Fast light, slow light, and left-handed light PDF
Preview Fast light, slow light, and left-handed light
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. (cid:1)c IOPPublishingLtd2005 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,recordingorotherwise,withoutthepriorpermission 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 agreementwithUniversitiesUK(UUK). BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary. ISBN0750309261 LibraryofCongressCataloging-in-PublicationDataareavailable CommissioningEditor:TomSpicer EditorialAssistant: LeahFielding ProductionEditor:SimonLaurenson ProductionControl:SarahPlenty CoverDesign:VictoriaLeBillon Marketing:LouiseHighamandBenThomas Published by Institute of Physics Publishing, wholly owned by The Institute of Physics,London InstituteofPhysicsPublishing,DiracHouse,TempleBack,BristolBS16BE,UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 929,150SouthIndependenceMallWest,Philadelphia,PA19106,USA TypesetinLATEX2ε byText2TextLimited,Torquay,Devon PrintedintheUKbyMPGBooksLtd,Bodmin,Cornwall Copyright © 2005 IOP Publishing Ltd. ToEnesNovelliBurns,myfavouriteteacher Mybooksarewater;thoseofthegreatgeniusesiswine. Everybodydrinkswater. MarkTwain NotebooksandJournals,VolumeIII(1883–1891) Copyright © 2005 IOP Publishing Ltd. Contents Preface xi 1 IntheBeginning 1 1.1 Maxwell’sequationsandthevelocityoflight 1 1.2 Refractiveindex 5 1.3 Causalityanddispersionrelations 9 1.4 SignalvelocityandEinsteincausality 16 1.5 Groupvelocity 17 1.6 Maxwell’sequationsandspecialrelativity:anexample 21 1.7 Groupvelocitycanbeverysmall—orzero 24 1.8 Therefractiveindexcanbenegative 25 1.9 Theremainderofthisbook 25 2 Fastlight 26 2.1 Frontvelocity 26 2.2 Superluminalgroupvelocity 29 2.3 Theoreticalconsiderationsofsuperluminalgroupvelocity 32 2.4 Demonstrationsofsuperluminalgroupvelocity 38 2.4.1 Repetitionfrequencyofmode-lockedlaserpulses 38 2.4.2 Pulsepropagationinlinearabsorbers 38 2.4.3 Photontunnellingexperiments 39 2.4.4 Gain-doubletexperiments 41 2.4.5 Otherexperimentsandviewpoints 44 2.5 NoviolationofEinsteincausality 45 2.6 Besselbeams 50 2.7 Propagationofenergy 51 2.8 Precursors 56 2.9 Sixvelocities 58 3 Quantumtheoryandlightpropagation 59 3.1 Fermi’sproblem 60 3.1.1 Heisenbergpicture 70 3.2 Causalityinphotodetectiontheory 73 3.2.1 Causality 78 Copyright © 2005 IOP Publishing Ltd. viii Contents 3.3 Microscopicapproachtorefractiveindexandgroupvelocity 81 3.4 EPRcorrelationsandcausality 87 3.5 Nocloning 88 3.5.1 Teleportation 91 3.6 AsuperluminalquantumMorsetelegraph? 92 3.7 MirrorswitchingincavityQED 95 3.8 Pre´cis 101 3.9 Appendix:OnEinsteinandhiddenvariables 101 4 Fastlightandsignalvelocity 108 4.1 Experimentsonsignalvelocities 108 4.2 Cantheadvanceofaweakpulseexceedthepulsewidth? 110 4.2.1 ApproximationleadingtotheARSfieldequation 117 4.2.2 Signalandnoise 118 4.2.3 Physicaloriginofnoiselimitingtheobservabilityof superluminalgroupvelocity 122 4.2.4 OperatororderingandrelationtoARSapproach 123 4.2.5 Limitofverysmalltransitionfrequency 124 4.2.6 Remarks 124 4.3 Signalvelocityandphotodetection 125 4.4 Absorbers 131 4.5 Whatisasignal? 131 4.6 Remarks 133 5 Slowlight 135 5.1 Someantecedents 135 5.2 Electromagneticallyinducedtransparency 136 5.3 SlowlightbasedonEIT 145 5.3.1 Slowlightinanultracoldgas 146 5.3.2 Slowlightinahotgas 147 5.4 Groupvelocitydispersion 150 5.5 Slowlightinsolids 152 5.5.1 Coherentpopulationoscillations 152 5.5.2 Spectralholeduetocoherentpopulationoscillations 155 5.5.3 Slowlightinroom-temperatureruby 157 5.5.4 Fastlightandslowlightinaroom-temperaturesolid 159 5.6 Remarks 162 6 Stopped,stored,andregeneratedlight 164 6.1 Controllinggroupvelocity 164 6.2 Dark-statepolaritons 165 6.3 Stoppedandregeneratedlight 172 6.4 Echoes 175 6.5 Memories 176 6.6 Somerelatedwork 178 Copyright © 2005 IOP Publishing Ltd. Contents ix 7 Left-handedlight: basictheory 180 7.1 Introduction 180 7.2 Negative(cid:1) andµimplynegativeindex 182 7.3 Dispersion 184 7.4 Maxwell’sequationsandquantizedfield 185 7.4.1 Radiativeratesinnegative-indexmedia 188 7.5 ReversaloftheDopplerandCerenkoveffects 190 7.5.1 Onphotonmomentuminadielectric 192 7.6 Discussion 194 7.7 Fresnelformulasandtheplanarlens 195 7.8 Evanescentwaves 199 7.8.1 Limittoresolutionwithaconventionallens 202 7.9 The‘perfect’lens 202 7.9.1 EvanescentwaveincidentonanNIMhalf-space 203 7.9.2 EvanescentwaveincidentonanNIMslab 204 7.9.3 Surfacemodes 206 7.10 Elaborations 208 7.11 Nofundamentallimittoresolution 209 7.12 Summary 209 8 Metamaterialsforleft-handedlight 211 8.1 Negativepermittivity 211 8.2 Negativepermeability 216 8.2.1 Artificialdielectrics 221 8.3 Realizationofnegativerefractiveindex 222 8.4 Transmissionlinemetamaterials 226 8.5 Negativerefractioninphotoniccrystals 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 possibilitythatthe(measurable)groupvelocityoflightcouldexceedcwith(2)the requirementofrelativitytheorythatnosignalcanbetransmittedsuperluminally. SommerfeldandBrillouinconcludedthatagroupvelocityisnot,ingeneral,the velocitywithwhichasignal,properlydefinedasacarrierofinformation,canbe 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 demonstrationsof superluminal group velocity would not have sparked so much discussion. The newsmedia,withthehyperbolecharacteristicofthetimes,haveoftenasnotbeen misleadingorwrongbutsohavethereportedcommentsofsomephysicists. TheprincipaldevelopmentsincethepublicationofBrillouin’smonographis theexperimentalstudyof‘abnormal’groupvelocities—groupvelocitiesthatare superluminal,infinite,negative,orzero. Theliteratureonthesubjecthasgrown substantially. Onepurposeof thisbookisto review, vis-a`-visthis development, themostbasicideasaboutdispersionrelations,causality,propagationoflightin dispersivemedia,andthedifferentvelocitiesusedtocharacterizethepropagation oflight. 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‘Fermiproblem’hasbeenrevisitedperiodicallyinthepastfewdecades. 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 ‘spookyactionatadistance’suggestedbyquantumcorrelationsoftheEinstein– 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 stimulatedemission. Thefactthatsuchschemesmust,ingeneral,beimpossible ledtotheno-cloningtheorem. Onepointthatisemphasizedhereisthatanymeasurableadvanceintimeof a‘superluminal’pulseisreducedbynoisearisingfromthefield,themediumin whichthefieldpropagates,orthedetector. The groupvelocity of light can also be extremelysmall. ‘Slow light’ with groupvelocitiesontheorderof10ms−1 wasfirstdirectlyobservedin1999and shortlythereafteritwasdemonstratedthatpulsesoflightcouldevenbebrought toafullstop,stored,andthenregenerated. Thesedevelopmentshavebeenbased largely on the quantum interference effects associated with electromagnetically inducedtransparency.Slowlightraiseslessfundamentalquestions,perhaps,than ‘fast light’ but it might have greater potential for applications. One application might be to quantum memories, as the storage and regenerationof light can be done without loss of information as to the quantum state of the original pulse: this informationis temporarilyimprintedin the slow-lightmedium. The ability to coherentlycontrollightin this way couldalso find applicationseventuallyin opticalcommunications. The third major topic addressed in this book is ‘left-handed light’—light propagationinmediawithnegativerefraction.Hereitisnotsomuchthevariation oftherefractiveindexwithfrequencythatmatters,asinthecaseoffastlightand slowlight,butrathertheindexitselfatagivenfrequency.Left-handednessrefers tothefactthat, whentherefractiveindexisnegative,the electricfieldvectorE, the magnetic field vector H, and the wavevector k of a plane waveform a left- handedtriad. Naturehasapparentlynotproducedmediawithnegativerefractive indices;however,so-calledmetamaterialswiththispropertyhavebeencreatedin thelaboratory. 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 spurredaveryrapidgrowthinthenumberofpublicationsinthisarea.Thelasttwo chaptersare an introductionto someof the foundationalworkonmetamaterials andleft-handedlight. My recent interest in these areas beganwith enlighteningdiscussions 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. physicistsincludingYAharonov,JFBabb,SMBarnett,PRBerman,HABethe, MSBigelow,RWBoyd,RJCook,GDDoolen,JHEberly,GVEleftheriades, H Fearn, M Fleischhauer, K Furuya, I R Gabitov, D J Gauthier, S A Glasgow, RJGlauber,DFVJames,PLKnight,PGKwiat,WELamb,Jr,ULeonhardt, R Loudon, G J Maclay, L Mandel, M Mojahedi, G Nimtz, K E Oughstun, JPeatross,JBPendry,EAPower,BReznik,MOScully,BSegev,DRSmith, A M Steinberg, L J Wang, H G Winful, E Wolf, and R W Ziolkowski. I have probablyleft out the names of many other people with whom I had helpful but long-forgottendiscussions. IapologizetothemanyauthorswhoseworkIhavenotcited.Thereisahuge literaturerelatingtothetopicscoveredinthisbook,andIhavenotcitedworkthat Ihavenotreadorunderstood,letalonepublicationsIhavenotevenseen. Thethreemajorsubjectsofthisbookhaveattractedparticularinterestinjust thepastfewyears.Theyarerelatedbythefactthattheyallinvolveunusualvalues orvariationsoftherefractiveindex. Ihavetriedtofocusonthebasicunderlying physics. Themanycitationstorecentworkdonotrepresentanattempttomake thisbookasup-to-dateaspossible;itdoesreflectmyopinionthatthisworkisof considerablefundamentalimportance. IthankTomSpiceroftheInstituteofPhysicsforsuggestingthisbookandfor hispatiencewhenIfailedtofinishitbythepromiseddeliverydate.DanGauthier ofDukeUniversitymadehelpfulsuggestionsforwhichIamgrateful. PeterWMilonni LosAlamos,NewMexico Copyright © 2005 IOP Publishing Ltd. Chapter 1 In the Beginning 1.1 Maxwell’s equations and thevelocity oflight 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 situationswherethesevariationsareunusualandnotyetofanypracticalutility. Ourconsiderationswillbebasedonthelawsofelectromagnetism: ∇· E =ρ/(cid:1) (1.1) 0 ∇· B =0 (1.2) ∂B ∇× E = − (1.3) ∂t ∂E ∇× B =µ J +(cid:1) µ . (1.4) 0 0 0 ∂t Theseequationsaresoincrediblyimportantthatwebeginwithabriefdiscussion of their conceptual foundations, even though this has been done thousands of timesbefore. The definite pattern formed by iron filings around a bar magnet, or by sawdustaroundanelectrifiedbody,ledFaradaytosuggestthatthespacearound suchobjectsisfilledwithlinesofforce. Electricandmagneticforces,fromthis point of view, are transmitted by the medium between the objects rather than arisingfrom‘actionatadistance’.Maxwellwasgreatlyimpressedandinfluenced bythisideaofwhathecalledanelectromagneticfield[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 theseatofthephenomenainrealactionsgoingoninthemedium,they weresatisfied thattheyhadfounditina powerofactionata distance ... Copyright © 2005 IOP Publishing Ltd.
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