PHYSICAL REVIE% B VOLUME 38, NUMBER 14 15 NOVEMBER 1988-I

Optimal structures for classical wave localization: An alternativeto the Iol'e-Regel criterion

Sajeev John and Raghavan RangarajanJoseph Henry Laboratories of Physics, Jadwin Hall, Princeton University

Princeton, Ne~ Jersey 08544(Received 30 June 1988)

Using the Korringa-Kohn-Rostoker method we compute the band structure for a classical sca-lar wave scattering from a periodic array of dielectric spheres in a uniform background. The op-timal volume filling fraction f of spheres for the creation of a total gap in the density of states,and hence localization, is found to be approximately 11%. This gap persists for refractive indexratios as small as 2.8.

The static structure factor of a disordered mediumplays a crucial role in the determination of transport prop-erties of waves in strongly scattering media. ' For weaklyscattering media, defined by the criterion that the classicalelastic mean free path I is long compared to the wave-length k, various approximation schemes adequately de-scribe wave propagation in terms of classical diff'usion onlong length scales. These approximations, in general, as-sociate no statistical correlation between scatterers atdifferent positions. For optical propagation in relativelydilute or low dielectric contrast microstructures, this"white-noise" approximation to the disorder captures theessential physics of coherent multiple scattering on lengthscales long compared to I.

This approximation has been successfully applied to de-scribe various correlation and fluctuation phenomena inthe weak-scattering regime. s 'n Among these are the op-tical analog of universal conductance fluctuationsobserved in electronic systems and the intensity time auto-correlation function for light scattering from dilute mobiledielectric spheres. s 'n Even in the absence of localizationphenomena, it has been suggested that time autocorrela-tions may be used as a spectroscopic tool to probe the stat-ic and dynamic structure of complex fluids exhibiting mul-tiple light scattering.

The possibility of photon localization, however, requiresthat light propagate through a relatively dense collectionof high dielectric constant scatterers of size comparable tothe optical wavelength. " The first correction to the sim-ple "white-noise" picture is to associate a form factor withindividual scatterers but to let their positions remainessentially uncorrelated. This allows the possibility ofsingle-scattering Mie resonances which can signi6cantlyreduce the elastic mean free path. ' ' It is, nevertheless,problematic to achieve the criterion 2trl/1, = 1 for localiza-tion proposed by IoAe and Regel' at densities sufficientlylow that the scatterers do not become optically connected.

It has been recently suggested' that the resolution tothis dilemma may be found by considering carefullyprepared dielectric superlattice structures. Here localiza-tion may occur even for very weak disorder relative to theotherwise periodic structure. This is based on the funda-mental assertion that the criterion for a mobility edge pro-

CO—V p(r)+U(r)p(r) ebp(r)C2

(la)

posed by Ioffe and Regel' is in fact inapplicable in

strongly correlated scattering media. Any significant per-turbation of the photon density of states by the scatteringmedium will significantly alter the criterion for localiza-tion. This is most easily seen in the extreme case of astrong periodic modulation which creates a gap in thephoton spectrum. At a band edge, light of wave vector kis Bragg scattered into k —G where G is a reciprocal lat-tice vector of the medium. The resulting superposition is astanding wave. At frequencies co slightly above the gap inthe photon density of states p(ro), the wave reacquires apropagating character which is expressed by a long-wavelength envelope which modulates this standing wave.As co approaches the band edge, the wavelength k' of theenvelope diverges. For sufficiently weak disorder that thegap persists, the criterion for localization is no longer2trl/k= 1 but rather 2trl/A, '= l. It is highly plausible thatthere is continuous crossover of the localization criterionfrom one form to the other as the static structure factorS(q) of the medium varies from being independent ofwave vector q (white-noise model) to having sharp Braggpeaks (periodic superlattice).

From an experimental point of view it is, therefore, ofparticular interest to determine the optimal scatteringstructure for the creation of either a gap or a pseudogap inp(ta). The above argument suggests that any significantdeparture of the photon group velocity from its phase ve-locity can significantly enhance the prospect for the obser-vation of a mobility edge.

We consider the scattering of a scalar classical wavefrom an fcc lattice of identical dielectric spheres of dielec-tric constant e, and radius R, embedded in a uniformbackground dielectric with eb 1. The method discussedbelow employs a variation-iteration method developed in-dependently by Korringa' and Kohn and Rostoker'(KKR) for computing the band structure of electrons in aperiodic lattice. The wave equation for a classical scalarwave of amplitude p(r) and frequency ro propagating insuch a dielectric medium can be expressed in the form of aSchrodinger equation:

10 101 1988 The American Physical Society

10 102 SAJEEV JOHN AND RAGHAVAN RANGARAJAN

where

Q)2

U(r) -—2 Xea( I r —R I )

C R

and

(lb)

Here kb = [(p) /c )eb] 'l and integration in (3) is over allspace. Since Bloch's theorem requires that (b(r+R)

exp(ik R)(b(r) for some photon "crystal momentum"k, the integral equation can be reduced to one involvingintegration only over a single unit cell of the lattice

Eb~ r (R~,e.(r) - ~

0, r~R, .

Here R runs over all translation vectors of the lattice andc is the vacuum speed of "light. "

Equation (la) can be formally reexpressed in terms ofthe associated free-photon Green's function

I (r, r') = — exp1kb I r r'I—

(2)4n Ir-r'I

as a linear integral equation:

p(r) d'r'I (r, r')U(r')(b(r') .

and

Rd'r'G(r, r') U(r')(b(r')

G(r, r') =+I (r, r'+R)e'"'R

(4)

(5)

Since the new Green's function G satisfies the Bloch con-dition, it is straightforward to verify that (b(r) does aswell.

As discussed by KKR, the integral Eq. (4) may be de-rived from a variational principle. The solution of (4) cor-responds to a stationary point of the functional

A[(I)]! !

(b U(b! ! ! !

(b ( r) U( r) G( rr')U(r')(b(r')

with respect to variations in (b(r'). As approximate solu-tion may be obtained by introducing a trial wave function

1m' l

(!)(r) g g i'C( jl(k, r)Yt" (e,y), ('7)l Om -I

where Ci are undetermined expansion coefficients,k, =[(cp /c )e,]',ji is a spherical Bessel function andYP is a spherical harmonic. Truncating this expansion at1,„2yields a 9x9 determinantal condition on the "ei-genvalues" (p) /c )eb of Eq. (la) for a fixed "crystalmomentum" k. This may be expressed as

Here primes denote derivatives, ni is a spherical Neumannfunction, and the square brackets denote the Wronskianevaluated on the surface of a scattering sphere:

[X,Y]—= Xd Y —YdXdr dr . R-

The coefficients Bl i are determined for a given k andp) entirely by the static structure factor of the mediumand are independent of the nature of the individualscatterers. They can be expressed in terms of a smallernumber of independent structure constants DLM.

detail, m;I', m'

where

(Sa) Bl,m;I'm' 4& Z DL, MCim;I'm';LM iL,M

where

(9a)

j[i(k,r)ji(kbr)]X {Blm;I'm' [Jl (ka r ),Jl (kb r )]

+kbbll b [jl(k,r),ni(kbr)]] .

(sb)

Ci,m;im;LM-„d& Yi YP'YL (9b)

Convergence of the infinite lattice sums is facilitated b~expressing DLM as the sum of reciprocal lattice sum DLMand a real-space sum DLt(r. Following Ham and Segall'we write DLM DLM+DLM+ b'L pDpg:

(» 4xk L kgi„~ I

6—+k I exp[ —(6+k)'/ril YM(6+k)V G (6+k) —kg

(IOa)

where V is the volume of the unit cell, )I is an arbitrary real positive convergence parameter (DLM is in fact independentof ri), and 6 runs over the entire reciprocal lattice:

L( 2) L+1 „Pao IcD(2) ~ g elk RRLYLM(R) d(( 2Lexp R2(2+i kf 4g 2 (lob)

and

(3)2ir, -p n!(2n —1)

(IOc)

OPTIMAL STRUCTURES FOR CLASSICAL WAVE. . . 10 103

fcc brillouin zone

C)::C)

Q~fcc lattice

08—

FIG. 1. Brillouin zone for an fcc lattice showing the extremalk points of the irreducible part: L(0.5,0.5,0.5), X-(1.0,0.0,O.O), K-(0.75,0.75,0), ~-(1.0,0.25,O. 2S), and W-(1.O,

0.5,0.0). The triangles LKW and XAW subtend the region ofthe irreducible Brillouin zone. The above wave vectors are givenin untis of 2ir/a where a is the length of the side of the cube ofthe fcc lattice.

For the choice g 0.6, we found that convergence couldbe achieved by summing over 125 points in reciprocalspace and 26 points in real space. '

Depicted in Fig. 1 is the first Brillouin zone for an fcclattice. For the dielectric contrasts considered 6&@',/cb ( 12 we find that the size of the photonic band gap isdetermined by the eigenvalue spectrum at points L and 8',these corresponding to the extremal points of the zone sur-face. For fixed e, =12 and eb 1, the band structure forclassical wave propagation was calculated for variousvalues of the volume filling fraction f of spheres. At thislarge dielectric contrast a complete gap in the spectrumoccurs over a wide range off with a maximum occurringat f= 1 l%%. This is depicted in Fig. 2. The band gap isexpressed in terms of the "energy" eigenvalueE—= (ru /c )eb of Eq. (la) and is measured in units of(2ir/a) where a is the side length of the elementary cubeof the fcc lattice (Fig. I). In Fig. 3 we show the frequen-cies [in units of (2n/a)l for the upper and lower bandedges as a function of f for e, /ab 12. Finally in Fig. 4we show the computed ratio of the band gap in frequencyto the center frequency of the gap. A maximum ratio of13/o is found at f=0.15. In all cases the lattice constant

F053

0 20 030 040Volume Filling Fraction f

FIG. 3. Plot of the upper boundary of the

(ai/c) Jab at L and the lower boundary at W asvolume filling fraction f (in units of 2ir/a).

050

band gap in

a function of

a was kept fixed and the radius R, of the spheres wasvaried to change the volume filling fraction.

At the optimum volume-filling fraction f 0.11 the re-fractive index contrast was reduced until the gap disap-peared. The band gap persists down to a ratio (e,/eb)'/of 2.8. Since high refractive index materials of this natureare available ' both in the microwave and optical regimesit is highly plausible that strong localization of photonsmay be observed in carefully prepared, weakly disorderedarrays of scattering spheres. In general, large refractiveindex spheres may be fabricated from semiconducting ma-terials with electron band gaps slightly larger than the en-ergies of the relevant electromagnetic waves to be scat-tered. A trade-off is nevertheless required between largereal part of the dielectric constant and an increasingimaginary part of the dielectric constant as the photonfrequency enters the regime of the absorption edge of thesemiconductor.

Application of these ideas to physical electromagneticwaves requires the generalization of the KKR techniqueto the case of a propagating vector field. It is likely that

0.10— Rotio of Frequency Gapto Center Fr equ ency

O

0.08 0 I2-

th

0.06

0.04-3lo008—

0.02— 004-

o.Io 0.20 0.30 0.40Volume F il ling Fraction f

FIG. 2. Plot of the band gap in (co /c ) bb in units of (2~/a)as a function of volume filling fraction f The largest gap occurs.at f=0.1 1 (e,/eb =12).

010 020 030 040Volume Filling Faction f

FIG. 4. Plot of the ratio of band gap hco in frequency to cen-

tral frequency of band gap ~.

10 104 SAJEEV JOHN AND RAGHAVAN RANGARAJAN

the existence of two polarization states will require aslightly higher refractive index ratio than that required toproduce a band gap for a scalar wave. ' However, the sca-lar wave calculations may provide a valuable guide indetermining the optimal volume filling fraction. This con-jecture is consistent with recent studies of photonic bandstructure by Yablonovitch using microwaves.

Recently, Economou ' has suggested that the photonband gap obtained for the periodic structure is in fact theremnant of a Mie resonance obtained for a single sphere.That is to say, although the single-scattering resonance isremoved by the optical connectivity of the spheres it reap-pears as a perturbation on the total photon density ofstates. This argument suggests the existence of a numberof higher frequency band gaps associated with each of thehigher angular momentum Mie resonances of the singlesphere. The precise volume-filling fraction required to op-timize the size of these higher gaps and the robustness ofthese gaps with the introduction of disorder remain as im-portant open questions. One immediate distinction be-tween scalar waves and electromagnetic waves is the ab-sence of an s-wave resonance due to the transverse natureof the electromagnetic wave. This may lead to an increasein the optimum volume-filling fraction from (0.1 &f& 0.15) for the observation of the lowest-order band gap

for true electromagnetic waves.In summary, the existence of a nontrivial static struc-

ture factor of a disordered medium can have profoundconsequences on the nature of wave transport in it. Sincethe traditional Ioffe-Regel criterion is almost attainablefor classical waves in an uncorrelated random medium, itis plausible that nearly any significant depression of thephoton density of states from its effective medium value(obtained by single-scattering theory) will considerablyenhance the prospects for the experimental observation oflocalization. The numerical results presented here strong-ly support the feasibility of producing the materials re-quired for such an experiment as well as the need for fur-ther band-structure calculations in the case of physicalelectromagnetic waves.

We are grateful to Dr. B. Segall for providing some un-

published tables of the coefficients CI;I,rsr and to Dr.A. R. Williams for advice concerning computationalmethods. We are also grateful to Dr. E. Yablonovitch forcommunicating some of his experimental results on pho-tonic band gaps prior to publication. This work was sup-ported in part by the National Science Foundation underGrant No. DMR 8518163.

'Present address: Department of Physics, University of Califor-nia, Santa Barbara, CA 91306.

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