Series in Optics and Optoelectronics Fast Light, Slow Light and Left-Handed Light P W Milonni Los Alamos, New Mexico Institute of Physics Publishing Bristol and Philadelphia Copyright © 2005 IOP Publishing Ltd. ⃝c IOP Publishing Ltd 2005 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 permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with Universities UK (UUK). British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0926 1 Library of Congress Cataloging-in-Publication Data are available Commissioning Editor: Tom Spicer Editorial Assistant: Leah Fielding Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Victoria Le Billon Marketing: Louise Higham and Ben Thomas Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 929, 150 South Independence Mall West, Philadelphia, PA 19106, USA A Typeset in LT EX2ε by Text 2 Text Limited, Torquay, Devon Printed in the UK by MPG Books Ltd, Bodmin, Cornwall Copyright © 2005 IOP Publishing Ltd. To Enes Novelli Burns, my favourite teacher My books are water; those of the great geniuses is wine. Everybody drinks water. Mark Twain Notebooks and Journals, Volume III (1883–1891) Copyright © 2005 IOP Publishing Ltd. Contents Preface xi 1 In the Beginning 1 1.1 Maxwell’s equations and the velocity of light 1 1.2 Refractive index 5 1.3 Causality and dispersion relations 9 1.4 Signal velocity and Einstein causality 16 1.5 Group velocity 17 1.6 Maxwell’s equations and special relativity: an example 21 1.7 Group velocity can be very small—or zero 24 1.8 The refractive index can be negative 25 1.9 The remainder of this book 25 2 Fast light 26 2.1 Front velocity 26 2.2 Superluminal group velocity 29 2.3 Theoretical considerations of superluminal group velocity 32 2.4 Demonstrations of superluminal group velocity 38 2.4.1 Repetition frequency of mode-locked laser pulses 38 2.4.2 Pulse propagation in linear absorbers 38 2.4.3 Photon tunnelling experiments 39 2.4.4 Gain-doublet experiments 41 2.4.5 Other experiments and viewpoints 44 2.5 No violation of Einstein causality 45 2.6 Bessel beams 50 2.7 Propagation of energy 51 2.8 Precursors 56 2.9 Six velocities 58 3 Quantum theory and light propagation 59 3.1 Fermi’s problem 60 3.1.1 Heisenberg picture 70 3.2 Causality in photodetection theory 73 3.2.1 Causality 78 Copyright © 2005 IOP Publishing Ltd. viii Contents 3.3 Microscopic approach to refractive index and group velocity 81 3.4 EPR correlations and causality 87 3.5 No cloning 88 3.5.1 Teleportation 91 3.6 A superluminal quantum Morse telegraph? 92 3.7 Mirror switching in cavity QED 95 3.8 Pre´cis 101 3.9 Appendix: On Einstein and hidden variables 101 4 Fast light and signal velocity 108 4.1 Experiments on signal velocities 108 4.2 Can the advance of a weak pulse exceed the pulsewidth? 110 4.2.1 Approximation leading to the ARS field equation 117 4.2.2 Signal and noise 118 4.2.3 Physical origin of noise limiting the observability of superluminal group velocity 122 4.2.4 Operator ordering and relation to ARS approach 123 4.2.5 Limit of very small transition frequency 124 4.2.6 Remarks 124 4.3 Signal velocity and photodetection 125 4.4 Absorbers 131 4.5 What is a signal? 131 4.6 Remarks 133 5 Slow light 135 5.1 Some antecedents 135 5.2 Electromagnetically induced transparency 136 5.3 Slow light based on EIT 145 5.3.1 Slow light in an ultracold gas 146 5.3.2 Slow light in a hot gas 147 5.4 Group velocity dispersion 150 5.5 Slow light in solids 152 5.5.1 Coherent population oscillations 152 5.5.2 Spectral hole due to coherent population oscillations 155 5.5.3 Slow light in room-temperature ruby 157 5.5.4 Fast light and slow light in a room-temperature solid 159 5.6 Remarks 162 6 Stopped, stored, and regenerated light 164 6.1 Controlling group velocity 164 6.2 Dark-state polaritons 165 6.3 Stopped and regenerated light 172 6.4 Echoes 175 6.5 Memories 176 6.6 Some related work 178 Copyright © 2005 IOP Publishing Ltd. Contents ix 7 Left-handed light: basic theory 180 7.1 Introduction 180 7.2 Negative ϵ and µ imply negative index 182 7.3 Dispersion 184 7.4 Maxwell’s equations and quantized field 185 7.4.1 Radiative rates in negative-index media 188 7.5 Reversal of the Doppler and Cerenkov effects 190 7.5.1 On photon momentum in a dielectric 192 7.6 Discussion 194 7.7 Fresnel formulas and the planar lens 195 7.8 Evanescent waves 199 7.8.1 Limit to resolution with a conventional lens 202 7.9 The ‘perfect’ lens 202 7.9.1 Evanescent wave incident on an NIM half-space 203 7.9.2 Evanescent wave incident on an NIM slab 204 7.9.3 Surface modes 206 7.10 Elaborations 208 7.11 No fundamental limit to resolution 209 7.12 Summary 209 8 Metamaterials for left-handed light 211 8.1 Negative permittivity 211 8.2 Negative permeability 216 8.2.1 Artificial dielectrics 221 8.3 Realization of negative refractive index 222 8.4 Transmission line metamaterials 226 8.5 Negative refraction in photonic crystals 230 8.6 Remarks 233 Bibliography 235 Index 243 Copyright © 2005 IOP Publishing Ltd. Preface It has been a century since R W Wood observed anomalous dispersion and Sommerfeld, Brillouin and others developed the theory of the propagation of light in anomalously dispersive media. The problem was to reconcile (1) the possibility that the (measurable) group velocity of light could exceed c with (2) the requirement of relativity theory that no signal can be transmitted superluminally. Sommerfeld and Brillouin concluded that a group velocity is not, in general, the velocity with which a signal, properly defined as a carrier of information, can be transmitted. The work of Sommerfeld and Brillouin, especially Brillouin’s Wave Propagation and Group Velocity (1960), is often cited. They focused attention on signal velocity, group velocity, and the velocity of energy propagation; and, according to Brillouin, ‘a galaxy of eminent scientists, from Voigt to Einstein, attached great importance to these fundamental definitions’. But apparently this classic work is not widely read, for otherwise the recent demonstrations of superluminal group velocity would not have sparked so much discussion. The news media, with the hyperbole characteristic of the times, have often as not been misleading or wrong but so have the reported comments of some physicists. The principal development since the publication of Brillouin’s monograph is the experimental study of ‘abnormal’ group velocities—group velocities that are superluminal, infinite, negative, or zero. The literature on the subject has grown substantially. One purpose of this book is to review, vis-a`-vis this development, the most basic ideas about dispersion relations, causality, propagation of light in dispersive media, and the different velocities used to characterize the propagation of light. Another aspect of the subject is the role of quantum effects. Fermi was among the first to discuss the problem of light propagation in quantum electrodynamics at the most basic level, namely the emission of a photon by an atom and its subsequent absorption by another atom. He obtained the right answer, or part of the right answer, for the time dependence of the excitation probability of the second atom. But his approach, based as it was on a certain approximation, did not provide proof of causal propagation and, consequently, the ‘Fermi problem’ has been revisited periodically in the past few decades. Quantum theory ‘protects’ special relativity from what might otherwise Copyright © 2005 IOP Publishing Ltd. appear to be superluminal communication. Thus, it is impossible to use the ‘spooky action at a distance’ suggested by quantum correlations of the Einstein– Podolsky–Rosen (EPR) type to devise a superluminal communication scheme. In one suggested scheme, it is the spontaneous emission noise that prevents superluminal communication when one photon of an EPR pair is amplified by stimulated emission. The fact that such schemes must, in general, be impossible led to the no-cloning theorem. One point that is emphasized here is that any measurable advance in time of a ‘superluminal’ pulse is reduced by noise arising from the field, the medium in which the field propagates, or the detector. The group velocity of light can also be extremely small. ‘Slow light’ with −1 group velocities on the order of 10 m s was first directly observed in 1999 and shortly thereafter it was demonstrated that pulses of light could even be brought to a full stop, stored, and then regenerated. These developments have been based largely on the quantum interference effects associated with electromagnetically induced transparency. Slow light raises less fundamental questions, perhaps, than ‘fast light’ but it might have greater potential for applications. One application might be to quantum memories, as the storage and regeneration of light can be done without loss of information as to the quantum state of the original pulse: this information is temporarily imprinted in the slow-light medium. The ability to coherently control light in this way could also find applications eventually in optical communications. The third major topic addressed in this book is ‘left-handed light’—light propagation in media with negative refraction. Here it is not so much the variation of the refractive index with frequency that matters, as in the case of fast light and slow light, but rather the index itself at a given frequency. Left-handedness refers to the fact that, when the refractive index is negative, the electric field vector E, the magnetic field vector H, and the wavevector k of a plane waveform a left- handed triad. Nature has apparently not produced media with negative refractive indices; however, so-called metamaterials with this property have been created in the laboratory. The propagation of light in metamaterials is predicted to exhibit various unfamiliar properties. For instance, the Doppler effect is reversed, so that a detector moving towards a source of radiation sees a smaller frequency than a stationary observer. Light bends the ‘wrong’ way when it is incident upon a metamaterial and it is theoretically possible to construct a ‘perfect’ lens in a narrow spectral range. The many potential applications of metamaterials have spurred a very rapid growth in the number of publications in this area. The last two chapters are an introduction to some of the foundational work on metamaterials and left-handed light. My recent interest in these areas began with enlightening discussions with R Y Chiao. I also enjoyed talking with other participants in a three-week workshop at the Institute for Theoretical Physics in Santa Barbara in 2002, and discussing related matters on that and other occasions with many excellent Copyright © 2005 IOP Publishing Ltd. physicists including Y Aharonov, J F Babb, S M Barnett, P R Berman, H A Bethe, M S Bigelow, R W Boyd, R J Cook, G D Doolen, J H Eberly, G V Eleftheriades, H Fearn, M Fleischhauer, K Furuya, I R Gabitov, D J Gauthier, S A Glasgow, R J Glauber, D F V James, P L Knight, P G Kwiat, W E Lamb, Jr, U Leonhardt, R Loudon, G J Maclay, L Mandel, M Mojahedi, G Nimtz, K E Oughstun, J Peatross, J B Pendry, E A Power, B Reznik, M O Scully, B Segev, D R Smith, A M Steinberg, L J Wang, H G Winful, E Wolf, and R W Ziolkowski. I have probably left out the names of many other people with whom I had helpful but long-forgotten discussions. I apologize to the many authors whose work I have not cited. There is a huge literature relating to the topics covered in this book, and I have not cited work that I have not read or understood, let alone publications I have not even seen. The three major subjects of this book have attracted particular interest in just the past few years. They are related by the fact that they all involve unusual values or variations of the refractive index. I have tried to focus on the basic underlying physics. The many citations to recent work do not represent an attempt to make this book as up-to-date as possible; it does reflect my opinion that this work is of considerable fundamental importance. I thank Tom Spicer of the Institute of Physics for suggesting this book and for his patience when I failed to finish it by the promised delivery date. Dan Gauthier of Duke University made helpful suggestions for which I am grateful. Peter W Milonni Los Alamos, New Mexico Copyright © 2005 IOP Publishing Ltd. Chapter 1 In the Beginning 1.1 Maxwell’s equations and the velocity of light The variations of the phase velocity or the group velocity of light in different media are of great practical importance. We will be concerned primarily with situations where these variations are unusual and not yet of any practical utility. Our considerations will be based on the laws of electromagnetism: ∇ · E = ρ/ϵ0 (1.1) ∇ · B = 0 (1.2) ∂ B ∇ × E = − (1.3) ∂t ∂ E ∇ × B = µ0 J + ϵ0µ0 . (1.4) ∂t These equations are so incredibly important that we begin with a brief discussion of their conceptual foundations, even though this has been done thousands of times before. The definite pattern formed by iron filings around a bar magnet, or by sawdust around an electrified body, led Faraday to suggest that the space around such objects is filled with lines of force. Electric and magnetic forces, from this point of view, are transmitted by the medium between the objects rather than arising from ‘action at a distance’. Maxwell was greatly impressed and influenced by this idea of what he called an electromagnetic field [1]: Faraday . . . saw lines of force traversing all space where the mathematicians saw centres of force attracting at a distance; Faraday saw a medium where they saw nothing but distance; Faraday sought the seat of the phenomena in real actions going on in the medium, they were satisfied that they had found it in a power of action at a distance . . . Copyright © 2005 IOP Publishing Ltd.