Physics Today Localization of LightSajeev John Citation: Physics Today 44(5), 32 (1991); doi: 10.1063/1.881300 View online: http://dx.doi.org/10.1063/1.881300 View Table of Contents:http://scitation.aip.org/content/aip/magazine/physicstoday/44/5?ver=pdfcov Published by the AIP Publishing

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LOCALIZATION OF LIGHTLight in certain dielectric microstructures exhibits localizedmodes, similar to the localized wavefunctions of electronsin disordered solids. These microstructures show new effectsin both classical optics and quantum electrodynamics.

Sojeev John

Since the beginning of scientific inquiry the nature of lighthas played a vital role in our understanding of the physicalworld. Physicists have marveled at the dual nature oflight as both corpuscle and wave; we have harnessed theremarkable coherent properties of light through the use oflasers; and the quantum mechanics of the interaction ofphotons with matter continue to provide fascinatingavenues of basic research. In essence, any alteration ofelectromagnetism, the fundamental interaction governingatomic, molecular and condensed matter physics, will leadto fundamentally new phenomena in all these areas.

At the microscopic level, ordinary matter exhibitsbehavior analogous to light waves. Electrons in a crystal-line solid produce electrical conductivity by a constructiveinterference of various scattering trajectories. The wavenature of the electron gives rise to allowed energy bandsand forbidden gaps for its motion in the solid. Disorder inthe crystal hinders electrical conductivity, and for someenergies the electronic wavefunctions are localized inspace. This remarkable phenomenon, first discussed byPhilip W. Anderson' in 1958 but fully appreciated only inthe last decade, arises from wave interference, an effectcommon to both photons and electrons. Yet it is onlyrecently that physicists have begun to ask whetherscattering and interference can give rise to an analogouslocalization of light in an appropriate dielectric micro-structure.2 The semiconductor is an electronic material offundamental importance in present day technology. Isthere an analogous photonic material that may performthe same functions with light that semiconductors do withelectrons? It is my aim in this article to describe such anew class of strongly scattering dielectric microstructures.Some of these microstructures will soon allow experimen-tal confirmation of localization of light.

Light localization is an effect that arises entirely fromcoherent multiple scattering and interference. It may be

Sajeev John is an associate professor of physics at theUniversity of Toronto.

understood purely from the point of view of classicalelectromagnetism. In traditional studies of electromag-netic wave propagation in dielectrics, scattering takesplace on scales much longer than the wavelength of light.Localization of light, much the same as that of electrons,occurs when the scale of coherent multiple scattering isreduced to the wavelength itself. This is an entirely newand unexplored regime of optical transport with bothfundamental and practical significance. Photons in alossless dielectric medium provide an ideal realization of asingle excitation in a static random medium at roomtemperature. (By contrast, studies of electrons in adisordered solid are hampered by the inevitable presenceof electron-electron and electron-phonon interactions.)High-resolution optical techniques also offer the uniquepossibility of studying the angular, spatial and temporaldependence of wave intensities near a localization transi-tion. On the practical side, multiple-scattering spectrosco-py is already proving to be a valuable tool in studying thehydrodynamics of colloids, suspensions and other densecomplex fluids. Multiply scattered electromagnetic wavesalready provide a valuable noninvasive probe in medicine.

Light localization, which occurs at the classical level,also has fundamental consequences at the quantum level.The quantum consequences are most readily seen inanalogy with the electronic energy band gap of asemiconductor: Certain dielectric microstructures haveno propagating modes in any direction for a range offrequencies and exhibit what is called a "completephotonic band gap." Such microstructures consist ofperiodic arrays of high-dielectric spheres or cylinders withdiameters and lattice constants comparable to the wave-length of light. Figure 1 shows one such microstructure inwhich a complete bandgap for microwave radiation hasbeen observed. Strictly speaking, such a structure has noallowed electromagnetic modes in the forbidden frequencyrange, although an impurity placed in the material willintroduce localized modes in the gap, an effect that is alsofamiliar in semiconductors. But in this photonic analog ofa semiconductor, an impurity atom with a transition

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o o o

O G G G G G Ci\j %J%.J %y %> V>Lt iy \J %J XJ %J %

a.

vy_

\ y

"HQ

AS& <

'i \i

W v;* VJ> v

G G € % j y j #QQQQQ %J QQQQQQQ t ! H <H O

Dielectric structure that exhibitsstrong localization ofelectromagnetic waves. Thepatterned structure, which has thepoint-group symmetry of an feelattice, is made by drillingcylindrical holes in a materialwhose refractive index is 3.6.The pattern of holes is bestunderstood by imagining aninverted cone with a 35.26°opening angle centered at eachcircle. The axes of the holes arethree equally spaced radial lineson the surface of the cone. Thecrisscrossing of cylindrical airholes beneath the surface appearsas small triangles inside thecircles. The holes are 8 mmapart. Eli Yablonovitch and hiscolleagues at Bellcore have foundthat this sample exhibits acomplete photonic bandgap formicrowaves. Figure 1

frequency in the band gap will not exhibit spontaneousemission of light; instead, the emitted photon will form abound state to the atom! Thus localization of light and theconcomitant alteration of quantum electrodynamics sug-gest new frontiers of basic research spanning the disci-plines of condensed matter physics and quantum optics.

Localized electrons and localized photonsNature readily provides a variety of materials for thestudy of electron localization. In disordered solids local-ization is more often the rule than the exception. This iseasily seen from the Schrodinger equation for an electronwith effective mass m* moving in a potential V(x) thatvaries randomly in space:

2m*V2 = Eipix) (1)

Electrons with sufficiently negative energy E may gettrapped in regions where the random potential V(x) is verydeep. The rate for electrons' tunneling out of the deeppotentials depends on the probability of finding nearbypotential fluctuations into which the trapped electron cantunnel. This rate increases as electron energy increases.

To quantify these ideas, consider F(x) to have a root-mean-square amplitude Vrms and a length scale a onwhich random fluctuations in the potential take place.The correlation length a for the disorder defines an energyscale ea =#7(2m*a2). For example, in an amorphoussemiconductor, a is the interatomic spacing, ea plays a roleanalogous to the conduction bandwidth of the semiconduc-tor and the zero of energy corresponds to the conductionband edge of the corresponding crystal. In the weakdisorder limit, Vrms4erl, a transition takes place at anenergy Ec ~ — Vrms

2/ea. Successive tunneling eventsallow an electron of energy greater than Ec to traverse theentire solid by a slow diffusive process and therebyconduct electricity, whereas electrons of energy lowerthan Ec are trapped and do not conduct electricity.Anderson first discussed this transition in 1958, and NevillMott called the critical value Ec the mobility edge.3 For

energies much greater than Ec, the scale on whichscattering takes place grows larger than the electron'sde Broglie wavelength and the electron traverses the solidwith relative ease. However, if the disorder becomesstronger, so that Vrms > ea, the mobility edge moves intothe conduction band continuum (E> 0), and eventually theentire band may be filled with states exhibiting Andersonlocalization. Since disorder is a nearly universal featureof real materials, electron localization is likewise aubiquitous ingredient in determining electrical, opticaland other properties of condensed matter.

In the case of monochromatic electromagnetic wavesof frequency <u propagating in an inhomogeneous butnondissipative dielectric medium, the classical waveequation for the electric field amplitude E may be writtenin a form resembling the Schrodinger equation:

- V2E + V(V • E) - ^- efluct (x)E = e0 ^- E (2)c2 c2

Here I have separated the total dielectric constant e(x) intoits average value e0 and a spatially fluctuating partffluct (x). The latter plays a role analogous to the randompotential V(x) in the Schrodinger equation; it scatters theelectromagnetic wave.

In a lossless material, in which the dielectric constante(x) is everywhere real and positive, several importantobservations based on the analogy between the Schro-dinger (1) and Maxwell (2) equations are in order. First ofall, the quantity e0co2/c2, which plays a role analogous toan energy eigenvalue, is always positive, thereby preclud-ing the possibility of elementary bound states of light indeep negative potential wells. It is also noteworthy thatthe laser frequency u multiplies the scattering potentialeMuct (x). Unlike an electronic system, where localization isenhanced by lowering the electron energy, lowering thephoton energy instead leads to a complete disappearanceof scattering. In the opposite high-frequency limit,geometric ray optics becomes valid and interferencecorrections to optical transport become less and lesseffective. In both limits the normal modes of the

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electromagnetic field are extended, not localized. Finally,the condition that e0 + eMuc, > 0 everywhere translates intothe requirement that the energy eigenvalue be alwaysgreater than the effective potential (&>2/c2)eMucl(x)|.Therefore, unlike the familiar picture of electronic local-ization, what we are really seeking when searching forlocalized light is an intermediate frequency windowwithin the positive energy continuum that lies at anenergy higher than the highest of the potential barriers!(See figure 2.) It is for this simple reason that ordinary die-lectrics appearing in nature do not easily localize light.

Independent scatterersThe physics underlying the high- and low-frequency limitsin the case of light can be made more precise byconsidering scattering from a single dielectric sphere.Consider a plane wave of wavelength A impinging on asmall dielectric sphere of radius a 4A of dielectric constantea embedded in a uniform background of dielectric

POSITION

Photon

POSITION

Scattering potential for electrons in a solid(black curve, top) and for photons in arandom dielectric medium (black curve,bottom). The effective scattering potential forphotons is (&>2/c2) eMuu, where eUuu is thespatially varying part of the dielectric. Theelectron (blue) can have a negative energy,and it can be trapped in deep potentials. Bycontrast, the eigenvalue (a>2/c2) e0 (gray line)of the photon (red) must be greater than thehighest of the potential barriers if the dielectricconstant (e0 + e,lua ) is to be real and positiveeverywhere. Figure 2

constant ef, in which the spatial dimension d = 3. Thescattered intensity 7sc.,tl at a distance R from the spherecan be a function only of the incident intensity /„, thedielectric constants ea and eh and the lengths R, A and a.In particular /scaU must be proportional to the square ofthe dipole moment induced in the sphere, which scales asthe square of the sphere volume, and by conservation ofenergy, it must fall off as R'1 ' with distance from thescattering center:

fM,e,,,eh)——I()hi

(3)

Since the ratio /scatt //„ is dimensionless, it follows that/"i W, £„, eb) = fkea> eb )i'A.'1 ' ', where f2 is another dimen-sionless function of the dielectric constants. The vanish-ing of the scattering cross section for long wavelengths asA "' ' ", obtained here by purely dimensional arguments,is the familiar result explaining the blue of the sky.

The weak A "'H " scattering is the primary reasonthat electromagnetic modes are extended in most natural-ly occurring three-dimensional systems. This behaviorholds also for a dense random collection of scatterers. Inthat case, the elastic mean-free path / is proportional toA'1 + ' for long wavelengths (see figure 3). This generaliza-tion of Rayleigh scattering to d spatial dimensions is alsoapplicable to anisotropic dielectric scattering systems.For example, a layered random medium in which scatter-ing is confined to directions perpendicular to the layerswould be described by setting d=l. Alternatively, acollection of randomly spaced uniaxial rods4 in whichscattering is confined to the plane perpendicular to theaxes of the rods would be described by setting d = 2. Aconsequence of the scaling theory of localization, whichapplies to both electrons in disordered solids and electro-magnetic waves in disordered dielectrics, is that all statesare localized in one and two dimensions. For electromag-netic waves in disordered dielectrics the localizationlength g|OC diverges due to Rayleigh scattering in the lowfrequency limit, behaving as £loc ~ / in one dimension andgiot. ~ / exp{(ol/c) in two dimensions.

It is likewise instructive to consider the opposite limit,one in which the wavelength of light is small compared tothe scale of the scattering structures. It is well known thatfor scattering from a single sphere, the cross sectionsaturates at a value of 2ira2 when A -4a. This is a result ofgeometric optics; the factor of two arises because of raysthat are weakly diffracted out of the forward directionnear the surface of the sphere. When discussing a denserandom collection of scatterers, it is useful to introduce thenotion of a correlation length a. On scales shorter than a,the dielectric constant does not vary appreciably exceptfor the occasional interface where the physics of refractionand diffraction apply. The essential point is that theelastic mean-free path never becomes smaller than thecorrelation length. This classical elastic mean-free path Iplays a central role in the physics of localization. Wave in-terference effects lead to large spatial fluctuations in thelight intensity in the disordered medium. If 1^>A, however,these fluctuations tend to average out to give a physicalpicture of essentially noninterfering, multiple scatteringpaths for electromagnetic transport. But when l-*A/2tr,interference between multiply scattered paths drasticallymodifies the average transport properties and a transitionfrom extended to localized normal modes takes place. Ifone adopts the most naive version of the Ioffe-Regel5

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Condition for localization is that theclassical transport mean free path / mustapproximately equal A/2ir (black curve),where A is the wavelength of light. In a

medium with small variations in the index ofrefraction (green curve), the photon states are

extended. In media with large variations inthe refractive index (blue curve), weak

scattering still dominates when A<i27ra orA^-2ira, where a is the characteristic length

scale of the medium. A window oflocalization opens at intermediate

frequencies. Figure 3

condition, 2vl/A~l for localization, with A being thevacuum wavelength of light or the wavelength in aneffective-medium theory for scattering, it follows thatextended states are expected at both high and lowfrequencies. For strong scattering, however, there arisesthe distinct possibility, depicted in figure 3, of localizationwithin a narrow frequency window when A Tin —a. It isthis intermediate frequency regime that we wish toanalyze in greater detail.

I will refer to the aforementioned criterion forlocalization as the free-photon Ioffe-Regel condition. Thecriterion is based on perturbation theory for multiplescattering of free photon states from point-like objects andon averaging over all possible positions of the scatterers.The latter assumes that all possible configurations of thescatterers are equally likely, or that the medium has an es-sentially flat structure factor on average.

A first correction to the above simple picture ariseswhen one associates some nontrivial structure to theindividual scatterers. For example, the scattering crosssection can become quite large when the size of thescatterer is close to a multiple of A, due to what are calledMie resonances in the scattering from dielectric spheres.These resonances have profound consequences for theelastic mean-free path. For spheres of dielectric constantea and radius a embedded in a background of dielectric con-stant efr such that ea fet, ~ 4, the first Mie resonance occursat a frequency given by co/c{2a)~l and yields a scatteringcross section a~6ira2. For a relatively dilute collection ofspheres of number density n, the classical elastic mean-freepath is l~l/na= 2a/9/*, where f is the volume fillingfraction of the spheres. Extrapolating this dilute scatter-ing result to higher density, it should be apparent that for afilling fraction f~l/9, the free-photon Ioffe-Regel condi-tion is satisfied on resonance. One might think thatincreasing the density of scatterers would further decreasethe mean-free path and localize light. The existence of theresonance requires, however, that the "spheres of influ-ence" of the scatterers do not overlap, whereas the fact thatthe cross section on resonance is six times the geometricalcross section indicates that a given sphere disturbs thewavefield over distances considerably larger than theactual sphere radius. Indeed, as the density increasesbeyond the value for the resonance, the spheres becomeoptically connected and the mean-free path begins toincrease rather than decrease. From the single-scatteringor microscopic-resonance point of view, the free-photoncriterion for localization is a very delicate one to achieve, aconclusion that Azriel Genack and Michael Drake" haveshown in recent experiments on light scattering fromrandomly arranged dielectric spheres.

Coherent scatterersThe familiar example of Bragg scattering of an electron ina perfectly periodic crystal tells us that the approach

<LU

WAVELENGTH (A/2*)

WAVEVECTOR (k)

Dispersion relation for photons scatteredcoherently by a face-centered cubic lattice ofdielectric spheres shows gaps for some valuesof k. The gap may persist over the entireBrillouin zone for a material composed ofhighly contrasting dielectrics. The two valuesof the gap shown (blue and black) stem fromthe two possible polarization states of thephoton. Figure 4

based on independent, uncorrelated scatterers outlinedabove overlooks an important aspect of the problem. Theexample underlies what may be regarded as a fundamen-tal theorem of solid state physics, namely, that certaingeometrical arrangements of identical scatterers can giverise to large-scale or macroscopic resonances. The statisti-cal average over all possible positions of the scatterersoverlooks the consequences of this theorem. As we shallsee, the theorem also has important consequences for lightscattering in disordered systems in the limit of a highdensity of scatterers exhibiting short-range spatial order.

Consider a medium with dielectric constant whose

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fluctuating part eHuct(x) = e,(x) + V(x) is a sum of twoterms, one e,(x) = e,iG UGe'G" that arises from a perfect-ly periodic Bravais superlattice of reciprocal vectors G,and a small perturbation V{x) arising from disorder. Herethe dominant Fourier component UG in the sum over thereciprocal lattice vectors is chosen so that the Braggcondition k-G = (1/2)G may be satisfied for a photon ofwavevector k. Such a structure is attainable, albeit for lowdielectric contrast between the scatterer and the back-ground, using a suspension of charged polystyrene balls inwater. The suspensions exhibit charge-induced face-centered cubic and body-centered cubic superlattice ar-rangements as well as a number of disordered phases.Setting V(x) = 0 for the time being, the modulation of thephoton spectrum by the periodic arrangement of scat-terers may be estimated within a nearly free-photonapproximation. Unlike electrons, which are described bya scalar wavefunction, the photon field has two polariza-tion states and they are degenerate. If the electric fieldvector is perpendicular to the plane defined by the vectorsk and k — G (optical s wave), the resulting photondispersion is the same as for scalar wave scattering. If, onthe other hand, the polarization vector lies in the plane ofBragg scattering, the scattering amplitude is diminishedby a factor of cos#, where 9 is the angle between k andk — G (optical p wave). The associated photon dispersionrelations are shown in figure 4.

The existence or near existence of a gap in the photondensity of states is of paramount importance in determin-ing the transport properties and localization of light.2 Thefree-photon Ioffe-Regel condition discussed above assumesan essentially free-photon density of states and completelyoverlooks the possibility of this gap and the concomitantmodification in the character of propagating states. Theelectric field amplitude of a propagating wave of energyjust below the edge of the forbidden gap is to a goodapproximation a linear superposition of the free-photonfield of wavevector k and its Bragg-reflected partner atk — G. As co moves into the allowed band, this standingwave is modulated by an envelope function whosewavelength is given by 2n/q, where q is the magnitude ofthe deviation of k from the Bragg plane. Under thesecircumstances the wavelength that must enter the local-ization criterion is that of the envelope. In the presence ofeven very weak disorder, the criterion 2irl/Aenvelope ~ 1 iseasily satisfied as the photon frequency approaches theband-edge frequency. In fact, near a band edge a>c,^envelope

In the presence of a complete photonic band gap, thephase space (or allowed momentum values) available forphoton propagation is restricted to a set of narrow,symmetry-related cones in the k space, analogous to thepockets of allowed electron states near a conduction bandedge well known in semiconductor physics. Randomnessin the positions of the dielectric scatterers leads to amixing of all nearly degenerate photon branches, evenwhen the disorder V(x) is weak and treated perturbatively.In complete analogy with semiconductors, the band gap isreplaced by a pseudogap consisting of localized states. (Seefigure 5.)

I have discussed in detail the two extreme limits: astructureless random medium for which the criterion2-rrl/A ~ 1 applies and a medium with nearly sharp Braggpeaks and a band gap for which 2nl/Aenvelope ~ 1 yieldslocalization. Ivariably a continuous crossover occursbetween these conditions as the structure factor of a high-dielectric material evolves from one limit to the other.

In the strong scattering regime required to produce asignificant depression of the photon density of states,important corrections to the nearly free-photon picture ofband structure emerge. Recent experimental studies7 byEli Yablonovitch on the propagation of microwaves in adielectric material with refractive index 3.6 containing anfee lattice of spherical air cavities show an almostcomplete photonic band gap when the solid volumefraction f~0.15. (See figure 1.) For a frequency rangespanning about 6% of the gap center frequency, propagat-ing electromagnetic modes are absent in all but a fewdirections. When the solid volume fraction is eitherincreased or decreased from 0.15, the magnitude of the gapdrops sharply. The existence of such an optimum value ofsphere density illustrates a very fundamental principleconcerning the origin of photonic band structures that isabsent for electronic band structures. Bands of allowedelectronic states arise in an ordinary semiconductor fromcoalescence of individual atomic orbitals. But there is noanalog of the atomic orbital in the propagation of lightthrough a periodic dielectric: A photon cannot be bound toa dielectric sphere. Instead, there are Mie scatteringresonances, which occur when the diameter of the sphereis an integral multiple of the wavelength of light. A largephotonic band gap arises when the density of dielectricspheres is chosen such that the Mie-resonances in thescattering by individual scatterers occur at the samewavelength as the macroscopic Bragg resonance of anarray of the same scatterers. This principle may be

Density of states for photons in a disorderedlattice of dielectric scatterers is dominated byRayleigh scattering at low frequencies and by

classical ray optics at high frequencies. Thephotonic gap of the perfect lattice of scatterers

is now replaced by a pseudogap. The statesin the pseudogap are strongly localized, in

analogy with pseudogaps in amorphoussemiconductors. The existence of a

localization window (shaded region) is highlysensitive to the static structure factor of the

dielectric material. Figure 5

Pseudogap, strong localization

Rayleigh

FREQUENCY

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Dielectric half space

X / v - 1

BACKSCATTERED INTENSITY

Coherent backscattering from a disordered dielectric, a precursor to localization, arisesbecause a path (red) and its time-reversed partner (green) interfere coherently, giving rise to abackscattered intensity larger (at small angles between the incident and backscattered wave)than the diffuse reflected background one expects in the absence of interference (left panel).The origin of the intensity is set at the level of the background here. The intensity scale is infractions of the background level. The higher of the two peaks shown (blue) describesbackscattering in which the incident photon helicity is preserved; the smaller peak (gray)describes backscattered intensity when multiple scattering has flipped the helicity. Figure 6

illustrated by a simple example of one-dimensional wavepropagation through a periodic array of square wells ofwidth a and spacing L. Suppose the refractive index is ninside each well and is unity outside. Then the Braggscattering condition is given by A = 2L, where A is thevacuum wavelength of light. The analog of a Mie-resonance in one dimension is a maximum in thereflection coefficient from a single well, and this occurswhen a quarter wavelength fits into the well: A/(4n) = a.Combining these two conditions yields the optimumvolume-filling fraction f=a/L= l/(2n). In analogy withthe formation of an electronic band, the photonic bandarises from the coalescence of Mie-scattering resonances ofindividual spheres.

Some very recent studies have suggested that the useof nonspherical scatterers or lattices with a nontrivialbasis might be even more effective in producing a completephotonic band gap rather than a pseudogap.8 Bandstructure calculations have shown that a complete gapmay be produced with refractive index n 2; 2.0 using adiamond lattice structure. Furthermore, Yablonovitchand his collaborators have found a complete microwavebandgap of width about 20% of the gap-center frequencyusing cylindrical rather than spherical dielectric micro-structures.8

Coherent backscatteringThe analogy between electrons and photons suggests thefabrication of a new class of dielectrics that are thephotonic analogs of semiconductors. In one such class—the periodic array of dielectric scatterers—a photonicband gap arises. Positional disorder of these scatterersalters this picture. As in a semiconductor, the band gap isreplaced by a pseudogap of localized states. In this section,I will describe the process by which a propagating photonbecomes localized and discuss experimental manifesta-tions of this phenomenon from the standpoint of classical

electrodynamics.A photon in a disordered dielectric propagates by

means of a random-walk process in which the length of eachrandom step is given by the classical transport mean-freepath I. On length scales that are long compared to I, it isconvenient to regard this as diffusion of light, with thediffusion coefficient given by D = '4 c/. Here c is someeffective speed of light in the dielectric medium. Unlike aclassical random walker, however, light is a wave, and thisdiffusion process must be described by an amplitude ratherthan a probability, so that interference between all possibleclassical diffusion paths must be considered in evaluatingthe transport of electromagnetic energy. In the case ofoptical waves propagating through a disordered dielectricmedium, this interference effect has been vividly demon-strated by a beautiful series of experiments initiated by Y.Kuga and A. Ishimaru, by Meint P. van Albada and Ad La-gendijk and by Georg Maret and Pierre Etienne Wolf.9 Inthese experiments, popularly known as observation ofcoherent backscattering, incident laser light of frequency coenters a disordered dielectric half-space or slab and theangular dependence of the backscattered intensity ismeasured. For circularly polarized incident light, theintensity of the backscattered peak for the helicitypreserving channel is a factor of two larger than theincoherent background intensity. Coherent backscatter-ing into the reversed helicity channel, however, yields aconsiderably reduced backscattering intensity. The angu-lar width of the peak in either case is roughly S8~A/{2TTI).

Figure 6 shows a scattering process in which incidentlight with wavevector k, = k(l is scattered at pointsx,,x2,... xN into intermediate (virtual) states with wave-vectors k,,k2,.. . kjv_ i and finally into the state kw = k^,which is detected. For scalar waves undergoing anidentical set of wavevector transfers, the scatteringamplitudes at the points x,. .. xN are the same for thepath (red) and the time-reversed path (dashed, green).

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The nature of the interference between the two paths isdetermined entirely by their relative optical path lengths.The result ing relative phase is given byexp[i(k, + ]£.r)-(xN - x,)]. In the exact backscatteringdirection q = k, + k/ = 0, there is constructive interfer-ence and a consequent doubling of the intensity over theincoherent background. If the angle between — k, andk^ is 0, then the coherence condition for small 9 becomesq-(xw — x,) = 2v0\xn — x, \A < 1. In the diffusion approxi-mation |x,, — x,|2~D(tN — <,)~/L/3, where D is thephoton diffusion coefficient and L is the total length of thepath. Thus paths of length L contribute to the coherentintensity for angles less than 9m = A/[2ir(lL/3)1/2].

At larger angles few paths contribute and thebackscattered intensity decreases rapidly. E. Akkermans,Wolf and Roger Maynard have given a detailed derivationof the backscattered lineshape for scalar waves; MichaelStephen and Gabriele Cwilich have extended the deriva-tion to electromagnetic waves."' Fred Mackintosh hasshown that the discussion of lineshapes becomes mosttransparent in the helicity representation."' In additionto time-reversal invariance, there is the parity symmetrybetween the right- and left-handed circular polarizationstates. The symmetries may be broken by the Faradayeffect and by natural optical activity, respectively. Thecalculated excess relative intensities with respect to theincoherent background, based on detailed calculations forthe lineshapes, are also shown in figure 6. In the figurethe intensity level is plotted relative to the diffusebackground level: A level of 1.0 corresponds to a precisedoubling of the light intensity in a particular directionrelative to the background. The angle 9 (measured inradians) is the angle that the wavevector of the back-scattered light makes with respect to the vector — k,,where k, is the incident direction.

The incorporation of coherent backscattering into thetheory of diffusion of wave energy leads to a simplerenormalization group picture of transport (see figure 7).When wave interference plays an important role indetermining transport, as it does in coherent backscatter-ing, the transport of wave energy is not diffusive in thesimple sense that a photon performs a classical randomwalk. Fortunately, there is a way of applying the conceptof classical diffusion even to this significantly morecomplicated situation, provided we make one majorconcession in our classical way of thinking: We must nolonger think of the diffusion coefficient as a local quantitydetermined by a classical mean-free path and a speed ofpropagation; rather, we must consider the macroscopiccoherence properties of the entire illuminated sample. Ina random medium it is reasonable to expect that scatterersthat are very far apart do not on average cause largeinterference corrections to the classical diffusion picture.(The word average here is very important. Changes indistant scatterers can give rise to significant fluctuationsabout the average.) Thus there exists a coherence length£coh 5; / that represents a scale on which we must verycarefully incorporate interference effects in order todetermine the effective diffusion coefficient at any pointwithin the coherence volume. As a specific example,consider a finite size sample of linear size L. By changingthe scale of the sample, the number of diffusion paths thatcan interfere changes, giving rise to an effective diffusioncoefficient D(L) at any point within the sample thatdepends on the macroscopic scale L of the sample. In thevicinity of a mobility edge, on length scales L in the rangel<L<gcoh, the transport of energy is subdiffusive innature as a result of coherent backscattering, which givesa significant wave interference correction to classicaldiffusion. In this range, the spread of wave energy may be

interpreted in terms of a scale-dependent diffusion coeffi-cient that behaves roughly as D (L) ~ (e//3) (IIL). On lengthscales that are long compared to groh , the photon resumesits diffusive motion except with a lower or renormalizedvalue (c//3) (//§,.„>,) of the diffusion coefficient.

The scaling theory of electron localization, formulatedby Elihu Abrahams, Anderson, Don Licciardello and T. V.Ramakrishnan" and based on the ideas of David Thouless,also summarizes the physical picture outlined above. Thetheory predicts that when the laser frequency co is close toa mobility edge co*, the scale-dependent diffusion coeffi-cient may be written in three dimensions as

(4)

The theory also predicts that £c.oh ~ \co — co*| ' diverges asco approaches co*.

The relevance of this result to an optical transmissionexperiment in the absence of dissipation or absorption isdepicted in figure 7. Consider first the case in which thecoherence length is short compared to the slab thickness.The time required for an incident photon to traverse thethickness L is given by T(L) = L'A/D(L). For l$£coh <L,the average displacement R of the photon as a function oftime is that of classical diffusion R~tw2. In the case of in-cipient localization, described by l<tL4gcoh, the diffusioncoefficient has the value D(L) = (cl/3)(l/L). The transittime from one face of the slab to the other now scales asT{L)~L3. In other words a photon near the mobility edgesuffers a "critical slowing down" and in time t traverses adistance R~-t'/:i rather than the longer distance t1'2traveled by a classical random walker.

Anomalies associated with incipient localization mayappear in the total intensity transmitted through a slab ofa disordered dielectric illuminated by a steady monochro-matic plane-wave source. For the case of classicaldiffusion, the transmission coefficient T, defined as theratio of the total transmitted intensity to the total incidentintensity, is given by the relation T = IIL, where I is theclassical elastic mean-free path. This may also be writtenas T = 3D/cL, from which one may infer a transmissioncoefficient T~P/(^c^hL) for Z5£coh 4L, but a new scaledependence T~ P/L2 appears in the incipient localizationregime /<<L<g<fcoh. Genack and Narciso Garcia haveobserved scale-dependent diffusion of precisely this naturein microwave scattering from a disordered collection ofteflon and aluminum spheres (see cover).

In the case of electrons, a conservation law preventstheir total number from changing. Photons, however,can be absorbed. The discussion of photon localization istherefore not complete without the analysis of wavepropagation in a weakly dissipative disordered medium.By weak dissipation I mean that the inelastic mean-freepath, or the typical distance between absorption events,is large compared to I but nevertheless smaller than thesample size L. This may be modeled by adding a small,constant imaginary part e2 to the dielectric constant, sothat eix) = eo + efluct(x) + ie2. The optical absorption co-efficient a is defined as the constant for the decay withdistance of the intensity Io from a source: / = Ioe ~~ax.The absorption coefficient describes the average absorp-tion on scales long compared to the transport mean-freepath /. It should not be confused with the scatteringlength of the incident beam, which may in fact be muchshorter than /. Classical electromagnetic theory yieldsa = (coe2/D)1/2 for radiation of frequency co. The effects ofcoherent wave interference may be incorporated into thispicture using the scaling theory of localization. For aninfinite medium (L = oo), the diffusion coefficient vanish-es when the coherence length diverges. If Zinel > £coh, it

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Incident light Transmitted light

Optical transport near a photon mobilityedge: For length scales L short compared tothe coherence length £ t l l h , the transport ofelectromagnetic energy is subdiffusive due tocoherent backscattering and must bedescribed in terms of a length-dependentdiffusion coefficient D~(c//3)(//Z.). On scaleslonger than £l ( )h, the photon resumes itsdiffusive behavior on average, but thediffusion coefficient has the renormalizedvalue C~(c//3)(//g\oh). Figure 7

follows that the increase in the absorption coefficient asthe mobility edge frequency co* is approached from theextended-state side can be obtained from the change inthe diffusion coefficient: Because D(w)~ \<o — w*\, a(o>)varies as (e.2/\oj — CJ*\)112 . On the other hand, if thecoherence length exceeds the inelastic length, then Zineiacts as a long-distance cutoff for coherent wave interfer-ence. In this case, there is a residual diffusivity given byZ>(^*) = (cZ/3)(Z/Zinel). Since Zinel = (Z)r,ntl )1/2 andrinei ~\/(e^fo), it follows that the residual diffusivityZ)(<y*)~e2

1/3- Substituting the value of the residualdiffusivity into the expression for a reveals that theabsorption coefficient exhibits an anomalous scalingbehavior with e2 a(co*)~e2^/J. The nontrivial criticalexponent value of V3 arises because of the critical slowingdown of the photon as it approaches localization, whichleads to a greater probability of absorption.

The anomalies in absorption associated with localiza-tion are a general indication of enhanced coupling of theelectromagnetic field to matter. This leads to someprofound new phenomena in atomic physics, which I willdescribe in the framework of quantum electrodynamics.

Quantum electrodynamics of localized lightWhen an impurity atom or molecule is placed in adielectric exhibiting photon localization, the usual lawsgoverning absorption and emission of light from theimpurity must be reexamined. This is most easily seen inthe strong localization limit, that is, in a dielectricexhibiting a complete photonic band gap. For a singleexcited atom with a transition energy to the ground stategiven by fuol} which lies within the band gap, there is notrue spontanteous emission of light. The photon such anatom emits will find itself within the classically forbiddenenergy gap of the dielectric. If the nearest band edgeoccurs at frequency coc, this photon will tunnel a distance£[oc ~c/(w,. \eu0 — o)c ])"2 before being Bragg reflected back,to the emitting atom. The result is a coupled eigenstate ofthe electronic degrees of freedom of the atom and theelectromagnetic modes of the dielectric. (See figure 8.)This photon-atom bound state is the optical analog of anelectron-impurity level bound state in the gap of asemiconductor." The atomic polarizability, which isnormally limited by the vacuum natural linewidth of thetransition, can in the absence of spontaneous emissiongrow sufficiently large near resonance to produce alocalized electromagnetic mode from the nearby propagat-ing band states of the dielectric.

The fundamental weakness of the vacuum photon-atom interaction, as expressed by the fine structureconstant a = V137, is completely offset by this nearlyunrestricted resonance. The alteration of the quantumelectrodynamic vacuum by the dielectric host also appears

in the spectroscopy of atomic levels. The ordinary Lambshift of atomic levels is dominated by the emission andreabsorption of high-energy virtual photons. In a photonicband gap, this self dressing is dominated instead by thereal, bound photon. In general, this will lead to someanomalous Lamb shift. If this level lies near a photonicband edge, a more striking effect is predicted to occur. Inthis case the atom is resonantly coupled to photons ofvanishing group velocity. The resultant self-dressing ofthe atom is sufficiently strong to split the atomic level intoa doublet. The atomic level is essentially repelled by itselectromagnetic coupling to the photonic band edge. Onemember of the doublet is pulled into the gap and retains aphoton bound state, whereas the other member is pushedinto the continuum and exhibits resonance fluorescence.In the nearly free-photon approximation to electromag-netic band structure, the splitting of a hydrogenic 2p]/2level is predicted to be as large as 10~6 fc0. This isanalogous to the observed atomic level splittings, usuallycalled Mollow splittings, which occur when an atom issubjected to an intense external laser field. For adielectric exhibiting photon localization, the same effectmay be achieved without any external field.

Further new phenomena are expected when a collec-tion of impurity atoms is placed into the dielectric. Asingle excited atom can transfer its bound photon to aneighboring atom by a resonance dipole-dipole interac-tion. For a frequency ratio A<u/«u0 = 0.05 between the bandgap and the band center, the photon tunneling distance f ]ocis on the scale of 10a, where the lattice constant a of the di-electric is itself on the scale of the photon wavelength. Forimpurity-atom spacings R = 10-1000 A, the suppression ofdipole-dipole interaction suggested by Gershon Kurizkiand Genack can be neglected.13 The matrix element Mdescribing the hopping of a bound photon from one atom toanother is given roughly by M~/J'2/R\ where the atomicdipole fi~ea,t is given by the product of the electroniccharge and the atomic Bohr radius an. This can beapproximately related to the transition energy fuott~e21'a0by writing M as (e2/al,)(a,)/i?)3. When impurity atomsseparated by R S 10 A have a finite density, it follows thatphotonic hopping conduction will occur through a narrowphotonic impurity band of width ~ (ikon){anlRf within thelarger band gap.

The occurrence of a photonic impurity band suggestsnew frontiers in nonlinear optics and laser physics. Thestrong coupling of light to matter may enhance nonlineareffects and cause them to be sensitive to the impurity-atom spacing. For example, when neighboring impurityatoms A and B are both excited, second harmonicgeneration may occur by the transfer of the bound statefrom atom A to atom B. Since atom B is already excited,the transferred photon creates a virtual state, and it may

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oio

a—I

FREQUENCY

then be emitted as a single photon of energy 2fuo() outsideof the photon band gap. The transfer can take place by di-pole emission from atom A followed by a quadrupole Qvirtual absorption by atom B. This process has anamplitude fiQ/R4. The resulting virtual excitation onatom B has odd parity, and it may then decay by a dipoleemission process. The rate of spontaneous second harmon-ic generation is given by the square of the correspondingamplitude and depends sensitively, as a>0a3(an/Rf, on theimpurity-atom spacing.

Another significant question is that of laser activitywithin the impurity band when many photons are present.The impurity band defines a novel quantum many-bodysystem in which the processes of spontaneous and stimu-lated emission of light are completely confined to andmediated by photonic hopping conduction between atoms.All photons are therefore more likely to contribute to thecoherent, cooperative response of the many-body system.This must of course be balanced against non-electromag-netic relaxation events. For example, if the atom isembedded in the solid fraction of the dielectric, phonon-assisted emission of light as well as other homogeneousline-broadening effects will reduce the lifetime of a singlephoton-atom bound state. For an atom in the vacuumpart of the dielectric, the dominant contribution to thislifetime is due to the finite absorption length Zabs of the or-biting photon. For a midgap atomic level the fraction oftime that the system spends as an orbiting photon is about10 " 7 . This corresponds to a lifetime on the scale of oneminute for each kilometer of absorption length Zabs. Theintegrity of the photonic impurity band requires that allsuch homogeneous line-broadenings for a single atom besmall compared to the overall frequency bandwidth of theimpurity band. Inhomogeneous line broadening effectssuch as random strain fields in the solid will also affect thenature of electromagnetic transport within the photonicimpurity band.

The implications of photon localization in quantumoptics appear numerous and are only beginning to beexplored. The fundamental challenge at present is one ofmaterials science, namely the fabrication of three-dimen-sional arrays of lossless, high-refractive index scatterers ofsize comparable to the wavelength of light in which somedegree of ordering can be induced. A possible solution to

Photon-atom bound state (blue) is predictedwhen an impurity atom is placed in adielectric such that the atom's transitionfrequency coo lies in the localization gap A<y ofthe dielectric. For A.CL>/OJO = 0.05, the boundphoton may tunnel a distance £,ot ~10a,where a is the dielectric lattice spacing, beforebeing Bragg reflected and reabsorbed by theatom. When several atoms are placed adistance R<£|nc apart in the dielectric, thephotons exhibit hopping conduction by meansof the atomic resonance dipole-dipoleinteraction, which leads to the formation of anarrow photonic impurity band in the (larger)photonic band gap (bottom panel). Figure 8

this fabrication challenge lies in the development ofcharged colloidal suspensions of high-index semiconduc-tors, such as TiO2. Even in the case of charged polystyrenespheres in water, which have a refractive index ratio of1.5:1.3, significant changes in the rate of spontaneousemission from atoms placed in solution has been ob-served.14 The charge-induced ordering of spheres leads tosignificant changes in the photon density of states even inthis weak scattering case. Another possible solution to thematerials problem is the refinement of etching techniquesto drill cylindrical holes with diameters comparable to thewavelenth of visible light in bulk semiconductors. Thesepossibilities suggest that the microwave experiments ofGenack and his collaborators, demonstrating mobilityedge behavior in a strongly disordered medium, as well asthe result from Yablonovitch's group, demonstrating acomplete microwave band gap in an ordered structure,may soon be extended to the optical wavelength regime. Aclear demonstration of light localization and a completeelucidation of its consequences appear imminent.

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vitch, T. J. Gmitter, Phys. Rev. Lett. 63, 1950 (1989).8. K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65,

3152 (1990). K. M. Leung, Y. F. Lui, Phys. Rev. Lett. 65, 2646(1990). Z. Zhang, S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).E. Yablonovitch, T. J. Gmitter, K. M. Leung (unpublished).

9. Y. Kuga, A. Ishimaru, J. Opt. Soc. Am. Al, 831 (1984). M. vanAlbada, A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985). P. E.Wolf, G. Maret, Phys. Rev. Lett. 55, 2696 (1985).

10. E. Akkermans, P. E. Wolf, R. Maynard, Phys. Rev. Lett. 56,1471 (1986). M. J. Stephen, G. Cwilich, Phys. Rev. B39, 7564(1986). F. C. Mackintosh, S. John, Phys. Rev. B37,1884 (1988).

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