Light diffraction from colloidal crystals with low dielectric constant modulation:Simulations using single-scattering theory

Alexander Tikhonov, Rob D. Coalson, and Sanford A. Asher*Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA

Received 19 October 2007; revised manuscript received 5 February 2008; published 3 June 2008

We theoretically characterized the diffraction properties of both closed-packed and non-closed-packed crys-talline colloidal array CCA photonic crystals. A general theory based on single-scattering kinematic approachwas developed and used to calculate the diffraction efficiency of CCA of different sphere diameters at differentincident light angles. Our theory explicitly relates the scattering properties of individual spheres calculated byusing Mie theory comprising a CCA to the CCA diffraction efficiency. For a CCA with a lattice constant of380 nm, we calculated the relative diffraction intensities of the fcc 111, 200, and 220 planes and deter-mined which sphere diameter gives rise to the most efficiently diffracting CCA for each set of crystal planes.The effective penetration depth of the light was calculated for several crystal planes of several CCAs ofdifferent sphere diameters at different angles of incidence. The typical penetration depth for a CCA comprisedof polystyrene spheres was calculated to be in the range of 10–40 CCA layers. A one-dimensional 1D modelof diffraction from the stack of 111 fcc crystal layers was developed and used to assess the role of multiplescattering and to test our single-scattering approach. The role of disorder was studied by using this 1Dscattering model. Our methodology will be useful for the optimization of photonic crystal coating materials.

DOI: 10.1103/PhysRevB.77.235404 PACS numbers: 42.25.Fx, 78.67.n, 42.72.g, 92.60.Ta

I. INTRODUCTION

Recently, there has been considerable interest in the fab-rication and the experimental and theoretical analysis of pho-ton propagation in the periodic dielectric systems known asphotonic crystals.1 These materials show promising applica-tions in optical devices, which may prove useful in the cre-ation of photonic logic chips, novel optical switches and op-tical filters, chemical sensors, as well as in numerous otheroptical technologies.2

A major class of photonic crystals is fabricated throughthe self-organization of individual particles, typically spheri-cal colloidal particles, organized in crystalline colloidal ar-rays CCAs. A CCA is comprised of a self-assembled peri-odic array of colloidal particles immersed in a dielectricmedium. Various methods have been developed to fabricatephotonic crystals such as, for example, by utilizing close-packing of spherical colloidal particles to create artificialopals.3

An alternative approach to the closed-packed system areCCAs where the colloidal particles self-assemble into non-close-packed crystal structure in a low ionic strength aque-ous solution.4,28 This procedure involves self-organization ofnegatively charged particles, which electrostatically repeleach other and adopt a minimum energy configuration, typi-cally an fcc lattice structure. By altering the concentration ofcolloidal particles in the solution, it is possible to manufac-ture CCAs with any desired lattice constant. CCAs obtainedby this self-assembly method generally form an fcc latticewhen the 111 planes orient along the surfaces of the con-tainer.

There are numerous theoretical approaches to calculatingthe interaction of light with photonic crystals. Band structuretheory, originally developed for electrons moving in an infi-nite periodic potential, has been used to solve the vectorMaxwell equations in periodic dielectric media.1 Band struc-ture calculations are most often implemented via the plane

wave expansion method, which allows the calculation of thephotonic band structure based on the expansion of the inter-nal electromagnetic field as a sum of many typically hun-dreds of plane waves. The total complex band structure ofphotonic crystals can be used to derive the boundary condi-tions for the electromagnetic field at the interface of a finitecrystal in order to determine the transmission and diffractionproperties.5

A more efficient and also more widely used method forcalculating diffraction and transmission properties is basedon relating the electromagnetic field components at the op-posite sides of a thin slab of dielectric material through aso-called transfer matrix.6,7 This is done by dividing thespace in each slab into parallelepiped cells with a couplingbetween these cells. Then, the whole system is represented asa stack of slabs by using the multiple scattering formulafamiliar in the theory of low-energy diffraction.7 The transfermatrix method is essentially a real-space finite-elementmethod of computational electrodynamics adopted for a sys-tem with a periodic dielectric function.

In the special case of dielectric spheres periodically ar-ranged within infinite slabs, the layer-multiple-scatteringmethod was developed.8 In this method, spherical vectorbasis functions were used to expand the electromagneticfield around each particle, and these fields were summedfor periodically spaced spherical scatterers in an infiniteslab. Transfer matrices were then utilized to couple fieldsbetween the different slabs. Recent extension of the methodenabled application to nonspherical particles, with scatteringproperties of individual particles calculated through theevaluation of the T matrix.9 All of these approaches treatmultiple scattering processes inside the crystal but requireeither a defined periodicity or infinite extent in some dimen-sion. An approximate perturbative behavior of electromag-netic wave packets was utilized in the envelope-function ap-proximation method.10,11

A number of computational techniques were developed tocalculate the light scattering properties of arbitrary shaped

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nonperiodic systems.12–14 These include the multiple multi-pole method,15 the finite difference time domain method,16

and the generalized field propagator17 method utilized to cal-culate transmission and reflection for photonic crystals con-taining various defects. Unfortunately, these methods are nu-merically expensive and are currently able only to treatsystems with relatively small numbers of scattering particles.

There are methods developed specifically for systemsconsisting of arbitrary located spheres, such as the T-matrixsuperposition method18 and the generalized multisphere-Mietheory.19 The total field scattered by a collection of spheres isrepresented as a superposition of individual sphere contribu-tions, where each contribution is expanded in vector spheri-cal harmonics. The multiple scattering between spheres istaken into account by representing the total field incident ateach sphere as a sum of the initial incident wave and scat-tering contributions from every other sphere of the system.To perform the required summation, the method utilizes thetranslation addition theorem, where a vector spherical wavecentered at one sphere is expressed through the sphericalwaves centered at other spheres. Currently, these methodsonly allow the analysis of hundreds of spheres,20 and hencecannot simulate realistic CCAs.

The interaction of light with a photonic crystal can beunderstood as a scattering process, where the total amplitudeof the scattered light is the result of interference of all scat-tering contributions from particles of the system. The totalscattering of incident light by a CCA can be represented as acombination of single and multiple scattering. The single-scattering approximation assumes that incident light interactsonly once with each scatterer. Multiple scattering, consistingof scattering events resulting from secondary waves rescat-tered by particles of the system, are disregarded.

The average distance that light travels between the con-secutive scattering events is called the mean free path. Mul-tiple scattering contributions are small when the mean freepath is much larger than the size of the whole system. Themean free path depends on how efficiently the individualparticles scatter, and the smaller this efficiency, the larger themean free path. There are many factors responsible for amagnitude of individual particle scattering efficiency, includ-ing the value of the dielectric contrast between the particleand the medium, the particle size and shape, and the direc-tion of scattering. Single scattering is the dominant scatteringmechanism for a low contrast dielectric modulation.

The opposite limit is a strongly scattering medium, wheremultiple scattering is important. For randomly and stronglyscattering media, setting aside the wave nature of light andinterference effects, the multiple scattering can be describedas a random walk; the light is said to be diffusely scatteredand is described by a diffusion equation.21 Interference inmultiple scattering of light in the random strongly scatteringmedia leads to such interesting effects as the localization oflight, speckles, enhanced backscattering, and Anderson local-ization. In a realistic photonic crystal with disorder, wheremultiple scattering is important, scattering of light results inan interplay between diffuse and Bragg-type scattering.22

Many of the theoretical methods briefly described above,such as the photonic band structure theory or transfer matrixderived methods, focus on numerically computing the total

scattered intensity by solving a master wave equation. Thisconceptual framework does not distinguish between singleand multiple scattering or between Bragg interference effectsfor collections of particles and the scattering phenomenafrom individual particles. Thus, the physical picture of theinterplay between these different processes is largely lost,and although the quantitative description is accurate, thephysical picture is unclear. Interpretation and prediction oflight interaction with photonic crystals is better described byusing a richer conceptual basis which describes the physicalprocesses involved. This description would benefit from theextensive use of such concepts as single and multiple scat-tering, Bragg interference, and individual particle scatteringform factors.

In this work, we developed a numerical method to analyzethe scattering of light and to numerically simulate the fullthree-dimensional 3D map of light intensities scattered by arealistically sized macroscopic CCA. Our method does notrequire the incident light to be a plane wave nor the distri-bution of particles inside the CCA to be periodic. We startwith a realistic size macroscopic system and utilize a simplesingle-scattering method, which treats a large system of ar-bitrary located spherical particles in a dielectric medium. Ourmethod produces accurate results for the case of photoniccrystals with relatively low dielectric contrast modulationwhere multiple scattering effects are small. It can be readilyemployed to investigate the roles of irregularities in size anddisorder in these systems.

Our method treats the diffraction of light from a system ofperiodically spaced scatterers in terms of the interplay be-tween the scattering properties of the individual particles andthe Bragg interference. The advantage of this approach isthat it explicitly relates the individual particle scatteringproperties to the CCA diffraction efficiency.

Our approach is inspired by the methods of x-ray diffrac-tion of atomic and molecular crystals,23,24 where light is scat-tered by the electrons of periodically spaced atoms in lowcontrast dielectric media. The main emphasis of x-ray dif-fraction theories is on the collective effects of constructiveand destructive interference from collection of scatterers.There are two main methods to model diffraction, the kine-matic theory, and dynamical diffraction theory DDT.

The DDT theory23–26 approximates the total electromag-netic field inside the crystal as the sum of a relatively smallnumber of plane waves. Multiple scattering effects are con-sidered through the interaction between these plane waves,each one propagating at the Bragg diffraction direction rela-tive to other waves. This set of internal plane waves forms asolution to Maxwell’s equations, and each plane wavecouples to other waves causing an energy exchange betweenall internal waves. In the situation when only two strongwaves dominate in the crystal, one incident and one dif-fracted, the DDT theory leads to just two equations describ-ing the diffraction by a particular set of lattice planes. Thedielectric function variation along this specific set of latticeplanes is described by only one Fourier component of thedielectric function. Thus, this model is essentially one di-mensional. When we consider diffraction from the 111planes, the two-wave DDT model is very similar to a systemof periodic one-dimensional 1D parallel slabs, with the

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main difference that in the case of DDT the refractive indexprofile is sinusoidal in the direction normal to these latticeplanes. We choose to utilize a 1D slab model instead of DDTbecause it allows us to analyze arbitrary sets of parallel slabs,while DDT requires a periodic crystal structure.

Kinematic theory does not assume that the total scatteredlight consists of just a single plane wave. Thus, it provides amore realistic description of the scattered light distribution.Also, kinematic theory can be used to model an arbitrarydistribution of scattering particles, not just the ideal crystalsto which DDT theory is restricted. Kinematic theory canclearly relate the diffraction properties of a CCA to the prop-erties such as diameter and refractive index of individualscattering spheres.

The kinematic theory is based on two approximations.1 Single-scattering approximation, where the total dif-

fracted light consists of interference between incident planewave scattered by all individual particles. This approach dis-regards multiple scattering.

2 Neglecting extinction of the incident wave—all par-ticles in the system are illuminated by an incident wave fieldof constant amplitude.

An obvious flaw in the standard kinematic theory“KNM” theory is the assumption of a constant electric fieldamplitude of incident light propagating through the CCAcrystal. This results in an unrealistically large value of thediffracted intensity for geometrically large CCAs. We modi-fied the kinematic theory to take into account extinction byincluding attenuation of the incident wave. We call this as the“extended kinematic” “EXKNM” theory. The incidentplane wave, after entering the crystal, gradually decays whiletransferring energy into the scattered light.

A similar method based on the KNM theory was previ-ously used27 to analyze light diffraction by a system ofstacked infinite slabs. The main difference between thatmethod and ours is that we treat arbitrary shaped finite 3Dsystems and take into account attenuation of the incidentlight.

In a recent paper,28 we briefly applied our method to ex-amine the differences in the integrated intensities of lightdiffracted by different crystal planes of a CCA. We also ap-plied the method to investigate the influence of stackingfaults on the scattering. In this work, we provide details ofthe method and calculate diffraction from realistic CCAs.

Here, we examine the importance of multiple scatteringby constructing an effective 1D system consisting of manydielectric slabs. We analyze diffraction from a stack of 111layers of an ideal fcc CCA. In Sec. II, we briefly describehow to apply the kinematic method to calculate and interpretthe scattering intensities from a finite perfectly ordered CCA.Extended kinematic theory is also presented. In Sec. III, aneffective 1D slab system is constructed and used to examinethe importance of multiple scattering. Section IV exploreshow the integrated intensity of specific Bragg peaks dependon factors, such as sphere size and the incident angle. Wealso study the effective penetration depth for the incidentlight.

II. METHODS

When light illuminates a collection of scattering particles,the overall amplitude of scattered light at any point in space

is simply the sum of contributions from individual particles.In a single-scattering approximation, we ignore multiplescattering between the particles by assuming that each par-ticle is excited by only the external incident field, but not bythe secondary fields scattered by other particles. Assumingthe external incident light wave to be a plane monochromatic

wave, the total scattered amplitude E sc at some distant pointr in the far-field approximation is23

E scr = j

E jr = j

AjF jrexpi jk , 1

where E j is the amplitude of light at r scattered from indi-vidual particle j with coordinates j and the summation isperformed over all particles of the system. We can expressevery individual contribution as a product of the absolutevalue of the electric field amplitude of the incident propagat-

ing wave Aj, a single sphere scattering form factor F jr, and

a phase factor expi jk , where k is the difference be-tween the wave vectors of the incident and scattered light.We assume our colloidal particles to be spheres of uniformdielectric constant embedded in a medium of another dielec-

tric constant. We can calculate the form factor F jr exactlyfrom Mie theory for scattering of a plane wave by a sphericalparticle.29

In standard KNM theory, the amplitude of the incidentpropagating wave is constant along the propagation directionin a media, Aj =const, and does not decay while propagatingthrough the crystal. This approximation assumes that the am-plitude of the scattered light is much smaller than that of theincident light.

In our EXKNM approach, we assume that the CCA me-dium can be represented as a stack of layers with each layerexperiencing a defined amplitude of the electric field of theincident light, which gradually attenuates while propagating.We determine the internal propagating wave amplitude bysubtracting from the initial incident light the amount of lightscattered. Specifically, we calculate the amplitude of the lightwhich propagates forward from the condition that the inten-sity of light after n layers equals the intensity of incidentlight minus the intensity of light scattered by all previous nlayers. We determine amplitude An+1 after layer n from thecondition An+12= A02− Rn2, where A0 is the amplitude ofthe incident wave before entering the crystal and Rn is thetotal amplitude of light diffracted by the first n layers. Wecalculate Rn by calculating the integrated intensity of all lightscattered by the first n layers, then determine Rn as thesquare root of this integrated intensity. We assume that theincident wave is completely attenuated with all energy trans-ferred to the diffracted light when Rn exceeds A0 after layern=Nef f. The effective number of layers which diffract essen-tially all light, Nef f, determines the penetration depth for theincident light. When this number is larger than the actualnumber of layers in CCA sample, then some of the lighttransmits through the CCA; otherwise, all incident light de-cays inside the crystal over the effective number of layersNef f.

The calculation procedure assumes that the CCA is a fi-nite crystal consisting of P layers. Unless specified other-

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wise, each layer has the shape of a parallelepiped and con-tains MN spheres, where M and N are the number of spheresalong each parallelepiped side. These spheres are periodi-cally arranged in ¯ABCABC¯ layers as the 111 planes ofan fcc crystal, although the method can be easily generalizedfor any possible arrangement of spherical particles.

For a finite CCA consisting of a stack of ideal crystallayers having the same shape and size, we can write thecoordinates of the CCA spheres as

= M p + ma + nb ,

where a and b are the layer lattice vectors and M p is thevector specifying the location of layer p. This vector speci-fies that the crystal is fcc with the ¯ABCABC¯ stacking of111 layers. Integers m and n define the locations of indi-vidual spheres inside the layer, and in the case of layersshaped as an identical parallelepipeds MN, these indices runthrough the set of integers m=0, . . . ,M, n=0, . . . ,N.

We calculate the total scattering amplitude by summingcontributions from all P layers stacked together to form theCCA,

E scr = p=1

P

E p111r . 2

Assuming that all spheres are identical and the amplitude ofthe incident wave Ap is the same for all particles in eachlayer, the scattering contribution from the individual layer pis

E p111r = Ap · F 0r ·

m=1

M

expi · m · a · k

· n=1

N

expi · n · b · k · expi · M p · k . 3

In this formula, both sums can be easily performed analyti-

cally, making calculations very fast for the case of layersshaped as parallelograms.

This computational procedure simulates diffraction fromany stacking pattern of ideal 111 layers. In particular, itallows the study of how stacking faults in finite CCAs affectdiffraction efficiencies, and it can be easily generalized toinvestigate other disorder in the CCA.

In Fig. 1, we show the scattering intensities for an inci-dent plane light wave diffracted by a perfect colloidal crystalconsisting of P=45 111 layers with each layer containing6050 spherical particles organized in a parallelogramplane layer. Here, the stacking sequence of 111 planes is oftype ¯ABCABC¯ corresponding to an fcc crystal.

We simulated the diffraction of an incident plane wavewith direction shown by large magenta arrow in Fig. 1awith wavelength of 367 nm by the crystal rotated such that itfulfills the Bragg condition for diffraction from the 200planes, shown by the red spot in the middle along the equatorof the scattering sphere. We define the Bragg angle as anincident glancing angle satisfying the Bragg condition. Thereare an infinite number of ways to orient the crystal such thatthe Bragg diffraction condition is satisfied for a particularcrystal plane; all directions of incident light satisfying theBragg condition occupy the surface of a cone whose axis isnormal to the crystal planes. The diffracted intensity dependson the specific direction chosen along this Bragg cone sur-face, so one has to specify the exact orientation of the CCArelative to the direction of incident light.

We choose a CCA orientation relative to the incident lightfor this calculation via the following procedure. The direc-tion of incident light occurs along the z coordinate lab axis.Initially, we orient the CCA such that the z axis is parallel tothe 111 direction and the x lab axis is parallel to one of thesides of the 111 layer parallelogram. Then, we perform tworotations of the CCA. First, we rotate around the z lab axissuch that the normal to a particular diffracting plane occursin the xz plane. A second rotation is done along the y labaxis until the glancing angle between the directions of inci-

(200)

(020)

(220)

Incident light (b)(a)

FIG. 1. Color online a Scattered light intensity from a CCA. The large red arrow indicates the direction of the incident light. Surfacelogarithmic scale color map yellow-green to orange indicates an intensity ratio of 105. The crystal was rotated about the z axis to achievediffraction by the 220 planes. The corresponding diffraction spot is shown at the center. Two other diffraction spots are shown, which resultfrom Bragg diffraction from the 020 and 200 crystal planes. Diameter of the colloidal spheres is 270 nm, the lattice constant L=805 nm, and the wavelength of incident light is 367 nm. b Reflection Ewald sphere in reciprocal space. Reciprocal lattice points areshown by the blue dots, and the reciprocal points marked by the magenta circles are located near the surface of the reflection sphere wherethe Bragg diffraction spots occur in a.

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dent light and the normal to the crystal plane fulfills theBragg condition. The two rotation angles for the Fig. 1 220calculation are /2 and 0.2546 rad, respectively.

Some incident light directions along the 220 Bragg conecan also be diffracted by other crystal planes. For the specificCCA orientation in Fig. 1, 367 nm incident light is simulta-neously Bragg diffracted by the 220, 020, and 200planes. The corresponding diffraction maxima are shown bythe bright red spots and indicated by red arrows. This simul-taneous diffraction by these crystal planes occurs only forthis specific wavelength and direction of incident light.30

The Bragg scattering directions are typically found by us-ing the Ewald sphere construction in reciprocal space Fig.1b. The yellow parallelepiped denotes the shape and ori-entation of the CCA crystal, while the blue dots are calcu-lated points of reciprocal space labeled by their Miller indi-ces. The gray cube indicates the cubic unit cell of the bccreciprocal lattice. The length of the red arrow along the ra-dius of the Ewald reflection sphere is equal to the wave vec-tor of the incident light. The magenta reciprocal lattice pointson the reflection sphere surface satisfy the Bragg diffractioncondition.

The red dots on the scattering sphere in Fig. 1a show thevalues of the scattered intensities. Each point of the surfacedenotes a specific 3D direction whose color represents thelogarithm of light intensity scattered into this direction by thecrystal. The intensity at each point of the scattering spherewas calculated for a sphere of radius r=1.

We analyze the calculated CCA light scattering resultsand identify Bragg bright spots on the scattering sphere byplotting the scattering sphere together with the reflectionEwald sphere in reciprocal space.

For a perfect or nearly perfect crystal, the diffraction pat-tern consists of discrete “diffraction spots” that arise fromthe Bragg diffraction conditions. Each Bragg diffraction spotcorresponds to constructive interference of light scattered byall individual particles. A Bragg maximum corresponds tothe simultaneous fulfillment of the three Laue equations for

three lattice vectors a , b , c and any integer number q, m, n,

a · k = 2q, b · k = 2m, c · k = 2n . 4

However, when only one or two of these equations aresatisfied, only some of the colloidal particles in the crystalscatter light in phase, resulting in partial constructive inter-ference. For a finite crystal, this “partial constructive inter-ference” is responsible for the appearance of bright lines andcircles on the scattering sphere in Fig. 1a.

The CCA consists of stacked 111 crystal parallelogramlayers, and each layer is represented by two two-dimensional

2D lattice vectors a and b within the parallelogram layerthese two vectors are not 3D fcc primitive translational lat-tice vectors since not every lattice translation can be formed

using these vectors. When either condition a ·k =2q or

b ·k =2m is met, we observe a bright circle on the surfaceof the scattering sphere as a result of partial constructiveinterference. When both of these conditions are simulta-neously true, constructive interference occurs for everysphere from the same layer and results in 2D diffraction

spots formed at the intersection of both lines. The identical111 layers stacked together results in the appearance of the“standard” 3D Bragg diffraction maximum.

The maximum intensity at the center of each Bragg peakresults from the condition that scattering from every particleconstructively interferes. Thus, the total amplitude in thisdirection depends on the single-scattering form factor calcu-lated from Mie theory and the total number of scatteringparticles in the diffracting volume of the crystal. The totalintegrated intensity over the diffraction spot depends notonly on the peak value in the center of the spot but also onthe Bragg peak angular width.

For ideal periodic large crystals, the diffraction spots arevery sharp peaks in specific Bragg directions, with very littlelight outside of these regions. We define the “integrated in-tensity of the diffraction from nml crystal planes” as theintegrated intensity calculated over a solid angle largeenough to contain the diffraction spot. The “intensity,” whichrefers to the square of the amplitude of the electric field at apoint on the scattering sphere, is proportional to the power ofelectromagnetic wave radiated into a specific direction.

We also define the “incident integrated intensity” as theintensity of the incident electromagnetic wave integratedover the area of the illuminated part of the CCA single 111layer. The ratio between the integrated intensity of the dif-fraction from nml crystal planes and the incident integratedintensity gives the “reflectance” by the nml planes.

III. VALIDITY OF THE METHOD: COMPARISON WITHEXACT SOLUTION FOR A ONE-DIMENSIONAL

SLAB SYSTEM, ROLE OF MULTIPLE SCATTERING

We tested the validity of our EXKNM approach and theimportance of multiple scattering by utilizing a 1D slabmodel. We compared the diffracted intensities calculated us-ing EXKNM theory with the exact results. First, we modeledour 3D CCA set of 111 layers as a 1D array of slabs. It isgenerally accepted that the scattering efficiencies of lightBragg diffracted by 111 CCA planes can be modeled byscattering of light by 1D slabs.31 To exactly determine thelight transmission and diffraction the diffracted wave is sim-ply a reflected wave in case of 1D slabs in the 1D system,we used the transfer matrix method Appendix B.

When the Bragg diffraction condition is approximatelysatisfied for the 111 layers of a weak dielectric contrast fccCCA crystal, we can model the diffraction by replacing each111 CCA layer with two slabs of different dielectric con-stant. The first slab with width L1 and an effective refractiveindex n1 marked as 1 in Fig. 2b represents the CCA par-ticles layer depicted as a layer of spheres in Fig. 2a, whilethe second “water” slab 2 has width L2 and represents awater layer of refractive index n2. The total width, the sum ofboth slabs, equals the distance between 111 layers.

We define the parameters of the effective 1D slab systemby comparing a single 1D slab scattering efficiency with thescattering efficiency of a single CCA 111 layer. The angulardistribution of the light scattered intensities by a single 111layer results from its 2D diffraction pattern, from its hexago-nally ordered spheres. Most of the light is concentrated in the

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narrow range of angles near the 2D Bragg maxima. Since weare modeling the Bragg diffraction from CCA 111 planes,we consider only the light specularly reflected from the 111planes at the zero order 2D maximum. It has previously beenshown32 that the scattering efficiencies in the specular direc-tion by a single plane layer of spherical particles can beapproximated by the reflection from a single 1D slab.

We adjust the 1D slab system parameters, such as thewidths and refractive indices of the slabs, until the intensitiesof light diffracted by a single 111 crystal layer were equalto those of a single slab of the 1D system. Specifically, thescattering by a CCA layer was calculated by using kinematictheory and compared to the exact result calculated for a 1Dslab. The refractive indexes n1 and n2 were constrained bythe condition that the average refractive index of the 1D slabsystem must be the same as the average refractive index ofthe CCA, and the slab widths L1 and L2 were constrained bythe condition that the L1+L2 was the same as the distancebetween CCA 111 planes.

Figure 3 shows the dependence of the reflectance on thewavelength of the incident light for a single 111 layer of aCCA blue line obtained in the kinematic approximation andby a corresponding slab of the effective 1D system redcurve obtained by exact calculation. Incident light normallyimpinges on the 111 plane in Fig. 3a and at a 30° glancingangle in Figs. 3b and 3c. The electric vector is polarizedperpendicular to the scattering plane in Fig. 3b and parallelto it in Fig. 3c. The reflectance was obtained by integratingover the diffraction spot and ratioed to the integrated incidentlight intensity. The incident light energy flux is the intensityof the incident light multiplied by the cross-sectional area ofthe crystal which is illuminated.

The CCA crystal parameters used in calculations include120 nm diameter colloidal particles with a 10% volume frac-tion, giving an fcc lattice constant of 330.8 nm. The refrac-tive index of the colloidal particles and the surrounding me-dium water are nc=1.6 and nw=1.33, respectively.

We vary two parameters of the 1D slab system, the waterslab refractive index n2 and width L1, until we obtain a goodfit between the curves corresponding to the 3D and 1D sys-tems in the spectral region of the first order Bragg diffractionfrom the 111 layers. The best match for a case of normalincidence was obtained for the L1=86 nm and n2=1.3312 nm, which fixes two other parameters as L2=105 nm and n1=1.3885 nm. For light incident at the glanc-

ing angle of 30°, the best fit values for the perpendicularpolarization were L1=89 nm and n2=1.3305 and, corre-spondingly, L2=103 nm and n1=1.3880. For parallel polar-ization, the best fit parameters were L1=115 nm and n2=1.3275 L2=76 nm and n1=1.3765.

At the Bragg incident angle for diffraction from the 111planes, all particles in the plane scatter coherently in phase.As a result, the single sphere scattering efficiency depen-dence on the wavelength of light is the main factor contrib-uting to the specific shape of the intensity curve in Fig. 3.Thus, the maximum of the reflectance occurs at the maxi-mum of the single particle scattering efficiency.

Figure 4 compares the reflectance by a CCA of 40 111layers, each consisting of 500500 colloidal particles, cal-culated by using the EXKNM theory. Light is incident nor-mal to the crystal surface in Fig. 4a, while Figs. 4b and4c show the results for a 30° glancing angle of incidencefor perpendicular and parallel polarizations. The results ob-

(a)

nc L1

L2

1

2

incident diffracted

n2

n1

incident diffracted

nw

(b)

FIG. 2. a CCA colloidal particles in a single 111 fcc CCAlayer and the corresponding two slabs of the modeled 1D slab sys-tem. Colloidal particles have a refractive index nc and are located ina water environment with a refractive index nw. The 1D slab systemconsists of a bilayer of slab 1 with refractive index n1 and the slab2 with the refractive index n2. The total thickness of the two 1Dslabs is the same as the distance between CCA 111 planes.

400400400400 500500500500 600600600600 7007007007001.21.21.21.2

1.31.31.31.3

1.41.41.41.4

1.51.51.51.5

1.61.61.61.6

1.71.71.71.7CCA layerCCA layerCCA layerCCA layer1D slab1D slab1D slab1D slab

200200200200 300300300300 400400400400 500500500500 6006006006000000

1111

2222

3333

4444

5555

6666 CCA layerCCA layerCCA layerCCA layer1D slab1D slab1D slab1D slab

200200200200 300300300300 400400400400 500500500500 6006006006000000

0.50.50.50.5

1111

1.51.51.51.5 CCA layerCCA layerCCA layerCCA layer1D slab1D slab1D slab1D slab

Wavelength (nm)

10-3Reflectance

Normal incidence

30° incidence,⊥⊥⊥⊥ polarization

30° incidence,|| polarization

b

c

a

10-3Reflectance

10-4Reflectance

FIG. 3. Color online a Reflectance for back-diffracted lightby a single 111 crystal plane of 500500 particles blue solidcurve and by the single slab of 1D system red dotted curve shownas a function of the wavelength of normally incident light. For thelight incident from vacuum at glancing angle of 30° for b perpen-dicular and c parallel polarizations.

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tained for the calculated 3D CCA are compared to the resultsfor the related 1D slab system. We examined the range ofwavelengths around the first order Bragg diffraction maxi-mum.

The reflectance for the 1D system was calculated by the1D exact and the 1D EXKNM methods Appendix A. Thered dashed curves show the EXKNM theory result for the 1Dslab system, where attenuation of the incident light in thecrystal is taken into account. The 3D and 1D EXKNM cal-

culations give almost identical results; the blue solid and thered dashed curves completely overlap. Since the EXKNMresults for 3D and 1D case are so close, we can examine theimportance of multiple scattering and compare the exact andEXKNM results only for the 1D slab system.

The exact solution for diffraction from the 1D slab systemcalculated by using the transfer matrix method Appendix Bis shown in Fig. 4 green dotted curves. Comparing theexact and EXKNM approximations allows us to examine thevalidity of kinematic theory and the relative importance ofmultiple scattering. The diffraction efficiency in Fig. 4b ismuch larger than that in Fig. 4c because the spheres scatterperpendicular polarized incident light more efficiently, re-sulting in a more efficient Bragg diffraction. The increase inthe single-scattering efficiency results in stronger multiplescattering. The result is a decreased calculated diffractionbandwidth, as compared to the exact calculations.

EXKNM works well when single spheres scatter rela-tively weakly such that the total diffracted integrated inten-sity is significantly less than the incident intensity. Thus,good agreement between the exact and EXKNM results wasobtained in Fig. 4c, especially in the wings of the mainBragg diffraction peak, where the diffracted intensity issmall. The agreement shown in Fig. 4a is better than that inFig. 4b because EXKNM theory works best when singlespheres scatter relatively weakly.

EXKNM results in a value of the Bragg peak width two-fold less than the exact result in the case of strong spherescattering Fig. 4b. In Figs. 4a and 4c, the Bragg peakwidths are similar for both the exact and EXKNM calcula-tions.

Disorder in the 1D slab system broadens the Bragg peaksand increases the intensity of diffuse scattering Fig. 5. TheBragg diffraction peak becomes more diffuse and the inci-dent wave is less attenuated. We compare the exact andEXKNM calculations for a disordered 1D slab system in Fig.5 for normally incident light. Disorder is modeled by varyingthe thickness of the slabs comprising the system. However,the refractive index of the slabs was maintained at the valueemployed for the perfectly periodic system shown in Figs.5c and 5d. The thickness of the slabs comprising the dis-ordered CCA was distributed according to the uniform ran-dom variations of 10% magnitude in a and c and 20%magnitude in b and d.

Figures 5a and 5b show the reflectance of a singleconfiguration of a disordered 1D slab system. We see that theBragg peak subdivides into a series of narrow peaks whichare spread over a wider spectral region and narrow featuresappear in the wings.

To simply model disorder, we calculated a random en-semble of configurations of 1D slab systems with randomwidth slabs and averaged the diffracted intensities over thisensemble. In Figs. 5c and 5d, we averaged the reflectanceover 80 random disordered configurations plotting the exactresult in blue and the EXKNM result in red. As a reference,the exact result for the perfectly periodic system is plotted inblue. Averaging the disorder broadens the Bragg peak anddecreases its maximum value, and reflectance peaks in thewings “smooth out.” The exact and EXKNM results are verysimilar. The major effects are that the diffracted bandwidth

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0.20.20.20.23D CCA EXKNM3D CCA EXKNM3D CCA EXKNM3D CCA EXKNM1D slab EXKNM1D slab EXKNM1D slab EXKNM1D slab EXKNM1D slab exac t1D slab exac t1D slab exac t1D slab exac t

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3D CCA EXKNM3D CCA EXKNM3D CCA EXKNM3D CCA EXKNM1D slab EXKNM1D slab EXKNM1D slab EXKNM1D slab EXKNM1D slab exac t1D slab exac t1D slab exac t1D slab exac t

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11113D CCA EXKNM3D CCA EXKNM3D CCA EXKNM3D CCA EXKNM1D slab EXKNM1D slab EXKNM1D slab EXKNM1D slab EXKNM1D slab exac t1D slab exac t1D slab exac t1D slab exac t

Normalincidence

30° incidence⊥⊥⊥⊥ polarization

30° incidence|| polarization

b

c

a

Reflectance

Reflectance

Reflectance

Wavelength (nm)

FIG. 4. Color online Reflectance from 50050040 particleCCA blue curve calculated by using 3D EXKNM theory and froma 1D system red which completely overlaps the blue curve andgreen curves as a function of the wavelength of incident light. Thered dashed curve was calculated by using the 1D EXKNM theory,while the green dotted curve is the 1D exact solution. a Normalincidence, b 30° glancing angle of incidence with perpendicularpolarization, and c 30° glancing angle of incidence with parallelpolarization.

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increases and the diffracted intensity decreases. For 10% dis-order, this system of 180 slabs continues to diffract all lightat the center of the band while the width increases by 50%.For 20% disorder, only 80% of light is diffracted in the cen-ter and the bandwidth triples.

IV. DEPENDENCE OF CRYSTALLINE COLLOIDALARRAY SCATTERING EFFICIENCY ON DIAMETER OF

COLLOIDAL SPHERES

The primary experimental quantity extracted is the inte-grated intensity we obtained by integrating over the solidangle containing the diffraction spot on the scattering sphere.The factors that could affect the integrated intensity over adiffraction spot include the magnitude of the single-scattering form factor, the shape and the size of the diffrac-tion spot, the effective number of crystal layers, and cross-sectional intensity profile of the incident light

Generally, the size of CCA in experiments is larger thanthe diameter of the incident beam. Upon changing the angleof incidence, we change the area of the CCA illuminated bythe incident beam and consequently change the number ofcolloidal particles participating in the scattering. However,the ratio between the integrated diffracted intensity and thisarea is constant for a large enough ideal crystal and will not

depend on the size of the illuminated CCA area.In what follows, we assume that the diameter of the inci-

dent beam is larger than the CCA size, but since we calculatethe reflectance the ratio between the diffracted and incidentintegrated intensities over the relevant area, our calculationsare also valid for the case when the CCA is larger than theincident beam. In our numerical simulations, we treat thecase of a macroscopic crystal; therefore, we choose the di-mension of the calculated CCA by increasing the lateral sizeof the CCA until the reflectance converges to a constantvalue.

In this section, we examine how the scattering propertiesof the CCA depend on the scattering properties of the indi-vidual spheres. We examine how the CCA scattering effi-ciency and Nef f depends on the diameter D of the colloidalsphere. The EXKNM method is well suited to analyze howthe diffraction properties of a CCA depend on x, the sizeparameter x=2Dn / where n is the refractive index of thesurrounding media. In this section, we show how the inten-sity of light Bragg diffracted by 111, 200, and 220 crys-tal planes depend on the size parameter of the spheres.

Our model examines a perfect periodic fcc crystal consist-ing of stacked 111 layers of a parallelogram shape anddimensions of 500500. This CCA is illuminated by a planewave monochromatic beam. We set the CCA fcc lattice con-stant at 380 nm and calculate the diffraction for different

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10% disorder,singleconfiguration

20% disorder,singleconfiguration

Perfectly periodic and10% disorder,average over 40configurations

Perfectly periodic and20% disorder,average over 40configurations

Wavelength (nm)

c

Wavelength (nm)

Reflectance

b

d

aReflectance

FIG. 5. Color online Wavelength dependence of reflectance for 1D slab system consisting of 180 slabs. a and b Single disorderedsystem configuration. The blue solid curve is the exact result. The red dashed curve is the EXKNM result. c and d Reflectance forperfectly ordered system and disordered system averaged over 40 random configurations. In c and d, the exact result for a periodic systemblue solid curve is compared to the exact and EXKNM results averaged over the disordered system red dashed and green dotted lines.

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diameters of colloidal particles ranging from 150 nm to themaximum diameter of 270 nm corresponding to a closedpacked system. We assume a colloidal particle refractive in-dex of 1.6 and a water refractive index of 1.33.

The single sphere Mie scattering efficiency can be veryangular dependent, as shown in Fig. 6, which considers a270 nm diameter sphere scattering 337 nm light. In this cal-culation, we also indicate the Bragg diffraction angles for anIR diffracting CCA with a lattice constant of 805 nm. Forthis 337 nm wavelength, the CCA was oriented to meet theBragg diffraction condition for the 111, 222, 200, 220,and 311 planes Fig. 6.

The single-scattering contribution of the colloidal par-ticles the form factor in kinematic theory depends on thescattering angle and the polarization of the incident light and,thus, will differ for diffraction from different crystal planeswith different Bragg diffraction angles at this single excita-tion wavelength.

The 3D Mie scattering diagram Fig. 6 for light scatteredby a single sphere indicates the scattering directions forBragg diffraction from specific crystal planes, for light po-larized perpendicular to the incident plane. The crystal wasrotated about the z axis to achieve diffraction from the dif-ferent crystal planes. Different colors on the scatteringsphere surface represent different scattered intensities, as la-beled by the linear color map. Much more light is scatteredin the forward direction than in the backward direction. Theratios of intensities scattered by a single sphere at the Braggangles for the 111, 200, 220, 311, and 222 CCAplanes have the relative intensities of 1, 0.75, 0.18, 0.04, and0.02, respectively.

The ratio between the forward and backscattered intensi-ties increases as the size parameter of the particle increases.For convenience, we give the well known exact formulas forMie single sphere scattering in the far-field approximation inAppendix C.

The diffraction efficiency of the CCA depends on theform factor of the spheres, which is determined by the sphere

diameter, the wavelength of incident light within the CCA,and the diffraction scattering angle. The form factor depen-dence is shown in Fig. 7 where we specifically examine thescattering of 599, 626, and 673 nm light from spheres withdiameters of 150, 210, and 270 nm and from a 111 CCAsingle layer. These wavelengths in vacuum were chosenbecause they meet the Bragg condition for 180° backscatter-ing from the 111 planes of the considered three CCAs ofidentical lattice constant, which utilized different sphere di-ameters. The resulting diffraction condition occurs for differ-ent wavelengths in vacuum of light because the CCA aver-

FIG. 6. Color online Mie scattering efficiency from a singlespherical particle shown as a color map linear in intensity. Weshow the Bragg diffraction directions for several crystal planes la-beled by their Miller indices. The incident wavelength is 337 nmand the sphere diameter is 270 nm.

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FIG. 7. Color online a The single sphere scattering crosssection is plotted vs incident light wavelength in vacuum for threesphere diameters of 150, 210, and 270 nm in a CCA. The sphererefractive index is 1.6 and the water refractive index is 1.33. b Thebackscattered light intensity from a single sphere is plotted for thesesame three spheres. c The reflectance from a single 111 CCAlayer is plotted for these three spheres.

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age refractive index increases with the sphere diameter,which decreases the actual wavelength of light propagatingwithin the CCA.

The scattering cross sections monotonically increase asthe size parameter increases as the wavelength decreases.The cross section monitors how much light is removed fromthe incident beam as a result of scattering in all directions.The total amount of light scattering monotonically increaseswith sphere diameter in the size parameter regime consideredhere. The relative intensity of forward versus back diffractionshows a more complex relationship Figs. 7b and 7c.

In Fig. 7b, we plot the wavelength dependence of thescattered intensity, where we plot the differential dependenceon solid angle dI /d in the exact backscattering directionfor sphere diameters of 150, 210, and 270 nm.

Although the Fig. 7a cross sections monotonically in-crease with the sphere size, resulting in an increasing totalscattering over all directions as the sphere diameter in-creases, the scattered intensity in specific directions does notmonotonically increase but oscillates as a function of wave-length. For example, for 626 nm light, the most efficientscattering with the largest form factor for back diffractionoccurs for the 210 nm sphere CCA. This backscattering ef-ficiency is approximately 20% larger than for a larger270 nm sphere.

We conclude that using smaller spheres in non-closed-packed systems can increase the CCA diffraction efficiencyin the Bragg direction. This has the desired advantage ofconsiderably decreasing the single sphere extinction crosssection and thus decreasing the amount of diffuse scattering,induced by disorder, for example. Diffuse scattering also oc-curs for perfect crystals when there is significant multiplescattering.22 In general, multiple scattering decreases as thesphere diameter decreases.

Figure 7c shows the wavelength dependence of reflec-tance by the single 111 CCA layer. Since the integratedintensity is calculated by integrating the scattered intensitiesover the diffraction spot, it depends on both the single sphereform factor and the solid angle subtended by the diffractionspot. The diffraction efficiency of the single 111 CCA layerfollows a similar dependence on the wavelength as we ob-serve for the single sphere diffraction efficiency Fig. 7b.The backscattering diffraction efficiency of a single 111layer is largest for the CCA comprised of 210 nm diameterspheres.

The effective depth of penetration of the incident lightNef f is defined as the number of 111 layers into the CCAis related to the diffraction efficiency of a single 111 layer.The higher the efficiency, the more the attenuation of theincident light within the CCA, and the smaller Nef f. We cal-culate Nef f to be 30, 19, and 20 layers for CCA with spherediameters of 150, 210, and 270 nm, respectively.

Figure 8 shows the light differential intensity backscat-tered by a single sphere as a function of the wavelength ofincident light divided by the average refractive index of theCCA prepared with different diameter spheres. The CCA lat-tice constant is 380 nm for all cases. The wavelength meet-ing the Bragg condition for normal incidence from the 111,200, and 220 planes in the CCA are indicated. HigherMiller index crystal planes have smaller spacings and, thus,

smaller Bragg wavelengths. The sphere diameter providingthe largest CCA diffraction efficiency will differ for differentMiller index crystal planes. Obviously, the CCA diffractionproperties can be controlled by controlling the sphere diam-eter.

Since the single sphere scattering efficiency into a specificdirection is an oscillating function of wavelength and spherediameter, the Bragg diffraction efficiency can sometimes bemuch larger for a CCA with smaller sphere sizes. For ex-ample, the largest sphere diameter of 270 nm shows thesmallest back diffraction for the 200 planes Fig. 9b,while spheres of 180 nm diameter diffract five times moreintensity into the Bragg direction.

The diffraction of light by photonic crystals is typicallydescribed in two related ways, each of which stresses differ-ent aspects of the physics involved. One is the photon banddispersion relations which rely on considering appropriateBloch waves. This approach is well suited to describe themultiple scattering of electromagnetic waves in infinite sizedperiodically modulated dielectric structures.

Another approach considers the scattering of light by pho-tonic crystals as the interplay between the single-particlescattering from individual colloidal particles comprising theCCA coupled with the macroscopic Bragg interference be-tween the scattering from all particles of the system. In anideal crystal, the particles are periodically arranged, and thescattering of plane waves by the crystal results in Bragg-typediffraction maxima. This approach, upon which we based ourEXKNM method, has the advantage of explicitly relating thescattering properties of single spheres to the CCA diffractionproperties. The stronger the single sphere scattering effi-ciency in the Bragg direction, the stronger the scattering ef-ficiency of each CCA 111 layer Fig. 7c, and, hence, thesmaller the effective depth of penetration of incident light,Nef f. In the thick crystal limit, this translates into a largerwidth of the diffraction peak and larger band gap for thecorresponding direction.

In Fig. 9, we examine the integrated intensity of lightnormally incident diffracted by a CCA consisting of 25 111

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400150180210240270

(111)

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λλλλ/nav (nm)

10-18dI/d

ΩΩ ΩΩ

Normalincidence

FIG. 8. Color online Differential backscattered intensity oflight from single spheres in a CCA for five sphere diameters. Thearrows show the wavelengths corresponding to Bragg diffraction atnormal incidence to the 111, 200, and 220 planes. The abscissaindicates the wavelength within the CCA /nav

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layers. In Fig. 9a, the light is incident normal to the 111planes, in Fig. 9b, the light is incident normal to the 200planes, while in Fig. 9c, the light is incident normal to the220 planes. The effective number of layers Nef f involved indiffracting essentially 100% of the incident light and thebackscattering intensity F from the single sphere are pre-sented in Table I. Comparing Fig. 8 to Table I, we see thatNef f is related to the single sphere scattering efficiency at thediffracted direction. Strong scattering translates to the larger

total diffraction efficiency and stronger attenuation of inci-dent light and a smaller Nef f.

For 150 nm diameter particles, Nef f =30 from the 111planes. Since in Fig. 9a the CCA consists only of 25 layers,some of the incident light is transmitted while 90% of thelight is diffracted. For the 210 nm diameter spheres, Nef f=19, and for the 270 nm spheres, Nef f =20. Since Nef f 25,all incident light is diffracted. Increasing the number of lay-ers above Nef f results in diffracting a broader range of wave-lengths; the Bragg peak broadens. Figure 9a shows similarintensity diffracting peaks for the 210 and 270 nm CCA,which results from their similar single sphere scattering effi-ciencies Fig. 8.

Figure 10 shows the incident glancing angular depen-dence of Nef f for three CCA containing 150, 210, and270 nm diameter particles diffracted by 111 crystal planesat the different Bragg wavelengths. The dotted curve showsthe intensity of single sphere scattering dI /d at the angle ofthe Bragg diffracted light. In general, an increasingly effi-cient single sphere scattering translates into a smaller Nef f.For perpendicular polarization, the single sphere scatteringefficiency monotonically increases with the incident angle,and Nef f correspondingly decreases. For parallel polarization,the single sphere scattering efficiency dips at an intermediatescattering angle, where almost no light is scattered by thespheres in the Bragg direction and the CCA becomes trans-parent.

Figure 11 shows the dependence of Nef f for Bragg diffrac-tion from the 200 planes. The glancing angle here is de-fined relative to the 111 CCA surface. The integrated inten-sity and Nef f are calculated only for specular diffracted lightfrom the 200 planes, and we do not take into account anyBragg diffraction from other crystal planes, which might alsofortuitously occur. The angular dependence of the singlesphere scattering efficiency from the 200 planes is shownby a dotted line. These calculations show that in addition tothe single-scattering efficiency, there are other factors affect-ing Nef f. For perpendicular polarization, the single-scatteringefficiency decreases with the glancing angle. However, Nef ffirst decreases and correspondingly CCA diffraction effi-ciency increases reaching a minimum at 73°. At this specificangle, light is most efficiently diffracted and propagateswithin the 111 plane. We will discuss this phenomenon in asubsequent paper.

Figure 12 shows the dependence of Nef f on angle for dif-fraction by the 220 planes. We see that Nef f has a compli-cated dependence on the angle of incidence and essentiallyfollows the shape of single sphere scattering efficiency.

V. CONCLUSION

We used the single-scattering approach to investigate lightdiffraction by a CCA photonic crystal with a small dielectricconstant mismatch between the spheres and the medium. Weextended standard kinematic theory by including attenuationof the incident light intensity during propagation through thecrystal. Our method EXKNM can be used to predict lightdiffraction from large but finite CCAs consisting of manyparticles. As the first step toward this aim, we model light

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FIG. 9. Color online Wavelength in vacuum dependence ofthe integrated intensity of light specularly diffracted by 25 111CCA layers each containing 500500 particles. The incident lightis normal to the a 111, b 200, and c 220 planes. The CCAconsist of spheres with diameters of 150 nm blue, 210 nm red,and 270 nm green.

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scattering by perfect fcc crystal CCAs. We examined thedependence of the diffraction efficiency on the sphere diam-eter and angle of incidence. The effective penetration depthof the incident light was calculated for several fcc planes forCCAs of different sphere diameters.

We studied both close-packed and non-close-packed sys-tems and compared their diffraction efficiencies as a functionof sphere diameter. We show that the diffraction efficiencydoes not always increase monotonically with the sphere di-ameter. Thus, a closed-packed photonic crystal system is notalways the most efficient Bragg diffracting crystal.33 For theCCA consisting of polystyrene spheres in water, the mostefficient Bragg diffraction by 111 planes at normal inci-dence is achieved at sphere diameters approximately 20%smaller than the close-packed case. At normal incidence tothe 200 planes, the single sphere scattering efficiency at theBragg condition is five times larger for spheres of a diameterapproximately 30% smaller than the close-packed case.

One of the simplest ways to model the diffraction fromstack of 111 layers is to replace each layer with a 1D di-electric slab. We show how to specify the dielectric modula-tion of such a 1D slab system by matching the diffractionintensity of a single 1D slab to a single 111 crystal layer.We tested the impact of multiple scattering by comparingresults obtained by EXKNM method to exact solutions for a1D slab system. For a low contrast modulation CCA, thediffracted intensities calculated by EXKNM are close to theexact result obtained for an effective 1D slab system. Al-though the EXKNM method does not take into account mul-tiple scattering effects, it gives good results in calculatingdiffracting intensities. For the perfect 1D slab system, theEXKNM bandwidth of the diffracted peak is smaller than theexact result by less than twofold.

Our calculation method can be used to predict the optimalproperties of photonic crystal films that have been used asphotonic crystal coatings and sensors.34 Our method can alsobe used to examine the impact of CCA disorder, such asvariations in particle position, diameter, and changes in di-electric constant.

ACKNOWLEDGMENTS

This work was supported by the PPG Grant No. 218363

and NIH Grant No. 2R01 EB004132. Work in R.D.C.’sgroup has been partially supported by NSF-ECS-0403865.

APPENDIX A: EXTENDED KINEMATIC ONE-DIMENSIONAL THEORY

We developed 1D analogs of EXKNM theory and usedthese formulas to investigate the role of multiple scattering inthe 1D case by comparing the result to the exact expressionsfor scattering intensities of a 1D slab system. Here, we as-sume that each unit cell consist of two layers, of high and oflow refractive index. These two layers form the repeatingunit cell of the 1D periodic structure.

We assume that our 1D slab system consists of N repeat-ing units each of thickness d. In EXKNM theory, we furtherassume that the incident propagating wave attenuates whilepropagating through the system. The total amplitude rn of thewave scattered by N units is the sum of contributions fromall N units,

rN = n=1

N

tnr1 expin ,

where r1 is the single unit scattering factor, and the phasefactor =2knav sin corresponds to the phase differencebetween the scattering contributions of two adjacent units.The incident wave with wave vector k propagates at a glanc-ing angle within the 1D slab system with an average re-fractive index nav.

The amplitude of the incident wave after propagatingthrough n units is tn=1−rn−1

2 . We calculate the single cellscattering factor r1 by using the exact transfer matrix 1Dtheory35 applied to a single dielectric layer.

APPENDIX B: EXACT SOLUTION BY THE TRANSFERMATRIX METHOD FOR THE PROBLEM OF

LIGHT SCATTERING BY ONE-DIMENSIONAL PERIODICLAYER SYSTEM

We solved Maxwell equations for 1D layered along zsystem by the standard transfer matrix method.35 Reflectionand transmission coefficients can be obtained by solvingMaxwell equations for either the H or E field. Here, we

TABLE I. EXKNM results for three different diameters D of colloidal particles in an fcc CCA with500500100 particles and lattice constant of 380 nm. Effective number of layers Nef f was calculated forthe light normally incident to the specified set of crystal planes. The Bragg wavelength of light inside theCCA is twice the lattice spacing, as indicated in the table. Backscattered intensity of the single sphere F isnormalized relative to the value obtained for the sphere diameter D=150 nm scattering from the 111 plane.

500500100,L=380 nm D=150 nm D=210 nm D=270 nm

Crystalplanes

nm=2dnml F Nef f F Nef f F Nef f

111 438.8 1 30 2.44 19 2.12 20

200 380 1.24 18 1.6 15 0.35 35

220 268.7 0.57 56 1.91 29 7.4 14

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examine it for the H field. In the case of TM modes mag-netic field H vector is parallel to the interface between lay-ers, we solve the wave equation for the Hz field,

−1

2

z2 +kx

2

Hz =

c2

Hz .

The solution for the magnetic field inside each layer can berepresented as a combination of two plane waves, one is inthe forward and another is in the backward direction,

Hnz = Anei1z + Bne−i1z.

For a layered 1D system consisting of a finite number ofidentical unit cells each unit cell contains two layers, wecan connect coefficients An and Bn for two arbitrary cells.

For example, we can connect cell n=0 and cell n=N sepa-rated by N cells with the matrix equation

A0

B0 = L=N · T*

=

1,NdAN

BN ,

whrere L= and T= are 22 matrices the exact expression forthese matrices is complex and can be found in Ref. 35,where d is the unit cell length and

1 = /c21 − kx2.

The exact result for the transmission r and reflection tamplitudes can then be obtained from

FIG. 10. Nef f solid line and the scattering efficiency of single sphere dotted for Bragg diffraction by 111 planes as a function ofglancing angle of incident light. a–c Incident light has perpendicular polarization and d–f parallel polarization. a and d TheCCA consists of 150 nm diameter colloidal particles, b and e 210 nm diameter colloidal particles, and c and f 270 nm diametercolloidal particles.

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1

r = L

N · T*=

1,Nd t

0 .

The reflection intensity r2 can be calculated by

r2 =L12L21

L12L21 + 1 − 2

2N − 2

N2 ,

where 1 and 2 are the eigenvalues of matrix L= . The reflec-tion coefficient r1 used in 1D kinematic theory for just onecell is obtained by

r12 = L21

L11 .

In the case of TE modes, we can solve a similar waveequation but for the E electric field vector, resulting inslightly different matrix elements for matrix L= but otherwisethe same formulas for the reflection and transmission coeffi-cients.

APPENDIX C: SCATTERING OF LIGHT BY A SINGLESPHERE

It is well known that when a plane electromagnetic waveis scattered by a dielectric sphere, it is possible to obtain theexact analytical solution for the scattered intensity by solvingMaxwell equations.29

When an incident plane wave of amplitude E0 is polar-ized along the x axis, we can calculate the single sphere

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Neff

Neff

10-18dI/d

Ω10

-18dI/d

Ω10

-18dI/d

Ω

10-18dI/d

Ω10

-18dI/d

Ω10

-18dI/d

ΩFIG. 11. Nef f solid line and intensity of single scattering dotted as a function of glancing angle of incident light. Light is diffracted

from 200 crystal planes. a–c Perpendicular polarization and d–f parallel polarization. a and d The CCA consists of 150 nmdiameter colloidal particles, b and e 210 nm colloidal particles, and c and f 270 nm colloidal particles.

TIKHONOV, COALSON, AND ASHER PHYSICAL REVIEW B 77, 235404 2008

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3D scattering amplitude E sr , , form factor F jr forsphere j in formula 1 as a function of spherical coordinatesr, , and with the coordinate origin at the sphere centeras

E sr, ,2 =E0

2

krS1

2 cos2 + S22 sin2 .

Asymptotic far-field expressions for parallel and perpendicu-lar polarization scattered effective intensities S12 and S22are given by

S1 = n=1

2n + 1

nn + 1an

Pn1cos sin

+ bn

dPn1cos

d ,

S2 = n=1

2n + 1

nn + 1bn

Pn1cos sin

+ an

dPn1cos

d ,

where Pnl are associated Legendre polynomials and the ex-

pressions for the scattering coefficients an and bn can befound, for example, in Ref. 29.

3040506070809035

40

45

50

55

3040506070809020

30

40

50

60

3040506070809050

100

150

200

304050607080900

10

20

30

3040506070809025

30

35

40

3040506070809040

60

80

100

304050607080900

100

200

304050607080900

200

400

3040506070809025

25.5

26

26.5

27

27.5

28

28.5

29

3040506070809090

92

94

96

98

100

102

104

106

304050607080908

10

12

14

16

30405060708090300

400

500

600

700

10-18dI/d

Ω

a

be

c f

d

Glancing angle (°) Glancing angle (°)

(220)

150 nm, ⊥⊥⊥⊥

210 nm, ⊥⊥⊥⊥

270 nm, ⊥⊥⊥⊥

150 nm, ||

210 nm, ||

270 nm, ||

Neff

Neff

Neff

Neff

Neff

Neff

10-18dI/d

Ω10

-18dI/d

Ω10

-18dI/d

Ω

1 0-18dI/d

Ω10

-18dI/d

ΩFIG. 12. Nef f solid line and intensity of single sphere scattering dotted line are plotted as a function of glancing angle. Light is

diffracted from 220 planes. a–c Perpendicular polarization of the incident light and d–f parallel polarization. a and d TheCCA consists of 150 nm diameter colloidal particles, b and e 210 nm colloidal particles, and c and f 270 nm colloidal particles.The glancing angle is defined relative to the 111 CCA surface.

LIGHT DIFFRACTION FROM COLLOIDAL CRYSTALS… PHYSICAL REVIEW B 77, 235404 2008

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