 257, 261, 263–265, 269 Nolte, S., 246 Nordholt, J.E., 405 Norris, D.J., 215, 218, 222, 223 Notomi, M., 252, 258, 259, 265, 267, 285 Notomi, N., 267 Nourrit, V., 10 Novikova, I., 150 Noz, M.E., 371 Nozaki, K., 251, 270 Nugent, K.A., 363 Numata, H., 201 Nye, J.F., 330, 349 O O’Brien, J., 236, 252 O’Brien, J.D., 254 Ochoa, D., 249 Oesterle, U., 184, 205, 209, 235, 243, 244, 246, 248, 249, 255, 257, 261–266, 269, 270 O’Faolain, L., 260 Ogawa, S., 219, 252 Ogawa, T., 136 Ohkouchi, S., 252

Author index for Volume 49 Ohtaka, K., 181, 201, 212, 217, 252, 284 Ohtera, Y., 220 Ohtsu, M., 366 Okano, M., 219, 252, 261 Okunev, O., 421 Olaofe, G.O., 3 Oliver, W.D., 420 Olivier, S., 235, 236, 243–245, 254, 255, 257, 261–265 Olson, J.V., 352 Olver, P.J., 346 O’Neill, E.L., 335 Oppenheim, J., 431, 435, 438 Orozco, L.A., 135, 136, 162 Orrit, M., 417 Ortigosa-Blanch, A., 276, 280, 281 Osher, S.J., 229, 283, 284 Ostrowsky, D.B., 418 Ottow, S., 238 Ou, Z.Y., 514 Overgaag, K., 215, 224, 225 Owens, P.C.M., 403, 411 Özbay, E., 202, 204, 205, 219, 260, 262 Ozin, G.A., 215, 218 P Padjen, R., 229 Painter, O., 195, 236, 243, 252 Painter, O.J., 242, 254 Palacios-Lidon, E., 218 Palamaru, M., 214 Palma, G.M., 411, 432 Pan, J.-W., 410, 522 Pan, J.W., 519 Pancharatnam, S., 330 Panev, N., 416 Pang, Y.K., 220, 221 Pardo, J., 88 Park, H.-G., 251, 255 Park, S., 191 Park, S.H., 217, 218 Parker, C.D., 216 Parker, G.J., 231, 245, 270 Parrent, G.B., 327 Patrini, M., 259 Pattanayak, D.N., 90 Pattantyus-Abraham, A.G., 215 Peacock, A., 422 Pearlmany, A., 421

561

Pearsall, T., 258 Pearsall, T.P., 258 Peev, M., 410 Pelant, I., 224 Pellat-Finet, P., 328 Pelliccia, D., 507, 508, 510 Pelton, M., 416 Pelucchi, E., 416 Pendry, J.B., 208, 211, 228 Peng, C.-Z., 410 Penney, W.G., 185 Penrose, R., 338, 345 Pepper, M., 416 Peres, A., 398, 400, 425, 432, 457, 519, 525 Peˇrina, J., 415 Perinotti, P., 469, 470, 491, 493 Peschel, U., 246 Peter, E., 193 Peterson, C., 407, 410 Peterson, C.G., 405 Petit, R., 213 Petranovskii, V.P., 217 Petrich, G.S., 270 Petriella, E., 136 Petroff, P.M., 250, 251 Petrov, A., 215, 246 Petrov, E.P., 215 Pfeiffer, L., 191, 277 Pfennigbauer, M., 446 Phillips, D.F., 101, 146, 150, 171 Piché, M., 182, 201, 212, 229 Picozzi, A., 348 Pinard, M., 127 Pincemin, F., 225 Pines, A., 330 Pistol, M.-E., 416 Platzer, H., 18 Plenet, J.C., 227 Plenio, M.B., 474 Plihal, M., 201 Pocas, S., 252 Podolsky, B., 401, 458 Pohl, D.W., 366 Poincaré, H., 325 Poingt, F., 254, 258 Poizat, J.-P., 405, 415, 416 Politi, A., 136 Polman, A., 226 Polzik, E., 408 Pommereau, F., 245, 252, 254, 257, 258, 265

562

Author index for Volume 49

Ponomarenko, S., 318, 331, 355, 374–376 Popa, I., 416 Popescu, S., 424, 466, 473, 497, 503, 519, 522 Popov, E., 212–214, 231 Poppe, A., 410 Pottage, J.M., 213 Prather, D.W., 268 Prati, F., 163 Preskill, J., 400, 437, 442, 443, 445 Proca, A., 364 Prokofiev, A.V., 222 Purcell, E.M., 194 Pustai, D.M., 268 Q Qi, B., 408 Qi, M., 219 Qian, L., 408 Qiu, M., 210, 236, 237, 243–246, 254, 257, 261–263 Qiu, Y., 254 Quang, T., 214, 224 R Racah, G., 351, 371 Raffaele, L., 244 Raikh, M.E., 223 Raineri, F., 284, 285 Raj, R., 284, 285 Rakhecha, V.C., 330, 339 Rakich, P., 270 Rakich, P.T., 219 Ralph, T.C., 366, 477, 539 Ram, J., 346 Ramachandran, G., 361, 371 Ramaseshan, S., 330 Rammuno, L., 194, 209, 210, 260, 286 Ramos-Mendieta, F., 206 Rao, R., 432 Rappe, A.M., 201, 202, 206, 211 Raring, J., 257 Rarity, J., 181 Rarity, J.G., 403, 405, 406, 411, 417 Rattier, M., 235, 236, 243, 244, 249, 251, 255, 257, 261, 262, 264, 265, 269, 270 Ravishankar, V., 361 Rayleigh Lord, 11

Reese, C., 250 Regreny, P., 252 ˇ Reháˇ cek, J., 415 Reid, D.T., 280 Reinecke, T.L., 249 Reithmaier, J., 249, 251 Reithmaier, J.P., 221 Reitzenstein, S., 249 Rempe, G., 417 Renner, R., 432, 435 Reynaud, S., 165 Rezakhani, A.T., 473, 526, 528 Rhodes, S.K., 363 Ribordy, G., 399, 408, 409, 412, 430, 513 Ricci, M., 514, 516, 518 Richartz, M., 328 Rickey, E.G., 100, 116, 153, 159, 161, 162 Riedmatten, H., 413 Rigneault, H., 213 Rikken, G.L.J.A., 184 Rindler, W., 338, 345 Ripin, D.J., 270 Risk, W., 408 Risken, H., 150 Ritchie, D.A., 416, 420 Rivacoba, A., 215 Rivest, R.L., 386 Robbins, D.J., 228 Robert, I., 416 Robert, J.-M., 427 Robert-Philip, I., 257, 416 Roberts, A., 363 Roberts, J.S., 243, 244, 284 Roberts, P.J., 208, 265, 266, 272, 276 Roberts, S.W., 245 Robertson, W.M., 206 Robinson, H.G., 101 Robson, B.A., 351, 362, 364, 371 Roch, J.-F., 405, 416, 417 Rochester, S.M., 100, 122 Rogers, T.J., 184 Rohrlich, F., 364, 371 Roizen-Dossier, B., 3 Rojo-Romeo, P., 252, 265 Roman, P., 327, 370 Romanov, S.G., 215, 217, 218 Rondi, D., 221 Rosa, T., 389 Rosen, N., 401, 458 Rosenberg, D., 422

Author index for Volume 49 Rosenberger, A.T., 135, 136, 162 Rostovtsev, Y., 100, 112, 122 Rottenberg, X., 476, 477, 534, 535 Rowson, S., 221, 228, 238 Roy, A., 464 Roy, R., 149–152 Roycroft, B., 244 Rozenberg, G.V., 327 Rubiera, A.I., 126, 127 Rupper, G., 249 Ruskai, M.B., 435 Russell, P.S.J., 186, 208, 209, 213, 265, 266, 271, 272, 276, 280, 281 Ryu, H.-Y., 251, 255, 265 Ryu, H.Y., 255 S Saastamoinen, T., 343, 358 Sabarinathan, J., 220 Sacchi, M.F., 464, 538 Saga, N., 26 Sagdeev, R.Z., 348 Sainson, S., 245 Sakai, A., 241 Sakaki, H., 255, 259 Sakoda, K., 181, 199, 204, 211, 252, 284 Sakuma, Y., 416, 417 Saleh, B.E.A., 185 Salehpour, M.R., 202 Salem, B., 245 Salieri, P., 136, 143 Salvail, L., 403, 408, 425–427, 440 Sambles, J.R., 227 Samoilovich, L.A., 217 Samson, J.C., 352 Samuelson, L., 416 Sanches-Mondragon, J., 368 Sanchez del Rio, C., 369 Sánchez-Dehesa, J., 217 Sanchis, P., 258 Sanders, B.C., 441 Sandle, W.J., 143 Sands, M., 338 Sano, D., 251, 270 Santori, C., 416, 417 Sarkar, G.G., 346 Sasaki, A., 219, 252 Sasaki, G., 252 Sasnett, M.W., 89

563

Satchell, J.S., 143 Sato, T., 217, 220, 267 Satpathy, S., 199, 202 Sautenkov, V.A., 100, 112, 122 Sauvan, C., 214, 251, 258, 263 Savage, C., 150 Savage, C.M., 136 Savona, V., 184, 193 Scalora, M., 184, 252 Scarani, V., 399, 408, 409, 504, 513 Schell, R.G., 3 Schelle, D., 246 Scherer, A., 227, 236, 242, 243, 249–252, 254, 258, 262, 269, 270 Scherf, U., 227 Schettini, V., 507, 508 Schindler, F., 227 Schlichter, C.P., 319, 345 Schmidt, H., 123, 125, 126, 156, 159, 160, 170, 171 Schmidt, M., 246 Schmitt, J.M., 329, 349 Schnabel, R., 366 Schneider, A., 238 Schneider, G.J., 268 Schnitzer, I., 269 Schori, C., 408 Schriemer, H.P., 215 Schrödinger, E., 364 Schubert, E.F., 269 Schubert, K., 227 Schuller, C., 221 Schultz, S., 228 Schumacher, B., 424 Schwach, C., 269 Schwarzacher, T., 422 Schwoob, E., 244, 252, 254, 257, 270 Sciarrino, F., 504, 507, 508, 510, 514, 516, 532 Scully, M.O., 99–102, 112, 122 Seassal, C., 221, 242, 252, 256, 258, 265, 284, 285 See, P., 420 Segal, I.E., 335 Segard, B., 163 Segura, J.-M., 417 Sek, G., 249 Self, S.A., 4, 83 Semenov, A., 421 Sen, S., 368

564

Author index for Volume 49

Senellart, P., 193 Sentenac, A., 213 Sercel, P.C., 195 Sergent, W., 163 Sergienko, A.V., 421, 422 Sermage, B., 194, 195, 251, 285, 417 Setälä, T., 318, 331, 352, 354, 357, 363 Settler, D., 260 Shahriar, M.S., 102, 130 Shambrook, A., 201 Shamir, A., 386, 387 Shamir, J., 317 Shank, C.V., 181, 184 Shannon, C.E., 384, 390 Sharfin, W.F., 147, 148 Sharkawy, A., 268 Sharma, R., 236, 270 Sharp, D.N., 215, 220 Sharpe, S.J., 132 Shashar, N., 321 Shaw, A., 244 Shchekin, O.B., 249 Shelby, R.A., 228 Sheng, P., 201 Shepherd, S., 102, 122 Shepherd, T.J., 272 Sherman, B., 214 Sherwood, M.H., 387 Shevchenko, A., 318, 331, 352, 354, 363 Shi, P., 261, 262 Shi, S., 268 Shields, A.J., 407, 416, 420 Shimoda, K., 513 Shimony, A., 401 Shin, D.-J., 251, 255 Shinya, A., 258, 259, 265, 285 Shor, P.W., 387, 435, 437, 442 Shore, B.W., 104, 106 Shuravlev, V.V., 217 Shurcliff, W.A., 321, 327, 339 Shverdin, M., 132 Siadatnejad, S., 473, 528 Sias, C., 504, 507, 508, 514, 516 Siegman, A.E., 4, 10, 72, 88 Sigalas, M., 202, 204 Sigalas, M.M., 210, 219, 225, 257, 260, 262 Sigmund, O., 261, 262 Silberhorn, S., 366 Silberstein, E., 214, 287 Silverman, R.D., 389

Silvestre, E., 213, 280, 281 Simmons, C., 407 Simmons, C.M., 405 Simon, C., 410, 504, 507–509, 512 Simon, R., 371 Sinclair, B.D., 102, 122 Sipe, J.E., 194, 209, 210, 260, 286, 346 Sklar, E., 4, 66 Skolnick, M.S., 231, 284 Skorobogatyi, M.A., 259 Skovgaard, P.M.W., 191, 227 Slusher, R.E., 252, 262 Slutsky, B., 432 Slyszyz, W., 421 Smajic, J., 262, 268 Smirnov, K., 421 Smith, B.K., 219 Smith, C., 262, 263 Smith, C.J., 244 Smith, C.J.M., 209, 235–237, 243, 244, 246, 248, 249, 251, 255, 257, 261, 264, 265, 269, 270 Smith, C.J.M.S., 254 Smith, D.A., 230, 260 Smith, D.R., 228 Smith, H.I., 219, 240 Smolin, J., 403, 424, 425, 427, 440 Smolin, J.A., 438, 462, 466, 500 Smolyaninov, I.I., 226 Sobcyk, G., 369 Sobolewski, R., 421 Soleillet, P., 327, 332 Soljacic, M., 284 Solomon, G., 251, 416 Solomon, G.S., 417 Someda, C.G., 227, 258, 261 Sondergaard, T., 236, 259 Song, B.-S., 251, 263–265 Song, D.-S., 251, 255 Song, D.D., 464 Song, Y.P., 242, 243 Sönnichen, C., 366 Sørensen, J.L., 408 Sotomayor-Torres, C.M., 215, 218 Soukoulis, C.M., 181, 186, 189, 201, 202, 204, 208, 210, 214, 225, 236, 238, 246, 254, 259–262 Sözüer, H.S., 201–203 Spanier, E., 345 Spillane, S.M., 195

Author index for Volume 49 Spisser, A., 221 Sprik, R., 222 Sprossig, W., 369 Srinivasan, A., 254 Stamnes, J.J., 4 Stamov, D.G., 327 Stanley, C.R., 249 Stanley, R., 194, 244, 269 Stanley, R.P., 184, 205 Staus, C., 284 Steane, A.M., 437 Stefanou, N., 262 Steffen, M., 387 Stehr, J., 227 Steinmeyer, G., 240 Stevenson, R.M., 231, 416 Stewart, W.J., 208, 228 Stijns, E., 88 Stinson, D.R., 384, 429 Stoicheff, B.P., 99, 101 Stokes, G.G., 323 Streetman, B.G., 184 Stringfellow, G.B., 269 Strogatz, S.H., 149–152 Stroud Jr., C.R., 102, 105 Stucki, D., 408, 409 Sturm, J.C., 218 Su, H.M., 220, 221 Sucha, G.D., 3 Suda, M., 410 Sugimoto, T., 426 Sugimoto, Y., 241, 252, 255, 259, 262 Sugitatsu, A., 257 Sun, H.B., 221 Sun, P.C., 432 Sundberg, E., 408 Suzuki, H., 252 Svirko, Y.P., 348, 366 Sweatlock, L.A., 226 Swillo, M., 237, 244, 245, 254, 257, 261–263 Swindell, H., 321, 327 Szapiel, S., 34, 66, 87 T Tabosa, J.W.R., 101 Taflove, A., 210 Tahraoui, A., 284 Taillaert, D., 241, 286

565

Takahara, J., 318 Takahashi, C., 258, 259 Takahashi, J., 258 Takahashi, J.-I., 258, 259 Takahashi, S., 269 Takahasi, H., 513 Takatsu, M., 416, 417 Takayama, O., 225 Takemori, T., 241 Takemoto, K., 416, 417 Takenaka, H., 371 Takenaka, T., 90 Takesue, H., 420 Takeuchi, S., 420 Talneau, A., 240, 245, 254, 255, 259, 261, 263 Tam, W.Y., 220, 221 Tamaki, K., 438, 440, 442 Tamamura, T., 220, 252, 267 Tamarat, P., 417 Tamir, T., 184 Tanabe, T., 265, 285 Tanaka, H., 262, 269 Tanaka, K., 25, 26, 255, 259 Tanaka, Y., 252, 255, 259, 262, 269 Tanzilli, S., 418 Tapster, P.R., 403, 405, 406, 411, 417 Tarng, S.S., 146, 148 Tartakovskii, I.I., 251 Tayeb, G., 214, 267 Tchelnokov, A., 181, 194, 199, 255 Teich, M.C., 185 Temelkuran, B., 205, 260, 262 Tervo, J., 318, 331, 343, 352, 357, 358, 363 Thedrez, B., 254 Theuer, H., 106 Thierry-Mieg, V., 194, 195, 251, 285, 416, 417 Thijssen, M.S., 222 Thikodeev, S.G., 227 Thio, T., 227 Thoen, E.R., 240 Thomas, E.L., 199 Thomas, J.A., 433 Thompson, R.I., 99, 101 Thoms, S., 242, 243 Thorhauge, M., 259 Thorwart, M., 151 Tian, B.-L., 410 Tiberio, R.C., 252

566

Author index for Volume 49

Tikhodeev, S.G., 213, 231, 233 Timko, A., 252 Tittel, W., 407, 408, 413, 418, 430, 504, 519 Toader, O., 215, 218, 221 Tocci, M.D., 184 Tokuda, T., 252 Tolkachev, E.A., 328, 364 Tombrello, T., 236 Tomita, A., 267, 407 Tomoda, K., 223 Torhauge, M., 261 Tortora, P., 362, 363 Townes, C.H., 513 Townsend, P., 406, 407 Townsend, P.D., 407 Tratnik, M.V., 346 Trautman, J., 366 Treps, N., 366 Treussart, F., 405, 416, 417 Tringides, M., 202 Tsegaye, T., 408, 416 Tu, C.W., 252 Tualle-Brouri, R., 534, 539 Tuchin, V.V., 366 Tünnermann, A., 246 Tuovinen, J., 90 Turberfield, A.J., 215, 220 Turitsyn, K.S., 366 Tuttle, G., 202, 204, 219 Tyras, G., 3 U Ueta, T., 252 Ujihara, K., 193, 214 Umerjee, R.K., 371 Urbantke, H., 319, 338, 345 Ursin, R., 410 Usadi, E., 100 Ushenko, A.G., 364 Ushenko, Yu.A., 364 Usikov, D.A., 348 Usuki, T., 416, 417 V Vaccaro, L., 362, 363 Vadhawan, V.K., 146 Vahala, K.J., 195, 251 Vahimaa, P., 318

Van Assche, G., 534, 535, 539 van Blaaderen, A., 222 Van Daele, P., 269 van der Linden, H.J.C., 150 van der Merwe, A., 457 van der Wal, C.H., 101, 171 van Driel, A.F., 215, 224, 225 van Driel, H.M., 215, 218, 222, 238, 284 van Hulst, N., 366 van Hulst, N.F., 287 van Loock, P., 477, 478, 533, 535, 537 van Soest, G., 223 Vanbésien, O., 228 Vandenberghe, L., 484 Vandersypen, L.M.K., 387 Vanmaekelbergh, D., 215, 224, 225 Vardeny, Z.V., 223 Varoutsis, S., 257, 416 Vawter, G.A., 241 Vecchi, G., 284, 285 Vedral, V., 474 Velichansky, V.L., 100, 101 Venkataraman, S., 268 Venkatesh, H.G., 346 Verevkin, A., 421 Verhoeve, P., 422 Verkerk, P., 127 Verma, N.K., 92 Vernam, G.S., 384, 389 Vernon, F.L., 338 Veronis, G., 286 Viana Ramos, R., 364 Viasnoff-Schwoob, E., 257, 265 Vigier, J.P., 364 Viktorovitch, P., 221, 252, 265, 284, 285 Villeneuve, P.R., 181, 182, 201, 204, 212, 219, 229, 231, 236, 238, 240, 241, 260, 265, 269, 284, 286 Villing, A., 405, 416 Vlasov, Y., 255 Vlasov, Y.A., 215, 218, 222–224 Vögele (Vogele), B., 243, 244, 249 Volkov, V.S., 287 Vollmer, M., 318 Volz, J., 415 von Plessen, G., 227, 366 von Poschingen, H., 366 Voronov, B., 421 Vos, W., 223

Author index for Volume 49 Vos, W.L., 193, 215, 217, 222, 224, 225 Vos, W.M., 215, 218, 223 Vrijen, R., 269 Vuckovic, J., 227, 250, 251, 254, 258, 270, 417 Vurgaftman, I., 254 W Wadati, M., 472, 480, 498, 527, 528 Wadsworth, W.J., 276, 280, 281 Wagh, A.G., 330, 339 Wai, P.K.A., 366 Waks, E., 251, 417, 420 Walls, D.F., 136, 150 Walsworth, R.L., 101, 146, 150, 171 Walther, H., 417 Wang, H., 100, 102, 103, 118, 119, 121–123, 136, 140–143, 148, 153, 155–159, 163, 165, 167, 170, 171 Wang, H.Z., 220, 221 Wang, K., 221 Wang, K.L., 221 Wang, L., 191 Wang, L.J., 100 Wang, X., 220, 221 Wang, X.-B., 399, 444 Wang, X.B., 472, 473, 480, 527, 528 Ward, A.J., 208, 211 Warrington, D.M., 143 Wasey, J.A.E., 194, 270 Watanabe, Y., 241, 262 Waterman, T.H., 321 Weaver, D.L., 337, 348, 369, 370 Weber, T., 457 Weber, W.H., 194, 227 Wegdam, G.H., 222 Wegman, M.N., 429 Wehner, R., 321 Wehrl, A., 335 Wehrspohn, R.B., 181, 193, 199, 215, 221, 231, 239, 283 Weihs, G., 410, 504, 508, 512, 513, 526 Weiner, J., 366 Weinfurter, H., 405, 410, 415, 419, 519 Weisbuch, C., 181, 184, 192, 193, 205, 209, 212, 235–237, 243, 244, 246, 248, 249, 251, 252, 254, 255, 257, 261–266, 268–270 Weiss, M.L., 335, 336, 338

567

Welch, G.R., 100, 112, 122 Wells, J.-P.R., 284 Wendt, J.R., 241, 261 Wenger, J., 534, 539 Werner, R.F., 466, 467, 478, 482, 483, 496, 500, 503, 504, 534, 537 Wheeler, J.A., 368 White, A., 410 White, I.H., 263 Whitley, R.M., 105 Whitney, C., 328 Whittaker, D.M., 213, 231, 233, 284 Whittaker, M., 224 Wiaux, V., 241 Wiegmann, W., 146, 148 Wiener, M., 388 Wiener, N., 326 Wiener, O., 325 Wiesner, S., 393, 459 Wiggins, T.A., 88 Wijnhoven, J.E.G.J., 193, 215, 222, 225 Wild, B., 236, 243, 245, 254, 260 Wild, U.P., 417 Wilkinson, C.D.W., 242–244 Williams, C.S., 3, 4 Williams, J., 250 Windish, R., 269 Winn, J.N., 181, 199, 206, 240 Wischmann, W., 246 Wittke, P., 193 Wolf, E., 3, 5, 15, 35, 194, 318, 326, 327, 331, 333, 334, 338, 349, 355, 356, 374–376 Wolf, K.B., 368 Wolf, M.M., 478, 534, 537 Wolf, M.O., 215 Wolf, S., 433, 434 Wolff, P.A., 227 Wong, S., 215 Wong, V., 102 Woods, G., 171 Wootters, W.K., 392, 400, 424, 425, 438, 457, 519 Worthing, P.T., 194, 270 Wrachtrup, J., 416 Wu, H., 148 Wu, L., 268 Wu, S., 191 Wu, S.L., 252 Wünsche, A., 90

568

Author index for Volume 49

X Xia, F., 259 Xia, H., 102 Xia, Y., 217, 218 Xiao, L.T., 417 Xiao, M., 99–103, 106–116, 118, 119, 121–123, 125, 127–130, 132, 133, 136, 140–159, 163–171 Xie, C., 102, 123, 148, 156 Xing, A., 257 Xu, J.F., 220, 221 Xu, Y., 193, 258, 262 Y Yablonovitch, E., 181, 201, 202, 206, 214, 229, 249, 269, 283, 284 Yablonskii, A.L., 213, 231, 233 Yabuzaki, T., 136 Yacomotti, A.M., 284, 285 Yamada, K., 258, 259 Yamada, M., 150 Yamamoto, N., 219, 223 Yamamoto, Y., 126, 153, 159–161, 170, 171, 193, 251, 416, 417, 420 Yamashita, M., 126 Yamazaki, K., 426 Yan, M., 100, 116, 153, 159, 161, 162 Yang, B., 410 Yang, G.-Z., 215 Yang, J., 410 Yang, J.-K., 255 Yang, T., 262, 410 Yang, W., 101, 103, 136, 143, 147–152, 164–169, 171 Yang, Y., 214 Yang, Y.P., 214, 224 Yannoni, C.S., 387 Yao, A., 409 Yao, J., 102, 123, 156 Yariv, A., 184, 189, 191, 193, 236, 242, 252, 254, 258, 262 Yarostky, D.A., 199 Yashchuk, V.V., 100, 122 Yee, K.S., 210 Yeh, P., 184, 189, 191 Yin, G.Y., 102, 132, 134 Yin, J., 410 Yodh, A., 366

Yokohama, I., 258, 259 Yokota, M., 90 Yokoyama, H., 193, 214 Yokoyama, I., 258 Yokoyama, M., 252 Yokoyama, N., 416, 417 Yonekura, J., 249, 255, 269 Yoshida, A., 4, 54 Yoshie, T., 249, 250, 254 Yoshimoto, S., 219 Young, J.F., 194, 209, 210, 260, 284, 286 Yu, F.T.S., 18 Yu, P.-C., 220 Yu, Q.L., 210 Yuan, Z.L., 407, 416 Z Zabala, N., 215 Zachariasen, W.H., 222 Zakharchenya, B., 319, 345 Zakhidov, A.A., 223 Zapasskii, V.S., 319, 345 Zarda, P., 405, 415, 419 Zaslavsky, G.M., 348 Zayats, V.Z., 226 Zbinden, H., 404, 407–409, 412, 413, 418, 430, 504, 513, 519 Zdetsis, A., 186 Zeilinger, A., 410, 425, 446, 504, 508, 512, 519 Zeng, Z.H., 220, 221 Zentel, R., 218 Zentgraf, T., 227 Zhang, A.-N., 522 Zhang, B., 251 Zhang, G.Z., 99, 101 Zhang, J., 421 Zhang, J.-P., 191 Zhang, J.P., 252 Zhang, Q., 410 Zhang, Y.S., 518 Zhang, Z., 199, 202 Zhang, Z.-Q., 218 Zhao, Y., 408 Zhao, Z., 522 Zheludev, N.I., 348, 366 Zheng, H., 214 Zhou, W., 220 Zhou, X.-Q., 522 Zhu, D., 220

Author index for Volume 49 Zhu, S., 99 Zhu, S.Y., 214, 224 Zhu, Y., 100, 102, 116, 124–127, 132, 133, 135, 153, 159, 161, 162, 170 Zhuang, Y.X., 261 Zibrov, A.S., 100, 101, 112, 122, 150, 171 Zoller, P., 136, 171, 424 Zoorob, M.E., 231, 245, 270

Zubairy, M.S., 214 Zubrzycki, W., 219, 241 ˙ Zukowski (Zukowski), M., 425, 473 Zumofen, G., 417 Zurek, W.H., 392, 457 Zutter, D.D., 286 Zwanziger, Z.W., 330 Zwiller, V., 194, 270, 416

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Subject index for Volume 49 Clifford algebra 328 cloner, continuous-variable 465 – , economical 466 – , entanglement 465 – , Fourier-covariant 465, 469, 471, 493 – , group-covariant 465, 471 – , Heisenberg 465, 492, 493 – , Pauli 464, 526 – , phase-covariant 465, 469, 494 – , universal 464 cloning machine 458, 465, 502 – – , continuous variable 476 – – , economical 479 – – , entanglement 474 – – , Heisenberg 468–470, 489, 491, 492 – – , highly-asymmetric 475 – – , Pauli 468 – – , probabilistic 478 – – , quantum 460, 461, 464, 466–480 – – , real 474, 481 – – , universal 466, 496, 527, 533 – of coherent state 533, 538, 540 – – optical continuous variables 533 – – quantum state 392 – , phase covariant 525, 528, 529, 532 coherency matrix 319, 326, 328, 365 coherent adiabatic population transfer 106 – population trapping (CPT) 102, 104–106, 132 color center 415, 416 comb frequency technique 282 completely positive map 480 Cooper pair 422 cross-phase modulation 122 cross-spectral density 327 cryptography 383 – , public-key 385, 386 – , quantum 390, 446 – , secret-key 387

A Abelian group 386 ac-Stark splitting 104 Airy pattern 11 apodization 3, 66, 70 astigmatism 43, 54, 58, 62, 86 – , Seidel 48 atomic coherence 101, 102, 170 Autler–Townes splitting 104, 138 B balanced homodyne detector 112 B92 protocol 397, 398, 442, 444 BB84 protocol 393, 398, 400, 402, 403, 405, 410, 437, 438, 439, 440, 442–444, 525 Bell’s inequality 400, 401, 410 – measurement 519 – state 487, 489, 519 – theorem 402 Berry phase 330 Bessel function 18 birefringence 276 – , stress-induced 404 bistable hysteresis 102 black-body radiation 374 Bloch equation 345 – state 262 – wave 208, 209, 237, 267, 284–285 Bravais lattice 200 Brillouin zone 197, 208, 267 C Caesar cipher 384 Carathéodory theorem 336 cavity ringdown effect 100 Cerenkov radiation 215 Clauser–Horne–Shimony–Holt inequality 402 571

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Subject index for Volume 49

cryptosystem – , public key 388 – , quantum 391 – , secret key 388 crystallography, X-ray 222 Csiszar–Korner formula 433 D data encryption standard 388 decoy-state protocol 399 decryption 383, 387, 389, 390 degree of polarization 328, 331, 339, 342, 343, 347, 352, 357, 358, 374, 375 distributed feedback laser 181 Doppler broadening 109, 114 – effect 99, 109, 153 – linewidth 99 – width 110, 114, 116 double refraction 321, 322 dressed state picture 106 Drude model 226 dynamic Stark splitting 104 E Eckert protocol 402, 410 electromagnetically induced absorption 101, 102 – – grating 101 – – transparency (EIT) 99–101, 103–112, 115, 116, 120, 123, 136–139, 142, 143, 145, 148, 150, 153, 155, 161, 171 – – – dip 100 encryption 383, 387, 389, 390 Enigma cipher 385 entangled state, maximally 438, 462 entanglement – , distillation of 424 – , energy–time 411 – , polarization 410 – purification protocol 438 – source 417 – swapping 425 – witness 436 entanglement-based protocol 409 EPR state 458 error correction 425 – – , quantum 437 Euler angles 369

F Fabry–Perot – cavity 239, 263, 268 – mirror 286 – resonator 136 Faraday rotator 407 fast light 100 Fermi golden rule 193 Floquet–Bloch theorem 199 Fock state 414 Fourier-modal method 213 Fourier–Stieljes transform 327 four-wave – mixing 103, 126, 161 – , nondegenerate 128 Fresnel number 4, 16, 17, 18, 24, 66, 74, 91 – zone 16, 21, 23, 27, 74 G gain without inversion 101 Gell–Mann matrices 350, 370 geometrical optics 233 Glauber P-representation 534 Gouy phase shift 194 group velocity dispersion 279 H Hanbury–Brown–Twiss experiment 414 Hankel transform 85 harmonic generation 102 Helmholtz equation 90, 185 Hilbert–Schmidt decomposition 464 Hong–Ou–Mandel interferometer 465 Hopf map 369 – vibration 340 Huygens–Fresnel principle 91, 194 I Ikeda instability 162 irradiance 3, 5, 7, 10, 24, 54, 65, 71 – axial 21–31, 63, 64, 73, 74, 76, 91 – distribution 10, 12 J Jaynes–Cummings Hamiltonian 512 Jones matrix 327, 328, 344, 348, 361, 366 – vector 319 Josephson junction 422

Subject index for Volume 49 K Kemmer algebra 370 Kerr nonlinearity 123, 125, 126, 136, 140, 142, 143, 156, 159, 160 Kerr-nonlinear – coefficient 102, 117 – effect 118, 120, 121 Korringa–Kohn–Rostoker method 212 L Laguerre polynomial 87 lasing without inversion 99 Lie algebra 320 – group in polarization optics 368–371 Liouville equation 106 lithography, e-beam 244 M Mach–Zehnder interferometer 112, 262, 405–407, 409, 411, 412, 504, 522, 531, 536 Madison convention 371 magneto-optical trap 125, 132 Majorana representation 360 Maxwell’s equations 160, 200, 233, 323, 324, 344 Minkowski space–time 332 modulation transfer function 15 Mueller matrix 319, 327, 330, 362, 366 N Navier–Stokes equations 323 nondegenerate optical parametric amplifier 536 O one-time pad 384, 385 optical bistability 102, 135–140, 142, 143, 145–148, 154 – – , dynamical hysteresis in 149–152 – fiber 422 – soliton generation 163 – transfer function 3, 5, 15, 31–34, 84 P parametric down-conversion 414, 417 paraxial approximation 90 – beam 4

573

partially polarized field 331, 333 Pauli matrices 402, 463, 470, 488 phase encoding 405 photonic bandgap 195, 201 – crystal 180, 181, 183, 206 – – fiber 271–283 – – laser 252 – – , three-dimensional 214–228 – – , two-dimensional 229–270 plasmon 270 Poincaré sphere 319, 339, 359, 365 point-spread function 3, 5, 10, 17, 72 polarization density matrix 334 – entropy 362 – of light 317, 321, 322 Poynting vector 267 pupil, aberrated Gaussian 66–71 – function 5, 6, 7, 16, 71 – , weakly-truncated Gaussian 71 Purcell effect 182, 237 – factor 194 Q quaternion algebra 328 quantum cloning 457, 495, 501 – – without signaling 463 – computation 400 – dot 224, 416, 417 – – detector 420 – entanglement 400, 458 – information 460 – – processing 364 – key distribution 391, 393, 397, 402, 403, 404, 407, 409–413, 416–420, 423, 424, 428–432, 434, 436, 439, 446, 462 – measurement 391 – no-cloning theorem 458, 459 – nondemolition measurement 124 – repeater 424 – teleportation 400, 490, 519 R Rabi frequency 99, 100, 104, 110, 113, 116, 124–126, 134 Raman scattering, coherent 102 – gain 100 Rayleigh range 79, 89 – scattering 223 RSA cryptosystem 386–388

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S

T

SARG protocol 399 Schrödinger equation 185 Schur’s lemma 370 Seidel coma 47, 48 self-focusing 122 Shannon entropy 335, 426 six-state protocol 398, 443 skytale cryptodevice 384 slow light 100, 132 Soleil–Babinet compensator 531 spinor 338 spontaneous emission 459 – , inhibition of 224, 230 squeezed state 364 stationary phase, method of 90 Stern–Gerlach apparatus 402 stimulated emission 457 Stokes law 323 – parameters 319, 322, 327, 334, 339, 340, 345, 346, 350, 351, 359, 364 – theorem 323 – vector 345 Strehl ratio 3, 4, 21, 34–36, 38, 43, 44, 45, 47–54, 60, 85 sub-Poissonian distribution 414 superconductor tunnel junction detector 422

transition edge sensor 421 TWINKLE device 387 two-photon absorption 161 U universal-NOT gate 503 V Vernam cipher 384, 385, 389 vertical cavity laser 285 von Neumann entropy 3, 39029 W Weyl–Heisenberg – group 468, 490, 491 – operators 488 Wigner function 533, 534 Wolf’s coherency matrix 326 Y Yablonovite 200, 202, 206, 215 Z Zeeman coherence 143 Zernike–Gauss polynomial 3, 34, 40, 41

Contents of previous volumes*

VOLUME 1 (1961) 1 2 3 4 5 6 7 8

The modern development of Hamiltonian optics, R.J. Pegis Wave optics and geometrical optics in optical design, K. Miyamoto The intensity distribution and total illumination of aberration-free diffraction images, R. Barakat Light and information, D. Gabor On basic analogies and principal differences between optical and electronic information, H. Wolter Interference color, H. Kubota Dynamic characteristics of visual processes, A. Fiorentini Modern alignment devices, A.C.S. Van Heel

1– 29 31– 66 67–108 109–153 155–210 211–251 253–288 289–329

VOLUME 2 (1963) 1 2 3 4 5 6

Ruling, testing and use of optical gratings for high-resolution spectroscopy, G.W. Stroke The metrological applications of diffraction gratings, J.M. Burch Diffusion through non-uniform media, R.G. Giovanelli Correction of optical images by compensation of aberrations and by spatial frequency filtering, J. Tsujiuchi Fluctuations of light beams, L. Mandel Methods for determining optical parameters of thin films, F. Abelès

1– 72 73–108 109–129 131–180 181–248 249–288

VOLUME 3 (1964) 1 2 3

The elements of radiative transfer, F. Kottler Apodisation, P. Jacquinot, B. Roizen-Dossier Matrix treatment of partial coherence, H. Gamo

1 2 3 4

Higher order aberration theory, J. Focke Applications of shearing interferometry, O. Bryngdahl Surface deterioration of optical glasses, K. Kinosita Optical constants of thin films, P. Rouard, P. Bousquet

1– 28 29–186 187–332

VOLUME 4 (1965)

* Volumes I–XL were previously distinguished by roman rather than by arabic numerals.

575

1– 36 37– 83 85–143 145–197

576 5 6 7

Contents of previous volumes The Miyamoto–Wolf diffraction wave, A. Rubinowicz Aberration theory of gratings and grating mountings, W.T. Welford Diffraction at a black screen, Part I: Kirchhoff’s theory, F. Kottler

199–240 241–280 281–314

VOLUME 5 (1966) 1 2 3 4 5 6

Optical pumping, C. Cohen-Tannoudji, A. Kastler Non-linear optics, P.S. Pershan Two-beam interferometry, W.H. Steel Instruments for the measuring of optical transfer functions, K. Murata Light reflection from films of continuously varying refractive index, R. Jacobsson X-ray crystal-structure determination as a branch of physical optics, H. Lipson, C.A. Taylor 7 The wave of a moving classical electron, J. Picht

1– 81 83–144 145–197 199–245 247–286 287–350 351–370

VOLUME 6 (1967) 1 2 3 4 5 6 7 8

Recent advances in holography, E.N. Leith, J. Upatnieks Scattering of light by rough surfaces, P. Beckmann Measurement of the second order degree of coherence, M. Françon, S. Mallick Design of zoom lenses, K. Yamaji Some applications of lasers to interferometry, D.R. Herriot Experimental studies of intensity fluctuations in lasers, J.A. Armstrong, A.W. Smith Fourier spectroscopy, G.A. Vanasse, H. Sakai Diffraction at a black screen, Part II: electromagnetic theory, F. Kottler

1– 52 53– 69 71–104 105–170 171–209 211–257 259–330 331–377

VOLUME 7 (1969) 1 2 3 4 5 6 7

Multiple-beam interference and natural modes in open resonators, G. Koppelman Methods of synthesis for dielectric multilayer filters, E. Delano, R.J. Pegis Echoes at optical frequencies, I.D. Abella Image formation with partially coherent light, B.J. Thompson Quasi-classical theory of laser radiation, A.L. Mikaelian, M.L. Ter-Mikaelian The photographic image, S. Ooue Interaction of very intense light with free electrons, J.H. Eberly

1– 66 67–137 139–168 169–230 231–297 299–358 359–415

VOLUME 8 (1970) 1 2 3 4 5 6

Synthetic-aperture optics, J.W. Goodman The optical performance of the human eye, G.A. Fry Light beating spectroscopy, H.Z. Cummins, H.L. Swinney Multilayer antireflection coatings, A. Musset, A. Thelen Statistical properties of laser light, H. Risken Coherence theory of source-size compensation in interference microscopy, T. Yamamoto 7 Vision in communication, L. Levi 8 Theory of photoelectron counting, C.L. Mehta

1– 50 51–131 133–200 201–237 239–294 295–341 343–372 373–440

Contents of previous volumes

577

VOLUME 9 (1971) 1 2 3 4 5 6 7

Gas lasers and their application to precise length measurements, A.L. Bloom Picosecond laser pulses, A.J. Demaria Optical propagation through the turbulent atmosphere, J.W. Strohbehn Synthesis of optical birefringent networks, E.O. Ammann Mode locking in gas lasers, L. Allen, D.G.C. Jones Crystal optics with spatial dispersion, V.M. Agranovich, V.L. Ginzburg Applications of optical methods in the diffraction theory of elastic waves, K. Gniadek, J. Petykiewicz 8 Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions, B.R. Frieden

1– 30 31– 71 73–122 123–177 179–234 235–280 281–310 311–407

VOLUME 10 (1972) 1 2 3 4 5 6 7

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. Dexter Elastooptic light modulation and deflection, E.K. Sittig Quantum detection theory, C.W. Helstrom

1– 44 45– 87 89–135 137–164 165–228 229–288 289–369

VOLUME 11 (1973) 1 2 3 4 5 6 7

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, O. Bryngdahl Production of electron probes using a field emission source, A.V. Crewe Hamiltonian theory of beam mode propagation, J.A. Arnaud Gradient index lenses, E.W. Marchand

1– 76 77–122 123–166 167–221 223–246 247–304 305–337

VOLUME 12 (1974) 1 2 3 4 5 6

Self-focusing, self-trapping, and self-phase modulation of laser beams, O. Svelto Self-induced transparency, R.E. Slusher Modulation techniques in spectrometry, M. Harwit, J.A. Decker Jr Interaction of light with monomolecular dye layers, K.H. Drexhage The phase transition concept and coherence in atomic emission, R. Graham Beam-foil spectroscopy, S. Bashkin

1– 51 53–100 101–162 163–232 233–286 287–344

VOLUME 13 (1976) 1

On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment, H.P. Baltes 2 The case for and against semiclassical radiation theory, L. Mandel 3 Objective and subjective spherical aberration measurements of the human eye, W.M. Rosenblum, J.L. Christensen 4 Interferometric testing of smooth surfaces, G. Schulz, J. Schwider

1– 25 27– 68 69– 91 93–167

578

Contents of previous volumes

5

Self-focusing of laser beams in plasmas and semiconductors, M.S. Sodha, A.K. Ghatak, V.K. Tripathi 6 Aplanatism and isoplanatism, W.T. Welford

169–265 267–292

VOLUME 14 (1976) 1 2 3 4 5 6 7

The statistics of speckle patterns, J.C. Dainty High-resolution techniques in optical astronomy, A. Labeyrie Relaxation phenomena in rare-earth luminescence, L.A. Riseberg, M.J. Weber The ultrafast optical Kerr shutter, M.A. Duguay Holographic diffraction gratings, G. Schmahl, D. Rudolph Photoemission, P.J. Vernier Optical fibre waveguides – a review, P.J.B. Clarricoats

1– 46 47– 87 89–159 161–193 195–244 245–325 327–402

VOLUME 15 (1977) 1 2 3 4 5

Theory of optical parametric amplification and oscillation, W. Brunner, H. Paul Optical properties of thin metal films, P. Rouard, A. Meessen Projection-type holography, T. Okoshi Quasi-optical techniques of radio astronomy, T.W. Cole Foundations of the macroscopic electromagnetic theory of dielectric media, J. Van Kranendonk, J.E. Sipe

1– 75 77–137 139–185 187–244 245–350

VOLUME 16 (1978) 1 2 3 4 5

Laser selective photophysics and photochemistry, V.S. Letokhov Recent advances in phase profiles generation, J.J. Clair, C.I. Abitbol Computer-generated holograms: techniques and applications, W.-H. Lee Speckle interferometry, A.E. Ennos Deformation invariant, space-variant optical pattern recognition, D. Casasent, D. Psaltis 6 Light emission from high-current surface-spark discharges, R.E. Beverly III 7 Semiclassical radiation theory within a quantum-mechanical framework, I.R. Senitzky

1– 69 71–117 119–232 233–288 289–356 357–411 413–448

VOLUME 17 (1980) 1 2 3

Heterodyne holographic interferometry, R. Dändliker Doppler-free multiphoton spectroscopy, E. Giacobino, B. Cagnac The mutual dependence between coherence properties of light and nonlinear optical processes, M. Schubert, B. Wilhelmi 4 Michelson stellar interferometry, W.J. Tango, R.Q. Twiss 5 Self-focusing media with variable index of refraction, A.L. Mikaelian

1– 84 85–161 163–238 239–277 279–345

VOLUME 18 (1980) 1 2

Graded index optical waveguides: a review, A. Ghatak, K. Thyagarajan Photocount statistics of radiation propagating through random and nonlinear media, J. Pe˘rina

1–126 127–203

Contents of previous volumes Strong fluctuations in light propagation in a randomly inhomogeneous medium, V.I. Tatarskii, V.U. Zavorotnyi 4 Catastrophe optics: morphologies of caustics and their diffraction patterns, M.V. Berry, C. Upstill

579

3

204–256 257–346

VOLUME 19 (1981) 1 2 3 4 5

Theory of intensity dependent resonance light scattering and resonance fluorescence, B.R. Mollow Surface and size effects on the light scattering spectra of solids, D.L. Mills, K.R. Subbaswamy Light scattering spectroscopy of surface electromagnetic waves in solids, S. Ushioda Principles of optical data-processing, H.J. Butterweck The effects of atmospheric turbulence in optical astronomy, F. Roddier

1– 43 45–137 139–210 211–280 281–376

VOLUME 20 (1983) 1 2 3 4 5

Some new optical designs for ultra-violet bidimensional detection of astronomical objects, G. Courtès, P. Cruvellier, M. Detaille Shaping and analysis of picosecond light pulses, C. Froehly, B. Colombeau, M. Vampouille Multi-photon scattering molecular spectroscopy, S. Kielich Colour holography, P. Hariharan Generation of tunable coherent vacuum-ultraviolet radiation, W. Jamroz, B.P. Stoicheff

1– 61 63–153 155–261 263–324 325–380

VOLUME 21 (1984) 1 2 3 4 5

Rigorous vector theories of diffraction gratings, D. Maystre Theory of optical bistability, L.A. Lugiato The Radon transform and its applications, H.H. Barrett Zone plate coded imaging: theory and applications, N.M. Ceglio, D.W. Sweeney Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity, J.C. Englund, R.R. Snapp, W.C. Schieve

1– 67 69–216 217–286 287–354 355–428

VOLUME 22 (1985) 1 2 3 4 5

Optical and electronic processing of medical images, D. Malacara Quantum fluctuations in vision, M.A. Bouman, W.A. Van De Grind, P. Zuidema Spectral and temporal fluctuations of broad-band laser radiation, A.V. Masalov Holographic methods of plasma diagnostics, G.V. Ostrovskaya, Yu.I. Ostrovsky Fringe formations in deformation and vibration measurements using laser light, I. Yamaguchi 6 Wave propagation in random media: a systems approach, R.L. Fante

1– 76 77–144 145–196 197–270 271–340 341–398

VOLUME 23 (1986) 1

Analytical techniques for multiple scattering from rough surfaces, J.A. DeSanto, G.S. Brown 2 Paraxial theory in optical design in terms of Gaussian brackets, K. Tanaka 3 Optical films produced by ion-based techniques, P.J. Martin, R.P. Netterfield

1– 62 63–111 113–182

580 4 5

Contents of previous volumes Electron holography, A. Tonomura Principles of optical processing with partially coherent light, F.T.S. Yu

183–220 221–275

VOLUME 24 (1987) 1 2 3 4 5

Micro Fresnel lenses, H. Nishihara, T. Suhara Dephasing-induced coherent phenomena, L. Rothberg Interferometry with lasers, P. Hariharan Unstable resonator modes, K.E. Oughstun Information processing with spatially incoherent light, I. Glaser

1– 37 39–101 103–164 165–387 389–509

VOLUME 25 (1988) 1

Dynamical instabilities and pulsations in lasers, N.B. Abraham, P. Mandel, L.M. Narducci 2 Coherence in semiconductor lasers, M. Ohtsu, T. Tako 3 Principles and design of optical arrays, Wang Shaomin, L. Ronchi 4 Aspheric surfaces, G. Schulz

1–190 191–278 279–348 349–415

VOLUME 26 (1988) 1 2 3 4 5

Photon bunching and antibunching, M.C. Teich, B.E.A. Saleh Nonlinear optics of liquid crystals, I.C. Khoo Single-longitudinal-mode semiconductor lasers, G.P. Agrawal Rays and caustics as physical objects, Yu.A. Kravtsov Phase-measurement interferometry techniques, K. Creath

1–104 105–161 163–225 227–348 349–393

VOLUME 27 (1989) 1 2 3 4

The self-imaging phenomenon and its applications, K. Patorski Axicons and meso-optical imaging devices, L.M. Soroko Nonimaging optics for flux concentration, I.M. Bassett, W.T. Welford, R. Winston Nonlinear wave propagation in planar structures, D. Mihalache, M. Bertolotti, C. Sibilia 5 Generalized holography with application to inverse scattering and inverse source problems, R.P. Porter

1–108 109–160 161–226 227–313 315–397

VOLUME 28 (1990) 1 2

Digital holography – computer-generated holograms, O. Bryngdahl, F. Wyrowski Quantum mechanical limit in optical precision measurement and communication, Y. Yamamoto, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa 3 The quantum coherence properties of stimulated Raman scattering, M.G. Raymer, I.A. Walmsley 4 Advanced evaluation techniques in interferometry, J. Schwider 5 Quantum jumps, R.J. Cook

1– 86 87–179 181–270 271–359 361–416

Contents of previous volumes

581

VOLUME 29 (1991) 1 2

Optical waveguide diffraction gratings: coupling between guided modes, D.G. Hall Enhanced backscattering in optics, Yu.N. Barabanenkov, Yu.A. Kravtsov, V.D. Ozrin, A.I. Saichev 3 Generation and propagation of ultrashort optical pulses, I.P. Christov 4 Triple-correlation imaging in optical astronomy, G. Weigelt 5 Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics, C. Flytzanis, F. Hache, M.C. Klein, D. Ricard, Ph. Roussignol

1– 63 65–197 199–291 293–319 321–411

VOLUME 30 (1992) 1 2 3 4 5

Quantum fluctuations in optical systems, S. Reynaud, A. Heidmann, E. Giacobino, C. Fabre Correlation holographic and speckle interferometry, Yu.I. Ostrovsky, V.P. Shchepinov Localization of waves in media with one-dimensional disorder, V.D. Freilikher, S.A. Gredeskul Theoretical foundation of optical-soliton concept in fibers, Y. Kodama, A. Hasegawa Cavity quantum optics and the quantum measurement process, P. Meystre

1– 85 87–135 137–203 205–259 261–355

VOLUME 31 (1993) 1 2 3 4 5 6

Atoms in strong fields: photoionization and chaos, P.W. Milonni, B. Sundaram Light diffraction by relief gratings: a macroscopic and microscopic view, E. Popov Optical amplifiers, N.K. Dutta, J.R. Simpson Adaptive multilayer optical networks, D. Psaltis, Y. Qiao Optical atoms, R.J.C. Spreeuw, J.P. Woerdman Theory of Compton free electron lasers, G. Dattoli, L. Giannessi, A. Renieri, A. Torre

1–137 139–187 189–226 227–261 263–319 321–412

VOLUME 32 (1993) 1 2 3 4

Guided-wave optics on silicon: physics, technology and status, B.P. Pal Optical neural networks: architecture, design and models, F.T.S. Yu The theory of optimal methods for localization of objects in pictures, L.P. Yaroslavsky Wave propagation theories in random media based on the path-integral approach, M.I. Charnotskii, J. Gozani, V.I. Tatarskii, V.U. Zavorotny 5 Radiation by uniformly moving sources. Vavilov–Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, V.L. Ginzburg 6 Nonlinear processes in atoms and in weakly relativistic plasmas, G. Mainfray, C. Manus

1– 59 61–144 145–201 203–266 267–312 313–361

VOLUME 33 (1994) 1 2 3 4

The imbedding method in statistical boundary-value wave problems, V.I. Klyatskin Quantum statistics of dissipative nonlinear oscillators, V. Peˇrinová, A. Lukš Gap solitons, C.M. De Sterke, J.E. Sipe Direct spatial reconstruction of optical phase from phase-modulated images, V.I. Vlad, D. Malacara 5 Imaging through turbulence in the atmosphere, M.J. Beran, J. Oz-Vogt 6 Digital halftoning: synthesis of binary images, O. Bryngdahl, T. Scheermesser, F. Wyrowski

1–127 129–202 203–260 261–317 319–388 389–463

582

Contents of previous volumes VOLUME 34 (1995)

1 2 3 4 5

Quantum interference, superposition states of light, and nonclassical effects, V. Bužek, P.L. Knight Wave propagation in inhomogeneous media: phase-shift approach, L.P. Presnyakov The statistics of dynamic speckles, T. Okamoto, T. Asakura Scattering of light from multilayer systems with rough boundaries, I. Ohlídal, K. Navrátil, M. Ohlídal Random walk and diffusion-like models of photon migration in turbid media, A.H. Gandjbakhche, G.H. Weiss

1–158 159–181 183–248 249–331 333–402

VOLUME 35 (1996) 1 2 3 4 5 6

Transverse patterns in wide-aperture nonlinear optical systems, N.N. Rosanov Optical spectroscopy of single molecules in solids, M. Orrit, J. Bernard, R. Brown, B. Lounis Interferometric multispectral imaging, K. Itoh Interferometric methods for artwork diagnostics, D. Paoletti, G. Schirripa Spagnolo Coherent population trapping in laser spectroscopy, E. Arimondo Quantum phase properties of nonlinear optical phenomena, R. Tana´s, A. Miranowicz, Ts. Gantsog

1– 60 61–144 145–196 197–255 257–354 355–446

VOLUME 36 (1996) 1 2 3 4 5

Nonlinear propagation of strong laser pulses in chalcogenide glass films, V. Chumash, I. Cojocaru, E. Fazio, F. Michelotti, M. Bertolotti Quantum phenomena in optical interferometry, P. Hariharan, B.C. Sanders Super-resolution by data inversion, M. Bertero, C. De Mol Radiative transfer: new aspects of the old theory, Yu.A. Kravtsov, L.A. Apresyan Photon wave function, I. Bialynicki-Birula

1– 47 49–128 129–178 179–244 245–294

VOLUME 37 (1997) 1 2 3 4 5 6

The Wigner distribution function in optics and optoelectronics, D. Dragoman Dispersion relations and phase retrieval in optical spectroscopy, K.-E. Peiponen, E.M. Vartiainen, T. Asakura Spectra of molecular scattering of light, I.L. Fabelinskii Soliton communication systems, R.-J. Essiambre, G.P. Agrawal Local fields in linear and nonlinear optics of mesoscopic systems, O. Keller Tunneling times and superluminality, R.Y. Chiao, A.M. Steinberg

1– 56 57– 94 95–184 185–256 257–343 345–405

VOLUME 38 (1998) 1 2 3

Nonlinear optics of stratified media, S. Dutta Gupta Optical aspects of interferometric gravitational-wave detectors, P. Hello Thermal properties of vertical-cavity surface-emitting semiconductor lasers, W. Nakwaski, M. Osi´nski 4 Fractional transformations in optics, A.W. Lohmann, D. Mendlovic, Z. Zalevsky 5 Pattern recognition with nonlinear techniques in the Fourier domain, B. Javidi, J.L. Horner 6 Free-space optical digital computing and interconnection, J. Jahns

1– 84 85–164 165–262 263–342 343–418 419–513

Contents of previous volumes

583

VOLUME 39 (1999) 1 2

Theory and applications of complex rays, Yu.A. Kravtsov, G.W. Forbes, A.A. Asatryan Homodyne detection and quantum-state reconstruction, D.-G. Welsch, W. Vogel, T. Opatrný 3 Scattering of light in the eikonal approximation, S.K. Sharma, D.J. Somerford 4 The orbital angular momentum of light, L. Allen, M.J. Padgett, M. Babiker 5 The optical Kerr effect and quantum optics in fibers, A. Sizmann, G. Leuchs

1– 62 63–211 213–290 291–372 373–469

VOLUME 40 (2000) 1 2 3 4

Polarimetric optical fibers and sensors, T.R. Woli´nski Digital optical computing, J. Tanida, Y. Ichioka Continuous measurements in quantum optics, V. Peˇrinová, A. Lukš Optical systems with improved resolving power, Z. Zalevsky, D. Mendlovic, A.W. Lohmann 5 Diffractive optics: electromagnetic approach, J. Turunen, M. Kuittinen, F. Wyrowski 6 Spectroscopy in polychromatic fields, Z. Ficek, H.S. Freedhoff

1– 75 77–114 115–269 271–341 343–388 389–441

VOLUME 41 (2000) 1 2 3 4 5 6 7

Nonlinear optics in microspheres, M.H. Fields, J. Popp, R.K. Chang Principles of optical disk data storage, J. Carriere, R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur Ellipsometry of thin film systems, I. Ohlídal, D. Franta Optical true-time delay control systems for wideband phased array antennas, R.T. Chen, Z. Fu Quantum statistics of nonlinear optical couplers, J. Peˇrina Jr, J. Peˇrina Quantum phase difference, phase measurements and Stokes operators, A. Luis, L.L. Sánchez-Soto Optical solitons in media with a quadratic nonlinearity, C. Etrich, F. Lederer, B.A. Malomed, T. Peschel, U. Peschel

1– 95 97–179 181–282 283–358 359–417 419–479 483–567

VOLUME 42 (2001) 1 2 3 4 5 6

Quanta and information, S.Ya. Kilin Optical solitons in periodic media with resonant and off-resonant nonlinearities, G. Kurizki, A.E. Kozhekin, T. Opatrný, B.A. Malomed Quantum Zeno and inverse quantum Zeno effects, P. Facchi, S. Pascazio Singular optics, M.S. Soskin, M.V. Vasnetsov Multi-photon quantum interferometry, G. Jaeger, A.V. Sergienko Transverse mode shaping and selection in laser resonators, R. Oron, N. Davidson, A.A. Friesem, E. Hasman

1– 91 93–146 147–217 219–276 277–324 325–386

VOLUME 43 (2002) 1 2 3

Active optics in modern large optical telescopes, L. Noethe Variational methods in nonlinear fiber optics and related fields, B.A. Malomed Optical works of L.V. Lorenz, O. Keller

1– 69 71–193 195–294

584

Contents of previous volumes

4

Canonical quantum description of light propagation in dielectric media, A. Lukš, V. Peˇrinová 5 Phase space correspondence between classical optics and quantum mechanics, D. Dragoman 6 “Slow” and “fast” light, R.W. Boyd, D.J. Gauthier 7 The fractional Fourier transform and some of its applications to optics, A. Torre

295–431 433–496 497–530 531–596

VOLUME 44 (2002) 1 2 3

Chaotic dynamics in semiconductor lasers with optical feedback, J. Ohtsubo Femtosecond pulses in optical fibers, F.G. Omenetto Instantaneous optics of ultrashort broadband pulses and rapidly varying media, A.B. Shvartsburg, G. Petite 4 Optical coherence tomography, A.F. Fercher, C.K. Hitzenberger 5 Modulational instability of electromagnetic waves in inhomogeneous and in discrete media, F.Kh. Abdullaev, S.A. Darmanyan, J. Garnier

1– 84 85–141 143–214 215–301 303–366

VOLUME 45 (2003) 1 2 3 4 5 6

Anamorphic beam shaping for laser and diffuse light, N. Davidson, N. Bokor Ultra-fast all-optical switching in optical networks, I. Glesk, B.C. Wang, L. Xu, V. Baby, P.R. Prucnal Generation of dark hollow beams and their applications, J. Yin, W. Gao, Y. Zhu Two-photon lasers, D.J. Gauthier Nonradiating sources and other “invisible” objects, G. Gbur Lasing in disordered media, H. Cao

1– 51 53–117 119–204 205–272 273–315 317–370

VOLUME 46 (2004) 1 2

Ultrafast solid-state lasers, U. Keller Multiple scattering of light from randomly rough surfaces, A.V. Shchegrov, A.A. Maradudin, E.R. Méndez 3 Laser-diode interferometry, Y. Ishii 4 Optical realizations of quantum teleportation, J. Gea-Banacloche 5 Intensity-field correlations of non-classical light, H.J. Carmichael, G.T. Foster, L.A. Orozco, J.E. Reiner, P.R. Rice

1–115 117–241 243–309 311–353 355–404

VOLUME 47 (2005) 1 2 3 4 5 6

Multistep parametric processes in nonlinear optics, S.M. Saltiel, A.A. Sukhorukov, Y.S. Kivshar Modes of wave-chaotic dielectric resonators, H.E. Türeci, H.G.L. Schwefel, Ph. Jacquod, A.D. Stone Nonlinear and quantum optics of atomic and molecular fields, C.P. Search, P. Meystre Space-variant polarization manipulation, E. Hasman, G. Biener, A. Niv, V. Kleiner Optical vortices and vortex solitons, A.S. Desyatnikov, Y.S. Kivshar, L.L. Torner Phase imaging and refractive index tomography for X-rays and visible rays, K. Iwata

1– 73 75–137 139–214 215–289 291–391 393–432

Contents of previous volumes

585

VOLUME 48 (2005) 1 2 3 4 5

Laboratory post-engineering of microstructured optical fibers, B.J. Eggleton, P. Domachuk, C. Grillet, E.C. Mägi, H.C. Nguyen, P. Steinvurzel, M.J. Steel Optical solitons in random media, F. Abdullaev, J. Garnier Curved diffractive optical elements: Design and applications, N. Bokor, N. Davidson The geometric phase, P. Hariharan Synchronization and communication with chaotic laser systems, A. Uchida, F. Rogister, J. García-Ojalvo, R. Roy

1– 34 35–106 107–148 149–201 203–341

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Cumulative index – Volumes 1–49* Abdullaev, F.Kh., S.A. Darmanyan, J. Garnier: Modulational instability of electromagnetic waves in inhomogeneous and in discrete media Abdullaev, F.Kh., J. Garnier: Optical solitons in random media Abelès, 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. Mandel, L.M. Narducci: Dynamical instabilities and pulsations in lasers Agarwal, G.S.: Master equation methods in quantum optics Agranovich, V.M., V.L. Ginzburg: Crystal optics with spatial dispersion Agrawal, G.P.: Single-longitudinal-mode semiconductor lasers Agrawal, G.P., see Essiambre, R.-J. Allen, L., D.G.C. Jones: Mode locking in gas lasers Allen, L., M.J. Padgett, M. Babiker: The orbital angular momentum of light Ammann, E.O.: Synthesis of optical birefringent networks Anderson, R., see Carriere, J. Apresyan, L.A., see Kravtsov, Yu.A. Arimondo, E.: Coherent population trapping in laser spectroscopy Armstrong, J.A., A.W. Smith: Experimental studies of intensity fluctuations in 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. Babiker, M., see Allen, L. Baby, V., see Glesk, I. Baltes, H.P.: On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment Barabanenkov, Yu.N., Yu.A. Kravtsov, V.D. Ozrin, A.I. Saichev: Enhanced backscattering in optics Barakat, R.: The intensity distribution and total illumination of aberration-free diffraction images Barrett, H.H.: The Radon transform and its applications Bashkin, S.: Beam-foil spectroscopy * Volumes I–XL were previously distinguished by roman rather than by arabic numerals.

587

44, 303 48, 35 2, 249 7, 139 16, 71 25, 1 11, 1 9, 235 26, 163 37, 185 9, 179 39, 291 9, 123 41, 97 36, 179 35, 257 6, 211 11, 247 34, 183 37, 57 39, 1 39, 291 45, 53 13,

1

29, 65 1, 67 21, 217 12, 287

588

Cumulative index – Volumes 1–49

Bassett, I.M., W.T. Welford, R. Winston: Nonimaging optics for flux concentration Beckmann, P.: Scattering of light by rough surfaces Benisty, H., C. Weisbuch: Photonic crystals Beran, M.J., J. Oz-Vogt: Imaging through turbulence in the atmosphere Bernard, J., see Orrit, M. Berry, M.V., C. Upstill: Catastrophe optics: morphologies of caustics and their diffraction patterns Bertero, M., C. De Mol: Super-resolution by data inversion Bertolotti, M., see Chumash, V. Bertolotti, M., see Mihalache, D. Beverly III, R.E.: Light emission from high-current surface-spark discharges Bialynicki-Birula, I.: Photon wave function Biener, G.: see Hasman, E. Bloom, A.L.: Gas lasers and their application to precise length measurements Bokor, N., N. Davidson: Curved diffractive optical elements: Design and applications Bokor, N., see Davidson, N. Bouman, M.A., W.A. Van De Grind, P. Zuidema: Quantum fluctuations in vision Bousquet, P., see Rouard, P. Boyd, R.W., D.J. Gauthier: “Slow” and “fast” light Brosseau, C., A. Dogariu: Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield Brown, G.S., see DeSanto, J.A. Brown, R., see Orrit, M. Brunner, W., H. Paul: Theory of optical parametric amplification and oscillation Bryngdahl, O.: Applications of shearing interferometry Bryngdahl, O.: Evanescent waves in optical imaging Bryngdahl, O., T. Scheermesser, F. Wyrowski: Digital halftoning: synthesis of binary images Bryngdahl, O., F. Wyrowski: Digital holography – computer-generated holograms Burch, J.M.: The metrological applications of diffraction gratings Butterweck, H.J.: Principles of optical data-processing Bužek, V., P.L. Knight: Quantum interference, superposition states of light, and nonclassical effects Cagnac, B., see Giacobino, E. Cao, H.: Lasing in disordered media Carmichael, H.J., G.T. Foster, L.A. Orozco, J.E. Reiner, P.R. Rice: Intensity-field correlations of non-classical light Carriere, J., R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur: Principles of optical disk data storage Casasent, D., D. Psaltis: Deformation invariant, space-variant optical pattern recognition Ceglio, N.M., D.W. Sweeney: Zone plate coded imaging: theory and applications Cerf, N.J., J. Fiurášek: Optical quantum cloning Chang, R.K., see Fields, M.H.

27, 161 6, 53 49, 177 33, 319 35, 61 18, 257 36, 129 36, 1 27, 227 16, 357 36, 245 47, 215 9, 1 48, 107 45, 1 22, 77 4, 145 43, 497 49, 315 23, 1 35, 61 15, 1 4, 37 11, 167 33, 389 28, 1 2, 73 19, 211 34,

1

17, 85 45, 317 46, 355 41, 97 16, 289 21, 287 49, 455 41, 1

Cumulative index – Volumes 1–49 Charnotskii, M.I., J. Gozani, V.I. Tatarskii, V.U. Zavorotny: Wave propagation theories in random media based on the path-integral approach Chen, R.T., Z. Fu: Optical true-time delay control systems for wideband phased array antennas Chiao, R.Y., A.M. Steinberg: Tunneling times and superluminality Choi, J., see Carriere, J. Christensen, J.L., see Rosenblum, W.M. Christov, I.P.: Generation and propagation of ultrashort optical pulses Chumash, V., I. Cojocaru, E. Fazio, F. Michelotti, M. Bertolotti: Nonlinear propagation of strong laser pulses in chalcogenide glass films Clair, J.J., C.I. Abitbol: Recent advances in phase profiles generation Clarricoats, P.J.B.: Optical fibre waveguides – a review Cohen-Tannoudji, C., A. Kastler: Optical pumping Cojocaru, I., see Chumash, V. Cole, T.W.: Quasi-optical techniques of radio astronomy Colombeau, B., see Froehly, C. Cook, R.J.: Quantum jumps Courtès, G., P. Cruvellier, M. Detaille: 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 Cruvellier, P., see Courtès, G. Cummins, H.Z., H.L. Swinney: Light beating spectroscopy Dainty, J.C.: The statistics of speckle patterns Dändliker, R.: Heterodyne holographic interferometry Darmanyan, S.A., see Abdullaev, F.Kh. Dattoli, G., L. Giannessi, A. Renieri, A. Torre: Theory of Compton free electron lasers Davidson, N., N. Bokor: Anamorphic beam shaping for laser and diffuse light Davidson, N., see Bokor, N. Davidson, N., see Oron, R. Decker Jr, J.A., see Harwit, M. Delano, E., R.J. Pegis: Methods of synthesis for dielectric multilayer filters Demaria, A.J.: Picosecond laser pulses De Mol, C., see Bertero, M. DeSanto, J.A., G.S. Brown: Analytical techniques for multiple scattering from rough surfaces Desyatnikov, A.S., Y.S. Kivshar, L. Torner: Optical vortices and vortex solitons De Sterke, C.M., J.E. Sipe: Gap solitons Detaille, M., see Courtès, G. Dexter, D.L., see Smith, D.Y. Dogariu, A., see Brosseau, C. Domachuk, P., see Eggleton, B.J. Dragoman, D.: The Wigner distribution function in optics and optoelectronics

589

32, 203 41, 283 37, 345 41, 97 13, 69 29, 199 36, 1 16, 71 14, 327 5, 1 36, 1 15, 187 20, 63 28, 361 20, 1 26, 349 11, 223 20, 1 8, 133 14, 1 17, 1 44, 303 31, 321 45, 1 48, 107 42, 325 12, 101 7, 67 9, 31 36, 129 23, 1 47, 291 33, 203 20, 1 10, 165 49, 315 48, 1 37, 1

590

Cumulative index – Volumes 1–49

Dragoman, D.: Phase space correspondence between classical optics and quantum mechanics Drexhage, K.H.: Interaction of light with monomolecular dye layers Duguay, M.A.: The ultrafast optical Kerr shutter Dušek, M., N. Lütkenhaus, M. Hendrych: Quantum cryptography 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 Eggleton, B.J., P. Domachuk, C. Grillet, E.C. Mägi, H.C. Nguyen, P. Steinvurzel, M.J. Steel: Laboratory post-engineering of microstructured optical fibers 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 Essiambre, R.-J., G.P. Agrawal: Soliton communication systems Etrich, C., F. Lederer, B.A. Malomed, T. Peschel, U. Peschel: Optical solitons in media with a quadratic nonlinearity Fabelinskii, I.L.: Spectra of molecular scattering of light Fabre, C., see Reynaud, S. Facchi, P., S. Pascazio: Quantum Zeno and inverse quantum Zeno effects Fante, R.L.: Wave propagation in random media: a systems approach Fazio, E., see Chumash, V. Fercher, A.F., C.K. Hitzenberger: Optical coherence tomography Ficek, Z., H.S. Freedhoff: Spectroscopy in polychromatic fields Fields, M.H., J. Popp, R.K. Chang: Nonlinear optics in microspheres Fiorentini, A.: Dynamic characteristics of visual processes Fiurášek, J., see Cerf, N.J. Flytzanis, C., F. Hache, M.C. Klein, D. Ricard, 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. Foster, G.A., see Carmichael, H.J. Françon, M., S. Mallick: Measurement of the second order degree of coherence Franta, D., see Ohlídal, I. Freedhoff, H.S., see Ficek, Z. Freilikher, V.D., S.A. Gredeskul: Localization of waves in media with one-dimensional disorder Frieden, B.R.: Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions Friesem, A.A., see Oron, R. Froehly, C., B. Colombeau, M. Vampouille: Shaping and analysis of picosecond light pulses Fry, G.A.: The optical performance of the human eye Fu, Z., see Chen, R.T.

43, 433 12, 163 14, 161 49, 381 31, 189 38, 1 7, 359 48,

1

21, 355 16, 233 37, 185 41, 483 37, 95 30, 1 42, 147 22, 341 36, 1 44, 215 40, 389 41, 1 1, 253 49, 455 29, 321 4, 1 39, 1 46, 355 6, 71 41, 181 40, 389 30, 137 9, 311 42, 325 20, 63 8, 51 41, 283

Cumulative index – Volumes 1–49 Gabor, D.: Light and information Gamo, H.: Matrix treatment of partial coherence Gandjbakhche, A.H., G.H. Weiss: Random walk and diffusion-like models of photon migration in turbid media Gantsog, Ts., see Tana´s, R. Gao, W., see Yin, J. García-Ojalvo, J., see Uchida, A. Garnier, J., see Abdullaev, F.Kh. Garnier, J., see Abdullaev F.Kh. Gauthier, D.J.: Two-photon lasers Gauthier, D.J., see Boyd, R.W. Gbur, G.: Nonradiating sources and other “invisible” objects Gea-Banacloche, J.: Optical realizations of quantum teleportation Ghatak, A., K. Thyagarajan: Graded index optical waveguides: a review Ghatak, A.K., see Sodha, M.S. Giacobino, E., B. Cagnac: Doppler-free multiphoton spectroscopy Giacobino, E., see Reynaud, S. Giannessi, L., see Dattoli, G. Ginzburg, V.L.: Radiation by uniformly moving sources. Vavilov–Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena Ginzburg, V.L., see Agranovich, V.M. Giovanelli, R.G.: Diffusion through non-uniform media Glaser, I.: Information processing with spatially incoherent light Glesk, I., B.C. Wang, L. Xu, V. Baby, P.R. Prucnal: Ultra-fast all-optical switching in optical networks Gniadek, K., J. Petykiewicz: 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 Gredeskul, S.A., see Freilikher, V.D. Grillet, C., see Eggleton, B.J. Hache, F., see Flytzanis, C. Hall, D.G.: Optical waveguide diffraction gratings: coupling between guided modes Hariharan, P.: Colour holography Hariharan, P.: Interferometry with lasers Hariharan, P.: The geometric phase 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. Hasman, E., G. Biener, A. Niv, V. Kleiner: Space-variant polarization manipulation Hasman, E., see Oron, R. Heidmann, A., see Reynaud, S. Hello, P.: Optical aspects of interferometric gravitational-wave detectors

591 1, 109 3, 187 34, 333 35, 355 45, 119 48, 203 44, 303 48, 35 45, 205 43, 497 45, 273 46, 311 18, 1 13, 169 17, 85 30, 1 31, 321 32, 267 9, 235 2, 109 24, 389 45, 53 9, 281 8, 1 32, 203 12, 233 30, 137 48, 1 29, 321 29, 1 20, 263 24, 103 48, 149 36, 49 12, 101 30, 205 47, 215 42, 325 30, 1 38, 85

592

Cumulative index – Volumes 1–49

Helstrom, C.W.: Quantum detection theory Hendrych, M., see Dušek, M. Herriot, D.R.: Some applications of lasers to interferometry Hitzenberger, C.K., see Fercher, A.F. Horner, J.L., see Javidi, B. Huang, T.S.: Bandwidth compression of optical images

10, 289 49, 381 6, 171 44, 215 38, 343 10, 1

Ichioka, Y., see Tanida, J. Imoto, N., see Yamamoto, Y. Ishii, Y.: Laser-diode interferometry Itoh, K.: Interferometric multispectral imaging Iwata, K.: Phase imaging and refractive index tomography for X-rays and visible rays

40, 77 28, 87 46, 243 35, 145 47, 393

Jacobsson, R.: Light reflection from films of continuously varying refractive index Jacquinot, P., B. Roizen-Dossier: Apodisation Jacquod, Ph., see Türeci, H.E. Jaeger, G., A.V. Sergienko: Multi-photon quantum interferometry 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. Joshi, A., M. Xiao: Controlling nonlinear optical processes in multi-level atomic systems

5, 247 3, 29 47, 75 42, 277 38, 419 20, 325 38, 343 9, 179 49, 97

Kastler, A., see Cohen-Tannoudji, C. Keller, O.: Local fields in linear and nonlinear optics of mesoscopic systems Keller, O.: Optical works of L.V. Lorenz Keller, U.: Ultrafast solid-state lasers Khoo, I.C.: Nonlinear optics of liquid crystals Khulbe, P., see Carriere, J. Kielich, S.: Multi-photon scattering molecular spectroscopy Kilin, S.Ya.: Quanta and information Kinosita, K.: Surface deterioration of optical glasses Kitagawa, M., see Yamamoto, Y. Kivshar, Y.S., see Desyatnikov, A.S. Kivshar, Y.S., see Saltiel, S.M. Klein, M.C., see Flytzanis, C. Kleiner, V., see Hasman, E. Klyatskin, V.I.: The imbedding method in statistical boundary-value wave problems Knight, P.L., see Bužek, V. Kodama, Y., A. Hasegawa: Theoretical foundation of optical-soliton concept in fibers Koppelman, G.: Multiple-beam interference and natural modes in open resonators Kottler, F.: The elements of radiative transfer Kottler, F.: Diffraction at a black screen, Part I: Kirchhoff’s theory

5, 1 37, 257 43, 195 46, 1 26, 105 41, 97 20, 155 42, 1 4, 85 28, 87 47, 291 47, 1 29, 321 47, 215 33, 1 34, 1 30, 205 7, 1 3, 1 4, 281

Cumulative index – Volumes 1–49

593

Kottler, F.: Diffraction at a black screen, Part II: electromagnetic theory Kozhekin, A.E., see Kurizki, G. Kravtsov, Yu.A.: Rays and caustics as physical objects Kravtsov, Yu.A., L.A. Apresyan: Radiative transfer: new aspects of the old theory Kravtsov, Yu.A., G.W. Forbes, A.A. Asatryan: Theory and applications of complex rays Kravtsov, Yu.A., see Barabanenkov, Yu.N. Kubota, H.: Interference color Kuittinen, M., see Turunen, J. Kurizki, G., A.E. Kozhekin, T. Opatrný, B.A. Malomed: Optical solitons in periodic media with resonant and off-resonant nonlinearities

6, 331 42, 93 26, 227 36, 179 39, 1 29, 65 1, 211 40, 343

Labeyrie, A.: High-resolution techniques in optical astronomy Lean, E.G.: Interaction of light and acoustic surface waves Lederer, F., see Etrich, C. Lee, W.-H.: Computer-generated holograms: techniques and applications Leith, E.N., J. Upatnieks: Recent advances in holography Letokhov, V.S.: Laser selective photophysics and photochemistry Leuchs, G., see Sizmann, A. Levi, L.: Vision in communication Li, L., see Carriere, J. Lipson, H., C.A. Taylor: X-ray crystal-structure determination as a branch of physical optics Lohmann, A.W., D. Mendlovic, Z. Zalevsky: Fractional transformations in optics Lohmann, A.W., see Zalevsky, Z. Lounis, B., see Orrit, M. Lugiato, L.A.: Theory of optical bistability Luis, A., L.L. Sánchez-Soto: Quantum phase difference, phase measurements and Stokes operators Lukš, A., V. Peˇrinová: Canonical quantum description of light propagation in dielectric media Lukš, A., see Peˇrinová, V. Lukš, A., see Peˇrinová, V. Lütkenhaus, N., see Dušek, M.

14, 47 11, 123 41, 483 16, 119 6, 1 16, 1 39, 373 8, 343 41, 97

43, 295 33, 129 40, 117 49, 381

Machida, S., see Yamamoto, Y. Mägi, E.C., see Eggleton, B.J. Mahajan, V.N.: Gaussian apodization and beam propagation Mainfray, 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, V.I. Mallick, S., see Françon, M. Malomed, B.A.: Variational methods in nonlinear fiber optics and related fields Malomed, B.A., see Etrich, C. Malomed, B.A., see Kurizki, G. Mandel, L.: Fluctuations of light beams

28, 87 48, 1 49, 1 32, 313 22, 1 33, 261 6, 71 43, 71 41, 483 42, 93 2, 181

42, 93

5, 287 38, 263 40, 271 35, 61 21, 69 41, 419

594

Cumulative index – Volumes 1–49

Mandel, L.: The case for and against semiclassical radiation theory Mandel, P., see Abraham, N.B. Mansuripur, M., see Carriere, J. Manus, C., see Mainfray, G. Maradudin, A.A., see Shchegrov, A.V. Marchand, E.W.: Gradient index lenses Martin, P.J., R.P. Netterfield: 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 Meessen, A., see Rouard, P. Mehta, C.L.: Theory of photoelectron counting Mendez, E.R., see Shchegrov, A.V. Mendlovic, D., see Lohmann, A.W. Mendlovic, D., see Zalevsky, Z. Meystre, P.: Cavity quantum optics and the quantum measurement process Meystre, P., see Search, C.P. Michelotti, F., see Chumash, V. Mihalache, D., M. Bertolotti, C. Sibilia: Nonlinear wave propagation in planar structures Mikaelian, A.L.: Self-focusing media with variable index of refraction Mikaelian, A.L., M.L. Ter-Mikaelian: Quasi-classical theory of laser radiation Mills, D.L., K.R. Subbaswamy: Surface and size effects on the light scattering spectra of solids Milonni, P.W., B. Sundaram: Atoms in strong fields: photoionization and chaos Miranowicz, A., see Tana´s, 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 Nakwaski, W., M. Osi´nski: Thermal properties of vertical-cavity surface-emitting semiconductor lasers Narayan, R., see Carriere, J. Narducci, L.M., see Abraham, N.B. Navrátil, K., see Ohlídal, I. Netterfield, R.P., see Martin, P.J. Nguyen, H.C., see Eggleton, B.J. Nishihara, H., T. Suhara: Micro Fresnel lenses Niv, A., see Hasman, E. Noethe, L.: Active optics in modern large optical telescopes Ohlídal, I., D. Franta: Ellipsometry of thin film systems Ohlídal, I., K. Navrátil, M. Ohlídal: Scattering of light from multilayer systems with rough boundaries Ohlídal, M., see Ohlídal, I.

13, 27 25, 1 41, 97 32, 313 46, 117 11, 305 23, 113 22, 145 21, 1 15, 77 8, 373 46, 117 38, 263 40, 271 30, 261 47, 139 36, 1 27, 227 17, 279 7, 231 19, 45 31, 1 35, 355 1, 31 19, 1 5, 199 8, 201

38, 165 41, 97 25, 1 34, 249 23, 113 48, 1 24, 1 47, 215 43, 1 41, 181 34, 249 34, 249

Cumulative index – Volumes 1–49 Ohtsu, M., T. Tako: Coherence in semiconductor lasers Ohtsubo, J.: Chaotic dynamics in semiconductor lasers with optical feedback Okamoto, T., T. Asakura: The statistics of dynamic speckles Okoshi, T.: Projection-type holography Omenetto, F.G.: Femtosecond pulses in optical fibers Ooue, S.: The photographic image Opatrný, T., see Kurizki, G. Opatrný, T., see Welsch, D.-G. Oron, R., N. Davidson, A.A. Friesem, E. Hasman: Transverse mode shaping and selection in laser resonators Orozco, L.A., see Carmichael, H.J. Orrit, M., J. Bernard, R. Brown, B. Lounis: Optical spectroscopy of single molecules in solids Osi´nski, M., see Nakwaski, W. Ostrovskaya, G.V., Yu.I. Ostrovsky: Holographic methods of plasma diagnostics Ostrovsky, Yu.I., V.P. Shchepinov: Correlation holographic and speckle interferometry Ostrovsky, Yu.I., see Ostrovskaya, G.V. Oughstun, K.E.: Unstable resonator modes Oz-Vogt, J., see Beran, M.J. Ozrin, V.D., see Barabanenkov, Yu.N. Padgett, M.J., see Allen, L. Pal, B.P.: Guided-wave optics on silicon: physics, technology and status Paoletti, D., G. Schirripa Spagnolo: Interferometric methods for artwork diagnostics Pascazio, S., see Facchi, P. 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. Peiponen, K.-E., E.M. Vartiainen, T. Asakura: Dispersion relations and phase retrieval in optical spectroscopy Peng, C., see Carriere, J. Pe˘rina, J.: Photocount statistics of radiation propagating through random and nonlinear media Peˇrina, J., see Peˇrina Jr, J. Peˇrina Jr, J., J. Peˇrina: Quantum statistics of nonlinear optical couplers Peˇrinová, V., A. Lukš: Quantum statistics of dissipative nonlinear oscillators Peˇrinová, V., A. Lukš: Continuous measurements in quantum optics Peˇrinová, V., see Lukš, A. Pershan, P.S.: Non-linear optics Peschel, T., see Etrich, C. Peschel, U., see Etrich, C. Petite, G., see Shvartsburg, A.B. Petykiewicz, J., see Gniadek, K. Picht, J.: The wave of a moving classical electron

595 25, 191 44, 1 34, 183 15, 139 44, 85 7, 299 42, 93 39, 63 42, 325 46, 355 35, 61 38, 165 22, 197 30, 87 22, 197 24, 165 33, 319 29, 65 39, 291 32, 1 35, 197 42, 147 27, 1 15, 1 1, 1 7, 67 37, 57 41, 97 18, 127 41, 359 41, 359 33, 129 40, 115 43, 295 5, 83 41, 483 41, 483 44, 143 9, 281 5, 351

596

Cumulative index – Volumes 1–49

Popov, E.: Light diffraction by relief gratings: a macroscopic and microscopic view Popp, J., see Fields, M.H. 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 Prucnal, P.R., see Glesk, I. Psaltis, D., Y. Qiao: Adaptive multilayer optical networks Psaltis, D., see Casasent, D.

27, 315 34, 159 45, 53 31, 227 16, 289

Qiao, Y., see Psaltis, D.

31, 227

Raymer, M.G., I.A. Walmsley: The quantum coherence properties of stimulated Raman scattering Reiner, J.E., see Carmichael, H.J. Renieri, A., see Dattoli, G. Reynaud, S., A. Heidmann, E. Giacobino, C. Fabre: Quantum fluctuations in optical systems Ricard, D., see Flytzanis, C. Rice, P.R., see Carmichael, H.J. Riseberg, L.A., M.J. Weber: Relaxation phenomena in rare-earth luminescence Risken, H.: Statistical properties of laser light Roddier, F.: The effects of atmospheric turbulence in optical astronomy Rogister, F., see Uchida, A. Roizen-Dossier, B., see Jacquinot, P. Ronchi, L., see Wang Shaomin Rosanov, N.N.: Transverse patterns in wide-aperture nonlinear optical systems Rosenblum, W.M., J.L. Christensen: Objective and subjective spherical aberration measurements of the human eye Rothberg, L.: Dephasing-induced coherent phenomena Rouard, P., P. Bousquet: Optical constants of thin films Rouard, P., A. Meessen: Optical properties of thin metal films Roussignol, Ph., see Flytzanis, C. Roy, R., see Uchida, A. Rubinowicz, A.: The Miyamoto–Wolf diffraction wave Rudolph, D., see Schmahl, G. Saichev, A.I., see Barabanenkov, Yu.N. Saito, S., see Yamamoto, Y. Sakai, H., see Vanasse, G.A. Saleh, B.E.A., see Teich, M.C. Saltiel, S.M., A.A. Sukhorukov, Y.S. Kivshar: Multistep parametric processes in nonlinear optics Sánchez-Soto, L.L., see Luis, A. Sanders, B.C., see Hariharan, P. Scheermesser, T., see Bryngdahl, O.

31, 139 41, 1

28, 181 46, 355 31, 321 30, 1 29, 321 46, 355 14, 89 8, 239 19, 281 48, 203 3, 29 25, 279 35, 1 13, 69 24, 39 4, 145 15, 77 29, 321 48, 203 4, 199 14, 195 29, 65 28, 87 6, 259 26, 1 47, 1 41, 419 36, 49 33, 389

Cumulative index – Volumes 1–49 Schieve, W.C., see Englund, J.C. Schirripa Spagnolo, G., see Paoletti, D. Schmahl, G., D. Rudolph: Holographic diffraction gratings Schubert, M., B. Wilhelmi: The mutual dependence between coherence properties of light and nonlinear optical processes Schulz, G.: Aspheric surfaces Schulz, G., J. Schwider: Interferometric testing of smooth surfaces Schwefel, H.G.L., see Türeci, H.E. Schwider, J.: Advanced evaluation techniques in interferometry Schwider, J., see Schulz, G. Scully, M.O., K.G. Whitney: Tools of theoretical quantum optics Search, C.P., P. Meystre: Nonlinear and quantum optics of atomic and molecular fields Senitzky, I.R.: Semiclassical radiation theory within a quantum-mechanical framework Sergienko, A.V., see Jaeger, G. Sharma, S.K., D.J. Somerford: Scattering of light in the eikonal approximation Shchegrov, A.V., A.A. Maradudin, E.R. Méndez: Multiple scattering of light from randomly rough surfaces Shchepinov, V.P., see Ostrovsky, Yu.I. Shvartsburg, A.B., G. Petite: Instantaneous optics of ultrashort broadband pulses and rapidly varying media Sibilia, C., see Mihalache, D. Simpson, J.R., see Dutta, N.K. Sipe, J.E., see De Sterke, C.M. Sipe, J.E., see Van Kranendonk, J. 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, V.K. 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 Soskin, M.S., M.V. Vasnetsov: Singular optics Spreeuw, R.J.C., J.P. Woerdman: Optical atoms Steel, M.J., see Eggleton, B.J. Steel, W.H.: Two-beam interferometry Steinberg, A.M., see Chiao, R.Y. Steinvurzel, P., see Eggleton, B.J. Stoicheff, B.P., see Jamroz, W. Stone, A.D., see Türeci, H.E. Strohbehn, J.W.: Optical propagation through the turbulent atmosphere Stroke, G.W.: Ruling, testing and use of optical gratings for high-resolution spectroscopy

597 21, 355 35, 197 14, 195 17, 163 25, 349 13, 93 47, 75 28, 271 13, 93 10, 89 47, 139 16, 413 42, 277 39, 213 46, 117 30, 87 44, 143 27, 227 31, 189 33, 203 15, 245 10, 229 39, 373 12, 53 6, 211 10, 165 10, 45 21, 355 13, 169 39, 213 27, 109 42, 219 31, 263 48, 1 5, 145 37, 345 48, 1 20, 325 47, 75 9, 73 2, 1

598

Cumulative index – Volumes 1–49

Subbaswamy, K.R., see Mills, D.L. Suhara, T., see Nishihara, H. Sukhorukov, A.A., see Saltiel, S.M. 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. Swinney, H.L., see Cummins, H.Z.

19, 45 24, 1 47, 1 31, 1 12, 1 21, 287 8, 133

Tako, T., see Ohtsu, M. Tanaka, K.: Paraxial theory in optical design in terms of Gaussian brackets Tana´s, R., A. Miranowicz, Ts. Gantsog: Quantum phase properties of nonlinear optical phenomena Tango, W.J., R.Q. Twiss: Michelson stellar interferometry Tanida, J., Y. Ichioka: Digital optical computing Tatarskii, V.I., V.U. Zavorotnyi: Strong fluctuations in light propagation in a randomly inhomogeneous medium Tatarskii, V.I., see Charnotskii, M.I. Taylor, C.A., see Lipson, H. Teich, M.C., B.E.A. Saleh: Photon bunching and antibunching Ter-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 Torner, L., see Desyatnikov, A.S. Torre, A.: The fractional Fourier transform and some of its applications to optics Torre, A., see Dattoli, G. Tripathi, V.K., see Sodha, M.S. Tsujiuchi, J.: Correction of optical images by compensation of aberrations and by spatial frequency filtering Türeci, H.E., H.G.L. Schwefel, Ph. Jacquod, A.D. Stone: Modes of wave-chaotic dielectric resonators Turunen, J., M. Kuittinen, F. Wyrowski: Diffractive optics: electromagnetic approach Twiss, R.Q., see Tango, W.J.

25, 191 23, 63

47, 75 40, 343 17, 239

Uchida, A., F. Rogister, J. García-Ojalvo, R. Roy: Synchronization and communication with chaotic laser systems Upatnieks, J., see Leith, E.N. Upstill, C., see Berry, M.V. Ushioda, S.: Light scattering spectroscopy of surface electromagnetic waves in solids

48, 203 6, 1 18, 257 19, 139

Vampouille, M., see Froehly, C. Vanasse, G.A., H. Sakai: Fourier spectroscopy Van De Grind, W.A., see Bouman, M.A. Van Heel, A.C.S.: Modern alignment devices

20, 63 6, 259 22, 77 1, 289

35, 355 17, 239 40, 77 18, 204 32, 203 5, 287 26, 1 7, 231 8, 201 7, 169 18, 1 23, 183 47, 291 43, 531 31, 321 13, 169 2, 131

Cumulative index – Volumes 1–49 Van Kranendonk, J., J.E. Sipe: Foundations of the macroscopic electromagnetic theory of dielectric media Vartiainen, E.M., see Peiponen, K.-E. Vasnetsov, M.V., see Soskin, M.S. Vernier, P.J.: Photoemission Vlad, V.I., D. Malacara: Direct spatial reconstruction of optical phase from phasemodulated images Vogel, W., see Welsch, D.-G. Walmsley, I.A., see Raymer, M.G. Wang, B.C., see Glesk, I. Wang Shaomin, L. Ronchi: Principles and design of optical arrays Weber, M.J., see Riseberg, L.A. Weigelt, G.: Triple-correlation imaging in optical astronomy Weisbuch, C., see Benisty, H. 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. Opatrný: 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. Woli´nski, T.R.: Polarimetric optical fibers and sensors 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, O. Wyrowski, F., see Bryngdahl, O. Wyrowski, F., see Turunen, J. Xiao, M., see Joshi, A. Xu, L., see Glesk, I. Yamaguchi, I.: Fringe formations in deformation and vibration measurements using laser light Yamaji, K.: Design of zoom lenses Yamamoto, T.: Coherence theory of source-size compensation in interference microscopy Yamamoto, Y., S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa: 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 Yeh, W.-H., see Carriere, J.

599

15, 245 37, 57 42, 219 14, 245 33, 261 39, 63 28, 181 45, 53 25, 279 14, 89 29, 293 49, 177 34, 333 4, 241 13, 267 27, 161 39, 63 10, 89 17, 163 27, 161 31, 263 40, 1 1, 155 10, 137 28, 1 33, 389 40, 343 49, 97 45, 53

22, 271 6, 105 8, 295 28, 87 28, 87 32, 145 41, 97

600

Cumulative index – Volumes 1–49

Yin, J., W. Gao, Y. Zhu: Generation of dark hollow beams and their applications 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

45, 119 11, 77 23, 221 32, 61

Zalevsky, Z., D. Mendlovic, A.W. Lohmann: Optical systems with improved resolving power Zalevsky, Z., see Lohmann, A.W. Zavorotny, V.U., see Charnotskii, M.I. Zavorotnyi, V.U., see Tatarskii, V.I. Zhu, Y., see Yin, J. Zuidema, P., see Bouman, M.A.

40, 271 38, 263 32, 203 18, 204 45, 119 22, 77