Photonic crystals have attracted much attentionduring the past decades due to the possibility of in-telligent control of electromagnetic waves and poten-tial development of new optical and optoelectronictechnologies [1–3]. A photonic crystal is a structurewith periodic modulation in its refractive index,which may present a gap in the band structure forelectromagnetic waves.
The properties of photonic crystals based on dielec-tric spheres have been broadly studied theoreticallyand experimentally in recent years [4–21]. The elec-tromagnetic behavior of arrays of dielectric spherescan be understood by analogy with the electronicband structure of crystalline solids, where the dielec-tric spheres play the role of “optical” atoms. Spheresare well-defined resonators that can be regarded asthe photonic version of individual atoms in electronic
crystals. A photon in an isolated dielectric sphereundergoes an attractive potential and it is confinedwithin the sphere, where this confinement is asso-ciated with the Mie modes of the individual spheres.The degree of optical confinement increases as re-fractive index of the sphere grows because a largerrefractive index induces a stronger attractive poten-tial for electromagnetic waves [4]. However, this con-fined photon can hop from one sphere to its neighborby the optical tunnel effect. This coherent motion ofthe photon in the periodic lattice gives rise to thephotonic band [14].
Monolayers of dielectric spheres are a much sim-pler system to study from a fundamental point ofview and do not lack practical applications, for exam-ple, filters or waveguide design [22]. Advances inmicrofabrication using several methods are rapidlyoccurring [10,23–27], although, in most cases, thelayers are made of low refractive index materials(polystyrene, polymethyl methacrylate, or SiO2) andclose-packed configurations. The use of higher refrac-tive index materials could open new possibilities in
designing more efficient photonic devices. In addi-tion, noncompact structures offer a new degree offreedom for optical property design [19]. Recentstudies have reported the synthesis of titanium di-oxide (TiO2) spheres with diameters of a few hundrednanometers with coefficients of variation in sizesranging from 5% to 20% and refractive indices ashigh as 2.9 [10,27], as well as noncompact singlelayer fabrication [28].
The perfect periodic structure of a photonic crystalcan be altered by unintentional local defects thatcan affect its electromagnetic performance. Thiskind of defect can appear in fabrication processesdue to lack of precision. Some studies of disordereffects in these systems have been published for one-dimensional [29], two dimensional [30–40], andthree-dimensional photonic crystals [41–47].
In this work, we have systematically studied theeffect of structural disorder in noncompact singlelayers of dielectric spheres of high refractive index.The obtained results help to understand the basicelectromagnetic behavior of these systems and, onthe other hand, enable us to establish practical limitson fabrication tolerances. The study was performedexperimentally in themicrowave range, with spheresthat were several millimeters in diameter and wavesranging from 10 to 30GHz. The scalability of theMaxwell equations guarantees that the obtained re-sults can be appropriately applied in the opticalrange, but the experiments can be performed in amuch more convenient way in the microwave range.However, it has to be taken into account that therefractive index of glass is very different in themicro-wave and the optical regimes. A suitable materialpresenting a refractive index for optic frequenciessimilar to the glass one for microwaves is titaniumdioxide (TiO2).
Samples were prepared starting from a regularlattice and applying to each position a random dis-placement whose magnitude was obtained from aGaussian distribution of a given variance (sigma)and an angular variation that was completely ran-dom. For variances up to 20% of the lattice param-eter, the transmission spectra remained very closeto the regular lattice one. Even for completely ran-dom distributions of the spheres, the transmissionspectra resemble that of the ordered lattice, butsignificantly smoothened and with a certain dipmissing.
2. Experimental
Transmission spectra were measured by using a net-work analyzer (HP 8722ES) spliced to rectangularhorn antennas 60mm × 40mm (Narda Model 639),aligned in the vertical direction and separated ap-proximately 300mm from each other. The incidentwave propagated in a direction perpendicular tothe plane of the photonic crystal. In all the measure-ments presented in this paper, the electric field waspolarized along the Γ–M direction of the lattice,although identical results were obtained for the
Γ–K orientation. More details on the experimentalsetup can be found in previous works [19–21].
Arrays of dielectric spheres were prepared withsoda-lime glass spheres with diameter (Φ) of 8mm.These spheres had a high dielectric permittivityε ¼ 7:0 (n ¼ 2:65) in the frequency range consideredfor the measurements, which extended from 10 to30GHz. The sphericity and monodispersion ofspheres were good; nonuniformity of diameter wasmeasured to be less than 1% of the nominal value.Samples were prepared by placing the spheres overa piece of cardboard (200mm × 200mm) on whichtheir desired position had been previously marked(see Fig. 1). The effect of the cardboard in the trans-mitted radiation was completely negligible. Regularlattices were always triangular [Fig. 1(a)] with a lat-tice parameter denoted as Λ. A simple parameter tomeasure the compactness of a particular arrange-ment is the filling factor (f f ), defined as the volumefraction occupied by spheres into a unit cell of aheight that equals the diameter of the sphere. Fromsimple geometrical considerations, it is easy to de-monstrate that, for triangular lattices, the f f valueis f f ¼ ðπ=3 ffiffiffi
3p ÞðΦ=ΛÞ2.
To study the effect of crystal disorder caused bydisruption of the lattice periodicity, disordered trian-gular lattices of dielectric spheres were designed,introducing random processes into the position of thespheres, as shown in Fig. 2. The radial aberration θwas taken from a Gaussian distribution with meanμ ¼ 0 and standard deviation σ. The angular aberra-tion θ was taken from a uniform distribution ½0; 2π�.The position of the spheres is thus defined as
x ¼ x0 þ ρ cos θ; ð1Þ
y ¼ y0 þ ρ sin θ: ð2Þ
Fig. 1. (Color online) Photography of single layers of glassspheres of f f ¼ 0:24 with different degrees of disorder included:(a) σ ¼ 4%, (b) σ ¼ 8%, (c) σ ¼ 12%, and (d) completely randomsample.
The standard deviation σ of the samples is a rela-tive dimensionless quantity and it is defined as theratio ζ=Λ to ensure the scaling of the absolute stan-dard deviation ζ when scaling the lattice. Thismagnitude (σ) measures the relative disorder of thelattice, being completely regular for σ ¼ 0. Figure 1presents a series of increasingly values of σ (4, 8,12%) and a completely random lattice, where the dis-order variation can be visually appreciated. This pro-cess of gradually increasing the σ value somehowresembles the increasing of the temperature in a so-lid, because in both cases, particles are allowed tomove farther from their equilibrium positions.
This kind of “thermally disordered” sample canonly be generated up to certain values of σ. Usingthe algorithm previously described to generate disor-dered positions, when σ increases, so does the prob-ability of getting a spheres overlapping with theprevious ones. When it happens, this position is dis-carded and another one calculated. So the number ofcalculated positions increases significantly (over thenumber of spheres to place) as σ grows. Therefore, welimited our study of this kind of sample up to a cer-tain value of σ (different for each compactness value)where the calculation process became too tedious.
Besides the thermally disordered samples, we alsoconsidered a completely random one, generated bya different calculation procedure. In this case, theappropriate number of spheres to have the desiredaverage compactness (f f ) in a determined surfacewas calculated, and their positions were randomlylocated. The only constraint was that the spheresdid not overlap; therefore, when the computer de-tected such condition, a new position was generated.By iterating the procedure, a final pattern was ob-tained with the originally desired filling factor. Inthis completely random sample there is no resem-blance of the regular lattice. As the random sampleis generated with a different algorithm that uses noGaussian distribution, it cannot be described by a va-lue of σ as we did for the thermally disordered ones.
Different arrays of spheres have been studied.These arrays are characterized by two magnitudes:
compactness, measured by f f, and disorder, indicatedby σ (except the case of completely random samples).However, as previously stated, these two factors arenot completely independent. Highly compacted lat-tices do not allow large standard deviation valuesdue to the proximity of neighboring spheres. The ex-treme case is clear: the maximum density of spheresis only possible for a completely regular lattice. Ascompactness of the lattice decreases, larger standarddeviation values can be introduced into the samples.
Different values of the lattice parameter (Λ) havebeen selected in order to study the effect of disorderfor different compactness values f f ¼ 0:3, 0.24, and0.16. These compactness values correspond to the re-gion of medium compactness of the triangular lattice[21], where the lowest frequency dips in the trans-mission spectrum are deeper and sharper, becomingmore interesting as candidates for applications suchas optical filtering. Other effects that could be re-garded as disorder (as sphere diameter or refractiveindex in homogeneities) have not been consideredhere, limiting the study only to structural disorder.
To have a better understanding of the disorder in-troduced by the above-mentioned procedures, wehave calculated the pair correlation function (PCF)gðrÞ [30] of each of the built samples using imagesof the fabricated structure. The PCF is a dimension-less quantity that provides a measure of local spatialordering in a two- or three-dimensional distribution,depicting the probability density of finding a particleat each position (r).
As the sample is not infinite, a slight broadening ofthe dips is found when comparing the ordered sam-ple fabricated (cyan curve in Fig. 3) with the ideal one(black straight lines). These lines are placed at dis-tances form a sphere easily deduced from the latticestructure: first neighbors at Λ, second ones at
ffiffiffi
3p
Λ,third ones at 2Λ, and so on. However, in the case ofdisordered lattices, the dips of the PCF reduce theirheight and broaden at the bottom clearly and moresignificantly as the degree of disorder increases(higher values of σ): the spheres are no longer at pre-cisely defined distances from each other. When the
Fig. 2. (Color online) Procedure of disordering of the regularlattice. Each sphere is displaced from its original position accord-ing to the randomly generated values of θ and ρ.
Fig. 3. (Color online) PCF calculated for samples of f f ¼ 0:16 andwith different degrees of disorder, measured by σ. The inset shows,magnified, the PFC for the first value of r.
allowed movement around the original position issmall (σ values 4% or 8%), the PCF dips do not merge;they are close enough to their original position. Thiscan be visually appreciated in Figs. 1(a) and 1(b). Forσ values of 12% (green curve in Fig. 3), the tails of thePCF pecks start to merge; disorder is high enough tochange a sphere from its original position to its firstneighbor’s one, although still with a low probability.
The pair correlation function of the completelyrandom sample is represented by the brown curve.As can be seen, it is completely flat, indicating that,in this case, no memory of order is present in thisarrangement.
This analysis of the resulting samples indicatesthat the procedure followed to construct the samplesis coherent and that the standard deviation of theprobability function (σ) used to compute the placeof the spheres really is a good measurement of thesamples’ disorder. It also confirms that the procedureto generalize the completely random sample pro-duces a lattice with no resemblance to the orderedlattice.
To get a spectrum representative of each sample,different measurements were taken and averaged.The microwave illuminated spot in the experimental
configuration used was approximately 100mm,significantly smaller than the sample size. There-fore, due to the disordered nature of the arrange-ments, slightly different spectra were measuredwhen focusing the spot over different parts of thesample. To have information representative of thewhole sample, different spectra were registered andaveraged. This procedure was standardized to bedone in the same way for all the samples: nine sam-ples were taken. This averaged spectrum would bethe one obtained directly in the optical regime be-cause even the smallest laser spots would illuminateareas with a number of spheres much higher thanin the microwave model used in this work.
3. Results and Discussion
Figure 4 shows the experimental transmission spec-tra of a number of samples with compactness valuesgiven by (a) f f ¼ 0:3, (b) 0.24, and (c) 0.16. The fre-quency values in the plots are presented normalizedto sphere size (Φ=λ), resulting in an adimensionalmagnitude. Each of the three plots includes differentlines representing the measurement of different dis-order situations. As previously mentioned, as the ar-rangements are more compact, less disorder can be
Fig. 4. (Color online) Transmission spectra of two-dimensional dielectric arrays with Φ ¼ 8mm for (a) Λ ¼ 11:35mm (f f ¼ 0:3),(b) Λ ¼ 13mm (f f ¼ 0:24), and (c) Λ ¼ 15:5mm (f f ¼ 0:16). Perfect sample with no disorder in the lattice parameter is represented withdifferent samples with slight and large disorder in the lattice parameter.
introduced in the sample. This is why, in the uppervalue, for f f ¼ 0:3 [Fig. 4(a)], the maximum studiedwas σ ¼ 8%, while in the cases of f f ¼ 0:12 [Fig. 4(b)]and f f ¼ 0:16 [Fig. 4(c)], it was σ ¼ 12% and σ ¼ 16%,respectively. Each of the three plots also includes theresult of the completely random sample for the samecompactness (thick red curve).
In samples with no disorder introduced, the spec-tra show complete transmission for low frequencies;the samples are completely transparent. Betweennormalized frequencies Φ=λ ¼ 0:32 and 0.64, it ispossible to identify a set of four well-defined dips thatare profound and relatively narrow. These dips corre-spond to electromagnetic resonances of the plane ofspheres, and their precise frequency values dependon the sample compactness as previously studied[19–21]. At higher frequencies, although more reso-nances are present [21], they are difficult to observewith our experimental setup and, therefore, they willnot be considered in this analysis.
For low disorder levels (up to 16%), all the dipsremain clearly observable, at the same frequency va-lues; however, their depth decreases with disorder.The depth decrease of the dips is more or less linear,and reaches 50% around σ ¼ 16%, as can be seen inthe insets of Figs. 4(b) and 4(c), where the percent ofdip depth reduction with respect to the ordered caseis represented against σ for each dip. In the case off f ¼ 0:16 [Fig. 4(c)], dips C and D are beginning tomerge and D has not been depicted. For these lowdisorder values, dip width does not increase signifi-cantly with disorder, and the regions of high trans-mission in the spectra, those between the dips,remain almost unaltered.
The thicker curves in Fig. 4 represent the trans-mission spectra of the completely random samples.The transparent zone of spectra, at low frequencies,is slightly lowered (between 1 and 2dB). It is fol-lowed by a lack of transmission band that seems tobe the result of the broadening and merging of thesame dips observed in previous cases. However,not all the dips are changed in the same way: dipsA and C remain clearly observable in the three com-pactness cases, dip D remains observable but eroded(more as compactness decreases), and dip B disap-pears completely. Previous works [6,21] suggest thatthese three observable dips (labeled A, C, and D)correspond to photonic bands associated with iso-lated sphere Mie modes TE11, TM11, and TM21respectively, while the relation of the other onewith Mie modes is not so clear, suggesting a highlevel of dependence of this dip with the structuralarrangement.
Besides detailed considerations about dips at fixedfrequencies, we can see that the spectra of completelydisordered arrangements present a band of signifi-cant lack of transmission in all cases. These spectracorrespond to low-pass filters. Details of the rejectedband of these filters are controlled by the compact-ness value of the spheres arrangement. The samplewith filling factor 0.3 [Fig. 4(a)] presents a rejected
band of at least −12dB extending for frequenciesΦ=λ from 0.40 to 0.62. The same rejected bandreduces its depth to around −7dB for f f ¼ 0:24[Fig. 4(b)] and down to −5dB in the lowest compactcase studied, f f ¼ 0:16 [Fig. 4(c)]. The cutoff fre-quency of these filters can be shifted to optic valuesby reducing the sphere diameters to the micrometerregime. The presence of these bands regardless ofthe positional disorder of the spheres foresees mucheasier fabrication procedures than for noncompactordered arrangements of micrometer spheres.
Summarizing, the presented results show that, fordisordered values up to σ ¼ 16%, the transmissionspectra are slightly distorted, and even in completelyrandom arrangements, spectra shapes keep manycharacteristics of the ordered case. This behavioris opposite of the one observed in [30], where the po-sitional disorder of holes in a dielectric slab destroysthe band effects of the system, as concluded fromtheir transmission properties. These results empha-size the strongly cooperative nature of the modes ofthe slab, typically called leaky modes or guided reso-nances. Such cooperative behavior is reminiscent ofthe sensitivity of modes to lattice disorder near theedges of a photonic crystal [30]. The origin of theguided resonances or leaky modes in a monolayer ar-ray of dielectric spheres is not equivalent because wehave to consider the spheres as resonators, which isnot the case for the holes in the slab.
In the samples studied here, two characteristicsof the sphere arrangements have been considered:compactness and disorder. The frequency positionof the dips in the spectra is determined exclusivelyby compactness, while disorder, especially at low le-vels, decreases only dip depth.
High refractive spheres, as the ones used in ourexperiments, present an attractive potential forphotons of appropriate energies. These photons areeasily trapped in localized states in the spheres;states that correspond to the Mie modes of the elec-tromagnetic field in a sphere. These modes are notconfined completely inside, allowing certain overlapof the states of neighboring spheres. Therefore, thephotons trapped in a sphere can hop to anotherone with a certain probability by the tunnel effect [5].
Considering spheres of the same refractive index,there are two factors affecting the photon jump prob-ability: the mode confinement and the distanceamong the spheres. As lowest order modes, those tak-ing place at lower frequencies, are less confined thanhigher ones, it is reasonable to expect these modes tobe more affected by the presence of neighbors. Thedistance between spheres is directly related to thefilling factor of the samples. As can be seen in Fig. 4,and was previously reported [19–21], the frequencyposition of the dips in the transmission spectrumchanges with filling factor, somehow reflecting thisdependence of the layer modes on the separationamong the individual resonators. We can conclude,therefore, that the frequencies of the dips depend
on the distance among the spheres as an averagevalue.
The electromagnetic modes of planes of noncom-pact highly refractive spheres, as a whole, are formedby weakly coupled Mie modes of the individualspheres, and this coupling is governed by the averagedistance among the spheres. Only when the disper-sion around this average is very high, resonancesat different frequencies begin to be significant, ascan be seen, for example, in the loss of transparencyat low frequencies (thick red curve in Fig. 4).
It is also interesting to note the practical valueof the results here presented. The transmissionspectra presented could be shifted to optical frequen-cies by appropriately decreasing the sphere size.Values between 100nm and 1 μm would be requiredfor different applications. The small influence ofpositional disorder on the transmission propertiesreduces the precision requirements on fabricationprocedures, which allows envisaging much simplerprocesses.
4. Conclusions
We have systematically studied the effect of disorderin the electromagnetic transmission of arrangementsof dielectric spheres of different compactness. Sam-ples of different values of disorder were computergenerated and constructed. Microwave transmissionspectra of the resulting arrangements, built withglass spheres, were measured.
The main conclusions that can be drawn from themeasured data can be summarized in the following.
• Single layers of dielectric spheres presentinghigh dielectric permittivity are highly tolerant topositional disorder.
• For low disorder (up to 16%) values, themain features of the spectra (four shallow dips)are conserved, and dip depth decreases linearly withdisorder.
• The collective modes of the sphere planes areformed by weakly coupled Mie modes of the indivi-dual spheres, and this coupling is governed by theaverage distance among the spheres.
• In completely random arrangements, the dipscorresponding to isolated sphere Mie resonancesTE11, TM11, and TM21 remain observable, but notthe B dip.
• The transmission spectra of fully disorderedarrangements present an opaque band that can beused for practical applications as optical and micro-wave filters.
• Fabrication procedures of planes of spheresof tenths of micrometers, interesting for optical ap-plications, could be eased by relaxing the precisionrequirements on the spheres position.
The authors acknowledge S. Tainta and M.Basterra for valuable comments and discussions.
References
1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062(1987).
2. S. John, “Strong localization of photons in certain disordereddielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489(1987).
3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, PhotonicCrystals: Molding the Flow of Light (Princeton UniversityPress, 1995).
4. T. Kondo, S. Yamaguti, M. Hangyo, K. Yamamoto, Y. Segawa,and K. Ohtaka, “Refractive index dependence of the transmis-sion properties for a photonic crystal array of dielectricspheres,” Phys. Rev. B 70, 235113 (2004).
5. T. Kondo, M. Hangyo, S. Yamaguchi, S. Yano, Y. Segawa,and K. Ohtaka, “Transmission characteristics of a two-dimensional photonic crystal array of dielectric spheres usingsubterahertz time domain spectroscopy,” Phys. Rev. B 66,331111 (2002).
6. Y. Kurokawa, Y. Jimba, and H. Miyazaki, “Internal electric-field distribution of a monolayer of periodically arrayed dielec-tric spheres,” Phys. Rev. B 70, 155107 (2004).
7. Y. Kurokawa, H. Miyazaki, and Y. Jimba, “Optical band struc-ture and near-field intensity of a periodically arrayed mono-layer of dielectric spheres on dielectric substrate of finitethickness,” Phys. Rev. B 69, 155117 (2004).
8. Y. Kurokawa, H. Miyazaki, and Y. Jimba, “Light scatteringfrom a monolayer of periodically arrayed dielectric sphereson dielectric substrates,” Phys. Rev. B 65, 2011021 (2002).
9. Y. Kurokawa, H. Miyazaki, H. T. Miyazaki, and Y. Jimba,“Effect of a semi-infinite substrate on the internal electric fieldintensity distribution of a monolayer of periodically arrayeddielectric spheres,” J. Phys. Soc. Jpn. 74, 924–929 (2005).
10. H. T. Miyazaki, H. Miyazaki, K. Ohtaka, and T. Sato, “Photo-nic band in two-dimensional lattices of micrometer-sizedspheres mechanically arranged under a scanning electron mi-croscope,” J. Appl. Phys. 87, 7152–7158 (2000).
11. K. Ohtaka, “Scattering theory of low-energy photon diffrac-tion,” J. Phys. C 13, 667–680 (1980).
12. K. Ohtaka, “Energy band of photons and low-energy photondiffraction,” Phys. Rev. B 19, 5057–5067 (1979).
13. K. Ohtaka and M. Inoue, “Light scattering from macroscopicspherical bodies. I. Integrated density of states of transverseelectromagnetic fields,” Phys. Rev. B 25, 677–688 (1982).
14. K. Ohtaka, Y. Suda, S. Nagano, T. Ueta, A. Imada, T. Koda,J. S. Bae, K. Mizuno, S. Yano, and Y. Segawa, “Photonicband effects in a two-dimensional array of dielectric spheresin the millimeter-wave region,” Phys. Rev. B 61, 5267–5279(2000).
15. K. Ohtaka and Y. Tanabe, “Photonic band using vector sphe-rical waves. I. Various properties of Bloch electric fields andheavy photons,” J. Phys. Soc. Jpn. 65, 2265–2275 (1996).
16. K. Ohtaka and Y. Tanabe, “Photonic bands using vector sphe-rical waves. II. Reflectivity, coherence and local field,” J. Phys.Soc. Jpn. 65, 2276–2284 (1996).
17. K. Ohtaka and Y. Tanabe, “Photonic bands using vectorspherical waves. III. Group-theoretical treatment,” J. Phys.Soc. Jpn. 65, 2670–2684 (1996).
18. R. Sainidou, N. Stefanou, I. E. Psarobas, and A. Modinos,“Scattering of elastic waves by a periodic monolayer ofspheres,” Phys. Rev. B 66, 024303 (2002).
19. A. Andueza, R. Echeverría, and J. Sevilla, “Evolution of theelectromagnetic modes of a single layer of dielectric sphereswith compactness,” J. Appl. Phys. 104, 043103 (2008).
20. A. Andueza and J. Sevilla, “Non compact single-layers ofdielectric spheres electromagnetic behaviour,” Opt. QuantumElectron. 39, 311–320 (2007).
21. A. Andueza, R. Echeverría, P. Morales, and J. Sevilla, “Geome-try influence on the transmission spectra of dielectric singlelayers of spheres with different compactness,” J. Appl. Phys.107, 124902 (2010).
22. K. Vynck, K. Cassagne, and E. Centeno, “Superlattice forphotonic band gap opening in monolayers of dielectricspheres,” Opt. Express 14, 6668–6674 (2006).
23. F. Jonsson, C. M. S. Torres, J. Seekamp, M. Schniedergers,A. Tiedemann, J. Ye, and R. Zentel, “Artificially inscribeddefects in opal photonic crystals,” Microelectron. Eng. 78–79,429–435 (2005).
24. W. Cai and R. Piestun, “Patterning of silica microspheremonolayers with focused femtosecond laser pulses,” Appl.Phys. Lett. 88, 111112 (2006).
25. J. Sun, Y. Li, H. Dong, P. Zhan, C. Tang, M. Zhu, andZ. Wang, “Fabrication and light-transmission properties ofmonolayer square symmetric colloidal crystals via controlledconvective self-assembly on 1D grooves,” Adv. Mater. 20,123–128 (2008).
26. S. Matsushita and M. Shimomura, “Light-propagationpatterns in freestanding two-dimensional colloidal crystals,”Colloids Surf. A 284–285, 315–319 (2006).
27. E. Mine, M. Hirose, D. Nagao, Y. Kobayashi, and M. Konno,“Synthesis of submicrometer-sized titania spherical particleswith a sol-gel method and their application to colloidal photo-nic crystals,” J. Colloid Interface Sci. 291, 162–168 (2005).
28. Z. Y. Koide, K. Fujisawa, and M. Nakane, “Preparation ofnon-contact ordered array of polystyrene colloidal particlesby using a metallic thin film of fused hemispheres,” ColloidsSurf. A 330, 108–111 (2008).
29. I. V. Ponomarev, M. Schwab, G. Dasbach, M. Bayer,T. L. Reinecke, J. P. Reithmaier, and A. Forchel, “Influenceof geometric disorder on the band structure of a photoniccrystal: Experiment and theory,” Phys. Rev. B 75, 205434(2007).
30. T. Prasad, V. L. Colvin, and D. M. Mittleman, “The effect ofstructural disorder on guided resonances in photonic crystalslabs studied with terahertz time-domain spectroscopy,” Opt.Express 15, 16954–16965 (2007).
31. M. Skorobogatiy, G. Bégin, and A. Talneau, “Statistical analy-sis of geometrical imperfections from the images of 2D photo-nic crystals,” Opt. Express 13, 2487–2502 (2005).
32. H. Y. Ryu, J. K. Hwang, and Y. H. Lee, “Effect of size nonuni-formities on the band gap of two-dimensional photonic crys-tals,” Phys. Rev. B 59, 5463–5469 (1999).
33. A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran,N. A. Nicorovici, and C. M. de Sterke, “Effects of disorder onwave propagation in two-dimensional photonic crystals,”Phys. Rev. E 60, 6118–6127 (1999).
34. M. A. Kaliteevski, J. M. Martinez, D. Cassagne, and J. P.Albert, “Disorder-induced modification of the attenuation oflight in a two-dimensional photonic crystal with completeband gap,” Phys. Status Solidi A 195, 612–617 (2003).
35. L. L. Lima, M. A. R. C. Alencar, D. P. Caetano, D. R. Solli, andJ. M. Hickmann, “The effect of disorder on two-dimensionalphotonic crystal waveguides,” J. Appl. Phys. 103, 123102(2008).
36. T. N. Langtry, A. A. Asatryan, L. C. Botten, C. M. de Sterke,R. C. McPhedran, and P. A. Robinson, “Effects of disorder intwo-dimensional photonic crystal waveguides,” Phys. Rev. E68, 026611 (2003).
37. R. Meisels and F. Kuchar, “Density-of-states and wave propa-gation in two-dimensional photonic crystals with positionaldisorder,” J. Opt. A 9, S396–S402 (2007).
38. B. Wang, Y. Jin, and S. He, “Effects of disorder in a photoniccrystal on the extraction efficiency of a light-emitting diode,”J. Appl. Phys. 106, 014508 (2009).
39. W. R. Frei and H. T. Johnson, “Finite-element analysis of dis-order effects in photonic crystals,” Phys. Rev. B 70, 165116(2004).
40. D. M. Beggs, M. A. Kaliteevski, S. Brand, R. A. Abram, D.Cassagne, and J. P. Albert, “Disorder induced modificationof reflection and transmission spectra of a two-dimensionalphotonic crystal with an incomplete band-gap,” J. Phys. Con-dens. Matter 17, 4049–4055 (2005).
41. L. A. Dorado and R. A. Depine, “Modeling of disorder effectsand optical extinction in three-dimensional photonic crystals,”Phys. Rev. B 79, 045124 (2009).
42. B. Auguié and W. L. Barnes, “Diffractive coupling in goldnanoparticle arrays and the effect of disorder,” Opt. Lett.34, 401–403 (2009).
43. S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos,“Theoretical investigation of fabrication-related disorder onthe properties of photonic crystals,” J. Appl. Phys. 78,1415–1418 (1995).
44. R. Rengarajan, D. Mittleman, C. Rich, and V. Colvin, “Effect ofdisorder on the optical properties of colloidal crystals,” Phys.Rev. E 71, 016615 (2005).
45. M. M. Sigalas, C. M. Soukoulis, C. T. Chan, R. Biswas, andK. M. Ho, “Effect of disorder on photonic band gaps,” Phys.Rev. B 59, 12767–12770 (1999).
46. A. V. Baryshev, V. A. Kosobukin, K. B. Samusev, D. E. Usvyat,and M. F. Limonov, “Light diffraction from opal-based photo-nic crystals with growth-induced disorder: experiment andtheory,” Phys. Rev. B 73, 205118 (2006).
47. C. Rockstuhl and F. Lederer, “Suppression of the local densityof states in a medium made of randomly arranged dielectricspheres,” Phys. Rev. B 79, 132202 (2009).
