anderson localization of light in layered dielectric ... · notice are the great number of meetings...

55

2.4Anderson Localization of Light in Layered Dielectric Structures

Yury BliokhTechnion—Israel Institute of Technology

Konstantin Y. BliokhNational University of Ireland

Valentin Freilikher Bar-Ilan University

Franco NoriAdvanced Science Institute, RIKEN

University of Michigan

CONTENTS

2.4.1 Introduction.......................................................................................................................562.4.2 LyapunovExponent,Localization,andTransmissionLengths.................................582.4.3 StatisticsoftheTransmissionin1DDisorderedSystems........................................... 61

2.4.3.1 TransportandLocalizationinRandomlyLayeredMedia........................... 612.4.3.1.1 NormalIncidence:TransferMatrixMethod................................. 612.4.3.1.2 ObliqueIncidence:ReductiontotheOscillatoryProblem.......... 62

2.4.3.2 TransportandLocalizationinContinuousActiveMedia............................ 622.4.3.2.1 InvariantEmbeddingMethod......................................................... 62

2.4.3.3 TransportandLocalizationin1DPeriodicStructureswithDisorder........642.4.4 Disorder-InducedResonancesin1DSystems...............................................................65

2.4.4.1 ExplorationbyAnalogy:DeterministicModelofRandomResonances....652.4.4.2 HowIsanEffectiveCavityBuiltUp?............................................................... 712.4.4.3 CouplingandLevelRepulsion..........................................................................722.4.4.4 BistabilityofAndersonLocalizedStatesinNonlinearRandomMedia....75

2.4.5 ExperimentalStudiesofResonances..............................................................................772.4.5.1 InverseScatteringProblemsandRemoteSensing ofDisorderedSamples.......................................................................................80

2.4.6 AndersonLocalizationinExoticMaterials................................................................... 812.4.6.1 SuppressionofAndersonLocalizationinDisorderedMetamaterials........ 812.4.6.2 TransportandLocalizationinDisorderedGrapheneSuperlattices............ 82

2.4.7 Conclusion..........................................................................................................................83Acknowledgments........................................................................................................................84References.......................................................................................................................................84

56 Optical Properties of Photonic Structures

2.4.1 Introduction

Wavepropagationindisorderedmediaisarichandlong-standingproblemthatattractsmany efforts, both theoretical and experimental. After almost a century of a completesway of radiative transport and diffusion approaches, it recently became clear thatthe interference of multiply-scattered fields (which is neglected in classical diffusiontheory)dramaticallyaffectsallwaveprocesses,especially insystemswithfluctuations.Themoststartlingmanifestationofthiseffectisthestronglocalizationofelectromagneticradiationinweakly-disorderedrandommedia.SinceAnderson’sseminalpaper,1localiza-tion has attracted ever-increasing attention from physicists and engineers. Withoutattempting an exhaustive review of available literature we note that the bibliographyrelatedtothisratheryoungareaalreadynumbersinhundredsoforiginalarticles,reviews,and books (see, e.g., the review,2 the monographs,3 and references therein). Worthy ofnoticearethegreatnumberofmeetingsdedicatedtothe50years’anniversaryofAndersonlocalization(half-a-yearnon-stopworkshopinCambridge,conferencesinParis,Dresden,SantaBarbara,etc.)wherephysicists,mathematicians,chemists,engineers,biologists,andeveneconomistsnotonlypresentedaplethoraofnewresultsbutalsoformulatedagreatmanychallengingquestions.

AboosttothestudiesofAndersonlocalizationindisorderedopticalandquantumsys-temswasgivenrecentlybythecreationofnewmaterialswithuniquepropertiesthathavespurred the rise of new conceptual challenges and high-tech applications. The mostimpressivelatestexamplesincludephotoniccrystals,plasmonics,left-handedmetamateri-als,Bose–Einsteincondensates,andgraphene.Yetitshouldberememberedthatmostoftheprospectsforpotentialtechnologicaluseofthesematerialsrestonthepredictedprop-ertiesofideal(e.g.,perfectlyperiodic)systems.Evenasmallamountofdisorder,however,whichisinevitablypresentinanyrealsample,couldaffectitspropertiesdramatically(seeFigure2.4.1).Therefore,whenitcomestorealapplications,acomprehensivestudyoftheeffectsofdisorderisamust.Moreover,theseinvestigationsareofinterestbyitselfbecausestrongly disordered (with no periodic component) systems possess further unexpectedphysicalproperties,whichmakethempotentiallyusefulasanalternativetothepureperi-odicconfigurations.

One-dimensional(1D)stronglocalizationhasreceivedthemoststudy,bothanalyticallyandnumerically.Inparticular,thelocalizationoftheeigenstatesinclosed1Ddisorderedsystems and the exponentially small (with respect to the length) transparency of opensystemswith1Ddisorderhavebeenscrutinizedwithmathematicalrigor(e.g.,seeRef.4andreferencestherein).

Themostcommonphysicalmanifestationoflocalizationisthefactthatsheetsofper-fectlytransparentpaperstackedtogetherinlargenumbersreflectlightasagoodmirror.5Muchlessevident(thoughlongpredictedtheoretically6)isthatforeachsufficientlylongdisordered1Dsample,thereexistsarandomsetoffrequenciesthatgoallthewaythroughthesamplealmostunreflected,thatis,withthetransmissioncoefficientclosetounit.Hightransparencyisalwaysaccompaniedbyarelativelylargeconcentration(localization)ofenergyaroundrandomly locatedpoints inside thesystem.Alongwith these“classical”trademarksofstronglocalizationthereisaplethoraofnot-less-amazingeffectsthatdisor-dercansetupinonedimension.Examplesare:randomlasing,7,8criticalcoupling,9neck-lacestates,10,11levelcrossingandrepulsion,9slowlightandsuperluminalgroupvelocities,12bistability and nonreciprocity of resonant transmission in nonlinear random media,13delocalizationinmetamaterials,14−16andingraphenesuperlatices,17andsoon.

57Anderson Localization of Light in Layered Dielectric Structures

Although,generally,adisordered1Dsystemisamathematicalabstraction,itcanprovideanadequatemodelformanyactualphysicalobjects.Forexample,randomlystratifiedmediaarefoundinnumerousgeologicalandbiologicalsettings,aswellasinfabricatedmaterials.Interference of waves in such systems determines the transport of seismic waves in theearth’s crust and sonic waves in the oceans; reflection and transmission from multilayerdielectricstacksusedasopticalreflectors,filters,andlasers;propagationandlocalizationinsingle-modeopticalfibersandmicrowavewaveguides,etc.Evenmoreimportant,itmaybepossibletoutilizehighlydisorderedsamplesformanyapplications.Forinstance,tunableswitchesornarrow-linelasersourcescanbecreatedusingrandomlystackedsystems.

Althoughthestronglocalizationofwavesin1Drandommediahasbeenwellstudiedtheoretically,mostoftheanalyticalresultshavebeenobtainedforvaluesaveragedoverensemblesof randomrealizations.These resultsarephysicallymeaningful for theself-averagingLyapunovexponent(inverselocalizationlength),whichbecomesnonrandominthemacroscopiclimit.Fornon-self-averagingquantities(fieldamplitudeandphase,inten-sity, transmissionand reflectioncoefficients, etc.), a systemofanysize is alwaysmeso-scopic,and,therefore,meanvalueshavelittletodowiththemeasurementsatindividual(usuallysmallinnumber)samples.Thisismostpronouncedwhenitcomestodisorder-inducedresonanceswhoseparametersareextremelysensitivetothefinestructureofaparticularsampleandstronglyfluctuatefromrealizationtorealization.Inparticular,theensembleaveragingwipesoutallinformationaboutthefrequenciesandlocationsofindi-vidual localized states within a particular sample, even though just this set of data isessentialforapplications.Anotherfrustratinginconsistencybetweenmostoftheexistingtheories and measurements is that in real systems, losses (absorption and leakage) areinevitably present, whereas mathematicians and theoreticians usually prefer lossless(Hermitian)modelsthataremucheasiertodealwith.

1.7200.0

0.2

0.4

0.6

0.8

1.0

1.725 1.730Frequency (a.u.)

Tran

smiss

ion

coeffi

cien

t

1.735

FIGURE 2.4.1Transmissionspectraofaregularperiodicsample(thickblackline)andofasamplewhoseperiodfluctuatesintherangeof1%.


Inthischapter,wepresentabriefoverviewofmethodsandresultsregardingthetrans-portandlocalizationindisordered1Dsystems,followedbyadetaileddescriptionofthecurrentstate-of-the-artintheoreticalandexperimentalstudiesoftheresonantpropertiesofrandomlylayeredmedia.

2.4.2 Lyapunov Exponent, Localization, and Transmission Lengths

Considerthe1DHelmholtzequation

′′ + + =u x k x u x( ) [ ( )] ( )20 0ε δε (2.4.1)

withtheself-adjoined(currentless)boundaryconditionatapointx0

u x au x( ) ( ) ,0 0 0+ ′ = (2.4.2)

whereaisanyrealnumber.ItiseasytoshowthatEquation2.4.2meansthatthemodulusofthereflectioncoefficientfromthepointx0 equalstoone.Thefunctionsξandφdeter-minedas

u x x k

u x k x k

x k

x k

( ) sin ( , ),

( ) cos ( , )

( , )

( , )

=

′ =

e

e

ξ

ξ

ϕ

ϕ

(2.4.3)

satisfythefollowing(nonlinear)equations:

′ = −

= +

ϕ δε ϕ

ξ ξ δε ϕ

( , ) [ ( )sin ( , )]

( , ) ( , ) ( )sin ( ,

x k k x x k

x k kk

y y k

1

02

2

2

)) dyx

0∫ .

(2.4.4)

Assumingthatδε(x)isastatisticallyhomogeneousrandomfunctionwithzeroaverage,δε( )x = 0 , and disappearing at infinity correlations, W x x

x( ) ( ) ( )= →→∞δε δε0 0 , the

following statement is true:4 in the limit |x|→∞, the ratio ξ(x,k)/x approaches a non-randomlimitthatispositiveforallk:

lim

( , )lim

( , )( )

x x

x kx

x kx k

→∞ →∞= ≡ > .ξ ξ γ 0

(2.4.5)

Inprinciple,thisresultfollowsfromtheFurstenbergtheorem,18whichholdsthat,undersomeconditions,thespecificlogarithmoftheproductofNtransfermatricesMjtendstoapositivelimitasNgoestoinfinity:

lim ln .N

j

N

jN

M→∞

=

∏ = >10

1

µ


Equation2.4.5presentstwofundamentalpropertiesofa1Drandomsystemsatisfyingtheabove-listedconditions:

1.The Lyapunov exponent, γ(k), is a self-averaging quantity; that is, at any singlerandom realization it tends to the ensemble-averaged nonrandom mean valuewhenthesizeoftherealizationincreasesinfinitely.

2.For a single random realization, the amplitude of the wave function increasesexponentiallywithnonrandomincrementγ(k)onbothsidesofthepoint,atwhichthecurrentlessboundarycondition(2.4.2)holds.

TheinverseoftheLyapunovexponent

lloc = 1

2γ (2.4.6)

iscalledthelocalizationlength.ThemeaningofthistermbecomesclearifoneconsiderstwosolutionsofEquation2.4.1,u1(x)andu2(x),intheinterval0≤x≤L(closed1Dsystem),eachsatisfyingtheboundarycondition(2.4.2)atx=0andx=L,respectively,andthereforeeach increasing exponentially away from these points. An eigenfunction, ψn(x), of thissystemcanbeconstructedfromu1(x)andu2(x)undertheconditionthatthesefunctionsandtheirderivativesmatchatsomepointxninsidetheinterval.Obviously,aneigenfunction,ψn(x),obtainedinthiswayislocalized;thatis,itsenvelop,A2(x),decreasesexponentiallyonbothsidesofxn:

A x u x

ku x

x xl

n2 22

21( ) ( ) ( ) exp≡ + ′

−−

~loc

(2.4.7)

Inthecaseofthewhite-noisedisorder[δε(x)isaδ-correlatedrandomprocess],theclosedFokker–Planckequationfortheprobabilitydensitydistribution,P(ξ,z),ofthequantityξ(x)canbederivedandsolvedusingaaveragingoverrapidrandomphase.4,19Thisisimpossi-ble if δε(x) has a finite correlation radius, and the random-phase approximation breaksdown (as well as the single-parameter scaling theory of localization). In this case, theordered cummulant method of Van-Kampen can be used to obtain the weak disorderexpansionoftheLyapunovexponentγ(k).20ThefirsttermofthisexpansioncanbeobtainedfromEquation2.4.4bysolvingtheequationforthephaseperturbativelyandsubstitutingtheresultintotheintegralforξ(x,k).Thelimit|x|→∞yieldsthefamousresult

γ( ) ( )cos ,kk

xW x kx=∞

∫2

04

2d

(2.4.8)

whichmeansthatthelocalizationisduetotheresonantBragbackscatteringprovidedbythe2kFouriercomponentof therandompotential.Higherordersof theweakdisorderexpansionhavebeencalculatedandcanbefoundintheliterature.20,21

ThenotionoftheLyapunovexponentisrelatedtotheeigenvaluesboundaryproblem,andthereforeiswelldefinedonlyforcloseddisorderedsystems.Fromthephysicalpointofview,notlessrelevantisthescatteringproblemthataddressesthetransmission,reflec-tion,andpropagationinopenstructureswithfluctuatingparameters.


Transportpropertiesofa1DsystemofafinitelengthLcanbedescribedbythetransfermatrix M̂ ,whichrelatestheamplitudesoftheincident(AL )andoutgoing(BL)wavesononesideofthesampletothoseontheotherside(ARandBR ):

B

AM

B

AR

R

L

L

= ˆ

(2.4.9)

Assumingtime-reversalinvariance,thetransfermatrixM̂ canbewrittenas

ˆ( )

(M t

rt

rt t

T

T

T

i i

i=

−

−

=− −

− −

∗

∗

− −

−

1

1

1 1

1

e e

e

t r t

r

φ φ φ

φ φφ φt te),

−i

(2.4.10)

wheret T i= exp( )φt ,andr T i= −1 exp( )φr arethetransmissionandreflectionampli-tudes,respectively;T≡|t|2isthetransmissioncoefficient.Inwhatfollows,thequantity

l L T LLtr( ) ln ( )= − −11

2 iscalledthetransmissionlength.Evidently,thesolutionoftheboundaryvalueproblem(2.4.1),(2.4.2)andthesolutionof

thescatteringprobleminthelimitL→∞areuniquelyrelated;thatis,

u L

u LM

u

u

( )( )

( )( )′

= ′′

� 00

where ˆ ′M isanelementarylineartransformationofthetransfermatrixM̂ .WhenthesizeLofthesystemismuchlargerthanthelocalizationlengthlloc,thetrans-

mission coefficient T becomes exponentially small (with the probability exponentiallyclosetoone),andstatisticallyindependentofthephasesϕtandϕr.Asaresult,eachmatrixM̂ and ˆ ′M factorizesintoaproductofalargefactor1/ T andamatrixoftheorderofunity,whichisstatisticallyindependentofT.Thismeansthat,asymptoticallyforlargeL,onecan write

γ( ) ln ( ) ( ) ln ( ) ( ),k

Lu L

ku L

LT L O L= + ′

= − +12

1 12

22

2 0

(2.4.11)

wherethesecondtermintheasymptoticexpansionisindependentofthefirstone,andtheircummulantsareoftheorderofunity.Apparently,inthelimitL→∞,thetransmis-sionlengthcoincideswiththeinverseLyapunovexponent(localizationlength):

l ltr loc= . (2.4.12)

This fact isgenerally recognized.However, itwasshownrecently14 that, surprisinglyenough, llocand ltr canbedifferent instacksmadeofalternatingright-and left-handeddielectriclayerswithrandomrefractiveindicesandthicknesses.


The equality Equation 2.4.12 means, in particular, that the localization length can beprobed noninvasively from the transmission coefficient, without measurements of thefieldamplitudeinsiderandomsamples.

2.4.3 Statistics of the Transmission in 1D Disordered Systems

2.4.3.1 Transport and Localization in Randomly Layered Media

2.4.3.1.1 Normal Incidence: Transfer Matrix Method

Oneofthemostefficienttheoreticalmethodsofstudyinggeneralpropertiesoftransmis-sionin1Ddisorderedsystemsisbasedonthecompositionruleforachainofstatisticallyidenticalandindependentrandomscatterers.22Forstratifiedmedia,themethodisstraight-forwardandinvolvesthecalculationofthetransfermatrix M̂ usingthefollowingexactrecurrencerelationsforthetransmissioncoefficients(fordetails,seeRef.14andreferencestherein):

T

T tR r

nn n

n n=

−−

−

1

11,

(2.4.13)

R r

R tR r

n nn n

n n= +

−−

−

12

11 (2.4.14)

HereTnandRnare,respectively,thetotaltransmissionandreflectioncoefficientsofastackofnlayers,tnandrnarethe(complex)transmissionandthereflectionamplitudesofasinglelayer.Equations2.4.13and2.4.14aregeneraland,takingintoaccountallmul-tiply-scattered fields, present exact solutions that can be used for direct numericalsimulations.

ForasamplecomposedofN statistically identicaland independentrandomlayersofnormal(withpositiverefractiveindex)dielectrics,thefollowingexpressionfortheaverageinversetransmissionlengthcanbederivedfromEquations2.4.13and2.4.14:14

1 1 1 1

1

22

22l N l Nd

rt

t

N

tr locRe

( ),= +

−

−( )

(2.4.15)

wheredistheaveragewidthofthelayers,andtheinverselocalizationlengthis

1 11

2

2l lt

r

tloc trRe=

∞= − | | −

−.

( )ln

(2.4.16)

Note that when N→∞, Equation 2.4.15 transforms into Equation 2.4.12, that is, thelocalizationandtransmissionlengthsbecomeequal.


Inthecaseofweakscattering,thereflectionfromasinglelayerissmall,|rn|<<1andEquation2.4.16yields:

lk

kloc

/

/=

= →

= → ∞.

122 0

32

2

2

2

2 2

d

dδ

λ π

λπ δ

λ π

, ,

,

(2.4.17)

In thederivationofEquation2.4.17, itwasassumed that thewidthof each layerwasdistributed uniformly over the interval [d−δ, d+δ], and the refractive index did notfluctuate.Nevertheless,thefunctionalλ-dependences,Equation2.4.17,ofthelocalizationlengthinweaklydisorderedsystemsarerathergeneral(e.g.,seeRef.23).

2.4.3.1.2 Oblique Incidence: Reduction to the Oscillatory Problem

AnoriginalefficientmethodofcalculatingthelocalizationlengthwasdevelopedinRef.24.Itusesthefactthatthereflectionfromanadequatelylong,randomlylayeredsamplediffersfromunitybyanexponentiallysmallnumber,1−R(L)~exp(−L/lloc),and,therefore,thefluxalongthesystemisalsoexponentiallysmall.Thisa prioriinformationenablesonetoassume(withanexponentialaccuracy)thatthefieldineachlayerinsidethesampleisastandingwave,andtoreducethewavepropagationproblemtotheoscillatoryone,withthereal-valuedwaveamplitudebeingasingleunknown.Thissimplifiestheproblemsig-nificantlyascomparedtotheconventionaltransfermatrixmethod,wheretheevolutionoftwoindependentwavesineachlayerisconsidered.Usingthismethod,theobliqueinci-denceofelectromagneticwavesonarandomlylayeredmediumwasstudied.24Twoeffectsnotfoundatnormalincidencewerepredicted:dependenceofthelocalizationlengthonthepolarizationandthedecreaseofthelocalizationlengthasaresultoftheinternalreflec-tionsfromlayerswithsmallrefractiveindices.Theattenuationrateforp-polarizedradia-tionisshowntobealwayssmallerthanthatofs-polarizedwaves,whichistosaythatanadequatelylong,randomlylayeredsamplepolarizestransmittedradiation.Thelocaliza-tionlengthforp-polarizationdependsnonmonotonicallyontheangleofpropagationand,undercertainconditions,turnstoinfinityatsomeangle,whichmeansthattypical(non-resonant) randomrealizationsbecometransparentat thisangleof incidence (stochasticBrewstereffect).

2.4.3.2 Transport and Localization in Continuous Active Media

2.4.3.2.1 Invariant Embedding Method

Analternativeapproachtothe1Drandomscatteringproblemistheinvariantembeddingmethod,25whichamountstofindingthesolutionofthefollowingsystemof(exact)first-orderLangevin-typeequationsforthereflectionandtransmissioncoefficients:

ddr LL

ik L r LikL ikL( )( ) ( )= +−

22

ε e e

(2.4.18)

ddt LL

ik L t L r LikL ikL( )( ) ( ) ( )= +−

2

2ε e e

(2.4.19)


Equations2.4.18and2.4.19canbetreatedstatisticallybymeansofexactnumericalcalcula-tionsandapproximateanalyticalmethodsaswell.AFokker–Planckequationcanbederived,which,inthecaseofthewhite-noisedisorder,yieldsthedistributionfunctionforthereflec-tioncoefficientR(L)=|r(L)|2.Intheabsenceofabsorptionoramplification,thisdistributionfunctionprovidescompleteinformationonthetransmissioncoefficientT=|t(L)|2.

The problem becomes much more complicated in the case of lossy media where theenergyconservationlawnotonlyconnectsTandRbutalsoinvolvesarandomamountoftheabsorbedintensity.Inthepaper,26theasymptoticallyexactexpressionsforallmomentsofthetransmissioncoefficienthavebeenobtainedbymappingtheFokker–Planckprob-lemontoaShrödingerequationwithimaginarytime.Inparticular,ithasbeenshownthatinthecaseofsmallabsorption,lloc<<la(laistheabsorptionlength),

− = +

ln ( ),

T LL l l

14

1

loc in

(2.4.20)

where

l l

ll

ain loc

loc=

ln2

(2.4.21)

isadisorder-inducedabsorptionlength,whichliesbetweenthelocalizationandabsorp-tionlengths,seeRef.26

l l laloc in� � .

Thismeansthatinthelocalizedregime,thedisordercausesdrasticenhancementoftheattenuationoftheaveragetransmissioncoefficientascomparedtothatinthecorrespond-ingpure(δε=0)sample(lin<<la).Notethatthedisorder-inducedabsorptionlengthforthelocalizedwaves,Equation2.4.21,isalsosignificantlysmaller(i.e.,theeffectofabsorptionismuchstronger)thanthatinthediffusiveregime:l l lain loc� .

IncontrasttoEquation2.4.20,thecontributionsfromscatteringandabsorptiontotheaverage decrement of the transmission coefficient (or to the Lyapunov exponent) areadditive:

− = +

> − .ln ( ) ln ( )T LL l l

T LLa

1 1

loc (2.4.22)

Itcanbeshownthatln⟨T(L)⟩and⟨lnT(L)⟩aredifferentnotonlyinlossymedia(compareEquations 2.4.20 and 2.4.22) but also in nonabsorbing (la→∞) systems. This is becauselnT(L)/L is a self-averaging quantity (see Equations 2.4.8 and 2.4.11), with very narrowdistribution (δ functionatL→∞) centeredat itsmeanvalue.Thismeans thatata ran-domlychosenrealization,lnT(L)willbefoundinasmallvicinityofitsaveragevaluewithaprobabilityexponentiallyclosetoone,andthereforethevalueofafunctionF[lnT(L)]willbeclosetoF[⟨lnT(L)⟩]withthesameprobability.Inparticular,thetransmissioncoefficienttypicallyisexponentiallysmall:

T L

T LL

Ll la

typloc

( ) expln ( )

exp=

= − +

.1 1

(2.4.23)


On the other hand, T(L) itself is a strongly fluctuating random variable with broaddistribution.Itturnsoutthatthemaincontributiontoitsaveragevaluecomesnotfromthetypical (nontransparent) realizations but from low-probable ones, so-called resonantrealizations(seeSection2.4.4),correspondingtothenon-Gaussiantailofthedistributionof lnT, where the transmission coefficient is of the order of unity. This is due to theseresonantlytransparentrealizationsthattheaveragetransmissionismuchlargerthanthetypicalone.Forexample,4inthelosslessmediawithdelta-correlateddisorder

T LT L

Ll

( )( )

exptyp loc

=

.34

1�

whilebothTtyp(L)and⟨T(L)⟩areexponentiallydecayingfunctionsofL(thedifferenceisintheattenuationrate).Morethanthat,therearequantitiesforwhichthetypicalandmeanvalueshavecompletelydifferentfunctionaldependencies.Anexampleofhowmisleadingaformallycalculatedmeanvaluecanbeistheenergyflux,J=2Im(u*u′),createdbyapointsourcelocatedattheperfectlyreflectingedge(x=0)ofadisorderedsampleoflengthL.AsshowninRef.2,themeanfluxdoesnotinteractwithdisorderandisequaltoitsvalueinthe homogeneous sample: ⟨ J(L)⟩=2/k. This result is physically meaningless because toobtainitexperimentally,averagingoveranexponentiallylargenumberofrealizationsisnecessary.Atthesametime,themeasurementatasinglerandomsamplewillgive(withaprobabilityexponentiallyclosetounity)thetypicalvalue,whichisexponentiallysmallasa result of the localization effect: Jtyp=exp(⟨lnJ⟩)~exp(−L/lloc). This is because lnJ is anadditiveself-averagingquantity.

Thetypical-mediumapproachinthetheoryofMott–AndersonlocalizationinelectronsystemsisdiscussedinRef.27.

Anoutstandingdistinctionbetweenthetransmissionattypicalandresonantconfigura-tionsofamplifyingrandommediahasbeenfoundinRef.28.Ithasbeenshownthatinrandom systems with complex dielectric permittivity, ε(x)=ε0+δε(x)+iΓ, the inverseLyapunovexponentisalwaysnegative,independentofthesignofΓ;29thatis,thetypicaltransmissionthroughafinitedisordereddielectricsampleisexponentiallysmallforbothabsorbing and amplifying disordered samples. To the contrary, the mean value of thetransmissioncoefficientinrandommediawithgain(Γ>0)diverges(becauseoftheinfi-nitely increasing resonant intensity) evenat samplesoffinite size.Toobtainphysicallymeaningfulfinitevaluesofthetransmission,thenonlineareffectofsaturationshouldbeincludedinthemodel.

2.4.3.3 Transport and Localization in 1D Periodic Structures with Disorder

Thestudyoftheeffectsofdisorderonthewavepropertiesofperiodicstructuresisessen-tialforbetterunderstandingthephysicsoftheinterplaybetweenperiodicityanddisorder,andalsoforpracticalapplications.Indeed,thoughconsiderableefforthasbeenexpendedtodevelophighlyperiodic structures,deviations fromperiodicity inevitablypresent inanymanufacturedphotoniccrystalcansignificantlymodifyitsopticalcharacteristics.Torevealthemostgeneraltransportpropertiesofdisordered1Dstructuresthatareperiodiconaverage(1Dphotoniccrystals),theHelmholtzEquation2.4.1canbeused,inwhichε0isaperiodic functionof thecoordinatex, and ⟨δε(x)⟩=0.Three typesofperiodic systemswithweakly-perturbedperiodicitywerestudiedinRef.30:(i)stacksofalternatingdiscretedielectric layers with constant permittivities, ε(1) and ε(2), and fluctuating width of eachlayer, di=d + δdi; (ii) samples of the same geometry but with constant di=d and


ε ε δε2 11

01

2 1i i− −= +( ) ( ) , ε ε δε22

02

2i i( ) ( )= + ; (iii) continuous periodic media with ε(x)=Acos qx +

δε(x). The quantities δdi, δεj, and δε are assumed to be random variables with knownstatistics.Fornumericalsimulationsofthepropagationindiscretesystems(i)and(ii),thetransfer matrix approach (Section 2.4.3.1) is appropriate, whereas for the continuousmodel (iii),theinvariantembeddingmethod(Section2.4.3.2)isbestsuited.

The followingproperty is found tobeuniversal, independentof thegeometryof thesystemandofthetypeofdisorderforthefrequenciesoftheincidentwavebelongingtoaband gap of the underlying periodic structure, weak disorder enhances (in contrast tohomogeneousinaveragerandom1Dsystems)thetransparency.Moreover,thelocalizationlengthandthetransmissioncoefficientgrowwhenthestrengthofthedisorderincreases.Thisisbecauseinthepresenceofdisorder,thechannelsofthepropagationthatareclosedintheperfectlyperiodicsystemopenupasaresultofthepartialfillingofthedensityofphotonicstatesinthegapbythetailsofthisdensityfromthetransparencyzonesborderingthegap.

Incontrasttothebandgap,thefeaturesofthetransmissionforfrequenciesinthetrans-parencyzonedependonthetypeofdisorder.Forthesefrequencies,thesurprisingnon-monotonicdependenceofthelocalizationlengthonthestrengthofdisorderwasobservedinstratifiedmediawithgeometricaldisorderandconstantdielectricpermittivities[type(i)]. In such a medium, the initial decrease of the transmission coefficient is a classicalmanifestationofAndersonlocalization,whichisusuallystrongerforlargerfluctuations(seeprevioussections).Whenthefluctuationsofthewidthbecomeadequatelylarge,thedecreasegivesway to theenhancementof the transmission for increasingdisorder.Toexplainthisrathercounterintuitiveresult,wenotethatinthestronglocalizationregime,theinverselocalizationlengthisapproximatelyequalto30

11

lR

loc≈ ,

(2.4.24)

whereR1isthereflectioncoefficientofasinglelayer.Thismeansthat,inthiscase,thetotaltransmissionofastackiscompletelydeterminedbythemeanvalueofthereflectioncoef-ficientofasingleelement.Foradielectriclayer,R1isaperiodicfunctionofthewidthdofthelayer,andtheaveraginginEquation2.4.24meanstheintegrationofR1overd inanintervalΔd,inwhichthefluctuationsδdiaredistributed.Evidently,ifthedisorderisstrong,Δd≃d, the increase of the interval of the integration of the periodic function causes adecreaseof⟨R1⟩.

Inasystemofthesecondtype(randomε),R1isproportionaltoδεandthegrowthofitsvarianceenhancesthestrengthofasinglescattering,leadingtoamonotonicincreaseof⟨R1⟩.Onfurtherincreaseofthedisorder,allthreetypesofrandomsystemsfinallylosealltracesoftheunderlyingperiodicity,thebandstructuredisappears,andwavesofallfrequenciesexperiencethesamedisorderedmediumthatbecomeshomogeneousinaverage.

2.4.4 Disorder-Induced Resonances in 1D Systems

2.4.4.1 Exploration by Analogy: Deterministic Model of Random Resonances

AsshowninSection2.4.1,disordercanstronglyaffectthetransportpropertiesofperiodicsystems,sometimestothepointwherethephotonicbandstructureiscompletelydestroyed.


Asfluctuationsofthedielectricandgeometricalparametersareinevitablypresentinanymanufacturedperiodicsample,thiscouldcreateaseriousobstacleintheefficientpracticaluseofphotoniccrystals.Therefore,nowadaysconsiderableeffortsofresearchesandpro-ducersgointothecontroloffluctuations.However,ifratherthancombatingimperfectionsofperiodicity,onefabricatedhighlydisorderedsamples,theycouldbeequallywellhar-nessed,forexample,forcreatingtunableresonantelements.Thisisbecause1Drandomconfigurationshaveauniquebandstructurethatforsomeapplicationshasobviousadvan-tagesoverthoseofphotoniccrystals.

The transparency spectrum of a typical 1D random sample consists of very narrowbandsseparatedbybroadgaps(Figure2.4.2).Foradequatelylongstructures,thebandsaresonarrowthattheycanbetreatedas(quasi)-resonancesthatarewellpronounced;thatis,theirwidthsaremuchsmaller thanthedistancesbetweenthem,both inthefrequencydomainandinrealspace.Physically,ateachresonantfrequency,anopenrandom1Dcon-figurationcanbeconsideredasanopenresonatorwithhighqualityfactor.Animportantadvantageofapplicationfeaturesofsuchasystemisthat,incontrasttoaregularresonatorwhose modes occupy all inner space, in a 1D random structure, each eigenfrequency(mode)islocalizedinsideitsowneffective“cavity”whosesizeismuchsmallerthanthatofthesample.Figure2.4.3showstheintensitiesofthefieldsgeneratedbyaresonantfre-quency(centralcurve),andbytwooff-resonance, typical frequencies (sidecurves)withexponentiallysmalltransmissioncoefficients.Anotherimportantadvantageofdisorderedsamples is that they are much easier to fabricate as they do not require preciseperiodicity.

The existence of disorder-induced resonances in 1D random media was predicted awhileago.6Therandomsetofresonantfrequenciesisasortof1Doptical“specklepattern,”whichisindividualforeachrandomconfigurationandrepresentsitsunique“fingerprint.”

10.000.0

0.2

0.4

0.6

0.8

1.0

10.02 10.04

Frequency (a.u.)

Tran

smiss

ion

coeffi

cien

t

FIGURE 2.4.2TransmissioncoefficientTasafunctionofthewavefrequency.


Althoughtheresonancesindirectlymanifestthemselvesinthedominantcontributiontothemeantransmissioncoefficient(seeSection2.4.3.2),theensemble-averagedquantitiesdonotprovideanyinformationaboutthefrequencies,locations,andspatialintensitydis-tributionoftheindividuallocalizedstates,yetjustthissetofdataisessentialforapplica-tions (e.g., random lasing). Unfortunately, an explicit, general analytical solution ofEquation 2.4.1 with arbitrary function δε(x) does not exist, and standard approximatemethods (e.g., small perturbations expansion) are of little help, because the adequatedescriptionofstronglocalizationcallsforthesummationofinfinitenumberofmultiply-scatteredfields.Ontheotherhand,directopticalmeasurementsofthefieldinsideagivendisorderedsamplearegenerallynotfeasible.

Insuchasituation,thequestionarisesastowhethertheoutgoingradiationbearsthenecessaryinformationonwhathappensinsidethesampleor,morespecifically,whethertheparametersandinternalstructureofindividualresonancescanberetrievedfromthestandardexternalmeasurementsofthetransmissionandreflectionamplitudes.

ThepositiveanswertothisquestionisgivenbymeansoftheapproachdevelopedbyBliokhet al.,31whichisbasedontheconceptthatthefundamentalpropertiesofresonancesareuniversalandindependentofthephysicalnatureofthesystem,regularorrandom,whetheritisaquantum-mechanicalpotentialwell,anopticalormicrowaveresonator,ora1Drandommedium.32Thedistinguishingfeatureofarandomstructureisthattherearenoregularwallsinit,andthestrongreflectionthatlockstheradiationinaneffectivereso-nant cavity is the result of Anderson localization. Moreover, different segments of thesampleturnouttobetransparentfordifferentfrequencies;thatis,eachlocalizedmodeisassociatedwithitsownresonator.

Intheframeworkofapproach,31theproblemofthetransportandlocalizationina1Drandommediumismappedontoanexactlysolvablequantummechanicalproblemoftun-nelingandresonanttransmissionthroughaneffectivedeterministicpotential.Withtheformulae derived on the basis of this mapping, the parameters of the individual reso-nances,inparticular,theirspectralandspatialwidths,fieldamplitudes,andtransmissioncoefficients,canbecalculated.Theseresultsalsoenablesolvinginverseproblems,namely,tofind(withsomeaccuracy)theabsorptionrateofthemediumandthepositionsoftheeffectivecavities,usingthemeasuredtransmissionandreflectioncoefficientsastheinputdata.Theanalyticalresultsdeducedfromquantum-mechanicalanalogyareincloseagree-mentwiththeresultsofdirectnumericalsimulations.

Random dielectric medium

FIGURE 2.4.3(See color insert.)Amplitudeofthefieldinsidethesampleasafunctionofthecoordinateforthreedifferentwavelengths.(ReprintedwithpermissionfromK.Bliokhetal.,Rev.Mod.Phys.80,1201,2008.Copyright2008bytheAmericanPhysicalSociety.)


The idea of this mapping was inspired by the well-pronounced similarity betweenFigures2.4.3and2.4.4obtainedforarandomstackoflayers,andthecorrespondingdepen-dencies calculated for a regular potential well bounded at both sides by two potentialbarriers(e.g.,seeRef.33).Althoughthephysicsofthepropagationineachsystemistotallydifferent (interference of the multiply-scattered random fields in a randomly layeredmedium,andtunnelingthrougharegulartwo-humpedpotential),theaffinitybetweenthem stands out. Indeed, in both cases, the transmission coefficients are exponentiallysmallformostofthefrequencies(energies)andhavewell-pronouncedresonantmaxima(sometimesoftheorderofunity)atdiscretepointscorrespondingtotheeigenlevelsofeachsystem.Theenergyatresonantfrequenciesislocalized,andthetransmissiondependsdrasticallyonthepositionoftheareaoflocalization.

Figure2.4.4presents thetransparenciesof threesegmentsof thesamplecut inaccor-dancewiththecentralcurve(resonantwave)inFigure2.4.3andseparatelyilluminatedbythesameresonant(forthewholesample)wave.Itisseenthatattheresonantfrequency,themiddlepartwheretheenergyisconcentratedisalmosttransparent,whereasthesidesec-tionsarepracticallyopaque.Thisqualitativelycorrespondstowhathappenstoquantumparticlesinapotentialwell.Moreover,evenquantitatively,theintensitydistributionspre-sentedinFigure2.4.3andthecorrespondingvaluesofthetransmissioncoefficientscom-pare favorably with those calculated quantum-mechanically by solving Equation 2.4.1,withtherandomfunctionδε(x)beingreplacedbyaregularpotentialprofilewithproperlychosenparameters.

Forany(nottoohigh)energy,thetransmissioncoefficientofatwo-humpedpotentialprofile (potential well) can be calculated in the Wentzel–Kramers–Brillouin (WKB)approximation,33 which yields expressions independent of the “fine structure” of theprofileanduniquelydeterminedbythesizeofthewellandbythetunnelingtransparen-ciesoftheboundingbarriers.Tousetheseformulaeforthequantitativedescriptionofthewavetransmissionthrougharandomsample,itisnecessarytoexpresstheparame-tersoftheeffectivepotentialthroughtheparametersofthedisorderedsysteminhand.For an ensemble of 1D random realizations, those parameters are the length L of thesamplesandthe(self-averaging)localizationlengthlloc(Equation2.4.6).Notethatinthelocalizedregime,L>>lloc>>λ.ThisinequalityjustifiesthevalidityoftheWKBapproxi-mation.Toestimate the lengthof theeffectivewell,wenote that theappearanceofatransparentsegment(effectivewell)insidearandomsampleistheresultofaveryspe-cific(andthereforelow-probable)combinationofphasesofthemultiply-scatteredfields.Obviously, the longer such segment the less the probability of its occurrence. On the

“Barrier”

(a) (b) (c)

“Barrier”“Cavity”

FIGURE 2.4.4(See color insert.)Amplitudeofthefieldasafunctionofthecoordinateinsidethewholesample,inthe(a),(b),and(c)partstakenasaseparatesampleeach.(ReprintedwithpermissionfromK.Bliokhetal.,Rev.Mod.Phys.80,1201,2008.Copyright2008bytheAmericanPhysicalSociety.)


otherhand,thetypicalscaleinthelocalizedregimeislloc.Hence,theminimalandthusthemostprobablesizeoftheeffectivewell,lres,isoftheorderofthelocalizationlength,whichweassume(asaresultofself-averaging)tobethesameinallrealizations.Underthisassumption,differentvaluesoftheresonanttransmissioncoefficientsanddifferentintensitiescanbereproducedbyvariationsofthelocationofthewellinthecorrespond-ingquantum-mechanicalformulae.

Ifthecenterofthetransparentsegmentofaresonantrealizationisshiftedadistancedfromthecenterofthesample,thelengthsofthenontransparentpartsoftheresonantreal-izationsbecome

L

L ld1 2 2, = − ±res ,

(2.4.25)

andthetransmissioncoefficientsoftheconfiningbarriersare

T

Ll

1 21 2

,,= −

exp .loc

(2.4.26)

SubstitutingEquations2.4.25and2.4.26intothecorrespondingWKBformulae,weobtain

T d

TTT T d l d l

resloc loc/ /

( )( ) [exp( ) exp( )]

,=+

=+ −

4 41 2

1 22 2

(2.4.27)

| | =+

= − −+

A dT

T TL l d l

d lres

loc loc

loc

/ /

/( )

( )exp( )

exp( ) e2 1

1 22 2

8 8 1

xxp( ),

− d l/ loc

(2.4.28)

whereTres(d)andAres(d)are,respectively,thetransmissioncoefficientandthepeakampli-tudeofthefieldpumpedintothecavitylocatedatapointdbyanincidentresonantmono-chromaticwaveofunitamplitude.Notethat|Ares|2isasymmetricwithrespecttoT1andT2,whichmeansthattheintensityinducedinaneffectivecavitybytheresonantincidentwavedependsonthedirectionofincidence.34Thewidthofhigh(Tres~1)resonancesinalongsampleisexponentiallysmall:

δk

lL

lres

loc loc∼

1exp −

.

(2.4.29)

Equation2.4.27showsthatthetransmissioncoefficientofadisorderedsampleatareso-nanceisindependentofthelengthofthesampleandisgovernedexclusivelybytheloca-tionoftheeffectivecavity.Thisrathercounterintuitiveresultistotallydifferentfromthatforthetypical(nonresonant)transmission,whichdecaysexponentiallywiththeincreaseofthelengthL.

From Equation 2.4.28 it follows that the amplitudes of the centrally located (d<<lloc)strongresonancesareanexponentiallyincreasingfunctionsofthelengthofthesample:

| |

.ALl

2 2 1� �exploc

.

(2.4.30)


Whereasthetransmissionismaximalwhentheeffectivecavityislocatedpreciselyinthemiddleofthesample,Tres(d=0)=1,thelargestamplitudecanbepumpedinaresonatorshiftedfromthecentertowardtheinputbythedistance

d lA = − .0 27 loc , (2.4.31)

whichisindependentofthelengthofthesample.Itshouldbeemphasizedthatinadisorderedsample,foreachresonantfrequency,ωres,

thereexistsitsown“strangeresonator,”which(unlikeregularpotentialwells)istranspar-entonlyforthisparticularωres.Theseresonatorsare locatedatdifferentrandompoints;therefore,itshouldberememberedthatthecoordinatedinEquations2.4.25through2.4.28isfrequency-dependent:d=d(ωres).Thestructureoftheeffectivecavitiesisdescribedbelow.

Totesttheabilityoftheabove-introduceddeterministicmodeltodescribequantitativelythe resonances in randomly layered samples, extensive numerical experiments werecarriedout.ThecomparisonoftheresultsofdirectnumericalsimulationswiththosegivenbyEquations2.4.27through2.4.31demonstratesthattheseeminglyroughanalogybasedonjustonefittingparameter(localizationlength)performssurprisinglywell.31Forexam-ple,notonlythecoordinateoftheeffectivecavitywiththehighestresonantamplitudeisindependent of the total length of the sample and proportional to lloc, as predicted byEquation2.4.31,butalsothecoefficientinEquation2.4.31coincideswiththatobtainedinnumericalsimulationswithanaccuracy~10%.

Toprovideanadequatedescriptionoftheresonancesinrealdielectricstructures,theabsorptionshouldbeincorporatedinthemodel.Althoughinquantum-mechanicalprob-lemslossesareratheruncommon,theabsorptionoflightinadielectricmediumcanbetakenintoaccountbyformallyaddingtothecorrespondingeffectivepotentialanegativeimaginarypartproportionaltotheimaginarypartofthepermittivity.Ifthespatialdecre-mentofthewaveenergyduetoloss,Γ=1/la~Imε0,issmallcomparedtotheinverselocal-ization length, la>>lloc, calculations of the resonant transmittance and intensity yield35(comparewithEquations2.4.27and2.4.28):

T d

TTl T T

resres Re

( )( )

,=+ +

4 1 2

0 1 22Γ ε

(2.4.32)

| | .A d

TT

resres( ) 2

2

2=

(2.4.33)

Theconnectionbetween the resonant reflectionand transmissioncoefficients followsfromtheenergyconservationlawandin,acaseofsmalllosses,takestheform

R T l Ares res res res Re= − − | |1 20Γ ε , (2.4.34)

wherethelasttermisduetoabsorptioninthemedium.Thewidthoftheresonantpeakinthereflectionisdeterminedbytheabsorptioninthemediumandbythetransmittanceoftheeffectivewallsthatformtheeffectivewells,andisequalto

δ

εk

T Tl

resres Re

= + + .Γ 1 2

0 (2.4.35)


Whentheparameter

b l

Ll

=

22

10Γ resloc

Reε exp ,�

(2.4.36)

even small absorption (la>>lloc or la>>L), which practically does not affect the typicaltransmission, suppresses dramatically both the reflected and transmitted fluxes at reso-nances.Interestingly,theresonantreflectioncoefficient,Equation2.4.34,isanonmonotonicfunctionofthedissipationrateΓ,andturnstozerowhenb = −2sinh(d/lloc).Thiseffectisknowninopticsandmicrowaveelectronicsascriticalcoupling.36Asnoenergyisreflectedfromthesampleatcriticalcoupling,itcorrespondstotheresonancewiththehighestintensity.

2.4.4.2 How Is an Effective Cavity Built Up?

Themodelpresentedabovegivesamorepenetratinginsightintothemechanismofforma-tionoflocalizedmodesandallowsanexplanationofhowandwhythecharacteristicsofaresonance(Q-factor,transmissioncoefficient,linewidth,etc.)dependontheparametersoftheeffectivecavity(location,size,andabsorptionrate)associatedwithit.Itisnowclearthattheresonanttransmissionthrougharandomsampletakesplaceatafrequencyωresifaround some (random) point inside this sample there exists an area of length lres ≳ llocwhichistransparentforthisfrequency.

Hereanotherquestionarises:whydoeffectiveresonantcavitiesexistin1Ddisorderedsamples?Inotherwords,whyaredifferentsegmentsofarandomconfigurationtranspar-entoropaqueforthewavewithagivenwavenumberk?Toanswerthisquestionrecallthatinthecaseofweakscattering,theso-calledresonancereflectiontakesplace;thatis,thereflectioncoefficientofanadequatelylongsegment[x − a,x + a]ofarandommediumisproportionaltotheamplitudeofthe2k-harmonicinthepowerspectrum, �ε( , ),2k x ofitsdielectricconstantε(x).37Thisamplitudecanbecalculatedas

�ε ε( , ) ( )2 2k x y yi ky

x a

x a

= .−

−

+

∫ e d

(2.4.37)

ThisexpressionisknownasawindowFouriertransformationwitharectangularwin-dowfunction.Toinvestigatethestructureofaresonantconfiguration,wefirstdeterminedtheresonantwavenumbersforwhichthetransparencyT(k)ofthisconfigurationexceeded,forexample,0.5.Foreachofthosewaves,thespatialdistributionsoftheamplitudeandofthecorrespondinglocalpowerspectrum(2.4.37)werejuxtaposed.AnexampleisshowninFigure2.4.5.Onecanseethatintheareaswherethefieldislocalized,thefunction �ε( , )2k x isstronglysuppressed,whereasinthenontransparentpartsithaswell-pronouncedmax-ima.Therefore,theresonantcavityforafrequencyωresarisesintheareaofadisorderedsamplewhere,accidentally,theharmonicwithwavenumberqres=2kresinthepowerspec-trumofthe(random)permittivityhassmallamplitude.

This result is of profound importance for the correct understanding of the resonanttransparencyof1Drandomsystems.Ithasbeencommonlyacceptedthat,inaccordancewiththecelebratedformula(2.4.8),high(not“typical,”exponentiallysmall)transparencyat a given frequency requires the suppression of the corresponding spatial spectralcomponentin the whole sample.Afundamentallynewoutcomeoftheaboveconsiderationis that for arising of a localized eigenmode, which provides resonant transmission,it suffices for the 2kres-harmonic to be suppressed in a much smaller area of size lres ≳ lloc.


Obviously,theprobabilityofsucharandomeventisexponentially(withrespecttoL/lres>>1)largerthanthatofthesametooccursimultaneouslyinthewholerandomconfiguration.Thisnotonlyexplainstheobservedhighspectraldensityoftheeigenstate(2.4.3)butalsoprovidesabackgroundforharnessingdisordered1Dsystemsindesigningtunablelight-tailoringdevices(seeSection2.4.7).

2.4.4.3 Coupling and Level Repulsion

Anadequatelylong,disordered1Dsamplecancontainseveral isolatedregionswherethespectralharmonicswithwavenumbersclosetoqresaresuppressed.Thespatialoverlapofthewavefunctionslocalizedinsuchregionscouplesthesemodesandleadstotheformationoftheso-callednecklacestates,whichhavebeenpredictedtheoreticallyinthestudiesbyLifshitsandKirpichenkov38andbyPendry11andobservedandstudiedexperimentallyinRefs.9and39. These states have broadened spectral lines7 and contribute substantially to the overalltransmissioninlocalizedregime.AnexampleofanecklacestateisshowninFigure2.4.6.

Necklacestatescanbeeasilyincorporatedinthemodelingschemeastwoormorepoten-tial wells coupled by the evanescent fields that tunnel through barriers separating thewells.9,32 The temporal dynamics of the field in such a chain of coupled resonators isdescribedbyasystemofoscillatorequationswithanexternalforce,damping,andcouplingcoefficient,whichaccountfortheincidentwave,thefiniteQ-factors,andthespatialoverlapofthemodes,respectively.Inthesimplestcaseoftwocavitieswithcoordinatesd1andd2,theequationsthatprovidedaneffectivedescriptionofcoupledmodescanbewrittenas:32

dd

dd

21

2 11 1

12

1 21ψτ

ψτ

ψ ψ+ + − ∆ = +−Q q f( ) ,

(2.4.38)

dd

dd

22

2 21 2

22

2 11ψτ

ψτ

ψ ψ+ + − ∆ = .−Q q( )

(2.4.39)

Wav

e int

ensit

y and

2k-

harm

onic

ampl

itude 1.0

0.8

0.6

0.4

0.2

0.0500 1000 1500 2000

Layer number

FIGURE 2.4.5Normalizedwave (black line)and2k-harmonic (grayarea)amplitudesas functionsof thecoordinate (layernumber).


Hereψi(τ) is thefield inthe itheffectiveresonator,τ=ω0t is thedimensionless time(ω0 isacharacteristiccentralfrequencyoftheproblem),and1−Δi(|Δi|<<1)isthedimen-sionless(inunitsofω0)eigenfrequencyoftheithresonator.Theeffectiveexternalforcefis

f

dl

i= −

−ψ ντ0

1exp ,loc

e

(2.4.40)

whereψ0andν, (|ν − 1|<<1),aretheamplitudeandthefrequencyoftheexternalfield,excitingthefirst(closetotheinput)resonator.Thecouplingcoefficientq<<1oftwocavi-ties,whichisduetothespatialoverlapofmodes,isequalto

q

dl

= −

exploc

(2.4.41)

withdbeingthedistancebetweentheeffectivecavities.TheQ-factorsdescribingthelossesofenergyintheithresonatorare:40

Q

v Tl

i ii

i

− = +1

021Γ g

ω� ,

(2.4.42)

whereΓi is thedissipationrateinthe ithresonator,vg is thewavegroupvelocityinsidethe resonator,andliisthecavitylength.ThelastterminEquation2.4.42accountsforthe

60

50

40

30

20

10

00 500 1000 1500 2000 2500 3000

Layer number

Wav

e int

ensit

y

FIGURE 2.4.6Necklacestate.Normalizedwaveintensityasafunctionofthecoordinate(layernumber).


leakageoftheenergyfromthesystem;therefore,analogouslytoEquation2.4.26,thetrans-missioncoefficientsT1,2aregivenby32

Tdl

TL dl

11

22

= −

= − −

.

exp ,

exp

loc

loc

(2.4.43)

Substitutionψi=Aiexp(−ivτ)reducesEquations2.4.38and2.4.39toacoupleofalgebraicequations,fromwhichtheeigenfrequenciesoftwocomplexindependenteigenmodesofthesystemcanbefound:

ν±

−

= − ∆ + ∆ − ± ∆ − ∆ + .12 2

12

1 21

1 22 2i

Qq( )

(2.4.44)

Wehaveassumedhere thatQ1=Q2=Q.Dependingon theparameters,Equation2.4.44describeseitherlevelanticrossing(levelrepulsion)orthecouplingofisolatedresonatorstoform collective eigenmodes. The anticrossing takes place when the eigenfrequencies oftwocavitiescomecloseandaretransformedintodouble-peaked,moreextendedmodes.Thegapbetweeneigenfrequenciesisequalto

G qe = ∆ − ∆ +( ) ,1 22 2

(2.4.45)

andisminimalat theresonance,Δ1=Δ2.Notethat the levelrepulsionofmodes, that isgenerallyagreedtobeaninherentfeatureofdiffusion,ariseshereintheregimeofstronglocalization.Awayfromresonance,|Δ1−Δ2|>>q, theeigenmodestendtothemodesofisolatedresonators.Theshapesof themodesareexchangedwhenpassing through theresonance;thatis,+(−)eigenmodescorrespondtothefirst(second)resonatoratΔ1<<Δ2,andtothesecond(first)resonatorwhenΔ1>>Δ2.

Thefrequency(ν)dependencesoftheamplitudesA1andA2oftheoscillationsinducedinthecavitiesbyincidentmonochromaticwavesaredeterminedbytheratiobetweenthecouplingcoefficientqandlossesQ−1.Bothamplitudesareatmaximumattwofrequencies

νmax

± −= − ∆ + ∆ ± − .12

12

1 2 2 2q Q

(2.4.46)

Therearetwodifferentregimesoftheexcitationofcoupledresonators,dependingonthevalueoftheqQfactor,asshowninFigure2.4.7forthecaseofidentical(Δ1=Δ2=0)cavities.32Whenlossesaresmall,sothattheconditionqQ>1holds,therearetwocollectiveanticross-

ingresonantmodeswiththefrequencygapG q QA = − −2 2 andequalfieldamplitudesinbothresonators.AsqQdecreases,theresonantpeaksinthespectraarelocatedneareachotherandmeetwhenqQ=1.IntheregimeqQ < 1,thereisonepeakatν=1.

The parameter qQ that appears in the model has a simple physical meaning: itdetermineswhetherthetworesonatorsshouldbeconsideredasessentiallycoupledorisolated. When Q−1<<q, the losses are negligible and the field characteristics are


essentiallydeterminedbythecoupling.Remarkably,inthiscasethefieldintensityinthefirst(incoming)resonatorisnegligibleatν=1,andalmostalltheenergyisconcentratedinthesecondresonator:A2>>A1.Onthecontrary,whenthelossesprevailoverthecou-pling,Q−1>>q,theincidentwaveonlyexcitesthefirstresonator,andtheenergyiscon-centratedmostlyinit:A1>>A2.

2.4.4.4 Bistability of Anderson Localized States in Nonlinear Random Media

A combination of disorder and nonlinearity offers a multitude of striking physicalphenomena,someofwhichstill remainenigmaticandcall for further investigation. Inparticular, nonlinear interactions between electromagnetic radiation and disorderinfluencestheinterferenceofthemultiply-scatteredwavesandcanaffectlocalizationinratherunusualways.Thisareahaslongbeenthesubjectofkeenscientificinterestthathasquickenedrecently,mostlyasaresultofthecreationofhigh-powerlasersandofthelatestadvancement in studies of Bose–Einstein condensates. Although publications on thetransportandlocalizationinnonlinearrandommediaarenumerous,theoverwhelmingmajorityoftheanalyticalresultshaveacommonshortcoming:theyarerelatedtoensem-ble-averagedcharacteristics,whichareofalittleusewhenindividuallocalizedstatesareconcerned.Abreakthroughinthetheoreticalstudyof thedisorder-inducedresonanceshas been made possible with the above-presented quantum-mechanical deterministicmodel, in which the nonlinearity was incorporated. Surprisingly, this rather simpleapproachnotonlyofferedaclearerinsightintothephysicsoftheresonancesinnonlinearrandommediabutalsoperformedwellintheirquantitativedescription.13

Accordingtothemodel,thetransmittancespectrumT(k)inthevicinityofaresonantwavelength,|k − kres|<<kresisgivenbytheLorentziandependence

T kII

T

Q k k( ) ,= =

+ −( )

out

in

res

res/1 2 12

(2.4.47)

1 1νν

νres±νres

±

(a)

1

0

1

0

qQ = 10, 3, 1, 0.5

(b)Ain| |Aout| |

FIGURE 2.4.7(See color insert.)Near-resonant transmissionofan incidentwavethroughtwocoupledopenresonatorsatdifferentvaluesofqQ.Thenormalized(i.e.,multipliedbythefactor2Q−1)absolutevaluesofthefieldamplitudesintworesonators,|Aout|(a)and|Ain|(b),areshown.(ReprintedwithpermissionfromK.Bliokhetal.,Rev.Mod.Phys.80,1201,2008.Copyright2008bytheAmericanPhysicalSociety.)


whereIinandIoutaretheintensitiesoftheincidentandoutgoingwaves,respectively.TheresonanttransmissioncoefficientTres=T(d)isgivenbyEquation2.4.32,whered = d(kres).

Obviously,thenonlinearitybecomesmostnoticeableatthepointswheretheresonancesarelocatedandtheintensityismaximal,I=Ires.Itchangestheeffectiverefractiveindexofthe medium leading to the intensity-dependent shift of the resonant wave number:k k Ires res res→ � ( )(Figure2.4.8).

AsthevaluesofIresandIoutareunambiguouslyconnected,theresonantwavenumberisafunctionoftheoutputintensity,andEquation2.4.47establishesarelationbetweentheinputandoutputwave intensities,which in thecaseofweakKerr-typenonlinearity iscubicwithrespecttoIout.13Ithasauniversalformtypicalofnonlinearresonatorswithopti-calbistability.41Theultimatedependence Iout(Iin) isof theS-type,and, insomerangeofparameters,thestationarytransmissionspectrumT(k)isathree-valuedfunction.Typically,oneofthesolutionsisunstable,whereastheothertwoformahysteresisloopintheIout(Iin)dependence.13 Figure 2.4.8 shows nonlinear deformations of the resonant transmissionspectraT(k),whichatlargevaluesoftheparameterχIin(χistheKerrcoefficient)exhibittransitions to bistability. The analytical dependence T(k), derived from Equation 2.4.47,withparametersfoundfromthenumericalexperiments,areinexcellentagreementwiththedirectnumericalsimulations.

Theadequacyofthemodelhasalsobeensubstantiatedbynumericalmodelingintimedomain.Inthesesimulations,thetransitionaloscillationsandreshapingofthetransmit-ted pulse that typically accompany switching between two regimes of transmission indeterministic bistable nonlinear structures42 have been found in disordered nonlinearsamples. Nonreciprocity (diode-like unidirectional propagation) of resonant tunneling

0.3Iin

0.2

Tran

smiss

ion

0.1

0.0618 619

Wavelength (nm)620

(3) Iin(2)

Iin(1)

FIGURE 2.4.8

NonlineardeformationsofthetransmissionspectrumoftheresonanceatdifferentintensitiesI I Iin in in(3)( ) ( )1 2< < of

theincidentwave.Thelight-graystripeindicatesthethree-valuedregion.Onlythelowerandupperbranchesofthetransmissionspectrumarestable.(ReprintedwithpermissionfromI.Shadrivovetal.,Phys.Rev.Lett.104,123902,2010.Copyright2010bytheAmericanPhysicalSociety.)


throughanonlinearrandomstructurethatstemsfromtheintrinsicasymmetryofdisor-derhasalsobeenobserved.

2.4.5 Experimental Studies of Resonances

Comprehensiveexperimentalstudiesoflocalizedstatesanddisorder-inducedresonanceswerecarriedoutinthemicrowavefrequencyrange(14GHz≤f≤20GHz).9,35Alongmetal-licsingle-mode(atthesefrequencies)waveguidefilledwithrandomlyarranged,weaklyabsorbing dielectric slabs were used as 1D disordered system. The experimental setupallowed measurements of the complex transmission and reflection amplitudes and thecomplex field inside the waveguide for different random configurations. Figure 2.4.9depictstheintensityI(x,f )≡|A(x,f )|2generatedinsideasamplebyanincidentmonochro-maticwavewithfrequencyf,asafunctionofcoordinatexandfrequencyf.35Althoughthe“finestructure”ofthefieldchangesdramaticallyfromsampletosample,thegeneralfea-turesintrinsicinall1Ddisorderedsystemsareclearlyrecognizedintheresultsofasinglemeasurement presented, as an example, in Figure 2.4.9. Localized states (resonances)excitedbytheincidentwaveareclearlyseeninFigure2.4.9.Whenb>>1(Equation2.4.36),thehighestofthemarelocatedintheleft(closetotheinput)partofthesample.Thetrans-mittedsignalissuppressedbylossesbelowtheexperimentalnoiseandisindiscernibleinFigure2.4.9evenatresonantfrequencies.Atthesametime,theresonancesmanifestthem-selves(andcanbeeasilydetected)bysharpdipsinthefrequencydependenceofthereflec-tioncoefficient (seeFigure2.4.10).Moreover,atdifferentvaluesofb (i.e.,ofΓ) localizedeigenstatesareexcitedanddetectableinreflectionindifferentregionsofthesystem,thusprovidingapossibilityforscanningthesamplethroughvariationsoflosses.Thismeansthatdissipation,whichusuallyimpairstheexcitationandobservationofresonancesin1Drandomsamplesimprovesessentiallythe“observability”ofthelocalizedstates.

120100

80604020

200

100

0 1416

1820

x (mm) f (GHz)

FIGURE 2.4.9Intensityversusfrequencyandpositioninsidethesample.(ReprintedwithpermissionfromK.Bliokhetal.,Phys.Rev.Lett.97,2439094,2006.Copyright2006bytheAmericanPhysicalSociety.)


InthestudybyBlikohet al.,9 thesamesetupwasusedtoexploreexperimentallythedynamics of formation of the necklace states and to study their spectral and transportproperties.Measurementsweremadeinasequenceofconfigurationsinwhichthespacingbetweentworandomlylocatedscattererscouldchangestepsinacontrolledway.Theposi-tionatwhichtheairgapwasintroducedwaschosentocorrespondtothepeakofasingleAndersonlocalizedmodeoftheunperturbedrandomsample.Thisallowedtomanipulatethe frequency of the selected mode inamanner similar to the tuningof adefect statethroughabandgapinaperiodicstructure.Indoingso,themodefrequencyshiftedandcrossedthefrequenciesofotherlocalizedstates,whichmadeitpossibletostudythecou-plingofmodes.Changingtheairspacingatpointswhereotherstateshavebeenlocalized,allowed to couple several localized modes, thereby creating necklace states extendedthroughoutthesample.Thespectralpositionsofthelocalizedstatesasfunctionsoftheairgap introduced into the sample are plotted in Figure 2.4.11. The frequencies of modeseithercrossoranticross,inreasonablecompliancewithEquation2.4.44.Directmeasure-ments of the electromagnetic field inside the samples have revealed that in the case ofanticrossing(regions1,2,4,5inFigure2.4.11),thecouplingwithinthesamplewasaccompa-niedbythetheoreticallypredictedexchangeofshapes.Incontrast,whenmodescrossed(region3inFigure2.4.11)theydidnotexchangeshapesandremainedpracticallyindepen-dent of each other. When the frequencies of the modes were closest, the two localizedstates coupled into double-peaked modes signifying the formation of quasi-extendednecklacestates.

Theminimumfrequencydifferenceswerecalculatedfortheinteractingpairs1,2,4,5in

Figure2.4.11,asG q QA = − −2 2 withqandQfoundfromEquations2.4.41and2.4.42,andthencompared to themeasuredvaluesof thegap.Acomparisonof themeasuredandcalculateddataispresentedinFigure2.4.12forthefollowingparametersofthesystem:9f0=ω0/2π=15.5GHz, lloc=12mm,ω0Γ=7×107s−1,andvg=c/2.4.Goodagreementexistsbetweentheexperimentandthemodel.

10

0.8

1.6

0.4

0.2

0.0

f (GHz)

R ( f

)

14 16 18 20

FIGURE 2.4.10SpectrumofreflectionforFigure2.4.9. (ReprintedwithpermissionfromK.Bliokhetal.,Phys.Rev.Lett.97,2439094,2006.Copyright2006bytheAmericanPhysicalSociety.)


Andersonlocalizationofmillimeterelectromagneticwaves(75–110GHz)hasbeenstud-iedexperimentally,12utilizing100-layerdielectricstacksofrandomlyshuffledquartzandTeflonwafers.Exponentiallysmalltransmissionattypicalfrequencies,resonanttransmis-sionateigenfrequencies,andenhancedabsorptionhavebeenobserved.Slow lightandsuperluminalgroupvelocities,whichincontrasttophotoniccrystalswerenotassociatedwithanyperiodicityinthesystem,havealsobeendiscovered.

16.5

Freq

uenc

ies o

f mod

es (G

Hz)

Air gap inside the sample (mm)

16.0

15.5

15.0

14.50 5 10 15

54

3

2

1

FIGURE 2.4.11Resonantfrequenciesofexcitedlocalizedmodesversusthedrivingparameter—theairgapinsidethesample.Fivepair-interactionregionsarecircled.(ReprintedwithpermissionfromK.Bliokhetal.,Phys.Rev.Lett.101,133901,2008.Copyright2008bytheAmericanPhysicalSociety.)

Theory0.2

0.1

0.0

15.0 15.5 16.0Frequency (GHz)

Experiment

Min

imal

freq

uenc

y gap

(GH

z)

FIGURE 2.4.12Experimentallymeasuredand theoretically calculatedminimal frequencygaps G q QA = − −2 2 forpairsofinteractingmodes1,2,4,5presentedinFigure2.4.11.(ReprintedwithpermissionfromK.Bliokhetal.,Phys.Rev.Lett.101,133901,2008.Copyright2008bytheAmericanPhysicalSociety.)


2.4.5.1 Inverse Scattering Problems and Remote Sensing of Disordered Samples

Microwaveexperiments9,12,35madeitpossibletotestimmediatelythevalidityofthemodelintroducedinSection2.4.4.Theoreticalpredictionsbasedonthemodelhavebeencheckedagainsttheresultsofmeasurementscarriedoutatalargenumberofrandomconfigurationsandinawiderangeofparameters.ItturnedoutthatEquations2.4.23and2.4.32through2.4.35notonlyprovidedanewinsightintothephysicsoftheexperimentallyobservedfea-turespresentedintheprevioussubsectionbutalsowereinagoodquantitativeagreementwiththemeasureddata.Thiswasmadepossibleusingtheseequationsasabasisforformu-lationandsolvingaclassicalwaveinverseproblem:retrievalofinternalcharacteristicsofamediumofpropagationfromparametersoftheexternalfields.Inpractice,analgorithmofremotesensingofrandomsampleshavebeendeveloped,whichhasenablednonintrusivedetection and monitoring of the disorder-induced resonances and determination of theabsorptionandlocalizationlengthsbymeasuringthereflectedandtransmittedfields.Thealgorithmisverysimple.Indeed,Equations2.4.23and2.4.32through2.4.35canbetreatedasfouralgebraicequationsforfourunknowns,andbysolvingthemonecanfindtheloca-tion,d,andthesize,lres,ofaneffectivecavityforeachresonantfrequency,andthelocaliza-tionandabsorptionlengthsofthesample.Then,theintensity,Ares,pumpedinacavitybytheincidentwavecanbecalculatedbyEquation2.4.30.

Insuchamanner,theseparametershadbeenretrievedinRef.12formanydisorderedconfigurations,usingthedirectlymeasuredvaluesofTres,Rres,δf = cδkres,andlloc.Anexam-pleispresentedinTable2.4.1.Shownincolumns2,3,and4are1−Rres,Tres,andδf,respec-tively, measured for the resonances indicated in column 1. The loss tangent,tan( )α ε= Γc f/ Re 0 ,isgivenincolumn6.Thevalueofthelosstangentaveragedoverthefiveresonancesequals8.35×10−4.Thegenuineweightedlosstangentforthedisorderedquartz/Teflon system was 5.2×10−4, so that the measured and retrieved values of theabsorptionagreedtowithintheaccuracyoftheexperiment.Similarexperimentsinthecentimeter-wavelength range35 also yielded retrieved data consistent with the truevalues.

Theremotesensingprocedurecanbealsoappliedformonitoringnonlineardisorderedsamples.Inthisinstance,Tres,kres,andQaredeterminedfromthetransmissionspectruminthelinearregimeasitwasdescribedabove,andtheadditionalexternalparameterofthemedium—Kerrcoefficient—isretrievedfromthemeasuredshiftofthetransmissionspec-tral linewhen the intensitychanges.Thisenablesone toobtain thewholedependenceIout(Iin,k)foranygivenresonanceperformingexternalmeasurementsofT(k)atonlytwodifferentintensitiesoftheincidentwave.

TABLE 2.4.1

MeasuredandCalculatedParametersAssociatedwithFiveResonances

Frequency (GHz) 1–Rres Tres δf (GHz) G e◊/ 102 tanα · 104

f1=83.5 0.978 0.75 0.40 0.83 4.77

f2=92.0 0.998 0.33 0.39 2.6 13.45

f3=105.7 0.993 0.31 0.34 2.25 10.14

f4=101.8 0.87 0.18 0.25 1.33 6.22

f5=99.8 0.77 0.30 0.45 1.5 7.16

Source: ReprintedwithpermissionfromJ.Scalesetal.,Phys.Rev.B76,085118,2007.Copyright2007bytheAmericanPhysicalSociety.

Note: Thelocalizationlengthis1cm(obtainedfromthenonresonanttransmissioncoefficient).


2.4.6 Anderson Localization in Exotic Materials

2.4.6.1 Suppression of Anderson Localization in Disordered Metamaterials

Thetheoreticalstudyofanynewlydiscoveredphysicalphenomenonorlaboratory-createdmaterialalwaysstartsfromasimplified,idealmodel,whichmakesitpossibletounder-standtheunderlyingprinciplesandtoexplainthebasicfeaturesobservedinthepioneer-ingexperiments.More in-depth investigationscall formorerealisticmodels.Soonerorlatter,inparticularwhenitcomestoapplications,takingaccountofdisorderbecomesnec-essary. Such is indeed the case in the current status of research on metamaterials andgraphene.

Unusualphysicalpropertiesofmetamaterialsopenupuniquepossibilitiesfornumer-ous applications in modern optics and microelectronics. As all real metamaterials arealwaysdisordered(mostlyasaresultofinevitablefabricationerrors),theinvestigationoftheeffectsofrandomscatteringontheirtransportpropertiesisnotonlyafundamentalacademicproblembutisalsoofsignificantpracticalimportance.

TheanalyticalandnumericalanalysesbasedonthetransfermatrixmethodpresentedinSection2.4.3showthatinstratifiedmediawithalternatinglayersofright-andleft-handed materials (mixed stacks), the localization properties differ dramatically fromthoseexhibitedbyconventionaldisorderedmaterials.Inparticular,atlongwavelengths,thelocalizationlengthofmixedstackswithrandomrefractiveindicesandnonfluctuat-ingthicknessesisproportionaltothesixthpowerofthewavelength,theresultthathasbeen neither predicted nor observed in conventional 1D random media.14,15 It meansthatleft-handedmetamaterialscansubstantiallysuppressAndersonlocalizationin1Ddisorderedsystems.Thesuppressionrevealsitselfalsointhevanishingofthedisorder-induced resonances when left-handed layers are added to a random stack of normaldielectrics.Thisisattributabletothelackofphaseaccumulationoveramixedsample,due to the cancellation of the phase across alternating left- and right-handed layers.Whenbothrefractiveindexandthicknessofthelayersconstitutingamixedstackfluctu-atethetransmissionlengthinthelong-waverangeofthelocalizedregimeexhibitsthewell-knownquadraticpowerwavelengthdependencewithdifferentcoefficientsofpro-portionalityformixedandhomogeneous(onlymetalayers)randomstacks.However,thetransmissionlengthofamixedstackdiffersfromthereciprocaloftheLyapunovexpo-nentofthecorrespondinginfinitestack,presentingauniqueexampleofa1Ddisorderedsystem,inwhichthelocalizationandtransmissionlengthsaredifferent.Incontrasttonormal disordered materials, the characteristic ballistic and localization lengths ofmixedstacksarealsodifferent,atleastintheweakscatteringlimit.Thecrossoverregionfromlocalizationtotheballisticregimeisrelativelynarrowforbothmixedandhomo-geneousstacks.

PolarizationeffectshavebeenconsideredinRef.36.Itisshownthatthetransportlengthstronglydependsontheangleofincidenceforbothvertical(p)andhorizontal(s)polariza-tionsof the incidentwave. Inparticular,when theangleof incidenceexceedsacriticalangle,anadditionalexponentialdecayarisesduetotheinternalreflectionfromtheindi-viduallayers.Inmixedstackswithonlyrefractive-indexdisorder,p-polarizedwavesarestrongly localized, whereas for the s-polarization the localization is substantiallysuppressed at all angles of incidence. The Brewster anomaly angle depends on boththe polarizationandthenatureofdisorder,thatis,disorderineitherthepermittivityorthepermeability.ForincidenceattheBrewsterangle,localizationissuppressed,and,in


contrast to the case of normal incidence, the localization length is proportional to thesquareofthevarianceofthefluctuationsratherthantothevarianceitself.

Theeffectsofabsorptionon1Dtransportandlocalizationhavebeenstudiedbothana-lyticallyandnumerically.15Itturnsoutthatthecrossoverregionisparticularlysensitivetolosses,sothatevensmallabsorptionnoticeablysuppressesfrequency-dependentoscilla-tions in the transmission length. The disorder-induced resonances, which present animportantsignatureofthelocalizationregime,arealsostronglyaffected(suppressed)byabsorption.

Thefrequencydependence(dispersion)ofthepermittivity,ε,and/orpermeability,μ,hasaprofoundeffectonAndersonlocalizationleadingtoratherunusual,sometimescounter-intuitivephenomena.Themostexoticbehaviorisobservedinmixedstacksatthefrequen-cies,atwhichεorμturnstozero.Inthisinstance,thewavesaredelocalizedfornormalincidence,whereasthelocalizationisenhancedwhendisorderispresentinbothpermit-tivityandpermeabilityofthelayers.

2.4.6.2 Transport and Localization in Disordered Graphene Superlattices

Shortlyafterthediscoveryofhighlyunusualphysicalpropertiesofgraphene,itwasreal-izedthattheelectrontransportinthismaterialhadmanycommonfeatureswiththeprop-agation of light in dielectrics. In mathematical terms, under some (rather general)conditions,Diracequationsdescribingthechargetransportinagraphenesuperlatticecre-ated by applying an inhomogeneous external electric potential could be reduced toMaxwell equations for the propagation of light in a dielectric medium. The role of therefractiveindexofthiseffectivemediumisplayedbythequantityneff=E − U,whereEandUare,respectively,thedimensionlessenergyofthechargecarrierandthescalarpotentialoftheexternalelectricfield.Itiseasytoseethatifthepotentialisapiecewiseconstantfunctionofonecoordinate,thecorrespondinggraphenesuperlatticereproducesalayereddielectricstructure.17Inparticular,alayer,inwhichthepotentialexceedstheenergyoftheparticle, U > E, is similar to a slab with negative refractive index (metamaterial). It isbecauseofthissimilaritythatajunctionoftworegionshavingoppositesignsofE − U(so-calledp–njunction)focusesDiracelectronsingrapheneinthesamewayasaninterfacebetweenleft-andright-handeddielectricsfocuseselectromagneticwaves.43However,theanalogy isnotcomplete:althoughtheequationsareakin, theboundaryconditionsare,generally,different.Comparingtheseconditionsonecaninferthatintheparticularcaseofnormalincidence,thetransmissionofDiracelectronsthroughajunctionissimilartothetransmissionof light throughan interfacebetween twomediawithdifferent refractiveindices but equal impedances. Such an interface is absolutely transparent to light andthereforetotheDiracelectronsingrapheneaswell.ThisexplainstheKleinparadox(per-fecttransmissionthroughahighpotentialbarrier)ingraphenesystems,andleadstothesurprisingconclusion thatDiracelectronsaredelocalized inadisordered1Dgraphenestructure,providingaminimalnonzerooverallconductivity,whichcannotbedestroyedbyfluctuations,nomatterhowstrongtheyare.44Nevertheless,manyfeaturesofAndersonlocalizationcanbefoundinrandomgraphenesystems.17Thereexistadiscreterandomsetofangles(oradiscreterandomsetofenergiesforeachgivenangle)forwhichthecorre-spondingwavefunctionsareexponentiallylocalized.Dependingonthetypeofunper-turbed system, the disorder could either suppress or enhance the transmission. Thetransmission of a graphene system built of alternating p–n and n–p junctions has ananomalouslynarrowangularspectrumand,insomerangeofdirections,itispracticallyindependentoftheamplitudeofthefluctuationsofthepotential.


Disordermanifestsitselfinvariousothersituations,includinggraphenedevicesconsid-eredinRef.48,andthelocalizationofacousticswavesindisorderedandpartiallydisor-deredone-dimensionalstructures.49–53

2.4.7 Conclusion

Theterm“disorder”usuallybearsanegativeconnotation.Itisdeemedobvious(bothineverydaylifeandinphysicsandengineeringaswell)thatirregularitiesarealwaysinjuri-ousanddetrimental.Contrarytothiswidelyheldview,herewearguethatiftreatedprop-erly,disordercanbetakenadvantageofinnumeroustechnicalapplications.Theuniquespectralpropertiesofwavetransportinthelocalizedregimepresentedinthischapterleadonetobelievethatexploitingrandomnesscanbeaneffectivestrategyforcreatinglight-tailoring devices, in particular switchable mirrors and tunable resonant micro- andnano-cavities.

Nowadaysphotoniccrystalsarethemostextensivelyusedforthesepurposes.Theabil-ityofperfectlyperiodicstructurestomanipulatelighthavelongbeendemonstratedwithregardtohigh-Qcavityresonances,spontaneousemissioncontrol,cavityquantumelec-trodynamicaleffects,andsoon.45However,althoughinlaboratorystudiesphotoniccrys-talsperformwhollysatisfactorily,practicalapplicationsarefrequentlyproblematicbecauseoftheheavydemandsontheaccuracyofmanufacturing:evensmalldeviationsfromperi-odicitycouldmodifydramaticallytheopticalcharacteristicsandhindertheperformanceofcrystal-baseddevices.Thatiswhysignificanteffortandfinancialresourcesareexpendedto eliminate disorder and to develop pure, ideally regular structures. Yet, a differentapproach is a possibility: rather than combat the imperfections in periodicity, one canattemptharnessinghighlydisorderedsamplesashigh-Qresonatorsinopticalandmicro-waveswitches,filters,andamplifiers.DespitetherandomcharacterofAndersonmodes,theirbehaviorandevolutionareratherdeterministic,and,therefore,thesemodescanbeusedforefficientcontroloflightsimilartoregularcavitymodes.

Tosuittheopticaldevicedesigner’srequirements,thetransmissionshouldbefast-tun-able.Resonantcavitiesinphotoniccrystalsarecreatedbyimplantingspeciallydesigneddefects.Then,thetransmissioncanbecontrolledbyvaryingthespacingbetweenthegivenfrequencyoftheincidentradiationandtheresonantfrequencyofthecavity,whichhastobeeasilytunable.However,anyshiftofthespectrallineusuallyrequiresstructuralchangesofthewholesample,45,46whichmakessuchmethodspracticallyunusable.

Asthelocalizationlength(andthereforethetypicaltransmissioncoefficient)ofa1Dran-domconfigurationisdeterminedbythepowerspectrumofdisorder,Equation2.4.8,itisobviousthatthefrequencyspectrumofthetransmissioncanbetailoredbyvaryingthespatialstructureofthecorrelationfunction.47Althoughthephysicalideaistrivial,itsimple-mentationfordesigningfast-tunableopticaldevicesisproblematicbecause,justasitisinthecaseofphotoniccrystals,itneedsarearrangementofthesampleasawhole.Thishow-everdoesnotmeanthatuniquetransportpropertiesofdisorderedsystemscannotbeuti-lized.Moresophisticatedanalysisofthenatureofthedisorder-inducedresonancesleadstotheconclusionthattheyareextremelysensitivetochangesoftheparametersofthemediumonlyinsidetheeffectivecavitieswhereeigenmodesarelocalized.Figure2.4.13presentsthenumericallycalculateddependenceoftheresonanttransmissioncoefficientonthevariationsofthedielectricconstant,T(δε),inanareaoccupying1/50ofthetotallengthofarandom


stackoflayers.Itisseenthat1%changeinthepermittivityonlyinthisareaalreadyresultsinadecreaseinTbythefactor103.Thisexamplegivesgoodgroundstobelievethatonecanswitchasamplefromreflectiontotransmissionortunetheemissionofasourcelocatedinsidethesamplebyexternalactions;forexample,illuminatingitbyelectromagneticradia-tionthatchangesthedielectricconstantofthematerialduetononlineareffects.

Acknowledgments

V.F.acknowledgespartialsupportfromtheIsraelScienceFoundation(GrantNo.894/10).K.Y.B.acknowledgespartialsupportfromtheEuropeanCommission(MarieCurieAction)andScienceFoundationIreland(GrantNo.07/IN.1/I906).F.N.acknowledgespartialsup-portfromtheNationalSecurityAgency(NSA),LaboratoryPhysicalSciences(LPS),ArmyResearchOffice(ARO),andNationalScienceFoundation(NSF)(GrantNo.0726909),Grant-in-AidforScientificResearch(S),MEXTKakenhionQuantumCybernetics,andFundingProgramforInnovativeR&DonS&T(FIRST).

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0.8

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0.2

0.0–1.0 –0.5

Tran

smiss

ion

coeffi

cien

t, T

0.0δε/ε (%)

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FIGURE 2.4.13Dependenceoftheresonanttransmissioncoefficientonthevariationsofthedielectricconstant,T(δε), intheareaoccupying1/50ofthetotallengthofarandomstackoflayers.


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A TAYLOR & FRANC IS BOOK

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Optical Properties ofPhotonic Structures

Interplay of Order and Disorder

Edited byMikhail F. Limonov

Richard M. De La Rue

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v

Contents

Preface...............................................................................................................................................ixEditors...............................................................................................................................................xiContributors.................................................................................................................................. xiii

Section 1 Introduction

1 Introduction..............................................................................................................................3Sajeev John

Section 2 Theory

2.1 Optical Properties of One-Dimensional Disordered Photonic Structures.................9Mikhail V. Rybin, Mikhail F. Limonov, Alexander B. Khanikaev, and Costas M. Soukoulis

2.2 Propagation and Localization of Light in Two-Dimensional Photonic Crystals.....23Mikhail A. Kaliteevski, Stuart Brand, and Richard A. Abram

2.3 Modeling the Optical Response of Three-Dimensional Disordered Structures Using the Korringa–Kohn–Rostoker Method................................................................. 39Gabriel Lozano, Hernán Míguez, Luis A. Dorado, and Ricardo A. Depine

2.4 Anderson Localization of Light in Layered Dielectric Structures..............................55Yury Bliokh, Konstantin Y. Bliokh, Valentin Freilikher, and Franco Nori

2.5 The Disorder Problem for Slow-Light Photonic Crystal Waveguides....................... 87Mark Patterson and Stephen Hughes

2.6 Quasicrystalline Photonic Structures: Between Order and Disorder...................... 131Alexander N. Poddubny and Eugeniyus L. Ivchenko

2.7 Multicomponent Photonic Crystals with Inhomogeneous Scatterers..................... 151Alexander B. Khanikaev, Mikhail V. Rybin, and Mikhail F. Limonov

Section 3 Experiment

3.1 Anderson Localization of Light: Disorder-Induced Linear, Nonlinear, and Quantum Phenomena................................................................................................. 171Yoav Lahini, Tal Schwartz, Yaron Silberberg, and Mordechai Segev

vi Contents

3.2 Optical Spectroscopy of Real Three-Dimensional Self-Assembled Photonic Crystals................................................................................................................. 197Juan F. Galisteo López and Cefe López

3.3 Photonic Glasses: Fabrication and Optical Properties................................................ 213Pedro David Garcia-Fernández, Diederik S. Wiersma, and Cefe López

3.4 Superdiffusion of Light in Lévy Glasses.......................................................................227Kevin Vynck, Jacopo Bertolotti, Pierre Barthelemy, and Diederik S. Wiersma

3.5 Optical Properties of Low-Contrast Opal-Based Photonic Crystals........................ 249Alexander A. Kaplyanskii, Alexander V. Baryshev, Mikhail V. Rybin, Alexander V. Sel’kin, and Mikhail F. Limonov

3.6 Light and Small-Angle X-Ray Diffraction from Opal-Like Structures: Transition from Two- to Three-Dimensional Regimes and Effects of Disorder........................................................................................................... 275Anton K. Samusev, Kirill B. Samusev, Ivan S. Sinev, Mikhail V. Rybin, Mikhail F. Limonov, Natalia A. Grigoryeva, Sergey V. Grigoriev, and Andrei V. Petukhov

3.7 Interplay of Order and Disorder in the High-Energy Optical Response of Three-Dimensional Photonic Crystals....................................................................... 301Gabriel Lozano, Hernán Míguez, Luis A. Dorado, and Ricardo A. Depine

3.8 Opal-Based Hypersonic Crystals..................................................................................... 323Andrey V. Akimov and Alexander B. Pevtsov

Section 4 Toward Applications

4.1 Strong Light Confinement by Perturbed Photonic Crystals and Photonic Amorphous Structures......................................................................................343Keiichi Edagawa and Masaya Notomi

4.2 Nonlinear Optics in Silicon Photonic Crystal Nanocavities...................................... 361Lucio Claudio Andreani, Paolo Andrich, Matteo Galli, Dario Gerace, Liam O’Faolain, and Thomas F. Krauss

4.3 Random Lasing Highlighted by π-Conjugated Polymer Films................................. 379Randy C. Polson and Z. Valy Vardeny

4.4 Self-Optimization of Optical Confinement and Lasing Action in Disordered Photonic Crystals........................................................................................... 395Alexey Yamilov and Hui Cao

4.5 Ultrafast All-Optical Switching in Photonic Crystals................................................. 415Valery G. Golubev

viiContents

4.6 Resonant Light Scattering in Photonic Devices: Role of Defects..............................429Andrey E. Miroshnichenko and Yuri S. Kivshar

4.7 Structural Features and Related Optical Responses of Magnetophotonic Crystals................................................................................................445Mitsuteru Inoue, Alexander V. Baryshev, Alexander M. Merzlikin, Hironaga Uchida, and Alexander B. Khanikaev

4.8 Inhomogeneous Hybrid Metal–Dielectric Plasmonic–Photonic Crystals............... 469Sergei G. Romanov

4.9 Light Propagation in Photonic Crystals Infiltrated with Fluorescent Quantum Dots or Liquid Crystal: Different Dimensionality Analysis.................. 487Rabia Moussa, Alexander Kuznetsov, Ryotaro Ozaki, and Anvar A. Zakhidov

anderson localization of light in layered dielectric ... · notice are the great number of meetings...

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