statistical theory of multiple scattering of waves applied to three-dimensional layered photonic...

1866 J. Opt. Soc. Am. B/Vol. 21, No. 10 /October 2004 Ponyavina et al.

Statistical theory of multiple scattering of wavesapplied to three-dimensional

layered photonic crystals

Alina Ponyavina, Svetlana Kachan, and Nikolaj Sil’vanovich

Institute of Atomic and Molecular Physics, National Academy of Sciences of Belarus, 70 F. Scorina Avenue,Minsk 220072, Belarus

Received November 12, 2003; revised manuscript received April 14, 2004; accepted May 5, 2004

On the basis of the statistical theory of multiple scattering of waves, we offer a numerical approach to calculatecoherent transmission and reflection for the three-dimensional (3-D) photonic crystals that consist of partiallydisordered dielectric spheres. With the proposed scheme, which we call the transfer-matrix (TM) method withquasi-crystalline approximation (QCA), we consider a quasi-regular 3-D assembly of particles as a stack ofclose-packed monolayers with a short-range ordering. Single-scattering characteristics are determined byMie theory. Lateral electrodynamic coupling between the particles of a monolayer is treated in the QCA.Multibeam interference between monolayers is described in a manner analogous to the TM technique. Weapply the TM-QCA calculation technique to study two revealed effects: (1) short-wavelength attenuation dueto particles of finite sizes and (2) nonmonotonic dependence of the pseudogap depth on the particle size,refractive-index contrast, and intermonolayer distances. © 2004 Optical Society of America

OCIS codes: 290.4210, 030.1670, 290.5850, 160.4760.

1. INTRODUCTIONDuring the past decade a novel class of optical materials,photonic crystals (PCs), has attracted the attention ofmany researchers.1,2 A periodic modulation of the refrac-tive index in such dielectric microstructures leads to a for-mation of the nontrivial photonic band structure.3–5 Themost dramatic modification of the photonic states in thesesystems occurs when suitably engineered three-dimensional (3-D) PCs exhibit a frequency range, the so-called complete photonic bandgap (PBG), over which thelight propagation is forbidden regardless of the directionof propagation and polarization. However, applicationsthat rely on the Bragg diffraction or strong photonic dis-persion in PCs without a complete PBG can also be essen-tially sufficient to provide a large number of practicalneeds.

For example, in the late 1980s the pseudogap effect(i.e., formation of frequency ranges in which the photondensity of states is strongly suppressed but does not van-ish) was used to create new infrared interference-scattering low-pass filters with high steepness.6 In addi-tion, modification of the spontaneous emission rate of dyesolutions impregnating colloidal crystals with apseudogap has been observed,7 and distributed-feedbacklasing has been realized.8

It is clear that any experimental exploration as well asa technological application of PCs has to be accompaniedby a quantitative theoretical analysis. Detailed numeri-cal simulations of PC structures allow for the interpreta-tion of experimental data and help to extract the relevantparameters. Perhaps more importantly, the theoreticaldesign of PC structures allows for the identification of themost interesting cases and for the investigation of stabledesigns for successful operation of devices. During the

0740-3224/2004/101866-10$15.00 ©

past decade there have been many different accurate andapproximate theoretical methods developed for these pur-poses.

Currently the most popular schemes to calculate opti-cal properties of PCs are the plane-wave expansion9

method, the finite-difference time-domain method,10 andthe transfer-matrix (TM) method.11 There are also anumber of studies based on the theory of multiple scatter-ing of waves that employ an analytical treatment of thediffraction problem on a single scatterer.5,12–15 Thesemethods allow for the calculation of the band structure aswell as the transmission and reflection coefficients of pe-riodic particle arrays.

However, a majority of the applications for PCs re-quires a defect injection, which gives rise to a disturbanceof their perfect periodicity. Of even greater importance ispartial topological or size disorder, which is inherent formodern technologies of PC fabrication for the visual andinfrared regions. So there are many reasons that suchmaterials are not really perfectly periodic, but are onlyhigh-ordered arrays of mesoscopic particles. The mostadequate formalism to describe the optical properties ofthese media, from our point of view, is the approach basedon the statistical theory of multiple scattering of waves(STMSW).16

The statistical approach to describe 3-D PCs is given inRef. 17. For example, for the artificial opal the influenceof the matrix refractive index and the geometric thicknessof a sample on the spectral position and the depth of thetransmission dip have been established. Good agree-ment with the experimental data has been shown. How-ever, to our knowledge the optimization of optical geomet-ric parameters for 3-D opallike PCs has still not beensufficiently explored. In addition, there has been no con-

2004 Optical Society of America

Ponyavina et al. Vol. 21, No. 10 /October 2004 /J. Opt. Soc. Am. B 1867

sideration given to the possibility of controlling thepseudogap properties by a change of interlayer distancesor a lateral ordering of monolayers forming 3-D layeredPCs.

In this paper we apply the STMSW approach to studythe direct transmittance and specular reflectance of 3-Dlayered PCs such as opals and periodic stacks of close-packed monolayers separated by thin solid films. In Sec-tion 2 we provide a detailed description of the applica-tions of the transfer-matrix method with quasi-crystallineapproximation (TM-QCA method).

2. TRANSFER-MATRIX METHOD WITHQUASI-CRYSTALLINE APPROXIMATIONFrom the point of view of optics of light-scattering media,formation of PBGs is a result of interference of wavesmultiply scattered by the ensemble of correlated particles.The higher the space ordering of the disperse medium,the more significant are the collective coherent phenom-ena such as coherent overirradiation between the par-ticles and interference of waves scattered coherently.There are two main types of high-ordered dispersed lay-ers consisting of spherical particles. They are (i) thestructures such as the periodic stack of close-packedmonolayers separated by subwavelength thin solid filmsand (ii) the so-called opallike structures, which consist inthe regular array of spheres with a face-centered cubiclattice.

Structure (i) is characterized by at least the one-dimensional regularity leading to the manifestation of thepseudogap that is sensitive not only to the distances be-tween monolayers but also to parameters of individualmonolayers18 (Fig. 1). Structure (ii) is ordered in allthree dimensions similar to a mesoscopic crystalline lat-tice. This is an example of a 3-D PC that can be regardedas the special case of (i) with the regular monolayers andintermonolayer distances equal to the particle size.

The STMSW operates with the field moments, whichare the values averaged over the statistical ensemble ofparticles. It means that all possible configurations of thegiven particle array are taken into account. The mainfield characteristics of the STMSW are the first and sec-ond moments, i.e., the average (or coherent) field Ê(r)&and the field covariation Ê(r) • E* (r)&.

The coherent field Ê(r)& is determined from the follow-ing chain of equations:

Fig. 1. Scheme of light interaction with the stratified system ofparticle monolayers.

Ê~r!& 5 E0 1 p0E t~r, ri!Ê~ri!& idri , (1)

Ê~ri!& i 5 E0 1 p0E g~ri , rj!t~ri , rj!

3 Ê~rj!& jidrj , (2)

and so on. Here Ê(r)& is the field averaged over all con-figurations of particles; Ê(ri)& i is the averaged field withone fixed particle; Ê(ri)& ij is the averaged field with twofixed particles; p0 is the average concentration of par-ticles; g(ri , rj) is the binary correlation function, whichdetermines the probability of the location of the centers oftwo particles at points ri and rj , correspondingly.

Some special procedures were developed to interruptthe chains of equations and determine the coherent fieldand coherent intensity Ic 5 uÊ(r)&u2. The most well-known calculation schemes of the STMSW are the single-scattering approximation, the effective field approxima-tion, and the quasi-crystalline approximation (QCA).They differ from each other in the manner of the effectivefield description.

The QCA seems to be the most advanced technique.The main assumption of the QCA, proposed by Lax,19 isthat Ê(r)& ij ' Ê(r)& i . This approximation supposesthat fixation of any particle specifies the spatial configu-ration of the whole assembly. For a random monolayer,the closer the structure is to the crystalline assembly, themore justifiable is this assumption.

The stratified and opallike structures can be consideredas layer periodic dispersed structures. In our opinion,the most natural scheme of their optical characterizationis based on the solution of the problem in two steps: (i)determination of the vector scattering amplitude of amonolayer and then (ii) consideration of the electromag-netic interactions between all monolayers of the samplethrough a self-consistent procedure such as the well-known T-matrix technique.11

The most principal step is the first stage that is appliedto the lateral electrodynamic coupling of particles in ahigh-ordered monolayer. For this first stage, we proposeusing the STMSW approach, especially thewell-developed20 QCA.

In Subsections 2.A and 2.B we briefly consider the fun-damental points of the TM-QCA method used in this pa-per to study the pseudogap properties of 3-D PCs contain-ing mesoscopic spherical dielectric particles.

A. Vector Scattering Amplitude of a Close-PackedMonolayerLet us consider an infinite monolayer lying along the X,O, Y plane and irradiated by a plane-polarized wave com-ing along the z direction. Then by use of the Green func-tion approach, the ensemble-averaged field Ê(r)& and theeffective field inside a particle Ê(r 1 R)&R are describedin the QCA by the following system of equations:

Ê~r!& 5 e exp~ikz ! 12pi

kp0f~6z!exp~ikz !,


Ê~r 1 R!&R 5 e exp~ikz ! 1 E drGI~r 1 R,r8 1 R!

3 Ê~r8 1 R!&R 1 p0E dR8g~ uR 2 R8u!

3 E dr8GI~r 1 R, r8 1 R8!

3 Ê~r8 1 R8!&R8 , (3)

where

f~6z! 5k2

4p~ n2 2 1 !E

Vdr8 exp~6ikz !~1I 2 zz!Ê~r8!&

(4)

is the vector scattering amplitude of a single particle; p0

is the surface concentration of particles; GI(r, r8) is thetensor Green function; and k is the wave number. Thespace integral of Eq. (4) is taken over a particle’s volume.The plus and minus signs correspond to the transmittedand reflected waves, respectively; z is a unit vector alongthe z axis.

According to Eqs. (3), we can introduce the vector scat-tering amplitude of a monolayer:

F~6z! 52pi

kp0f~6z!. (5)

The monolayer scattering amplitude F(6z) is ex-pressed in terms of the vector scattering amplitude of asingle particle f(6z), which is determined with the over-irradiation between the particles taken into account.When a monolayer comprises particles of a sphericalshape, one can solve Eqs. (3) by expanding all fields, eventhe incident one, into the vector spherical functions andthen determine the corresponding coefficients from a setof linear algebraic equations.18,20 Then the expressionfor the single-scattering amplitude takes the form

f~6z! 5 ei

2k (i51

`

~61 !l~2l 1 1 !~b l 6 a l!.

Use of the additional theorem for the vector sphericalfunctions when we integrate Eqs. (3) for the unknown co-efficients b l and a l yields the following system of theequations:

b l 5 bl 1 p0bl(l

~Plb l 1 Qla l!,

a l 5 al 1 p0al(l

~Pla l 1 Qlb l!.

Here al and bl are the Mie coefficients; functions Pland Ql are expressed in a complicated manner18,20

through the Legendre polynoms and the characteristicfunctions Hp 5 2p*d

`dRRg(R)hp(1)(kR), where hp

(1) is theHankel function. Calculation of the radial distributionfunction for the dense particle monolayer is carried out inthe Percus–Yevick approximation for solid spheres21

when the degree of particle ordering and the subsequentfeatures of the radial distribution function are deter-mined by the overlap parameter h 5 p0pd2/4. Numeri-

cal simulation of g(r), b l , and a l is performed with theiteration method. As the Mie coefficients are deter-mined, we employ the reverse recursion technique.

B. Multilayer Scattering SystemTo calculate the coefficients of coherent transmission andreflection of a multilayer scattering system we use thetechnique described in this subsection.

Let the distances between the neighboring parallelmonolayers be equal and the propagation vector of the in-cident plane wave be transverse to the monolayer sur-faces and parallel to the z unit vector. To describe the co-herent field of a single monolayer we apply theexpressions obtained by the QCA for the vector scatteringamplitudes F6 5 F(6z) [see Eqs. (3)–(5)]. Then the co-herent field of the whole sample consisting of N monolay-ers can be determined as

Ê~z!& 5 exp~ikz !S e 1 (j51

N

Gj1D ,

Ê~2z!& 5 exp~ikz !H(j51

N

Gj2 exp@~ j

2 1 !2iklm#J . (6)

Here Gj6 5 G(6z) are the scattering amplitudes in the

forward and backward directions for the jth monolayer inthe presence of the other monolayers of the system; lm isthe interlayer distance, i.e., the spatial interval betweenthe centers of neighboring monolayers (see Fig. 1).

The method of a self-consistent field, which allows us toconsider electrodynamic interactions between the layers,is an effective technique to find Gj

6. It yields the follow-ing system of equations:

Gj1 5 F1 1 F1(

p51

j21

Gp1 1 F2 (

p5j11

N

Gp2

3 exp@~ p 2 j !2iklm#,

Gj2 5 F2 1 F2(

p51

j21

Gp1 1 F1 (

p5j11

N

Gp2

3 exp@~ p 2 j !2iklm#. (7)

The sums in these expressions take into account the co-herent irradiation of the jth monolayer by the othermonolayers. Solving the set relative to Gj

6 and puttingit in Eqs. (6), one can find the coherent field of themultilayer structure composed of equidistant monodis-perse layers. However, this technique is also effective forthe case of polydisperse monolayers18 as well as for mono-layers separated by films distinguished by thickness.

At this second stage the method proposed is close to theTM approach with only the monolayer amplitude functiondetermined in the QCA of the STMSW. That is why werefer to the scheme as the TM-QCA method.

Note that at a plane-wave incident on the plane-parallel dispersed layer, spectral coefficients of coherenttransmission and reflection can be defined through the co-


herent field as T 5 u^E(z)&u2 and R 5 u^E(2z)&u2, respec-tively. It is worthwhile to remember that coherent T andR correspond to the coefficients of direct transmission andspecular reflection measured in the case of the small an-gular detection.

3. RESULTS AND DISCUSSIONIn this section we use the TM-QCA calculation scheme tostudy the dependence of pseudogap characteristics of 3-Dlayered PCs on the refractive-index contrast, particlemean size, and intermonolayer distances. We focus onthe nature of the enhanced attenuation over the bluerange off a pseudogap and analyze the revealed nonmono-tonic dependence of this transmission interference dip onthe named parameters. To separate the role of systemordering in the third direction (stacking), we also considerspectral dependences of monolayer transmittance and re-flectance.

A. Photonic Crystal Transmission SpectraCharacterizationA typical transmission spectrum of a 3-D layer-periodicassembly of mesoscopic spheres is presented in Fig. 2.As one can see, the main feature is the presence of atransmission dip over the range of wavelengths compa-rable with particle sizes. It is caused by the suppressionof wave propagation due to the spatial ordering of meso-scopic particles. In addition, there is a strong attenua-tion of transmission in the high-frequency region. Thiseffect is in agreement with numerous experimental data(see, for example, Refs. 17 and 22–24), and its nature isdiscussed in Subsection 3.B.

To characterize the transmission spectra quantita-tively, let us introduce the following parameters (see Fig.2): the spectral position (l0), the residual transmittance(T0), and the bandwidth (Dv) of the PBG (or pseudogap),as well as short-wavelength attenuation characteristics—transmittances in the high-frequency (T,) and low-frequency (T.) gap edges and the width (D,) of the high-frequency transmission range (i.e., transmission band,high-frequency relative to the first PBG).

The principal characteristic of a PC is the spectral po-sition l0 of the PBG. Experimental data17 show the

Fig. 2. Coherent transmission spectra of the multilayer systemof dielectric spheres (N 5 8, h 5 0.6, n 5 1.26, lm 5 1.23d).

monotonic increase of l0 when the particle refractive in-dex grows. As can be seen from the solid curves in Fig. 3,our calculations confirm this dependence.

The main tendencies of the dependence of l0 on opticaland geometric parameters of the 3-D layered PCs can beconcisely described by the simple analytical expressionderived from the analog of the Bragg conditions for a PC.From the Bragg approximation for the interference mini-mum in forward direction neff lm 5 l0/2, one can obtainthe simple expression connecting the gap position, par-ticle and matrix refractive indices, particle sizes, and fill-ing fraction in a PC:

l0 ' 2lm@cnp 1 ~1 2 c !nm#. (8)

Here we define the effective refractive index of the PCneff as a sum of the refractive indices of particles np andmatrix nm in regard to the particle filling fraction c:neff 5 cnp 1 (1 2 c)nm . As one can see by approximation(8), the increase of intermonolayer distances or the par-ticle refractive index gives rise to the long-wavelengthshift of the bandgap. Note that for opals or inverse opalsthe interfacial spacing lm is equal to the mean particlesize d. Thus l0 ' 2d@cnp 1 (1 2 c)nm# (see, for ex-ample, Ref. 25).

The quantitative agreement between the TM-QCA andthe Bragg approximation calculations is illustrated inFig. 3. In our opinion, this good agreement is the conse-quence of the fact that the above-described modeling ofthe 3-D PC as a stack of monolayers is in some relationclose to the classical treatment of x-ray Bragg diffractioninto a crystal. Both the TM-QCA and the Bragg approxi-mation consider an interference of waves scattered bycrystal planes.

However, the TM-QCA method also solves two prob-lems of great importance for wave propagations in a spa-tially ordered system of strong scatterers (such as par-ticles with a relative refractive index n . 1.5). For thefirst, the TM-QCA method involves the rigorous solutionof a diffraction problem on a single sphere (the Mietheory) that allows us to describe light interaction withthe nonpoint scatterers. Second, we take into accountelectrodynamic coupling between the particles in a planar

Fig. 3. Dependence of the pseudogap spectral position l0 on therefractive-index contrast for the multilayer system of close-packed dielectric spheres with the different interlayer distanceslm . Calculations were performed by the QCA and the Bragg ap-proximation (N 5 8, h 5 0.6, d 5 200 nm).


monolayer. Both of these topical points are extremelyimportant for a description of the full set of PC character-istics mentioned at the beginning of this subsection.

B. High-Frequency Attenuation in Direct TransmittanceThe PC spectral coefficients of coherent transmission andreflection calculated over the range from 100 to 1400 nmfor different particle sizes are presented in Fig. 4. Thereservation of the spectral position of the first pseudogapwas achieved by the appropriate choice of interlayer dis-tances [see approximation (8)].

As one can see, for small particles there is a set of sec-ondary gaps in the short-wavelength range relative to thepseudogap position. In general, the dependence of thepseudogap depth on its order is nonmonotonic (see, for ex-ample, the transmission spectrum for d 5 100 nm). Inaddition, a comparison of the data in Fig. 4(a) also showsthe nonmonotonic dependence of the first pseudogapdepth on particle size. The common cause underlyingthese effects is discussed in detail in Subsection 3.D.

However, the other feature is more interesting. Whenthe particle size increases and becomes comparable to thestack period lm (and consequently to the wavelength ofthe first pseudogap), the attenuation in the short-wavelength region grows. It leads to a decreasing width(D,) of the high-frequency transmission range. Theusual explanation of experimental evidence of this effectis imperfect ordering in real PCs, which produces mul-tiple incoherent (so-called diffuse) scattering and so di-minishes the transport mean free path.26 But we shouldemphasize that, in the given case, the degree of spatial

disordering of samples with different sizes was the samebecause of the same monolayer overlap parameter. Thatis why we can conclude that this additional short-wavelength attenuation for samples containing large par-ticles can be connected with the spectral features of singlelight scattering.

It is well known that the cross section of single scatter-ing grows with an increase of the size parameter r5 pd/l. Attenuation of radiation connected with itsscattering on a single particle becomes stronger as thefrequency increases.27 Correspondingly, the high-frequency barrier appears in the direct transmission spec-tra of disperse media with any type of arrangement (evenfor the perfect spatial array). The larger the particlesizes, the longer the wavelength in this transmission bar-rier. For 3-D layered PCs the particle enlargement atfixed lm makes the blue transmission cutoff closer to thefirst pseudogap position. In addition, the cutoff steep-ness factor rises at the higher particle refractive index.So the growth of the particle relative refractive index re-sults in the narrowing high-frequency transmission rangeD, . Because the efficiency of a PC depends on both thebandgap depth and the width of the transmission bandsbeyond it, an increase in n is advisable only up to somevalue determined by practical necessity. Also, to observethe pronounced bandgap in the structures with the highrefractive-index contrast, it is necessary to meet the se-vere demand for packing quality and particle monodisper-sity.

The effect of coherence loss for scattered waves due tothe particle size comparability with the wavelength is of

Fig. 4. (a) Coherent transmission and (b) specular reflection spectra of the multilayer systems of close-packed monolayers with differentparticle sizes (N 5 8, h 5 0.6, n 5 1.6). All multilayer systems are stacked so that the pseudogap positions are coincident (for d5 10 nm, lm 5 347.6 nm; for d 5 100 nm, lm 5 326 nm; for d 5 280 nm, lm ' d).


principal importance, and not just for the treatment ofhigh-frequency attenuation in direct transmittance.From the same point of view we can provide answers tothe problem as to why it is so difficult to experimentallycreate the complete bandgap and absolute transparenceof a PC in the spectral ranges out of the bandgap. Thereis an additional physical reason other than such evidentreasons as size disorder of particles and destruction of thePC’s periodicity. This additional reason is the loss of co-herence and energy dissipation due to the finite sizes ofthe particles.

As was mentioned above, every particle in a PC can beconsidered as the source of scattered waves, and a band-gap results because of the interference summing of thesescattered waves. It seems apparent that a decrease inthe degree of the mutual coherence of scattered waveswill change the interference pattern. It is commonknowledge that the finite size of the light source gives riseto a decrease in the spatial coherence of waves. In itsturn, this circumstance leads to the degradation of thecontrast of the interference pattern created by two ormore sources of a finite size. From the above reasoning,in our opinion, an interference destruction of the bandgapmay be the result not only of some spatial disordering ofreal photonic structures, but also of such fundamentalcauses as the particle finite (compared with wavelengthand interparticle distances) size.

C. Collective Attenuation Resonance in a DenseMonolayer and Efficiency of the Interlayer InterferenceTo clarify the possible mechanisms initiating the forma-tion of PBGs, let us consider a simplified situation inwhich the layered structure includes only two monolayersof close-packed spherical particles (see Fig. 5). In thiscase the main contribution in the interference sum is pro-vided by the light rays passing through both monolayerswith no reflection (1) and by the light rays experiencingtwofold reflection (2).

To possess the efficient interference, the intensities(amplitudes) of light rays of these two types should be sig-nificantly large. That is, the transmission coefficient Tm

Fig. 5. Interference scheme for light rays passing throughneighboring monolayers with no reflection (1) and experiencingtwofold reflection (2).

of one monolayer should have a high value. Also, bothintensities should be comparable: I1 ' I2 . The lattercondition is satisfied when Tm(1 2 Rm

2) vanishes.Thus, combining the conditions of large Tm and smallTm(1 2 Rm

2), we conclude that the monolayer reflectioncoefficient Rm should also be significant. Let us intro-duce the function f 5 1 2 Tm(1 2 Rm

2), which we referto as the function of the efficiency of interlayer interfer-ence. We can say that the deepest pseudogap is reachedat maximal f while Tm and Rm are kept relatively large.

To establish the range of parameters for which theseconditions are satisfied simultaneously, let us considerthe spectral dependence of transmission and reflection co-efficients for the close-packed monolayer. Figure 6 showsTm , Rm , and f as functions of the size parameter r forthree values of the refractive-index contrast n (1.26, 2.0,3.5). The overlap parameter in all cases is unchangedand is equal to h 5 0.6. The dashed curves in the middlepanels of Fig. 6 show the transmission spectra for sparsesystems equivalent to monolayers with an appropriatenumber and type of particles.

As the lower panels of Fig. 6 illustrate, there are twomaxima of Rm over the size-parameter range under con-sideration. From a comparison of the middle and lowerpanels, it can be seen that, in the region of the second(short-wavelength) resonance of reflection, the Tm valueis decreasing because of the strong single scattering (i.e.,the Mie resonance). Apparently (see the upper panels ofFig. 6) it prevents the efficient interlayer interference oflight in the corresponding frequency range. At the sametime, the above-formulated conditions of the efficient in-terference are obviously satisfied over the range of thefirst (long-wavelength) Rm resonance, for which thetransmission coefficient remains large. Note that whenthe refractive index of particles increases, this optimalrange shifts to the smaller size parameter’s values. Forn 5 1.26, 2.0, and 3.5 we obtain, respectively, r ' 1, 0.7,and 0.5.

Note that the first maximum in the reflection spectrumas well as the coincident minimum in transmission of amonolayer have a fundamentally collective nature. Thiscollective nature is proved by the absence of the appropri-ate dip in transmission spectra of the sparse system (seethe middle panels in Fig. 6). Most likely the additionaltransmission attenuation, which arises in this spectral re-gion for the close-packed systems, is related to the short-range ordering of particles, resulting in strong lateralelectrodynamic couplings on the scales of the particles’correlation. Qualitatively, the origin of this resonanceattenuation can be associated with some effective scatter-ers (agglomerates) of the size larger than the size of asingle particle, which emerge in the close-packed mono-layer. The larger the overlap parameter of the partiallyordered monolayer, the higher the probability of forma-tion for such agglomerates. The growth of the refractiveindex of particles leads to the enhanced lateral electrody-namic coupling, which manifests in increasing the corre-sponding collective attenuation [compare the middle pan-els of Figs. 6(a)–6(c)].

The considered collective attenuation resonance in theregion r , 1 has already been revealed both in experi-mental and in numerical data.20 Its emergence and col-


lective nature are indirectly demonstrated by the detec-tion of reflection resonance for a close-packed monolayer28

and concentrative transformation of the angular distribu-tion of radiation scattered by it over the certain spectralrange.29

Thus, in our opinion, the first (long-wavelength) reso-nance attenuation of the close-packed monolayer is essen-tially collective in itself and is determined by the lateralelectrodynamic coupling of near-ordered scatterers.

In its turn, the second (short-wavelength) resonance ofreflection as well as the coincident minimum in transmis-sion of a monolayer is principally related to the light scat-tering on single spherical particles. This Mie resonanceis exhibited both in a sparse random system of noninter-acting particles and in a close-packed two-dimensionallayer. Some distinction in these appearances arises be-cause of the above-discussed concentration effect based on

coherent electrodynamic coupling of spatially correlatedparticles in a monolayer. It consists in the formation ofthe effective field inside the layer, which is essentially dif-ferent from the incident field. Namely, this change in thefield interacting with a particle gives rise to some spectralshift and modification of the Mie-scattering resonanceband when a sparse system turns into the close-packedmonolayer (see Fig. 6, middle panels).

In Fig. 7(a) we present the transmission spectra of PCsand of the corresponding sparse system of random distrib-uted particles. One can clearly see from the comparisonof these spectra that a feature of the PC spectrum is theexistence of a narrow and deep attenuation band in thevicinity of r ' 0.8. It is just the band, which is absent intransmission spectra of a sparse random system. It hasa coherent nature and is determined by interlayer inter-ference. In contrast, the deep gap in the region, which is

Fig. 6. Spectral dependence of coherent transmission Tm , specular reflection Rm , and the function of the efficiency of interlayer in-terference f 5 1 2 Tm(1 2 Rm

2) at light interacting with the close-packed (h 5 0.6) monolayers of spherical particles (d 5 200 nm)with different n. The dashed curves correspond to transmission coefficients for the sparse systems of the same number and type ofscatterers.

Fig. 7. Coherent transmission spectra of (a) the multilayer (N 5 8, lm 5 1.23d) and (b) the monolayer structures of dielectric sphericalparticles (d 5 200 nm, h 5 0.6, n 5 3.0). The dashed curves correspond to the sparse systems of the same number and type of scat-terers.


Fig. 8. Dependence of coherent transmission spectra of the multilayer system (N 5 8, h 5 0.6) of dielectric spheres (a) on therefractive-index contrast (d 5 200 nm, lm 5 1.23d); (b) on the particle diameter (n 5 1.6, lm 5 282 nm); (c) on the interlayer distance(d 5 200 nm, n 5 3.5). Arrows indicate the pseudogap spectral position.

present in the spectra of both systems, is related to thestrong resonances of single scattering. It is not causedby the periodicity of the refractive index, which is crucialfor PCs.

This conclusion is also supported by the data presentedin Fig. 7(b), which show the transmission spectra of boththe monolayer of close-packed particles and the sparsesystem with the corresponding number of particles. It isevident that the transmission spectra of the monolayerand sparse random system have the same fine structurefeatures over the Mie resonance region (r . 1.1).

D. Dependence of Pseudogap Depth on Optical andStructural Parameters of Three-DimensionalLayered Photonic CrystalsAs we mentioned in the discussion of the results pre-sented in Fig. 4, the spectral position of the pseudogap inthe structures under consideration can be tailored by achange in the particles’ size or interlayer distance. InFig. 6 this region is hatched. The right demarcation lineof the hatched region corresponds to the minimal (from allthe possible) value of interlayer distance lm 5 d. In-creasing the ratio lm /d (for example, by decreasing d)leads to the shift of the pseudogap to the range of lowervalues of the size parameter. In this case the function ofthe efficiency of interlayer interference f is changed non-monotonically (see the upper panels in Fig. 6). It first in-creases with decreasing d to some critical value d0 , butthen starts to diminish. Also, as one can see from Fig. 6,the larger the refractive index of the particles, the largerthe maximal value of the function f, and accordingly bet-ter optimal conditions for the pseudogap opening can bereached. Such behavior of function f is connected withthe collective attenuation resonance in a monolayer withshort-range ordering.

Thus the existence of the additional collective attenua-tion band, which is sensitive to such parameters of themonolayer as the refractive index, size, and packing den-

sity of particles, can cause the nonmonotonic dependenceof the pseudogap depth on several parameters of the PC.Such consideration provides a qualitative explanation ofthe nonmonotonic dependence of the pseudogap depth,presented in Fig. 4, on the particles’ size for the 3-D lay-ered PC structures with l0 5 const. The same explana-tion can also be provided for the results, presented in Fig.8, which demonstrate nonmonotonic dependence of thepseudogap depth on (a) the refractive-index contrast, (b)the particles’ size (in the case of structures with lm5 const), and (c) the interlayer distances.

A deeper insight into these effects is presented in Fig.9. We start with the consideration of the pseudogapdepth on the refractive-index contrast. As one can see,when the refractive index increases, the minimum of thefunction Tm(1 2 Rm

2) (see curves 1–3) shifts to the lower

Fig. 9. Dependence of the function Tm(1 2 Rm2) (curves 1–3)

and the relative interlayer distance lm /d (curves 4–6) on thepseudogap spectral position for the multilayer systems of close-packed (h 5 0.6) monolayers with different n.


Fig. 10. Dependence of the residual transmittance T0 (a) on the refractive-index contrast (N 5 8, d 5 200 nm, h 5 0.6, lm 5 1.23d)and (b) on the number of layers in the multilayer system @d 5 200 nm, h 5 0.6, lm 5 d (solid curves), lm 5 1.23d (dashed curves)].The two upper curves correspond to n 5 1.26 and the two lower curves correspond to n 5 2. The inset shows the logarithm of theresidual transmittance T0 by the number of layers.

values of the size parameter r. For the analyzed range ofthe refractive indices, the pseudogap depth increasesmonotonically with increasing n only for r , 0.65. Incontrast, as illustrated by spectra presented in Fig. 8(a),for r . 0.65 this dependence becomes nonmonotonic.

The origin of the nonmonotonic dependence of thepseudogap depth on particle size d was discussed abovewhen we considered Fig. 4. In Fig. 8(b) we plot anotherdependence, now keeping lm /d constant. As we can see,in this case the change in particle size results in the shiftof its spectral position (together with the nonmonotonicdependence of the depth of the interference dip).

Interpretation of the data presented in Fig. 8(c) is morecomplicated. Here we can see that for a PC with n5 3.5 the pseudogap depth at first decreases and thengrows as lm varies from 440 to 200 nm. Referring to Fig.9 (see curves 3 and 6), we note in this case that the valuesof the size parameter, corresponding to the pseudogap,are 0.55, 0.74, and 0.82 whereas lm /d changes from 2.2 to1. This range of r corresponds to a nonmonotonic changeof the function of efficiency of interlayer interference, butits values at r 5 0.82 and 0.74 are equal ( f 5 0.98) andone can expect the equivalent pseudogap depths. How-ever, it can be concluded from Figs. 6 and 9 that the valuer 5 0.82 is close to the transition region from the collec-tive attenuation band to the strong resonance of singlescattering. As one can see from Fig. 8(c), in this case thenarrow pseudogap is absorbed by the broad band ofstrong single attenuation (see the spectrum for lm5 200 nm). Thus, as we discussed in Subsection 3.C,this broad attenuation band is essentially the result of theMie resonance scattering and does not reflect the interfer-ential nature of PBGs.

In Fig. 9 we also illustrate how the parameters d andlm can be optimized to reach the maximal pseudogapdepth for some given values of the refractive-index con-trast and l0 . Let us demonstrate this action by the ex-ample of particles with the relative refractive index equalto 3.5. Fixing the required wavelength l0 , one can de-termine d from the size parameter value r 5 0.49, which

corresponds to the minimum of curve 3. Farther, whenwe specify from curve 6 the value of lm /d 5 2.2, appro-priate to r 5 0.49, it is easy to define the interlayer spac-ing lm .

As Fig. 9 shows, the optimal ratio of lm /d is close tounity only for small values of the refractive index, andthus the opallike structures (for which lm 5 d) becomeoptimal. However, when the refractive index of the par-ticles increases, the maximal pseudogap depth is reachedat lm , which is large compared with d.

Figures 10(a) and 10(b) depict the dependence of the re-sidual transmission in the pseudogap range on the refrac-tive index of particles and on the number of layers, re-spectively. As one can see from Fig. 10(a), the optimalvalues of the refractive index lie in the region n5 1.75–2.75. However, it should be emphasized thatthe higher the refractive index or the ratio lm /d, thesmaller the number of layers required to reach theasymptotic value of the residual transmission and toachieve the asymptotic linear dependence of ln(T0) by N,experimentally detected for artificial opals24 [see Fig.10(b)].

It should be noted that in all the presented estimationswe assumed the constant overlap parameter. Its changecauses a transformation of the monolayer spectrum and,as a consequence, changes the optimal values of the struc-ture parameters.

4. CONCLUSIONSWe have studied 3-D layer periodic dispersed arrays ofspherical particles applying the TM-QCA method andhave revealed several qualitative features of their opticaltransmittance spectra. The main feature consists in thenonmonotonic dependence of the pseudogap depth onsuch parameters as particle size, refractive-index con-trast, and intermonolayer distance. The origin of this ef-fect lies in the existence of the limited spectral rangewhere the lateral electrodynamic coupling in a singleclose-packed monolayer of spherical particles is maxi-


mized. As a consequence, the coherent reflectance of asingle monolayer is also maximized, creating optimal con-ditions for the interference of waves scattered by differentmonolayers.

The second feature is that there is a fundamentally in-herent attenuation of the direct transmittance in theshort-wavelength region. For 3-D PCs made of mesos-copic particles the attenuation appears at wavelengthsshorter than the position of the photonic pseudogap. It iscaused by the energy dissipation due to strong wave scat-tering on single particles for the wavelengths comparableor shorter than the particles’ size. So the single scatter-ing gives rise to the appearance of the transmittance bar-rier at frequencies higher than the pseudogap. This dis-sipation cannot be excluded even by perfect ordering ofthe particles’ assembly until their sizes are comparablewith the wavelength of the incident light. The existenceof this type of loss is the important distinction betweenlight diffraction in 3-D PCs made of mesoscopic particlesand Bragg diffraction of x rays in atomic or molecularcrystals, where the sizes of the scattering centers are neg-ligible in comparison with the x-ray wavelength and theinteratomic spacing.

A related fact is that the complete PBG is hardlyachieved for real 3-D PCs composed of particles of an ar-bitrary shape, but with the particles size comparable tothe light wavelength and the interfacial spacing. Thecause is strong losses of coherence in the process of lightinteraction with a single scatterer.

From this point of view it is preferential for some prac-tical applications to use periodical structures with thescatterers smaller than the light wavelength and interfa-cial spacing. The example of such PC structures for themost attractive visible and infrared ranges is provided by3-D layer periodic systems of dielectric nanoparticles withinterfacial solid subwavelength films.

ACKNOWLEDGMENTSThis research was supported by the INTAS from grant 01-0642 and partially by the International Scientific andTechnical Centre from grant B-276-2.

REFERENCES1. C. M. Soukoulis, ed., Photonic Crystals and Light Localiza-

tion in the 21st Century (Kluwer, Dordrecht, The Nether-lands, 2001).

2. T. Krauss and T. Baba, eds, ‘‘Feature section on photoniccrystal structures and applications,’’ IEEE J. QuantumElectron. 38, 724–963 (2002).

3. E. Yablonovitch, ‘‘Inhibited spontaneous emission in solid-state physics and electronics,’’ Phys. Rev. Lett. 58, 2059–2062 (1987).

4. S. John, ‘‘Strong localization of photons in certain disor-dered dielectric superlattices,’’ Phys. Rev. Lett. 58, 2486–2489 (1987).

5. K. Ohtaka, ‘‘Energy-band of photons and low-energy photondiffraction,’’ Phys. Rev. B 19, 5057–5067 (1979).

6. V. G. Vereshchagin and V. V. Morozov, ‘‘New type of cuttingfilters for the long-wave infrared spectrum region,’’ Zh.Prikl. Spektrosk. 49, 317–320 (1988) (in Russian).

7. J. Martorell and N. M. Lawandy, ‘‘Observation of inhibited

spontaneous emission in a periodic dielectric structure,’’Phys. Rev. Lett. 65, 1877–1880 (1990).

8. J. Martorell and N. M. Lawandy, ‘‘Distributed feedback os-cillation in ordered colloidal suspensions of polystyrene mi-crospheres,’’ Opt. Commun. 78, 169–173 (1990).

9. H. S. Sozuer, J. W. Haus, and R. Inguva, ‘‘Photonic bands:convergence problems with the plane-wave method,’’ Phys.Rev. B 45, 13962–13972 (1992).

10. A. Taflove, ‘‘Review of the formulation and applications ofthe finite-difference time-domain method for numericalmodeling of electromagnetic-wave interactions with arbi-trary structures,’’ Wave Motion 10, 547–582 (1988).

11. P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, ‘‘Aprogram for calculating photonic band structures andtransmission coefficients of complex structures,’’ Comput.Phys. Commun. 85, 306–322 (1995).

12. L.-M. Li and Z.-Q. Zhang, ‘‘Multiple-scattering approach tofinite-sized photonic band-gap materials,’’ Phys. Rev. B 58,9587–9590 (1998).

13. G. Tayeb and D. Mayste, ‘‘Rigorous theoretical study offinite-size two-dimensional photonic crystals doped by mi-crocavities,’’ J. Opt. Soc. Am. A 14, 3323–3332 (1997).

14. X. D. Wang, X. G. Zhang, Q. L. Yu, and B. N. Harmon,‘‘Multiple-scattering theory for electromagnetic-waves,’’Phys. Rev. B 47, 4161–4167 (1993).

15. N. Stefanou, V. Yannopapas, and A. Modinos, ‘‘Heterostruc-tures of photonic crystals: frequency bands and transmis-sion coefficients,’’ Comput. Phys. Commun. 113, 49–77(1998).

16. A. Ishimaru, Wave Propagation and Scattering in RandomMedia (Academic, New York, 1978).

17. V. N. Bogomolov, S. V. Gaponenko, I. N. Germanenko, A. M.Kapitonov, E. P. Petrov, N. V. Gaponenko, A. V. Prokofiev, A.N. Ponyavina, N. I. Silvanovich, and S. M. Samoilovich,‘‘Photonic band gap phenomenon and optical properties ofartificial opals,’’ Phys. Rev. E 55, 7619–7625 (1997).

18. A. N. Ponyavina and N. I. Sil’vanovich, ‘‘Interference effectsand spectral characteristics of multilayer scattering sys-tems,’’ Opt. Spectrosc. 76, 581–589 (1994).

19. M. Lax, ‘‘Multiple scattering of waves. 2. The effectivefield in dense systems,’’ Phys. Rev. 85, 621–629 (1952).

20. K. M. Hong, ‘‘Multiple scattering of electromagnetic wavesby a crowded monolayer of spheres: application to migra-tion imaging films,’’ J. Opt. Soc. Am. 70, 821–826 (1980).

21. J. Ziman, Models of Disorder (Cambridge U. Press, Cam-bridge, UK, 1979).

22. S. G. Romanov, N. P. Johnson, A. V. Fokin, V. Y. Butko, H.M. Yates, M. E. Pemble, and C. M. S. Torres, ‘‘Enhancementof the photonic gap of opal-based three-dimensional grat-ings,’’ Appl. Phys. Lett. 70, 2091–2093 (1997).

23. H. Miguez, C. Lopez, F. Meseguer, A. Blanco, L. Vazquez, R.Mayoral, M. Ocana, V. Fornes, and A. Mifsud, ‘‘Photoniccrystal properties of packed submicrometric SiO2 spheres,’’Appl. Phys. Lett. 71, 1148–1150 (1997).

24. J. F. Galisteo-Lopez, E. Palacios-Lidon, E. Castillo-Martinez, and C. Lopez, ‘‘Optical study of the pseudogap inthickness and orientation controlled artificial opals,’’ Phys.Rev. B 68, 115109 (2003).

25. S. G. Romanov, A. V. Fokin, and R. M. De La Rue, ‘‘Stop-band structure in complementary three-dimensional opal-based photonic crystals,’’ J. Phys.: Condens. Matter 11,3593–3600 (1999).

26. W. L. Vos, M. Megens, C. M. van Kats, and P. Bosecke,‘‘Transmission and diffraction by photonic colloidal crys-tals,’’ J. Phys.: Condens. Matter 8, 9503–9507 (1996).

27. C. Bohren and D. Huffman, Absorption and Scattering ofLight by Small Particles (Wiley, New York, 1983).

28. K. Ohtaka and M. Inoue, ‘‘Light-scattering from macro-scopic spherical bodies. 2. Reflectivity of light and elec-tromagnetic localized state in a periodic monolayer of di-electric spheres,’’ Phys. Rev. B 25, 689–695 (1982).

29. V. P. Dick, V. A. Loiko, and A. P. Ivanov, ‘‘Angular structureof radiation scattered by monolayers of particles: experi-mental study,’’ Appl. Opt. 36, 4235–4240 (1997).

statistical theory of multiple scattering of waves applied to three-dimensional layered photonic...

Documents