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1394 J. Opt. Soc. Am. B/Vol. 27, No. 7 /July 2010 Lozano et al.

Angular dependence of the intensity of lightbeams diffracted by colloidal crystals

Gabriel Lozano,1 Javier E. Mazzaferri,2 Luis A. Dorado,3 Silvia Ledesma,2

Ricardo A. Depine,3,4 and Hernán Míguez1,*1Instituto de Ciencia de Materiales de Sevilla, Consejo Superior de Investigaciones Científicas, Sevilla, Spain

2Laboratorio de Procesado de Imágenes, Departamento de Física,Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina

3Grupo de Electromagnetismo Aplicado, Departamento de Física,Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina

4E-mail: rdep@df.uba.ar*Corresponding author: hernan@icmse.csic.es

Received January 27, 2010; revised April 9, 2010; accepted April 27, 2010;posted April 30, 2010 (Doc. ID 123403); published June 17, 2010

An experimental and theoretical analysis of the angular dependence of the diffracted light beams emergingfrom three-dimensional colloidal photonic crystals is herein presented. Diffracted beams are identified accord-ing to their associated reciprocal-lattice vectors, and their intensities are obtained as a function of the zenithaland azimuthal incidence angles. Significant changes in the beam intensities are observed for large zenithalincidence angles as the azimuthal angle is varied. This phenomenon is related to the excitation of new reso-nant modes inside the photonic crystal which cannot be observed under normal incidence conditions. © 2010Optical Society of America

OCIS codes: 050.1940, 050.5298, 050.6875, 160.4670, 160.4760.

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. INTRODUCTIONince photonic crystals [1,2] were proposed as new mate-ials to mold the flow of light two decades ago, they havettracted much attention in diverse fundamental and ap-lied fields. Among all fabrication techniques developedp to date to prepare three-dimensional (3D) photonicrystals, those based on self-assembly are some of theost frequently employed and thoroughly analyzed [3,4].he introduction of the evaporation induced self-assemblyethods has largely improved the quality of the lattices

5,6] and made it possible to study for the first time fineptical phenomena [7–10]. In recent works, it has been re-orted that disorder in 3D photonic crystals dramaticallyffects the optical response in the so-called high-energyange, in which the lattice constant is equal or greaterhan the incident wavelength [11]. In fact, the origin ofhe fine features observed in this range has been the sub-ect of an exciting debate [9,11–19]. It is precisely in thisegion where the most appealing phenomena occur. Onef these is the optical diffraction. Although its observationas first reported and incipiently analyzed some yearsgo [13], the spectral dependence of the intensity of theiffracted beams emerging from a colloidal crystal re-ained unexplored. A full experimental and theoretical

nalysis of this has recently been reported for the case oformal incidence, that is, for an incident light beam per-endicular to the surface of a photonic crystal slab [20].In this paper, we present a complete description of the

ngular dependence of light beams diffracted by 3D col-oidal crystals, both theoretically and experimentally. We

easured the relative intensity of each diffracted spot asoth the zenithal and azimuthal incidence angles are var-

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ed when a laser beam impinges on a self-assembled 3Dolloidal crystal. We use a Korringa–Kohn–RostokerKKR) approach to calculate the expected optical responsef this kind of lattices. The effect of disorder is modeled bydding an imaginary part to the dielectric constant of thepheres [21], with a fairly precise reproduction of the ex-erimental observations being attained. We report signifi-ant changes in the beam intensities for large zenithal in-idence angles as the azimuthal angle is varied, which arexplained in terms of the excitation of resonant modes in-ide the photonic crystal that cannot be observed underormal incidence conditions.

. SAMPLE PREPARATION ANDXPERIMENTAL SETUPelf-assembled 3D photonic crystal films were prepareds it is described in [4] by an evaporation induced self-ssembly technique. In fact, the colloidal crystals wereade by the deposition onto flat glass substrates of 750m polysterene spheres (IKERLAT, polydispersity below%, density of �=1.1 g/cm3, and refractive index of n1.58) suspended in water with particle volume fractionanging from 0.05% to 0.20% and evaporated at tempera-ures ranging from 30°C to 60°C. As the suspensionvaporates, a crystalline film is deposited on the sub-trate at the contact line with the suspension meniscus.he model structure is therefore a close-packed face-entered-cubic (FCC) lattice of spheres of dielectric con-tant �s=2.5+ i�i, embedded in a medium of �m=1, whichould correspond to latex spheres in air. The imaginaryart of the dielectric constant is introduced to account for

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Lozano et al. Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. B 1395

he presence of disorder in the structure [21] since theaterial the spheres are made of is transparent in the op-

ical range analyzed [22]. Measurements have to be per-ormed with the photonic crystal slab deposited on a glassubstrate with a refractive index of 1.51, so this substrates also included in the theoretical model.

In order to analyze the angular dependence of the fluc-uations of the diffracted beams, we used the blue linewavelength of �=473 nm) of a solid state laser (Laser-low) as the light source. In the experiments performed,e varied the direction of the incident wavevector k� i with

espect to the surface of the photonic crystal, whose exactrystalline orientation was known, both from the patternf diffracted spots observed and the examination of itsurface and cross section under the scanning electron mi-roscope [4]. The experimental setup is drawn in Fig. 1(a).n this scheme, all relevant geometrical parameters arentroduced. The relative orientation of the photonic crys-al film with respect to the source and the screen or the

ig. 1. (Color online) (a) Sketch of the experimental setup. (b)cheme of the diffracted beams emerging from the colloidal crys-al. Diagrams showing the two primitive vectors used to describehe (c) real and (d) reciprocal lattices.

etector is described using a Cartesian reference systems shown in Fig. 1(a). The laser beam is linearly polarizedith the electric field vector always contained in the inci-ence plane (xz plane in our reference system). We ana-yzed the intensity of the diffracted beams for each inci-ent zenithal angle ��i�, varying the incident azimuthalngle ��i�. The former is defined as the angle formed be-ween the normal to the photonic crystal outer surfacend the incident beam, which is parallel to the z axis inur reference system, while the latter is the angle formedetween the longitudinal axis of the photonic crystal filmdotted line in Fig. 1(a)] and the incidence plane. The the-retical values of the zenithal and azimuthal angles weredjusted to those measured by fitting the positions ofaxima and minima in the efficiency spectra and by

nowing the crystalline orientation with respect to theupporting glass edges using a scanning electron micro-cope, as we mentioned. The colloidal crystal growth di-ection is indicated by the arrow tip drawn in the longi-udinal axis in Fig. 1(a). The sample is mounted in twotages that allow rotations in the so defined zenithal andzimuthal directions. The longitudinal axis of the colloi-al film coincides with the (1,�1) direction of the surfaceattice of the crystal, as described in Fig. 1(c), which isarallel to the x axis, for �i=0° and �i=0°, in our refer-nce system. In front of the crystal a diffusing screen islaced in such a way that its normal coincides with theirection of the specularly reflected channel. The spots onhe screen were imaged by a lens on the charge-coupledevice array of a camera (Sony XC-75), and the pictureas digitalized and stored in a computer. The magnifica-

ion of the imaging system was adjusted in order to obtainhe highest quality image that contains all the diffractedrders to be measured. The set of acquired images wasrocessed by digital image techniques to extract the infor-ation of the efficiency for the different angles of inci-

ence; in all cases the azimuthal angle �i is varied from165° to 135° in steps of 5°. The efficiency can be calcu-

ated by integrating the intensity in the correspondingpot area. Some further processing is necessary in ordero reduce the noise background produced by the diffuseight. In our case we have used a threshold criterion. Ac-ording to this, the efficiency of each nonspecular dif-racted channel was calculated as the addition of all theixels within each spot whose intensity values were abovehe assumed noise threshold. Furthermore, since effi-iency of the specularly reflected beam was to be mea-ured, too, the intensities of both the incoming and dif-racted beam were measured using a silicon photodiode.

Specular reflectance measurements were also per-ormed at normal incidence for a wide range of wave-engths in order to precisely determine the crystal param-ters from the fitting of the optical response. We used aourier transform infrared spectrophotometer (BrukerFS-66) attached to a microscope. A 4� objective with aumerical aperture of 0.1 (light cone angle of 5.7°) wassed to irradiate the lattices and collect the reflected lightt quasinormal incidence with respect to its surface. Thisquipment allowed us to obtain absolute values for thepecular reflectance at normal incidence, being straight-orward the comparison between measured and calcu-ated spectra.

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1396 J. Opt. Soc. Am. B/Vol. 27, No. 7 /July 2010 Lozano et al.

. DETERMINATION OF SIMULATIONARAMETERShe intensities of diffracted beams were calculated usinghe vector KKR method in its layer version [23,24]. Therystal slab is first divided into layers parallel to a givenrystallographic plane (111), with each layer containing awo dimensional (2D) lattice of identical spheres. A mul-ipole expansion in vector spherical waves is used to cal-ulate the multiple scattering in each 2D layer. Next, alane wave expansion is used to calculate the multiplecattering between layers. In a numerical calculation, aaximum angular momentum LMAX must be chosen in

rder to retain a finite number of terms in the series ofpherical waves, while a maximum number of planeaves must also be defined. In all the calculations that

ollow, numerical convergence was obtained using a maxi-um value of the angular momentum LMAX=9 in the

pherical wave expansion and 41 plane waves. Since were interested in FCC lattices and photonic crystal slabsith sphere layers piled up along the [111] direction, the

pheres in each layer are ordered in a triangular lattice. Ifhe sphere layers are parallel to the xy plane for �i=0°nd �i=0°, a set of 2D primitive lattice vectors is a�1

d�x+�3y� /2 and a�2=d�−x+�3y� /2, where d is the dis-ance between spheres in the same layer [see Fig. 1(c)]. Aet of primitive reciprocal-lattice vectors can be chosen as

�1=4�y / ��3d� and b�2=2��y /�3− x� /d [see Fig. 1(d)], sony reciprocal-lattice vector can be written as g� =pb�1

qb�2, where �p ,q� is a pair of integers. Since it is custom-ry to use the lattice constant a of a classical cubic cell,e have a=�2d, so the photon energy is expressed in re-uced units a /�, where � is the wavelength of the incidentight.

The wavevector of a diffracted beam emerging from thelab can be written as K� g�

±=g� +k� i�±�k2− �g� +k� i��2z, wherehe + (�) sign corresponds to a transmitted (reflected)eam [23], k� i� is the component of the incident wavevectorarallel to the surface of the slab (xy plane), k=2�nd /�,nd nd is the refractive index of the diffraction medium.ach diffracted beam corresponds to a propagating wave

f the z-component of K� g�± is purely real, so a diffraction

hannel g� =pb�1+qb�2 is open when �g� +k� i��k and we havediffraction cutoff whenever �g� +k� i��=k. The efficiency of aiffraction channel �p ,q� is denoted by R�p,q�, so the totaleflectance is given by R=��p,q�R�p,q� [11]. Therefore, R�p,q�s the relative intensity of the diffracted beam �p ,q� and its the parameter obtained in our measurements.

The calculated total reflectance spectrum of a colloidalrystal is shown in Fig. 2(a). As Checoury et al. reported,he different stacking patterns can be unambiguouslydentified by analyzing the optical response in the high-nergy range [25]. Hence, the measurements presented inig. 2(b) correspond to the simulated spectrum of a FCCtructure. Given the geometry of the lattice, we can cal-ulate the diffraction cutoff as explained above, so wenow that the opening of diffraction channels in air takeslace at a /�=1.63. Such a diffraction cutoff is indicatedith a dashed line in Fig. 2(a). Specular reflectance mea-

urements, presented in Fig. 2(b), were fitted using a codeased on the vector KKR method [11]. From the optimum

tting [see Fig. 2(b)], the crystal thickness (seven layers)nd the �i value (0.13) [26], related to the amount of dis-rder [21], were extracted. In order to fit the completepecular reflectance spectrum it is necessary to considerhat the interparticle distance of actual colloidal crystalsoincides with the expected diameter for spheres belong-ng to the same close-packed (111) plane but differs sig-ificantly in directions oblique to the [111] one. This leapas something to do with the fact that the spheres flattenlong directions oblique to the normal to the substratehey are deposited on [27]. We find an excellent agree-ent between the theoretically simulated and the mea-

ured reflectance spectra. The lattice parameters (filmhickness, lattice constants, or dielectric constants) ob-ained from this fitting are used in all ulterior calcula-ions performed to analyze the efficiency of diffractedeams showed in next sections.

. IDENTIFICATION OF DIFFRACTEDHANNELShotographs of different diffraction patterns of reflectedeams projected on a screen parallel to the xy plane takenhen the blue laser impinges perpendicularly (zenithalngle of �i=0°) onto the outer (111) plane of the colloidalrystal are shown in Fig. 3(a). Notice that the energy ofhe incident beam, in reduced units, is a /�=2.23, wellbove the predicted cutoff at a /�=1.63. Figure 3(b) showsiffracted spot patterns observed at the incident zenithalngle of �i=5°. For both zenithal angles, the series of pic-ures illustrate the evolution of the diffraction pattern ashe azimuthal angle increases.

The surface of the sample presents a triangular latticetructure with C symmetry and subsequent layers are

ig. 2. (Color online) (a) Calculated total reflectance spectrum.b) Measured (red thick line) and calculated (gray thin line)pecular reflectance spectra. Vertical dashed line indicates thenset of diffraction for reflected modes.

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Lozano et al. Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. B 1397

rdered following the well-known ABCABC. . . sequence,o diffraction pattern must show actually six spots with3 symmetry as it is described in [28,29]. In the case oformal incidence, the specularly reflected beam passeshrough the hole perforated in the screen to allow the la-er illuminating the photonic crystal. The set ofeciprocal-lattice vectors b�1 and b�2, shown in Fig. 1(d), issed to label this first set of diffracted beams as indicated

n Fig. 3(a). The spots associated with these modes rotateith the sample. An arrow pointing at the (1,0) diffraction

pot is plotted in each picture to easily follow the changesntroduced by varying the azimuthal incident angle. Inhe clockwise direction, the other diffracted modes aredentified as (1,�1), (0,�1), (�1,0), (�1,1), and (0,1).

Since the diffracted wavevector K� g�± depends on the com-

onent of the incident wavevector parallel to the surfacef the slab, the angular distribution of diffracted beamsepends strongly on the tilt and rotation of the sampleith respect to the incident beam. Also, just by looking at

he two series of pictures shown in Fig. 3, it can be seenhat the spot intensity R�p,q� fluctuates as the azimuthalngle is varied. This phenomenon is the subject of theext section.

. ANALYSIS OF DIFFRACTED INTENSITYet us first analyze the dependence of the intensity of thepecularly reflected beam as we vary the relative orienta-ion of the photonic lattice with respect to the incidentlectric field. This diffracted channel, which we label as0,0), forms an angle �d that is equal to the incident beamngle �i. This can be deduced by calculating K� g�

± for g� =0.hen the azimuthal angle is varied, the (0,0) mode keeps

ts angular position constant but, interestingly, not its ef-ciency R�0,0�. This is shown in Fig. 4, in which we plototh the measured (blue squares) and the calculatedgray line) intensity variations of the (0,0) mode as wehange �i. Fair agreement between theory and experi-ent is found. The typical intensity variations as the azi-uthal angle is modified are just a few percents in this

ase. Such intensity oscillations will be considerably moreronounced for the rest of diffraction modes, as we

ig. 3. (Color online) (a) Diffraction patterns of reflected beamsrojected on a screen parallel to the xy plane, when light illumi-ates the slab at (a) �i=0° and (b) �i=5°. Photographs 1–4 corre-pond to the azimuthal angles �i=0°, 15°, 30°, and 60°, respec-ively. Each spot is labeled according to its associated reciprocal-attice vector. In all cases the arrow points at the (1,0) mode. Theavelength of the blue laser is 473 nm.

resent in this section. Figure 4(b) shows some of the im-ges taken at different azimuthal angles used to obtainhe data plotted in Fig. 4(a).

Although there is no angular or energy restriction forhe propagation of the diffracted wave (i.e., the diffractedhannel is open), the reflectance strongly varies and canven be zero (i.e., the mode is switched off). This cannot bexplained using momentum transfer equations, buthould be attributed to electromagnetic resonances occur-ing within the ordered array [30]. We select the dif-racted channel (1,�1) to illustrate the dependence of theiffracted intensity with zenithal and azimuthal angleariations. In these experiments, the azimuthal angle isaried for two different zenithal angle illuminations,amely, �i=30° and 50°. In Fig. 5(a), we show the resultsf the measurements (blue squares) and calculationsgray line) of, respectively, the intensity and the reflectionfficiency of the diffracted channel (1,�1), when the blueaser impinges at �i=30° and the photonic crystal slab isotated from �i=−42° to 103°. Experimental data are ob-ained from the image processing, whereas the theoreticalnes are obtained using the parameters extracted fromhe fitting shown in Fig. 2(b). Figure 5(b) shows in detailhree of the images acquired for this analysis performedt �i=30°. Only the (1,�1) mode is colored proportionallyo its intensity for the sake of clarity. Images 1, 2, and 3orrespond to angular positions for which the intensity ofhis diffracted channel reaches a minimum, a maximum,nd a minimum again, respectively. This behavior waslso confirmed for other incident zenithal angles. In Fig.(a), we show the measured intensity (blue squares) and

ig. 4. (Color online) (a) Measured (blue squares) and calcu-ated (gray line) reflection efficiencies of the diffracted channel0,0) when the photonic crystal is illuminated at �i=10° and thezimuthal angle ��i� is varied. (b) Diffraction patterns of re-ected beams. Photographs 1–4 correspond to the azimuthalngles �i=0°, 15°, 30°, and 60°, respectively. Each spot is labeledccording to its associated reciprocal-lattice vector. The arrow islways pointing at the (1,0) mode. The wavelength of the blue la-er is 473 nm. Dashed vertical lines in (a) indicate the azimuthalngles at which the photographs in (b) were taken. The connect-ng solid blue line in (a) is only a guide for the eye.

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1398 J. Opt. Soc. Am. B/Vol. 27, No. 7 /July 2010 Lozano et al.

he theoretical reflection efficiency (gray line) of the sameeflected diffracted channel (1,�1), when the blue laser il-uminates the slab at �i=50°. In this case, the colloidalrystal is rotated from �i=−60° to 120°. Experimentalata are obtained from the image processing, whereas thealculations are done using the parameters extractedrom the fitting parameters obtained in Section 3. Again,ood agreement between theory and experiment is foundor the number of peaks, their positions, and relative in-ensities. Two minor peaks at azimuthal angles of 15° and5° are obtained in the calculated efficiency curve, whichannot be observed in the measurements. As we men-ioned in Section 2, the noise background produced by theiffuse light was reduced by using a threshold criterionnd the efficiencies were calculated by integrating theeasured beam intensity in the corresponding spot area.nfortunately, this procedure for post-processing theeasured data smoothed out the minor peaks observed in

he calculated spectrum in Fig. 6(a). Figure 6(b) showshree of the acquired images for this zenithal angle takent different azimuthal angles. It is possible to clearlydentify two maxima, near �i�−30° and �90° [see Fig.(b), panels 1 and 3], and observe that this channel is al-ost completely switched off at �i�30° [see Fig. 6(b),

anel 2].

ig. 5. (Color online) (a) Measured intensity (blue squares) andalculated reflection efficiency (gray line) of the diffracted chan-el (1,�1) when the photonic crystal is illuminated at �i=30° andhe azimuthal angle ��i� is varied. (b) Images of the emerged dif-racted beams at �i=−17° (1), 8° (2), and 78° (3); the (1,�1) modes artificially colored to highlight its intensity. All the spots areabeled according to their associated reciprocal-lattice vectors.ntensity color scale is also indicated. Dashed vertical lines in (a)ndicate the azimuthal angles at which the images in (b) were ob-ained. The connecting solid blue line in (a) is only a guide for theye.

By comparing Fig. 5(a) ��i=30°� and Fig. 6(a) ��i=50°�,e can see that the number of intensity peaks of channel

1,�1) increases as the zenithal angle �i increases. Weound that this phenomenon occurs regardless of the par-icular diffraction channel we are observing. In the case oformal incidence, it has been shown that peaks in the to-al reflectance as a function of the photon energy are dueo the excitation of resonant modes inside the photonicrystal slab [30]. In this case, we can observe the appear-nce of resonances by analyzing the diffracted beam in-ensities as a function of the zenithal incidence angle.herefore, for a constant photon energy above the diffrac-

ion cutoff, we can vary the zenithal and azimuthal anglesf the incident beam in order to extract information abouthotonic resonant modes.

. CONCLUSIONSn summary, we have observed and measured the angularependence of the reflection efficiency of modes diffractedy a colloidal crystal. Diffracted intensities have beeneasured as a function of both the incident zenithal and

zimuthal angles. We have theoretically found the angu-ar conditions at which diffracted modes are able to

ig. 6. (Color online) (a) Measured intensity (blue squares) andalculated reflection efficiency (gray line) of the diffracted chan-el (1,�1) when the photonic crystal is illuminated at �i=50° andhe azimuthal angle ��i� is varied. (b) Images of the emerged dif-racted beams at �i=−32° (1), 33° (2), and 93° (3); the (1,�1)ode is artificially colored to highlight its intensity. All the spots

re labeled according to their associated reciprocal-lattice vec-ors. Intensity color scale is also indicated. Dashed vertical linesn (a) indicate the azimuthal angles at which the images in (b)ere obtained. The connecting solid blue line in (a) is only auide for the eye.

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merge and propagate, as well as identified modes thatwitch off even though their respective diffracted chan-els are open. We have reproduced the features measuredsing the vector KKR method; good agreement betweenheory and experiments is found. We have observed an in-rement in the number of diffracted intensity peaks forarge zenithal incidence angles, which occurs since newesonances are excited inside the photonic crystal slab.herefore, we have presented an alternative method fornalyzing resonances at a constant photon energy, whichs a complementary approach to the study of reflectances a function of energy at a constant incident beam direc-ion.

CKNOWLEDGMENTShis research has been funded by the Spanish Ministry ofcience and Innovation under grant MAT2007-02166 andonsolider HOPE CSD2007-00007, Junta de Andalucíander grant FQM3579, the Consejo Nacional de Investi-aciones Científicas y Técnicas (CONICET, Argentina),nd the Agencia Nacional de Promoción Científica y Tec-ológica (PICT-11–1785). G. Lozano acknowledges theonsejo Superior de Investigaciones Cientificas (CSIC,pain) for funding through an I3P scholarship.

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