radiative energy transfer in disordered photonic...

IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 21 (2009) 175401 (16pp) doi:10.1088/0953-8984/21/17/175401

Radiative energy transfer in disorderedphotonic crystalsM V Erementchouk1,3, L I Deych2, H Noh1, H Cao1,4 andA A Lisyansky2

1 Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA2 Physics Department, Queens College, City University of New York, Flushing, NY 11367,USA

E-mail: [email protected]

Received 13 January 2009, in final form 1 March 2009Published 24 March 2009Online at stacks.iop.org/JPhysCM/21/175401

AbstractThe difficulty of description of the radiative transfer in disordered photonic crystals arises fromthe necessity to consider on an equal footing the wave scattering by periodic modulations of thedielectric function and by its random inhomogeneities. We resolve this difficulty byapproaching this problem from the standpoint of the general multiple scattering theory in mediawith an arbitrary regular profile of the dielectric function. We use the general asymptoticsolution of the Bethe–Salpeter equation in order to show that for a sufficiently weak disorder thediffusion limit in disordered photonic crystals is presented by incoherent superpositions of themodes of the ideal structure with weights inversely proportional to the respective groupvelocities. The radiative transfer and the diffusion equations are derived as a relaxation of longscale deviations from this limiting distribution. In particular, it is shown that in general thediffusion is anisotropic unless the crystal has sufficiently rich symmetry, say, the square latticein 2D or the cubic lattice in 3D. In this case, the diffusion is isotropic and only in this case canthe effect of the disorder be characterized by a single mean free path depending on frequency.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The interest in structures with periodic modulations of thedielectric function (photonic crystals [1]) was motivatedinitially by the possibility to modify substantially thespontaneous emission in such media. One of the main motivesof the study, therefore, has been the existence of the completeband-gap [2] when the propagation of light is completelyinhibited inside some frequency region. Only relativelyrecently has it been realized that the periodicity of the refractiveindex in photonic crystals by itself results in a number ofunusual properties even in the absence of the complete band-gap [3–8]. These properties do not necessarily require astrong contrast of the periodic modulation, i.e. the ratio ofthe minimum and the maximum values of the refractive index,and, particularly, can be observed even if the contrast is muchweaker than needed for the gap to open. One of the illustrative

3 Present address: NanoScience Technology Center, University of CentralFlorida, Orlando, FL 32826, USA.4 Present address: Department of Applied Physics, Yale University, NewHaven, CT 06520, USA.

examples is provided by dark modes [9–11], which are notcoupled to plane waves propagating outside the structure. Theexistence of these modes is related to the point symmetry of thephotonic crystal and, therefore, the dark modes present evenif the contrast is very low. The dark modes, as well as otherphenomena such as self-collimation and negative refraction,demonstrate that periodic spatial modulations of the dielectricfunction have more ways to affect the propagation of light thanjust producing a gap in the spectrum.

Real photonic crystal structures always contain oneor another type of disorder regardless of manufacturingprocedure. It is crucially important, therefore, to understand towhat extent disorder affects properties of these structures. Thisissue is of great interest because an interplay between periodicand random variations of the refractive index creates newchallenges for a theory of light propagation in inhomogeneousmedia, and promises new and unusual effects in the radiativetransport.

The problem of the disorder in photonic crystals can beapproached from two perspectives. On one hand, there is an

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J. Phys.: Condens. Matter 21 (2009) 175401 M V Erementchouk et al

issue of effects of disorder on spectral features of photoniccrystals and their manifestation in such characteristics asreflection or transmission spectra. This research directioninvolves, for instance, studying such problems as a dependenceof the width of the photonic band-gap on the degree of thedisorder [12–14]. A different type of question arises whenone is concerned with effects of disorder on propagationof light inside the photonic structure within the frameworkof the transport theory. Problems considered in this caseinclude diffusion in the photonic crystals [15, 16] or enhancedbackscattering [17–19].

The main objective of the current paper is to develop ageneral theoretical approach to wave transport in disorderedphotonic crystals, which would systematically, from firstprinciples, take into account the periodic nature of the averagerefractive index. The microscopic nature of our approachdistinguishes it from earlier papers, which relied either onad hoc modifications of results obtained within the multiplescattering theory in statistically homogeneous media [19, 18],or on phenomenological assumptions regarding the distributionof the field in the bulk of the photonic crystal [16, 17]. Forinstance, it was assumed in the latter papers that propagationof light in the bulk of the photonic structure is the sameas in statistically homogeneous random media, and the bandstructure of the photonic crystal only manifests itself withina narrow surface layer of the sample. This idea is obviouslybased on the assumption that multiple scattering destroysphotonic modes of an ideal periodic structure so that thestructure of the electromagnetic field in the bulk of the photoniccrystal becomes indistinguishable from that of a regulardisordered medium. A number of experimental and numericalresults, however, cast doubts on this assumption. For instance,it was shown in [20] that photonic modes are completelydeveloped in samples with linear dimensions as small as justa few periods. It was also determined experimentally that themean free path, � (see equations (60) and (72)), due to disorderin real photonic crystals (see. e.g. table I in [21]), substantiallyexceeds the lattice constant of the photonic crystal. Even forrelatively high frequencies �a/2πc = 1.6 it was found that�/a ≈ 4. Thus, even in this least favourable case thereare about a hundred elementary cells in a volume with thelinear dimensions of the order of �. Comparing these tworesults and invoking ideas of the separation of length scalesit is reasonable to expect that the underlying periodicity of thephotonic crystals must still manifest itself even in the presenceof disorder. In our paper we show that this expectationis indeed justified, and demonstrate explicitly the effects ofperiodicity on wave transport in the diffusive regime.

The standard multiple scattering theory of wave transportin disordered media depends significantly on the plane waverepresentation of the scattered field. The role of theplane waves is explicitly emphasized by the use of theWigner function in derivations of the radiative energy transferequations [22–25]. In photonic crystals, however, approachesbased on plane waves encounter significant difficulties becauseplane waves are not normal modes of the underlying idealperiodic structure. While the Wigner representation of thefield–field correlation function itself remains, of course, valid

even in such structures, the Wigner function, however, ceasesto be a smooth function of coordinates, which is essential forthe derivation of the radiative transfer and diffusion equations.

Qualitatively, one of the difficulties of the plane wavebased approaches is due to the fact that the plane waves arescattered not only by the random fluctuations of the refractiveindex but also by its periodic modulation. The latter is a purelydeterministic process and is responsible for the formation ofphotonic crystal modes, which can be considered as coherentsuperpositions of the plane waves. Thus, in order to describewave transport in disordered photonic crystals one has to beable to separate the deterministic contribution to the scatteringfrom the one caused by the disorder. This can be achievedby developing the transport theory on the basis of photonicmodes of an ideal crystal, but as we will see even within thisapproach the discrimination between coherent and incoherentprocesses is highly non-trivial. The second, more technical,difficulty in adapting the standard multiple scattering theory tophotonic crystals arises from the fact that the theory developedfor statistically uniform media heavily depends on certainassumptions (e.g. the translational invariance of the averagedGreen’s function) that are not valid in disordered photoniccrystals.

In the present paper we resolve these difficulties anddevelop a consistent multiple scattering theory of lighttransport in media with the periodic-on-average dielectricfunction. Some of the results obtained are actually validalso in media with arbitrary modulation of the background(average) dielectric functions. We introduce the interpretationof the field–field correlation function as the density matrix andshow how it can be used to separate coherent and incoherentcontributions in the transport. Using this idea we generalize theconcept of specific intensities to the case of photonic crystalsand derive respective radiative transfer equations. We alsofind an asymptotic solution of the Bethe–Salpeter equationdescribing a steady state intensity distribution in an infinitemedium far away from sources, which we use to derive adiffusion equation in a steady state regime describing longscale spatial relaxation of the intensity toward the limitingdistribution.

2. Multiple scattering in disordered photonic crystals

In the framework of the scalar model, the spatial distribution ofthe wave field at frequency ω in a disordered photonic crystalis governed by the Helmholtz equation

�Eω(r) + ω2ε(r)Eω(r) = j (r), (1)

where j (r) is the external source. Here and henceforth weuse units with c = 1. In (1) we have introduced the dielectricfunction

ε(r) = ε(r) + �ε(r), (2)

which consists of two components. The periodic part, ε(r +a) = ε(r), constitutes the photonic crystal with a being thevector of lattice translations. The zero-mean random term,�ε(r), describes the deviation of the dielectric function fromthe ideal periodic form. We assume that the random part of

2


the refractive index is a zero-mean Gaussian random field,i.e. its statistical properties are completely characterized bythe covariance K (r1, r2) = 〈�ε(r1)�ε(r2)〉. In appendix Awe provide a model of such inhomogeneities relevant fordisordered photonic crystals. With this assumption (1) canbe readily analysed by standard diagrammatic techniquesbased on the Born series representation of the solutions ofthe integral (Lippmann–Schwinger) formulation of (1), whichwere developed in the theory of multiple wave scatteringin statistically uniform media [26–28]. This approach isvirtually model independent and can be applied for structureswith arbitrary spatial profile of the deterministic part of therefractive index.

The Dyson equation for the Green’s function of (1)averaged over realizations of the disorder, G ≡ 〈G〉, isobtained using the standard diagrammatic technique and hasthe form

Gω(r, r′) = G0(r, r′)

+∫

dr1 dr2 G0(r, r1)�ω(r1, r2)Gω(r2, r′), (3)

where �ω is the self-energy ‘defined’ as a sum of all irreduciblediagrams. We use somewhat cumbersome coordinaterepresentation in order to emphasize its generality andindependence of a particular form of the regular modulationε(r). The latter determines the non-perturbed Green’sfunction, G0(r, r′), which is assumed to be known. Inparticular, in disordered photonic crystals G0(r, r′) is theGreen’s function of the ideal periodic structure.

The main difference between statistically uniform mediaand disordered photonic crystals is reflected in this equationthrough symmetry properties of the unperturbed Green’sfunction. In periodic-on-average systems, this function isinvariant with respect to lattice translations and the group ofpoint symmetries of the underlying photonic structure, whilein the uniform disordered systems it has full translational androtational symmetry. In appendix B we demonstrate that theaveraged Green’s function and the self-energy possess the sametranslational and point symmetries as G0(r, r′). We utilizethese properties by expanding all related quantities in termsof photonic Bloch modes,

n,k(r) = eik·ruk,n(r), (4)

where un,k(r) is periodic with the period of the photoniclattice, k is the Bloch wavevector lying inside the first Brillouinzone, and n enumerates photonic bands. Using these functionswe introduce the matrix representations of the quantitiesappearing in (3) according to

G0(r1, r2) =∑

k,n

gn(k)k,n(r1)∗k,n(r2),

Gω(r1, r2) =∑

k,n,m

Gn,m(k)k,n(r1)∗k,m(r2),

�ω(r1, r2) =∑

k,n,m

�n,m(k)ε(r1)k,n(r1)∗k,m(r2)ε(r2).

(5)

The respective matrix elements in these expansions are

gn(k) = 1

ω2 − ω2n(k)

,

Gn,m(k) = 1

V2

∫

Vdr1 dr2 ε(r1)u

∗k,n(r1)

× Gω(r1, r2)uk,m(r2)ε(r2),

�n,m(k) = 1

V2

∫

Vdr1 dr2 u∗

k,n(r1)�ω(r1, r2)uk,m(r2),

(6)

where ω2n(k) is the dispersion law of the nth band, the

integration is performed over the elementary cell of theideal structure, and V is the volume of the elementary cell.Deriving (6) we use the orthogonality condition of the Blochfunctions

∫

dr ε(r)∗n,k(r)m,q(r) = δmnδ(k − q). (7)

Using (6) we can rewrite (3) in the matrix form

Gmn(k) = gm(k)δmn +∑

l

gm(k)�ml(k)Gln(k). (8)

This equation emphasizes the fact that the translationalinvariance of the self-energy prevents modes with differentBloch vectors being mixed while states corresponding to thesame Bloch vector but belonging to different bands are coupledby non-diagonal elements of the self-energy �ml(k).

In order to analyse the general effect of the disorder, weseparate the diagonal, �d(k), and the off-diagonal, �o(k),parts of the self-energy, representing the latter in the form

�(k) = �d(k) + �o(k). (9)

The diagonal part �d(k) modifies each band independently.It can be accounted for by introducing a modified Green’sfunction

˜G0(k) = 1

G−10 (k) − �d(k)

, (10)

which is determined by a Dyson equation similar to the onewritten for the standard case of a statistically homogeneousmedium. In terms of the modified Green’s function (8) takesthe form

G = ˜G0 + ˜G0�oG. (11)

Equations (10) and (11) show the twofold role of the disorderin disordered photonic crystals. The disorder not only modifieseach band separately, similar to the case of the statisticallyhomogeneous media, but also couples these modified bands.It is important to note that the band coupling is a subject ofvarious selection rules. First, the most restrictive rule comesfrom the translational symmetry of the self-energy. As hasbeen noted, it prevents states characterized by different Blochvectors being coupled. In other words, from the perspective ofa band diagram one can have only ‘vertical’ coupling. Thesecond rule follows from the point symmetries. The self-energy transforms according to identity representation of thesymmetry group of a given point in the reciprocal space ofthe photonic crystal. As a result, its matrix elements between

3


states corresponding to different irreducible representationsvanish. This selection rule has important implications forhigh symmetry points and directions. In particular, it meansthat disorder does not lift the degeneracy at the pointswhere the degenerate states are described by different or bymultidimensional irreducible representations. The first can beshown by direct calculations of the respective matrix elementsof the self-energy. The second follows from the followingargument. The states corresponding to a representation witha dimension higher than 1 can be coupled only with thestates that transform according to the same presentation. Asa result the modified state also transforms according to thispresentation and, hence, the respective state should also bedegenerate. This implies, in particular, that the degeneracy isnot lifted at high frequency �-points.

In order to analyse the effect of the band coupling in moredetail, we consider a two-band model when all bands but two(denoted by 1 and 2) remain uncoupled. The solution of theDyson equation describing the coupled bands has the form

G = 1

D12

(

g−12 − �22 �12

�21 g−11 − �11

)

, (12)

where gi and �i j are the respective matrix elements of theunperturbed Green’s function and the self-energy, respectively.The zeros of the function

D12 = (

g−11 − �11

) (

g−12 − �22

) − �12�21 (13)

give the spectrum of averaged excitations. Introducing ω1,2(k),the unperturbed dispersion laws of the interacting bands, thepoles of the averaged Green’s function can be written in theform

ω2i = 1

2

(

ω21 + ω2

2

) ± �12

2

√

1 + η12. (14)

Here ω21,2 = ω2

1,2 − �11,22 are the band frequencies after thediagonal modification, �12 = ω2

1 − ω22, and

η12 = 4�12�21

�212

(15)

is the parameter characterizing the strength of the bandcoupling. As one could expect, this parameter is proportionalto the matrix elements of the self-energy between the coupledbands and is inversely proportional to the frequency separationof the modified bands. When this parameter is small,η12 � 1, the effect of the band coupling can be neglectedand the photonic crystals can be described in a single bandapproximation. It should be noted, however, that in spite offormal similarity between the averaged Green’s function inthe single band approximation and the respective expressionfor the uniform disordered medium, the transport properties ofthe two systems remain substantially different. The conditionη12 � 1 can be more easily satisfied at lower frequencies,when the band separation is of the order of magnitude of thefundamental band-gap. For higher frequency bands, however,the band coupling may play a significant role even in the caseof photonic crystals with weak disorder and strong contrast ofthe refractive index.

3. Transport in disordered photonic crystals

3.1. General formalism

The transport properties of disordered media are characterizedby the field–field correlation function

ρω1,ω2(r1, r2) = 〈Eω1(r1)E∗ω2

(r2)〉, (16)

which can be used to describe transfer of energy and, generally,spatial distributions and time evolutions of any quantityquadratic in the field. A relation between ρω1,ω2(r1, r2) andthe external sources is provided by the intensity propagator�(r1, r2; r′

1r′2) = 〈Gω1(r1, r′

1)G∗ω2

(r2, r′2)〉 according to

ρω1,ω2(r1, r2) =∫

dr′1dr′

2 �(r1, r′1; r2, r′

2) jω1(r′1) j∗

ω2(r′

2).

(17)Using the standard diagrammatic technique one can show thatthe intensity propagator satisfies the Bethe–Salpeter equation,which can be written as

�(r1, r2; r′1r

′2) = Gω1(r1, r′

1)G∗ω2(r2, r′

2)

+∫

dr3 dr′3 dr4 dr′

4Gω1(r1, r3)G∗ω2(r2, r4)

× Uω1,ω2(r3, r4; r′3, r′

4)�(r′3, r′

4; r′1r

′2). (18)

The kernel Uω1,ω2 is the irreducible vertex presentedformally as a sum of irreducible diagrams [26]. We wouldlike to emphasize that, similarly to the Dyson equation,the Bethe–Salpeter equation holds for an arbitrary regularspatial modulation of the dielectric function, not necessarilyperiodic. We show in appendix B that, regardless of thespatial dependence of the regular part of the dielectric function,the irreducible vertex possesses an important property ofreciprocity

Uω1,ω2(r1, r2; r′1, r′

2) = Uω1,ω2(r′1, r′

2; r1, r2). (19)

Additionally, in disordered photonic crystals this quantity isinvariant with respect to lattice translations (see appendix B),suggesting the following representation for the vortex:

Uω1,ω2(r1, r2; r′1, r′

2) =∑

v1,2,l1,2,qi

ε(r1)ε(r2)ε(r′1)ε(r

′2)

× U v1,v2l1,l2

(q1, q2; q3, q4)δ(

q1 + q4 − q2 − q3)

× q1,v1(r1)q4,l2 (r′2)

∗q2,v2

(r2)∗q3,l1

(r′1), (20)

where the bar over a vector denotes the vector reduced to thefirst Brillouin zone.

Using the Bethe–Salpeter equation, equations (18)and (17), one can derive an equation for the field–fieldcorrelation function ρω1,ω2(r1, r2). We present it in an integro-differential form, which is the most convenient for furtheranalysis. To derive such an equation one can apply operators

1

ε(r1,2)�1,2 + ω2

1,2, (21)

where indices 1 and 2 indicate a coordinate acted upon by theLaplacian, to both sides of (17). As a result, in the region of

4


space free of external sources, one has[

1

ε(r1)�1 − 1

ε(r2)�2 + ω2

1 − ω22

]

ρω1,ω2(r1, r2)

=∫

dr′1 dr′

2 Fω1,ω2(r1, r2; r′1, r′

2)ρω1,ω2(r′1, r′

2), (22)

where

Fω1,ω2(r1, r2; r′1, r′

2) = 1

ε(r1)�ω1(r1, r′

1)δ(r2 − r′2)

− 1

ε(r2)�∗

ω2(r2, r′

2)δ(r1 − r′1)

+∫

dr′′1 dr′′

2

[

δ(r1 − r′′1 )

ε(r1)G∗

ω2(r2, r′′

2 )

− δ(r2 − r′′2 )

ε(r2)Gω1(r1, r′′

1 )

]

× Uω1,ω2(r′′1 , r′′

2 ; r′1, r′

2). (23)

This equation has been a subject of numerous investiga-tions [26, 29], most of which were concerned with the radiativetransfer or diffusion regimes in statistically homogeneousmedia. In many of those works, diffusion was understoodas a transport process characterized by asymptotically slow(both in time and space) changes of the field intensity. In thespectral domain this behaviour manifests itself in the form ofthe characteristic ‘diffusion’ pole of the intensity propagator,which is proportional to (i� − DQ2)−1 in the limits � → 0,Q → 0. Here frequency and wavevector transfers, � =ω1 − ω2 and Q = q1 − q2, characterize slow spatiotemporaldynamics of intensity, and wavevectors q1 and q2 arise from theplane wave representation of the scattered field. This relianceon the plane waves significantly complicates generalizationof standard microscopic derivations of radiative transfer ordiffusive equations to the case of disordered photonic crystals.

In order to better recognize the source of these difficultiesand find a way to circumvent them, it is necessary to re-examine basic physical ideas about diffusion of light indisordered media. As a first step in this direction, in the currentpaper we consider time independent spatial distribution of thewave intensity in an infinite medium far away from the sources.This regime arises in the case of a monochromatic source,when the field in the structure harmonically depends on time,∝ exp(−iωt), so that � = 0. In this case, the equation forthe field–field correlation function ρ = 〈Eω E∗

ω〉 (hereafterwe omit the lower index corresponding to frequency) can beobtained from (22) by setting ω1 = ω2 = ω:[

1

ε(r1)�1 − 1

ε(r2)�2

]

ρ(r1, r2)

=∫

dr′1 dr′

2 F(r1, r2; r′1, r′

2)ρ(r′1, r′

2), (24)

where F ≡ Fω,ω. Using (24), the optical theorem (the Wardidentity) can be derived [26] in the form F(r, r; r′

1, r′2) ≡ 0,

which is especially useful from the technical point of view.Integrating this equation over r we obtain the Ward identityin the form

�(r2, r1) − �∗(r1, r2)

=∫

dr′′1 dr′′

2 [G(r′′2 , r′′

1 ) − G∗(r′′1 , r′′

2 )]U(r′′1 , r′′

2 ; r1, r2).

(25)

The physical picture of the transport in disordered mediais developed using close relation of the function ρ(r1, r2)

to both transport characteristics and coherence properties ofthe field. The description of transport (e.g. the energydensity and the flux) is based on the property that theaveraged values of any quantity quadratic in field can beexpressed in terms of convolution of respective operators withρ(r1, r2) (see equations (33) and (39) below). The coherenceproperties are described considering ρ(r1, r2) as the coherencefunction [30, 31].

A quantity that simultaneously describes such transportrelated characteristics as energy density and flux and coherenceproperties of the system is well known in quantum statistics.It is called the density matrix [32]. Indeed, on the onehand the density matrix can be used to calculate currentand energy densities, and on the other hand the densitymatrix describes mixed states, which can be characterized asincoherent superpositions of pure or coherent states. Usingthe density matrix analogy one can think of the field–fieldcorrelation function as a characteristic of such a mixed state ofthe wave field in disordered media. Separation of coherent andincoherent properties of the field would then involve findingstates whose incoherent superposition would reproduce thefield at the level of function ρ(r1, r2). However, beforedeveloping this idea any further, we need to demonstrate thatthe field–field correlation function has indeed all the formalproperties of the density matrix.

We expand ρ(r1, r2) in terms of the eigenstates of the non-perturbed system, say the modes of the ideal photonic crystal,writing

ρ(r1, r2) =∑

μ,ν

ρμ,νμ(r1)∗ν (r2), (26)

where summation over indices μ and ν enumerating theeigenstates can involve integration. It is easy to see fromthe definition of the correlation function that coefficientsρμ,ν constitute a Hermitian matrix. This matrix can bediagonalized, which corresponds to the spectral representationof the statistical operator in quantum mechanics, by means ofa unitary transformation to another basis

κ = Bκμμ (27)

so that one has (Mercer’s theorem [30, 33])

ρ(r1, r2) =∑

κ

ρκ κ (r1)∗κ (r2) (28)

with ρκ � 0, which follows from the fact that ρ(r1, r2) isnon-negatively defined [34]. Following the standard quantummechanical interpretation, ρ in the diagonal form represents anincoherent superposition or mixture of pure states κ , which,according to (27) are coherent superpositions of states μ. Inorder to clarify the exact meaning of this expression let us showthat the correlation function can be used to calculate the energydensity of the field and its Poynting vector in much the sameway as the density matrix is used in quantum statistics.

The energy density of the field in a steady state can bepresented as

w(r) = 12

[

ω2ε(r)|E(r)|2 + |∇E(r)|2] , (29)

5


where ε(r) is the total dielectric function including bothregular and the random components. By introducing theoperator

w = −(∇1 − ∇2)2/4, (30)

one can show that the averaged energy density can be expressedin terms of the function

w(r1, r2) = wρ(r1, r2) (31)

as〈w(R)〉 = w(R,R), (32)

where we used the Helmholtz equation to arrive at (32). Onecan see that this equation can be presented in the form typicalfor quantum statistics as

〈w(R)〉 = Tr[ρ wR], (33)

where wR = δ(r1 − R)δ(r2 − R)w.As an example let us consider the case when the density

matrix has the form of an incoherent superposition of theeigenstates μ of the ideal system. Then from (33) we find

〈w(R)〉 =∑

μ

ρμwμ(R), (34)

where we have introduced the energy density of the μth mode

wμ(R) = w μ(r1)∗μ(r2)

∣

∣

r1=r2=R. (35)

This expression provides a clear explanation of the notionof incoherent superposition. Indeed, the average energy ofthe field in this expression is presented as a sum of energiesof individual modes μ(r) as expected for the addition ofincoherent fields as opposed to the sum of the field amplitudesexpected for the coherent fields.

Similar expressions can be obtained for the average valueof the Poynting vector, S = iω[E∗∇E − E∇E∗]/2, which canbe calculated using the operator

S = − iω

2(∇1 − ∇2), (36)

which gives〈S(R)〉 = S(R,R), (37)

whereS(r1, r2) = Sρ(r1, r2). (38)

In the case when the density matrix is diagonal in the basis ofthe eigenfunctions of the ideal system we see again that theaverage Poynting vector is a sum of Poynting vectors of eachmode, which indicates the absence of any interference effectsin the superposition of modes μ(r):

〈S(R)〉 = Tr[

ρ SR

] =∑

μ

ρμSμ(R), (39)

where SR = δ(r1 − R)δ(r2 − R)S and Sμ(R) is thedistribution of the Poynting vector in the μth mode

Sμ(R) = S μ(r1)∗μ(r2)

∣

∣

r1=r2=R. (40)

It is seen from equations (34) and (39) that ρμ havethe meaning of the weights of the incoherent superposition.More generally, if we normalize ρ(r1, r2) in such a way thatTr[ρ] ≡ 1, we can interpret ρκ , the eigenvalues of the matrixρμ,ν , as the distribution function in the space of the states κ .The values of any quantity quadratic in field averaged overthe disorder realizations, therefore, can be calculated as theaverage over the distribution function ρκ in a similar way asis done in quantum mechanics. We would like to emphasizehere that the emergence of the incoherent superposition in theproblem of wave propagation is directly related to averagingover realizations of disorder. Without averaging, the densitymatrix defined in (28) would have a form ρμ,ν ∝ aμaν , andmatrices of such form have a single non-zero eigenvalue. As aresult, the sums in (34) and (39) would consist of just one term,indicating that ρ represents a pure or coherent state.

The notion of incoherent superposition expressed byequations (34) and (39) also allows one to provide a physicalmeaning for the states κ diagonalizing the density matrix.The spatial field distribution in a random medium can be inprinciple presented as a linear combination of functions fromany full system: plane waves, Bloch waves, etc. The conceptof normal modes as well defined spatial distributions of thefields that can be excited separately one from another is notvery useful here. Indeed the distribution of the field in arandom medium is so complex that at any given frequency itis impossible to excite a single mode out of infinitely manydegenerate modes. In this situation, functions κ play aspecial role as such distributions of the field whose linearcombination is purely incoherent in the sense of equations (34)and (39). The form of these functions is determined byremaining coherence effects in the scattered waves, and thusby using the density matrix formalism we achieve a separationbetween coherent and incoherent contributions to the energytransport in disordered systems. In other words, one can saythat the functions diagonalizing the density matrix describe themodes which provide the energy transfer.

In order to illustrate these general ideas, let us considera case of wave scattering in a homogeneous random medium.In the case of an infinite medium and asymptotically far awayfrom the sources the field–field correlation function restores itstranslational invariance: ρ(r1, r2) → ρ(r1 − r2). Using planewaves as a basis we can rewrite (26) for this particular case as

ρ(r1, r2) =∫

dq dk ρun(k)δ(k − q). (41)

One can see that the density matrix in this case is diagonal inthe basis of the plane waves so that they are responsible for theincoherent transport.

3.2. The field–field correlation in an infinite photonic crystal:asymptotic behaviour

In this section we consider a solution of the steady state Bethe–Salpeter equation (24) for the correlation function ρ(r1, r2)

valid in an infinite photonic crystal asymptotically far awayfrom the sources. Besides providing an important non-trivialexample of the application of our formalism, this solution

6


shows how the periodicity of the underlying photonic structureaffects the asymptote of the intensity distribution in thedisordered structure and provides a starting point for derivingthe diffusion equation. Using equations (19) and (25) one cancheck that (24) is solved by

ρ(∞)(r1, r2) = 1

2 iN[G∗(r2, r1) − G(r1, r2)], (42)

where N is a normalization constant independent ofcoordinates. Using the reciprocity theorem [26] this isrewritten as

ρ(∞)(r1, r2) = −Im[

G(r1, r2)]

/N. (43)

This solution is valid in a general case regardless of thespecific form of the regular modulation of the dielectricfunction and the distribution of the disorder. In the case ofstatistically uniform media it is reduced to a form found in [35]and [36]. Thus we can consider this function as the asymptoticdistribution of the field–field correlator, which can be called forthis reason an equilibrium density matrix.

In the case of disordered photonic crystals, the functionρ(∞)(r1, r2) can be expanded in terms of normal modes of theunderlying periodic structure. According to (5) we can presentthis expansion in the following form:

ρ(∞)(r1, r2) =∑

k,n,m

ρ(∞)m,n (k)k,n(r1)

∗k,m(r2), (44)

where

ρ(∞)m,n (k) = [

G∗n,m(k) − Gm,n(k)

]

/2 iN

= 1

NV2

∫

Vdr1 dr2 ε(r1)u

∗k,n(r1)

× Im Gω(r1, r2)uk,m(r2)ε(r2). (45)

This expression shows that in the basis of normal modes ofan ideal periodic structure the density matrix is diagonal withrespect to the quasi-wavevector k, but is not diagonal withrespect to the band indices. Following equations (26)–(28) wediagonalized the matrix ρ(∞)

m,n (k) by a unitary transformation

k,m =∑

n

Bmn(k)k,n . (46)

Since the diagonalization procedure involves only band indicesand leaves the Bloch vector intact, functions k,m(r) can alsobe presented in the Bloch form similar to (4)

k,m(r) = eik·ruk,m(r). (47)

Transformation (46) preserves the scalar product because ofunitarity so that

∫

dr ε(r)∗k,m(r)q,n(r) = δ(k − q)δnm . (48)

Using this property we normalize ρ(∞) defining

N = −π

∫

Vdr ε(r)Im [G(r, r)], (49)

where the integration is performed over the elementary cell ofthe ideal structure. Equations (42) and (49) define ρ(∞) asthe asymptotic form of the density matrix, which according toequations (44) and (46) is an incoherent superposition of thestates k,m with respective weights.

As was discussed, the eigenvalues of ρ(∞) define theprobability distribution function on the space of the statesk,m(r) parametrized by the number of the band m and by thepoint in the first Brillouin zone k. As follows from the Dysonequation the singularities of the averaged Green’s function and,respectively, of the eigenvalues of ρ(∞)

m,n are determined bythe dispersion law of the average excitations (see e.g. (13)).For more detailed analysis we consider the situation when wecan neglect the band coupling so that the averaged Green’sfunction is given by (10). In this case, the eigenvalues of thedensity matrix as functions of the quasi-wavevector have thelargest values when k obeys the dispersion equation for a givenfrequency. In other words, these eigenvalues reach maximumvalues on equifrequency surfaces, Fm(ω), corresponding todifferent bands of the ideal photonic crystal. The width of thesemaxima is proportional to Im[�dm(k)] (see (9)). If the disorderis weak in the sense of the Ioffe–Regel criterion (see below),then for frequencies which do not lie at band edges we canobtain

ρ(∞)(r1, r2) ≈ π

2Nω

∑

m

∫

Fm (ω)

dk1

|vm(k)|× e−|γ m(k)·(r1−r2)|k,m(r1)

∗k,m(r2), (50)

where the integration runs along the equifrequency surfaces,vm(k) = ∇kωm(k) is the group velocity, and

γ m(k) = − Im[�dm(k)]vm(k)

2 ω v2m(k)

= 1

2�−1

m (k)vm(k). (51)

Here vm(k) is the unit vector along the direction of the groupvelocity and we have expressed the imaginary part of the self-energy in terms of the respective mean free path �m(k), whichis defined later in (60).

As follows from (50) in the limit of vanishingdisorder, which corresponds to the on-shell approxima-tion [26, 29, 37–39] in the standard theory of transport instatistically homogeneous media, the density matrix takes theuniversal limit

ρ(∞)0 (r1, r2) = π

2Nω

∑

m

∫

Fm (ω)

dk1

vm(k)k,m(r1)

∗k,m(r2).

(52)The magnitude of the group velocity is, in general, not constantalong the equifrequency surfaces. As a result, different statesare not equally presented in the equilibrium distribution, as isseen from equations (50) and (52). In particular, if there are flatbands [40] with low group velocity near the frequency ω, thenthese bands would give the main contribution to ρ

(∞)

0 . Anotherexample of a highly inhomogeneous distribution is providedby the frequencies when an equifrequency surface touches theboundary of the Brillouin zone. In this case, the magnitude ofthe group velocity becomes very low at the points of contact,resulting in an increased weight of the respective states in theequilibrium distribution (see figure 1).

7


Figure 1. The dependence of 1/vm(k) along the equifrequencysurface corresponding to ωa/2π = 0.41. The horizontal plane is thefirst Brillouin zone. Here and below the calculations are presentedfor a square lattice 2D PC made of dielectric cylinders with thecontrast of the refractive indices n = 2 and with the radius–periodratio r/a = 0.4.

It should be noted that the on-shell approximation ispoorly suited for studying the correlation properties of the field.Indeed, the function ρ

(∞)

0 (r1, r2) does not tend to any limit as|r1 − r2| → ∞ while ρ(∞)(r1, r2) in (50) obviously vanishesin this limit. Thus, for this purpose one has to use (42) or inthe case of weak disorder the simpler version (50), as has beenpractically done in [36], where additionally the approximationof spherical equifrequency surfaces was used.

However, for studying transport properties the on-shellapproximation may by appropriate because, as follows fromequations (32) and (37), the energy density and the Poyntingvector are determined by the behaviour of ρ(r1, r2) near thediagonal r1 = r2. We estimate the effect of the exponentialterm in (50) by calculating the average energy density. Thedensity matrix in the on-shell approximation yields the energydensity as a sum of contributions of different modes (see (34))

〈w(R)〉 =∑

k,m

ρ(∞)k,m wk,m(R), (53)

where wk,m(R) is the energy density of the mode k,m of theideal photonic crystal. The exponential term in (50) can beshown to modify each contribution by the factor

∼1 − 1

[4ω�m(k)]2. (54)

In order to evaluate the correction, we note that ω�m(k) is aparameter of the Ioffe–Regel type [27], which is expected tobe much larger than unity far from the localization regime, sothat the deviation of the equilibrium density matrix from (52)can be neglected in the analysis of transport quantities.

The problem of special interest is what happens inthe immediate vicinity of the complete band-gap. Formalapplication of (60) gives vanishing mean free path for themodes lying at the edge of the gap. This means thatapproximation (50) for the density matrix is no longer valid

and that the modes of the ideal photonic crystal are not agood approximation for the modes μ. Such non-perturbativereconstruction of the modes constitutes a special problem andis not considered in the present paper. Therefore, in whatfollows we restrict ourselves to the frequencies which are nottoo close to the complete gap so that the condition ω�m(k) > 1can always be fulfilled if the magnitude of the inhomogeneitiesis not too high.

Let us conclude the analysis of the equilibrium distributionby outlining two significant differences between the transportin statistically homogeneous media and disordered photoniccrystals. The most essential difference is that the radiativetransport in photonic crystals is provided not by plane wavesbut by the modes which reflect properties of the ideal structure.Indeed, these are the modes of the photonic crystals thatconstitute ρ(∞)(r1, r2) rather than pure plane waves. As aresult, even in the asymptotic distribution of the energy densitythe disorder does not wash out the features specific for the idealstructure. In the case of weak disorder when the energy densitycan be approximated by (53), one can see explicitly that forfrequencies near the fundamental gap and higher, the spatialprofile of wk,m(R) is highly inhomogeneous on the lengthscale of the elementary cell [41]. This inhomogeneity is stable,to some extent, with respect to averaging over the equilibriumdistribution function. Most clearly this effect should be seenfor frequencies near the lower edge of the fundamental gap.Indeed, as follows from the variational principle [1] at suchfrequencies the field distribution takes higher values in theregions with higher values of the refractive index for each pointon the equifrequency surface. The same obviously holds for theenergy density of each mode. Thus, we can conclude that eventhe long scale asymptotic behaviour of intensity in disorderedphotonic crystals is highly inhomogeneous. This propertyis unexpected from the point of view of the transport instatistically homogeneous media, where the distinctive featureof this limit is the homogeneous distribution of the energydensity [42].

The second important feature of the field distribution inphotonic crystals is its significant anisotropy as a function ofdirection of the Bloch vector. Anisotropy itself is not, ofcourse, the specific feature of disordered photonic crystals.The similar effect can be expected in regular anisotropicmedia, where the group velocity depends on the directionof the wavevector. In disordered photonic crystals, however,this anisotropy can be very pronounced, especially near thefrequencies corresponding to the edges of (partial) band-gaps.

3.3. The radiative transfer equation

Applying (37) to the equilibrium state described in the previoussubsection,

S(∞)(r1, r2) = Sρ(∞)(r1, r2), (55)

and using the symmetry properties of the equilibriumdistribution given as ρ(∞)(r1, r2) = ρ(∞)(r2, r1), wecan immediately see that the flux in this state vanishes:S(∞)(R,R) = 0. This is of course a result expected foran asymptotic regime in the infinite medium. In order to

8


describe states with a non-zero flux, which are most relevantto experimental situations, we have to consider the correlationfunction at a finite distance from the sources or in a finite-sizedmedium. In this case, we cannot expect that the same modes

will diagonalize the correlation function, but if the deviationsfrom the equilibrium state are small we can assume that non-diagonal elements decay very fast when one moves away fromthe main diagonal of the density matrix. Then, instead of tryingto find the true diagonalizing states we will follow the sameapproach as in the standard transport theory in a statisticallyhomogeneous medium and convert the non-diagonal densitymatrix into a function slowly changing in real space. The field–field correlator in this representation takes the form of

ρ(r1, r2) =∑

k,m

Ik,m(R)k,m(r1)∗k,m(r2), (56)

where R = (r1 + r2)/2 and Ik,m(R) are smooth functions ofthe coordinate R. The functions Ik,m(R) can be interpretedas specific intensities of the respective modes k,m . In orderto show this let us calculate the average energy density andthe average Poynting vector. Using representation (56) inequations (32) and (37) we obtain

〈w(R)〉 =∑

k,m

Ik,m(R)wk,m(R),

〈S(R)〉 =∑

k,m

Ik,m(R)Sk,m(R),(57)

where we have used equations (35) and (40) to define formallythe energy density, wk,m , and the Poynting vector, Sk,m ,corresponding to the states k,m . Equations (57) agree withthe intuitive concept of the specific intensity, justifying ouridentification for functions Ik,m .

In order to derive an equation for the specific intensities,we calculate the matrix element of both sides of (24) betweenthe states ∗

k+q/2,n and k−q/2,n and then use the assumptionof smooth Ik,m(R) as expressed by (C.8). Calculations basedon the separation of length scales (see details in appendix C)lead to the radiative transfer equation in the form

ω Vm(k) · ∇RIk,m(R) = Im[�m,m(k)]Ik,m(R)

+∫

dq∑

n

σm,n(k, q)Iq,n(R). (58)

Here and below the matrix elements for the self-energy and theirreducible vertex in the basis provided by the states μ arerelated to the matrix elements defined in equations (6) and (20)through transformation (46).

In equation (58) we have introduced several importantquantities. The velocity Vm(k) determines the direction of the‘ray’ corresponding to the mode k,m and is expressed in termsof the group velocity of the photonic crystal modes as

Vm(k) = 1

ω

∑

n

|Bm,n(k)|2ωn(k)vn(k). (59)

The spatial decay of the initial distribution Ik,m(R)

concentrated in a single mode towards equilibrium is quantifiedby the mean free path

�−1m (k) = − Im[�m,m(k)]

ωVm(k). (60)

As follows from the optical theorem, the decay is a result ofthe redistribution of energy between the modes specified bydifferent points in the first Brillouin zone and different bandnumbers. Making use of representation (20) the scatteringkernel, which describes the redistribution, can be shown tohave the form

σm,n(k, q) = Nρ(∞)m (k)U m,m

n,n (k,k, q, q). (61)

Initially, in relatively small samples or near the boundaryof a big sample, the distribution of energy can be arbitrarycomplex. For example, almost all energy can be concentratedin only a few modes. The distribution changes along thesample according to (58) and eventually Ik,m(R) tends toρ

(∞)k,m , which cancels both sides of (58). This confirms

our interpretation of ρ(∞)

k,m as the asymptotic equilibriumdistribution.

If we can neglect the interband coupling and use the on-shell approximation, which is justifiable in the limit of weakand short scale disorder, the picture significantly simplifiessince in this approximation Bm,n(k) become just identitymatrices and the density matrix can be approximated by (52).This means that the transport is provided by the modes ofthe ideal photonic crystal corresponding to the frequency ω

and the specific intensity Ik,m has the transparent meaning ofthe intensity of these modes. The spatial distribution of thespecific intensity is governed by the same equation (58) withthe velocity Vm(k) substituted by the respective group velocityVm(k) → vm(k) as follows from (59).

In order to illustrate the specific features of the radiativetransfer equation in disordered photonic crystals, we calculatethe scattering kernel σm,n(k, q) for a 2D structure madeof dielectric discs in the limit of weak disorder. Weapproximate the intensity propagator by the sum of the ladderdiagrams and additionally assume the δ-correlated disorderwhen K (r1, r2) = V 2δ(r1 − rS)δ(r1 − r2), where rS denotespoints lying in a thin shell near the surface of the idealdisc (see appendix A). In this case the specific intensityis concentrated at the equifrequency surfaces Fm(ω) so thatit can be presented as Ik,m(R) = ˜Ik,m(R)δ[ω2 − ω2

m(k)].The amplitudes ˜Ik,m(R) satisfy the radiative transfer equationof the same structure as (58) with the scattering amplitudebetween the states on the equifrequency surface given by

σm,n(k, q) = ω3V 2

vn(q)V2

∫

Sdr |uk,m(r)|2|uq,n(r)|2, (62)

where the integration is performed in a thin shell near thesurface of the ideal disc. We show the dependence ofσm,n(k, q) on the direction of the final Bloch wavevectorin figure 2(b) for two cases when the frequency ω is wellbelow the frequency of the fundamental gap ωG (ωG ≈0.28 for the structure used in the numerical calculations) andwhen ω � ωG. As one can see, for low frequencies thescattering amplitude is close to isotropic (the variation ofthe scattering amplitude is not visible on this scale), whichis the consequence of slow spatial variation of the Blochamplitudes uk,m(r) and the shape of the equifrequency surfaceclose to spherical. However, when ω approaches ωG both

9


Figure 2. (a) The band structure of the ideal photonic crystal with the same parameters as in figure 1. Only two bands used in calculations forfigures 1 (b) and 3 are shown. The horizontal lines correspond to the frequencies ωa/2π = 0.21 and 0.41. (b) The scattering amplitudeσ(k, q)/ω3V 2 between the states belonging to the first (ωa/2π = 0.21, the internal points) and the second (ωa/2π = 0.41, the pointscorresponding to higher values) bands as a function of the angle between the vectors k and q. The direction of the initial Bloch wavevector qis taken along the �X direction for both frequencies, thus the angle in (b) is counted from the �X direction.

the spatial variation of the Bloch amplitudes and the shapeof the equifrequency surface become non-trivial, resulting instrongly anisotropic scattering. The interesting feature of thisdependence is that the scattering rates between the states,which belong to the same band (the points correspondingto ωa/2π = 0.41 near the horizontal line), are noticeablyhigher than the scattering amplitudes connecting the states inthe different bands (points near the vertical axis). It shouldbe noted that since the gap is not complete the scatteringamplitude remains finite for all directions.

The expression for the mean free path �m(k) in termsof the scattering amplitude σm,n(k, q) is found from therequirement for ρ(∞) to solve the radiative transfer equation,

�−1m (k) = 1

ωvm(k)

∑

n

∫

dq σm,n(k, q), (63)

where the integral runs over the respective equifrequencysurfaces. Substituting (62) into this expression we find �−1

m (k)

in terms of the Bloch amplitudes. The same result canbe obtained from (60) using for the self-energy the sameapproximation as for deriving (62).

The evaluation of the mean free path for the samefrequencies as in figure 2 is presented in figure 3. It is seenthat for the relatively low frequencies the mean free path onlyweakly depends on the direction of the Bloch wavevector,thus reproducing the well known results for statisticallyhomogeneous media. In a vicinity of the partial band-gapthe anisotropic scattering is manifest in strong directionaldependence of the mean free path. This is especially evidentfor an intermediate frequency ωa/2π = 0.31, for which nearthe �M direction the mean free path drops by a factor of five.

3.4. The diffusion equation

The right-hand side of (58) describing spatial variations of thespecific intensities consists of the sum of two terms. In order to

Figure 3. The directional dependence of the mean free path�(ω)ω4/V 2 for frequencies ωa/2π = 0.21 (curve 1), 0.31 (curve 2)and 0.41 (curve 3). The angle is counted from the �X direction.

appreciate their physical significance let assume that at smalldistances from the source only a single mode is excited so thatthe specific intensity can be presented in the following form:

Ik,m ∝ I(R)δm,m0 δ(k − k0). (64)

The first of the right-hand terms in (58) describes exponentialspatial decay of this specific intensity due to scattering andredistribution of the energy among other modes. At smalldistances this term dominates and the solution of (58) can bepresented in the same form as (64) with exponential spatialdependence. However, at distances from the source exceedingthe mean free path this term diminishes and the second,integral, term starts determining the spatial distribution of thespecific intensities, which is no longer exponential. One can

10


describe this situation at asymptotically large distances takinginto account that in this limit the density matrix ρ(r1, r2)

must be close to its limiting value ρ(∞)(r1, r2) so that onecan present ρ(r1, r2) using an appropriate expansion nearρ

(∞)k,m . Drawing an analogy with the derivation of the diffusion

approximation in the standard case [29, 42, 27] and assumingthat the photonic crystal has the centre of symmetry we presentspecific intensities in the following form:

Ik,m(R) = 1

vm(k)

[

W−1wd(R) + vm(k)T−1Sd(R)]

, (65)

where wd(R) and Sd(R) are assumed to be slowly changingfunctions of R, while W and T are scalar and tensorrespectively, which do not depend on spatial coordinates. Withan appropriate choice of these quantities one can show thatwd(R) and Sd(R) in (65) can be presented as

wd(R) = 1

V∫

V(R)

dr w(r, r),

Sd(R) = 1

V∫

V(R)

dr S(r, r),

(66)

where the integration is performed over the elementary cellwith the coordinate R; w(r1, r2), and S(r1, r2) are definedby equations (31) and (38), respectively. These expressionsclarify the physical meaning of wd(R) and Sd(R) as longscale envelopes of the averaged energy density and the flux,respectively. W and T in (65) are found to be

W = ω

2λ(ω), T = ω

2

∑

m

∫

dk vm(k) ⊗ vm(k). (67)

where ⊗ denotes the tensor product, (v ⊗ u)i j = vi u j andλ(ω) is the density of states of the photonic crystal defined asλ(ω) = ∑

μ δ(ω − ωμ).The equations with respect to wd(R) and Sd(R) can be

derived from the radiative transfer equation. Summation overall states on the respective equifrequency surface yields theenergy conservation law

∇ · Sd(R) = 0. (68)

Multiplying both sides of (58) by vm(k) and summing over allstates we obtain

∇D∇wd(R) = 0, (69)

where the diffusion tensor D(ω) up to a constant factoris defined by D(ω) = TM−1T with the current relaxationkernel [43]

M = ω2λ(ω)

4

∑

m

∫

dk

[

�−1m (k)

−∑

n

∫

dqσmn(k, q)

ωvm(k)vm(k) ⊗ vn(q)

]

. (70)

It is important to note that despite the scalar character ofthe initial Helmholtz equation (1) the diffusion in photoniccrystals is characterized by a diffusion tensor rather thanby a single diffusion constant. In particular, if the

periodic modulation is characterized by a rectangular (non-cubic) elementary cell then the eigenvalues of the tensorcorresponding to different principal directions will be different,resulting in anisotropic diffusion.

This diffusion anisotropy, however, appears only in lowsymmetry photonic crystals. If the point symmetry group of aparticular structure is such that irreducible representations ofthe full rotation group remain irreducible for the point groupof the crystal, the diffusion tensor is reduced to a scalar form.Indeed, crystals with such symmetries, e.g. crystals with squareand hexagonal lattices in 2D and cubic and fcc lattices in 3D,do not allow non-trivial tensors of the second rank [44]. In thiscase the diffusion, despite the strong directional dependence ofthe scattering kernel σmn(k, q), is isotropic and is characterizedby the diffusion constant

D = F(ω)

dλ(ω)

�(ω)

1 − 〈cos(θ)〉 , (71)

where d is the dimensionality of the problem, F(ω) =∑

m

∫

dk 1 is the total area of the equifrequency surface,

�−1(ω) = 1

F(ω)

∑

m

∫

dk �−1m (k) (72)

and

〈cos(θ)〉 ≡ �(ω)

F(ω)

∑

m,n

∫

dk dqσmn(k, q)

ωvm(k)vm(k) · vn(q).

(73)The expression for the diffusion coefficient has the samegeneral structure as for statistically homogeneous media [29]with the transport velocity vE (ω) = F(ω)/λ(ω) andthe ‘transport mean free path’ �(ω)/(1 − 〈cos θ〉). Thetransport velocity is independent of the details of the disorderdistribution owing to the δ-functional approximation forthe correlation function K (r1, r2), which is valid if thefrequencies corresponding to morphological resonances at thetypical inhomogeneity size are much higher than the range offrequencies under consideration [45–47]. Figure 4 shows thefrequency dependence of �(ω) for a disordered photonic crystalwith a square lattice. It should be noted that the minimumof �(ω) is reached at the band edge in the �X direction (seefigure 2(a)) and is due to the significant drop of the groupvelocity when the equifrequency surface touches the boundaryof the first Brillouin zone.

It should be emphasized, however, that the possibility tointroduce a single scalar diffusion constant and, respectively,to describe transport by a single transport mean free path isnot a result of the disorder destroying the effect of periodicitybut rather is the consequence of the underlying symmetry ofthe photonic crystal. In the case of low symmetry structuresthe diffusion is described by a tensor, and is characterizedby different effective mean free paths for different principaldirections of the diffusion tensor.

4. Conclusion

In this paper we have developed a systematic approach basedon multiple scattering analysis to the theoretical description of

11


Figure 4. The dependence of the effective mean free path �(ω)/V 2

(dashed line, squares) on frequency in a log–log scale. The solid line(circles) shows the dependence ∝ ω3 (this asymptotics for lowfrequencies is due to the dimensionality of the problem).

incoherent transport properties of disordered photonic crystalsin the steady state regime. The main difficulty in developingsuch a theory using a standard plane wave based multiplescattering approach lies in the separation between wavesscattering due to periodic modulations of the dielectric constantand scattering due to random deviations from the periodicity.One might think that this difficulty is readily resolved byincorporating modes of the ideal photonic crystal instead ofplane waves into the standard transport theory. However,such a straightforward approach fails because of the inherentpresence of two spatial scales, the period of the crystal and themain free path, both of which have to be retained in the theory.As a result, a formally constructed Wigner distribution, whichis the main technical tool in deriving the radiative transferequation in the regular case of a statistically homogeneousmedium, loses its smooth spatial dependence and, hence, itsmethodological usefulness.

We have shown that these difficulties can be overcomeif one incorporates the concept of the radiative transfer asan incoherent process into the foundation of the theory. Inorder to achieve this, we showed that the field–field correlationfunction, ρ(r1, r2) = 〈E(r1)E∗(r2)〉, has all the properties ofa density matrix defined in quantum statistics, and hence can beinterpreted as such. This interpretation allowed us to separatecoherent and incoherent contributions to the transport andeventually obtain radiative transfer and diffusion equations formedia with periodically modulated average dielectric function.

We found an exact asymptotic solution, ρ(∞), of theBethe–Salpeter equation (42), which describes the limitingform of the correlation function in an infinite mediumasymptotically far away from the sources. This functiondetermines the asymptotic distribution of the intensity of thefield, which was shown to be highly spatially non-uniform andanisotropic. This result should be contrasted with the case ofstatistically homogeneous medium, in which the asymptoticdistribution of intensity is spatially uniform. It is importantthat the properties of the intensity distribution even inside an

infinite photonic crystal are determined by the normal modesof the underlying periodic structure. This result casts seriousdoubts on the assumption made in [15, 16] that the fielddistributions deep inside a photonic crystal and a statisticallyhomogeneous media do not differ from each other, and thatthe anisotropy of the light emerging from photonic crystals isformed within a narrow layer at the boundary.

The asymptotic character of ρ(∞) is expressed by theabsence of flux in this state. Using analogy with the statisticalphysics we can interpret ρ(∞) as an equilibrium distribution. Inorder to describe actual energy transfer, one needs to considersmall deviations from equilibrium. We derived generalizedforms of radiative transfer and diffusion equations valid fordisordered photonic crystals and obtained general expressionsfor the scattering cross-section, mean free path, and diffusioncoefficient. As an example, we calculated the cross-section forone particular model of a photonic crystal and demonstratedhow the underlying periodic structure effects disorder inducedscattering of photonic modes. In particular, we found thatthe scattering cross-section describing the redistribution ofthe energy between modes becomes highly anisotropic athigh frequencies. The mean free path of a particular mode,which describes the rate of redistributing the correspondentspecific intensity, also depends strongly on the direction ofthe respective Bloch vector. The numerical calculations showthat for a chosen frequency near the (partial) band-gap themean free path may vary almost by an order of magnitude.Such significant variation presents especial interest from theperspective of the problem of the Anderson localization.

The interesting feature of the transport in the disorderedphotonic crystals is that even if scattering between differentmodes may be highly anisotropic it does not necessarily implyan anisotropic diffusion. Indeed, if the point symmetry ofthe crystal is sufficiently rich, e.g. in 2D this is rotationalsymmetry of the third order or higher, then all tensors of thesecond rank describing the intrinsic macroscopic propertiesare proportional to the unit tensor. That is, the diffusion isnecessarily isotropic and is characterized by a single diffusionconstant. The mean free path appears in this constant averagedover the respective equifrequency surface.

The derivation of the transport equations can begeneralized to describe the radiative transport in more generalsituations than considered in the paper. Principally, theequilibrium distribution given by ρ(∞) holds in the presenceof the boundary as well. The latter is accounted for through theboundary conditions determining Green’s functions. However,whether the respective density matrix can be described near theboundary as a long scale perturbation of ρ(∞) or not presents aspecial interesting problem.

Finally, we would like to comment on the possibility toextend the consideration provided in the paper to a non-steady-state regime. By a direct analogy with the standard case, aslow dynamics in the diffusion regime can be accounted for byassuming wd in (65) to be time dependent. However, a rigoroustheory which would yield this limit is yet to be developed.From the perspectives of the presented consideration, the mainformal obstacle is the fact that ρω1,ω2 is generally not non-negatively defined if ω1 �= ω2. This problem, of course, exists

12


and is recognized in the standard theory of multiple scatteringin homogeneous media as well. In order to ensure positivity ofthe specific intensities, it was suggested to consider a coarse-grained Wigner distribution [48]. A possibility to generalizeour approach to time-dependent correlation functions and theirpossible relation with the density matrix formalism is an openproblem.

Acknowledgments

The work of the Northwestern University group is supportedby the National Science Foundation under grant No DMR-0093949, and the work of the Queens College group issupported by AFOSR under grant No FA9550-07-1-0391 andby PCS-CUNY grants.

Appendix A. A model of Gaussian inhomogeneities inphotonic crystals

A Gaussian random field �ε(r) can be defined in twoequivalent ways. The functional definition is based on therequirement of an integral

I f =∫

dr f (r)�ε(r) (A.1)

to be a Gaussian random variable for an arbitrary functionf (r). The statistical definition states that a multi-pointcorrelation function of a zero-mean Gaussian field is expressedin terms of the two-point correlation function

〈�ε(r1) · · ·�ε(r2n)〉 =∑

pairs

K (rα, rβ) · · · K (rγ , rδ), (A.2)

where the summation is taken over all possible pairings of thepoints r1, . . . , r2n and K (r1, r2) = 〈�ε(r1)�ε(r2)〉.

The property of integral (A.1) being a Gaussian randomvariable is consistent with the intuitive idea about a Gaussianrandom field, while property (A.2) is convenient from thetechnical point of view for developing the perturbationtheory [26]. Both these properties, however, might seemsomewhat artificial from the perspective of a disorderedphotonic crystal. Here we would like to present a simple modelof the disorder in a photonic crystal that naturally results insimulation of the inhomogeneities by a Gaussian random field.

Let the ideal structure be constituted by spheres withthe dielectric constant ε1 at the sites of a periodic lattice.Furthermore, we assume that the disorder in this system isconstituted by slight variations of the size of the spheres, whiletheir positions are fixed. Thus, we can represent the spatialmodulation of the dielectric function in the disordered crystalin the form

ε(r) = ε0 +∑

R

δεR(r + R), (A.3)

where R are the lattice vectors, ε0 is the background dielectricconstant, and δεR(r+R) is the deviation from the backgroundvalue in the elementary cell with the coordinate R. By theconstruction δεR(r) = ε1 − ε0 inside the sphere and is equal

to 0 elsewhere. We represent the spatial distribution of thedielectric function in the form adopted in (1), i.e. ε(r) =ε(r)+�ε(r), where ε(r) describes the distribution in the idealstructure and

�ε(r) =∑

R

(δεR(r + R) − δε0(r + R)) (A.4)

with δε0(r) being the deviation from the background value inthe ideal structure.

Clearly �ε(r) is not zero in shells near the boundariesof the spheres constituting the ideal structure. It is positive inthe spheres with the radius larger than the radius of the idealspheres and is negative in smaller spheres. We simulate thisfunction by presenting it in the form

�ε(r) =∑

R

�εRu(r + R), (A.5)

where u(r) is a non-random function different from zero ina thin shell near the boundary of the ideal sphere and therandom amplitudes �εR have the Gaussian distribution withzero mean value and are independent in different elementarycells. Using (A.5) in the integral in (A.1), one has a Gaussianrandom variable and, hence, �ε(r) is the Gaussian randomfield.

This model, obviously, can be generalized by allowingthe random variables �εR to be a (usual) Gaussian randomfield rather than a constant inside each elementary cell. Sucha model of the disorder would account not only for the sizedispersion of the spheres but also for the roughness of theirsurfaces. Assuming that �εr in different elementary cells arenot correlated we obtain

K (r1, r2) = 〈�εr1 �εr2〉u(r1)u(r2) (A.6)

if r1 and r2 are situated inside the same elementary cell andK (r1, r2) = 0 otherwise. For calculations in the main textwe have used the simplest model of spheres with uniform sizesand with centres at the sites of an ideal lattice but with roughsurfaces. Thus we take u(r) = 1 in a thin shell around theideal sphere (respectively, the circle in 2D) and

〈�εr1 �εr2〉 = V 2δ(r1 − r2). (A.7)

Appendix B. Symmetries of the averaged Green’sfunction, the self-energy and the irreducible vertex

Leaving the problem of convergence of the perturbationalseries aside, the symmetry properties of the averaged Green’sfunction 〈G(r1, r2)〉, the self-energy �(r1, r2), and theirreducible vertex U(r1, r2; r′

1, r′2) can be proven on a diagram

by diagram basis [26]. We demonstrate the typical lineof arguments by proving that if the correlation function ofinhomogeneities is invariant with respect to lattice translationsthen so is the averaged Green’s function.

Any diagram in the perturbational expansion of〈G(r1, r2)〉 has the form of a line with 2n internal (n � 0)and 2 terminating vertices. The internal vertices divide the lineon 2n + 1 segments, which are the graphical representations

13


of the non-perturbed Green’s function of the ideal structure.The internal vertices are connected pair-wise by the lines ofthe correlation function.

Clearly, the value of a diagram with fixed values of thecoordinates corresponding to the vertices does not change ifwe shift the coordinates of all points (internal and terminating)by the vector of the lattice translation. Indeed, the Green’sfunction lines and the lines corresponding to the correlationfunctions do not change because of the translational invarianceof these functions. Furthermore, we note that in order toobtain the contribution of the diagram to 〈G(r1, r2)〉 we needto integrate with respect to coordinates of the internal vertices.Finally, we observe that the uniform shift of the coordinatesof all vertices produces the diagram which corresponds to theperturbational series for 〈G(r1 + R, r2 + R)〉.

This line of argument proves the translational invarianceof the averaged Green’s function, the self-energy and theirreducible vertex. In order to prove the reciprocity of theGreen’s function and the self-energy

〈G(r1, r2)〉 = 〈G(r2, r1)〉, �(r1, r2) = �(r2, r1),

(B.1)we need the version of the arguments sketched above, whichincludes the directionality of the lines constituting the diagram.The reciprocity would follow then from invariance of thediagram with respect to the reversion of the directionality ofall lines. For the irreducible vertex we additionally need to takeinto account that the correlation function connecting upper andlower lines is mapped to itself upon reverting. Thus, we have

U(r1, r2; r′1, r′

2) = U(r′1, r′

2; r1, r2). (B.2)

We complete the consideration of the symmetry propertiesby proving the invariance of the self-energy with respect to thepoint symmetries of the photonic crystal. More specifically,we show that if the correlation function transforms accordingto the identity representation of the respective group, thenso does the self-energy. The proof is based on the propertyof the unperturbed Green’s function G0(r1, r2) to transformaccording to the identity representation

TG0(r1, r2) ≡ G0(T−1 r1, T

−1 r2) = G0(r1, r2), (B.3)

where T is an element of the point symmetry group. The proofof the invariance of the self-energy uses the same argumentsas above with the only difference that now instead of shift orinversion operators we act with T on all points of the diagram.

It follows from this consideration that the eigenfunctionsof the Dyson equation can be classified according tothe irreducible representations of the group of the pointsymmetries of the ideal structure. We use this fact in the paperwhen we discuss the effect of the disorder on the degeneratepoints in the spectrum of the photonic crystal.

Appendix C. Separation of spatial scales inclose-to-equilibrium regimes

The main assumption used for the derivation of the radiativetransfer equation and the diffusion equation in the paper is

that the spatial scales of variation of the specific intensityor the envelope of the energy density and of the functionsdescribing the modes of the ideal photonic crystal can beseparated. Physically this assumption implies that the slowlyvarying functions do not lead to coupling between differentBloch modes. As a result, one can neglect the smooth functionswhile calculating the respective scalar products.

In order to give a formal expression for the separation oflength scales we consider a photonic crystal mode k,m(r)

modulated by a smooth function f (r)

hk,m(r) = f (r)k,m(r). (C.1)

The smoothness of the function f (r) can be quantified by 〈p〉 f

and 〈p2〉 f , where

〈p〉 f =∫

dpp f (p) (C.2)

and f (p) is the Fourier image of f (r). The limit of smoothf (r) can, thereby, be formalized as 〈p〉 f → 0 and 〈p2〉 f → 0.In this limit the weighted scalar product of hk,m(r) with themode q,n(r) is written as

(

q,n, hk,m) ≡

∫

dr ε(r)∗q,n(r)hk,m(r)

= f (k − q)Un,m(q,k), (C.3)

where

Un,m(q,k) =∫

Vdr ε(r)u∗

q,n(r)uk,m(r). (C.4)

The inverse Fourier transform of (C.3) with respect to k–q

gives for m = n∫

dp(

k−p/2,m, hk+p/2,m)

e−ip·r = f (r) + O( f ′(r)),

(C.5)where the remainder is small together with the derivatives off (r).

Similarly, the notion of the smooth modulation can beapplied for functions of the form

ρ(r1, r2) =∑

k,m

Ik,m[

(r1 + r2)/2]

k,m(r1)∗k,m(r2),

(C.6)where Ik,m(r) are smooth functions. The weighted matrixelement of ρ(r1, r2) between p+q/2,n and p−q/2,n is(

p+q/2,n|ρ|p−q/2,n)

=∑

m

Ip,m(q)Um,n(p + q/2,p)Un,m(p,p − q/2). (C.7)

The inverse Fourier transform with respect to q gives∫

dq e−iq·R (

p+q/2,n

∣

∣ ρ∣

∣p−q/2,n) = Ip,m(R) (C.8)

up to terms vanishing with the derivatives of Ik,m(r).Additionally, for derivation of the radiative transfer

equation one needs to compare the scales related to theirreducible vertex. Analysing the diagrammatic expansion forU(r1, r2; r′

1, r′2) one can see that there are two typical spatial

14


scales determining the decay of the irreducible vertex with thedistance between the first and second pairs of the points. The‘short’ scale is proportional to the correlation radius of theinhomogeneities. This spatial decay is characteristic for ladderand maximal crossed diagram approximations. Assuming thatthe correlations vanish at the scale of the elementary cell onecan approximate∫

dr′1 dr′

2 Uω1,ω2(r1, r2; r′1, r′

2)I[

(r′1 + r′

2)/2] · · ·

≈ I [

(r1 + r2)/2]

∫

dr′1 dr′

2 Uω1,ω2(r1, r2; r′1, r′

2) · · · .(C.9)

This approximation has been used for derivation of (58).There are additional terms in the diagrammatic expansion

of the irreducible vertex that go beyond the ladder andmaximally crossed diagram approximations and lead to thespatial decay on the scale of the mean free path. Theseterms would result in a non-local term in the radiativetransfer equation. The effect of the non-local scattering onthe radiative transfer in disordered photonic crystals will bestudied elsewhere.

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