Table Of ContentSeries 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
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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
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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
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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
...
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