PHYSICAL REVIEW E 71, 046605 s2005d Photonic band gap templating using optical interference lithography Timothy Y. M. Chan, Ovidiu Toader, and Sajeev John Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, M5S 1A7, Canada sReceived 23 June 2004; published 7 April 2005d We describe the properties of three families of inversion-symmetric, large photonic band-gap sPBGd template architectures defined by iso-intensity surfaces in four beam laser interference patterns. These templates can be fabricated by optical interference sholographicd lithography in a suitable polymer photo-resist. PBG materials can be synthesized from these templates using two stages of infiltration and inversion, first with silica and second with silicon. By considering point and space group symmetries to produce laser interference patterns with the smallest possible irreducible Brillouin zones, we obtain laser beam intensities, directions, and polarizations which generate a diamond-like sfccd crystal, a novel body-centered cubic sbccd architecture, and a simple-cubic sscd structure. We obtain laser beam parameters that maximize the intensity contrasts of the interference patterns. This optimizes the robustness of the holographic lithography to inhomogeneity in the polymer photo-resist. When the optimized iso-intensity surface defines a silicon to air boundary sdielectric contrast of 11.9 to 1d, the fcc, bcc, and sc crystals have PBG to center frequency ratios of 25%, 21%, and 11%, respectively. A full PBG forms for the diamond-like crystal when the refractive index contrast exceeds 1.97 to 1. We illustrate a noninversion symmetric PBG architecture that interpolates between a simple fcc structure and a diamond network structure. This crystal exhibits two distinct and complete photonic band gaps. We also describe a generalized class of tetragonal photonic crystals that interpolate between and extrapolate beyond the diamond-like crystal and the optimized bcc crystal. We demonstrate the extent to which the resulting PBG materials are robust against perturbations to the laser beam amplitudes and polarizations, and template inhomogeneity. The body centered cubic structure exhibits the maximum robustness overall. DOI: 10.1103/PhysRevE.71.046605 PACS numberssd: 42.70.Qs I. INTRODUCTION Photonic band-gap sPBGd materials f1,2g are periodically ordered dielectric microstructures that facilitate the localization of light f3g. This is a new frontier in quantum electrodynamics f4g, and provides a robust platform for integrating active and passive devices in an all-optical micro-chip f5g. For example, by introducing line defects into the periodic dielectric structure of the PBG material, it is possible to guide light through micron-scale, single-mode air-waveguide channels with sharp, low-loss bends f6–8g. PBG materials can also facilitate frequency selective control over spontaneous emission of light from individual atoms in the circuit, through control of the local electromagnetic density of states. As a result, active devices such as zero threshold lasers f9g and, possibly, all-optical transistors f10g may be included on-chip. In order to realize the goal of a photonic band gap material-based optical microchip, it is necessary to have high quality, three-dimensional s3Dd PBG materials. The design and efficient, low-cost micro-fabrication of such materials have been a major scientific challenge over the past decade. Novel experimental methods for templating, inversion, and replication f11,12g have made this paramount goal a near term reality. The optical properties of PBG materials scale with their lattice constants. PBG materials destined for use in optical telecommunications circuits must have submicron lattice constants. At the same time, the practical benefits of a PBG material-based optical circuit necessitate low-cost manufacturing of such materials with long range order sLROd on the scale of hundreds of lattice constants. Optical interference lithography f13–15g provides a unique opportunity to satisfy both requirements while maintaining a high 1539-3755/2005/71s4d/046605s18d/$23.00 degree of flexibility in the detailed PBG architecture. Early considerations of the photonic band structure focused on the face centered cubic sfccd lattice because the first Brillouin zone sBZd of this lattice exhibits the least anisotropy. In other words, the difference between the shortest and longest distances from the center to the surface of the first BZ is the smallest. This, it was reasoned f1,2g, increased the likelihood that one-dimensional stop-gaps in all directions would overlap and produce a 3D PBG. It was shown f16g, however, that a spherical scatterer within the fcc unit sWigner-Seitzd cell prevents the creation of a PBG between the lowest possible bands due to the polarization degree of freedom of the electromagnetic field. The diamond structure, which can be viewed as an fcc lattice with a two “atom” basis, breaks this spherical symmetry while retaining the fcc BZ. Indeed, a diamond lattice of overlapping air spheres was theoretically shown in f16g to possess a sizeable 3D PBG, approximately 27% of the center frequency when the air spheres are in a background material with a dielectric constant of 11.9, corresponding to Si. Although the diamond lattice has proven difficult to fabricate on the micron scale, the possibility of a large PBG has led to several theoretical blueprints and subsequent fabrication attempts for photonic crystals based on “diamond-like” structures which employ nonspherical bases on an fcc lattice. This began with the fabrication f17g of an fcc lattice of crisscrossing pores exhibiting a 3D PBG in the microwave spectrum. The straightforward but tedious fabrication on the centimeter scale has spurred many attempts to replicate this structure on the submicron scale as required for a PBG in the optical regime. However, this has proven to be very challenging. Methods involving electron beam lithography fol- 046605-1 ©2005 The American Physical Society PHYSICAL REVIEW E 71, 046605 s2005d TIMOTHY Y. M. CHAN, OVIDIU TOADER, AND SAJEEV JOHN lowed by reactive ion etching f18,19g were successful in producing submicron pores, but the resulting slabs were only a few periods thick and severe imperfections occurred at the pore crossing points. Experiments with deep x-ray lithography sLIGAd patterning of an x-ray sensitive resist f20g have produced several periods of the criss-crossing pores, even allowing for the incorporation of controlled defects by multiple exposures f21g. However, structures patterned in this fashion, to date, have lattice constants which are too large for a PBG in the optical regime. A similar outcome has been reported using a hybrid scheme involving photoelectrochemical etching followed by focused ion beam etching in macroporous silicon. This method was used to synthesize a five-periods thick slab with a photonic band gap centered around 3mm f22g. Photo-electrochemical etching methods alone f23g have also been used to produce an elongated version of the fcc lattice of criss-crossing pores f24g with high aspect ratios, but as a result of the elongation, the photonic band gap is reduced in size f25g. Recently, new “slanted pore” architectures have been introduced f26g whose simpler geometries may facilitate their fabrication by various pore etching methods. Another diamond-like structure is the layer-by-layer “woodpile” architecture consisting of stacked twodimensional photonic crystals f27,28g. There have been several approaches to address the fabrication of woodpile architectures, but no large scale swith more than three unit cells normal to the substrated woodpile photonic crystals operating at optical frequencies have been mass-produced as yet. The use of established silicon micromachining techniques such as repetitive deposition and etching have produced high-quality structures with relevant feature sizes, but the structures are only about one period sbetween two and five layersd thick f29,30g. Similarly, experiments using wafer-fusion and laserassisted alignment f31g produced only two to three periods of the woodpile in the third dimension. An inexpensive but tedious method involving nanofabrication of two-dimensional layers followed by microassembly of the separate layers f32g provides a simple way to introduce defects into the woodpile structure because each layer can be fabricated uniquely. Unfortunately only up to four layers have been assembled so far. On the other hand, woodpiles synthesized by laserinduced chemical vapor deposition f33g are limited only in the number of layers by the size of the deposition chamber, but due to the laser spot size, the resulting PBG is centered around 75 mm. Recently, “direct laser writing” processes involving two-photon absorption scausing polymerizationd in resins have been used to produce woodpile structures as a proof of concept f34,35g. An alternative PBG design, suitable for large-scale microfabrication, is an architecture involving glancing angle deposition methods f36g leading to the formation of silicon square spiral posts onto a silicon substrate. For suitably architectured spiral posts, a PBG as large as 24% of the gap center frequency has been predicted f37,38g. Optical reflectivity in a weakly disordered version of these silicon square spiral crystals has revealed a 3D PBG of roughly 10% relative to the center frequency f39g. While diamond and diamond-like structures may exhibit large PBGs, difficulties in fabrication have led to the consid- eration of other blueprints which sacrifice theoretical gap size in favor of feasibility of large-scale synthesis. One such blueprint is the fcc lattice of close-packed spheres sopal latticed, which does not have a PBG between the lowest possible bands, but does have a small PBG between the higher bands f40g. Moreover, the concept of fabricating opal structures by colloidal self-assembly has a long established history f41,42g. However, the materials suitable for the selfassembly process do not have large dielectric constants and low absorption at optical frequencies. Instead, selfassembled silica spheres are used as a template which is inverted by chemical vapor deposition of silicon, followed by selective etching of the template f11,43,44g. This process can now be performed at large scales. It has been shown f45g that by only partially infiltrating the template with silicon, a structure with a PBG of nearly 9% of the gap center frequency can be obtained. However, the small PBG of the inverted opal structure is fragile and vulnerable to disorder f46g, requiring that fabrication methods must yield very high quality structures. Many previous attempts to synthesize PBG materials have struggled to adapt various fabrication methods to emulate theoretical blueprints ill suited to fabrication at submicron length scales. Other attempts have focused on theoretically adjusting the fabrication parameters of a well-established technique in search of a PBG. The optical interference lithography method, described in this paper, offers a unique opportunity for efficient large-scale micro-fabrication of 3D PBG structures with large gaps in the optical regime. By employing the 3D interference pattern of four or more laser beams to expose a photopolymerizable material such as a photo-resist or polymerizable resin f13–15,47–52g, nearly perfect LRO can be maintained over length scales much larger than the lattice constant of the photonic crystal sPCd. At the same time, this technique allows straightforward control over the periodic properties sFourier coefficientsd of the resulting structure through the laser beam parameters. The photo-resist material undergoes a chemical alteration when the total light intensity due to the interference pattern, IsrWd, at position Wr, is maintained over a time interval dt such that the “exposure” IsrWddt exceeds a specified threshold, T. For a negative photo-resist, the “underexposed” region can be selectively removed using a developer substance which leaves the “overexposed” regions intact. sFor a positive photoresist, overexposed regions are removed, leaving the underexposed regions intact.d The developed photo-resist can then be infiltrated at room temperature with SiO2 f12g and burned away, leaving behind a daughter “inverse” template. Finally, the daughter template is inverted by high temperature infiltration with silicon f11,53g and selective chemical etching of the SiO2. As a result, a 3D silicon PC is formed, in which the silicon-air boundary is defined by the original, optical, isointensity surface IsrWddt = T. Alternatively, the photo-resist template can be infiltrated with a high refractive index material such as CdSe f54g, which results in an inverted PC after the photo-resist is burned or chemically etched away. A third option is to use a photopolymerizable film which contains titanium f55g, producing a direct PC after exposure to an intensity pattern and removal of underexposed material. In this paper, we discuss the formation of 3D PCs using holographic lithography based on the interference of four 046605-2 PHYSICAL REVIEW E 71, 046605 s2005d PHOTONIC BAND GAP TEMPLATING USING OPTICAL… laser beams. Other proposed configurations include the use of multiple exposures and phase shifts between exposures f56,57g and a hybrid five beam configuration with three beams forming a two-dimensional pattern and two other beams forming a one-dimensional pattern f15g in the out of plane direction. However, these configurations require careful manipulation of the relative phases between beams and between exposures. In contrast, in a four beam, single exposure configuration, the relative phases between laser beams does not change the shape of the the interference pattern. In previous theoretical reports, four beam holographic lithography configurations were used f56,58,59g to create structures which emulate particular cases of triply periodic minimal surfaces f60g, which have been shown f61,62g to produce large 3D PBGs. The resulting PCs included a diamond-like structure, a “gyroid” structure with body-centered cubic sbccd Bravais lattice symmetry, and a simple cubic sscd structure. Using simpler symmetry arguments, holographic beam configurations for the diamond-like structure and a novel bcc structure have been derived f63g. Here, we describe our approach in detail and find alternative beam configurations for producing the diamond-like, bcc, and sc structures in which the intensity contrast is maximized. We also explore the robustness of the resulting PBGs against perturbations in the beam polarizations and amplitudes. Both the diamond and bcc structures are robust against deviations in the experimental control parameters. In Sec. II we introduce our framework for describing structures created by holographic lithography. Section III presents our design process for achieving 3D photonic crystals using this method, using the three cubic Bravais lattices. Section IV describes a generalized class of tetragonal PBG materials that encompass both the diamond network and bcc structures. Section V describes the properties of our holographic photonic crystals relevant to robust micro-fabrication and error-tolerant PBG formation. II. STRUCTURE DESCRIPTION The interference of N monochromatic plane waves of freW , polarization vectors eW , phases u , quency v, wave vectors G i i i and real amplitudes Ei creates an electric field given by S N−1 W EW srW,td = e−ivteisG0·rW+u0d E0eW 0 + EieW ieisK Wr+g d o i=1 W i i D , where W ;G W −G W K i i 0 s1d and gi ; ui − u0. The corresponding, stationary intensity pattern is given by N−1 W *srW,td · EW srW,td = IsrWd ; E o i=0 N−1 E2i + 2 W · Wr E0EiueW *0 · eW iucossK o i i=1 N−1 o i.j=1 W · Wr + g − g + f d, EiEjueW *i · eW jucossK ij i j ij expt expt SsrW,Ithr d = Q„IsrWd − Ithr …, expt is a threshold value and Q = 1 for x ù 0 and zero where Ithr otherwise sHeaviside step functiond. By convention, we assume the high intensity regions in Eq. s2d become the silicon component of the photonic crystal f«srWd = 1 where IsrWd expt expt , Ithr g. This corresponds to and «srWd = 11.9 where IsrWd ù Ithr the case in which a negative photo-resist is used to make the photonic crystal template. It will be shown in Sec. V that for the optimized structures considered here, equivalent structures can be obtained with a positive photo-resist by simply expt . adjusting the threshold Ithr In the case N = 4 swhich provides the minimum number of non-collinear beams required to produce a nontrivial 3D intensity patternd, Eq. s2d becomes IsrWd = I0 + 2DIsrWd, where I0 ; E20 + E21 + E22 + E23and W · Wr + g + f d + c cossK W · Wr + g + f d DIsrWd ; c1 cossK 1 1 1 2 2 2 2 W · Wr + g + f d + c cossK W · Wr + g − g + c3 cossK 3 3 3 12 12 1 2 W · Wr + g − g + f d + f12d + c13 cossK 13 1 3 13 W · Wr + g − g + f d, + c23 cossK 23 2 3 23 s2d s3d with ci = E0EiueW *0 · eW iu;cij = EiEjueW *i · eW ju. s4d The spatial modulation of the intensity pattern given by Eq. s3d is periodic with a lattice whose primitive vectors, aW i, satW · aW = 2pnd , where n is an integer. As a consequence, isfy K i j ij the lattice constants are inversely proportional to the frequency of the laser beams. The phase factors gi in Eq. s3d, which result from the relative phases of the laser beams, can be simultaneously W 1 + g2hW 2 eliminated by a translation of the origin by rW = g1h W W 3, where hW i satisfy Ki · hW j = dij. For configurations with + g 3h more than four beams, the relative beam phases impart more than a simple translation in the intensity pattern, and cannot be easily ignored. On the other hand, the phase factors fi and fij, arising from the dot products between beam polarization vectors, cannot in general be simultaneously eliminated. In the case of linearly polarized beams, fi and fij are zero for all i and j and the intensity pattern is always inversion symmetric relative to position rW . However, in the case of elliptically polarized beams it is possible to obtain an interference pattern that is not inversion symmetric. By rewriting the “experimental” intensity threshold as expt = I0 + 2Ithr we arrive at the following simplified shape Ithr function: expt Q„IsrWd − Ithr … = Q„DIsrWd − Ithr… + g i + f id +2 W ;K W −K W , f ; argseW * · eW d, and f ; argseW * · eW d. The where K ij i j i ij j 0 i i holographic structure is a two phase medium whose phase boundary is defined by the “shape” function s5d In what follows, we focus our discussion of the resulting shapes on the simplified intensity pattern in Eq. s3d. fThe intensity background I0 is nevertheless important, as a large background intensity will overwhelm the variation DIsrWd.g 046605-3 PHYSICAL REVIEW E 71, 046605 s2005d TIMOTHY Y. M. CHAN, OVIDIU TOADER, AND SAJEEV JOHN III. DESIGN OF HOLOGRAPHIC STRUCTURES WHICH EXHIBIT PHOTONIC BAND GAPS We consider the design of holographic structures with inversion symmetry. A sufficient condition for this requirement to be fulfilled is that linearly polarized beams are used, or sin the case that elliptically polarized beams are usedd that the phases fi and fij vanish for all i and j. In this case, the design problem consists of identifying the set C = hci,cijji,j=1,3 s6d of six polarization and amplitude coefficients ssome possibly W vectors, and suitable threshold I , that generates 0d, three K i thr a photonic crystal fthrough the shape function given by Eq. s5dg with a complete PBG. Given the parameters of the inW , the laser beam directions, ampliterference pattern C and K i tudes, and polarizations are determined by Wj 1. Finding the four beam directions, hG i i=0 , 3, which satisfy Eq. s1d. W from stage 1 to find the four amplitudes and 2. Using G i polarization vectors, hEi , eW iji=0 , 3, which satisfy the nonlinear set of six equations given by Eq. s4d. W are obtained The identification of C and the choice of K i by two guiding principles: sid choose a given Bravais lattice, and siid identify architectures leading to the smallest possible irreducible Brillouin zones sIBZd, then incrementally relax this condition if the resulting structure is too symmetric for a fundamental gap to open. In order to satisfy criterion sid, it is W vectors to be necessary and sufficient to choose the three K i linear combinations with integral coefficients of vectors from the set hbW ij, where we denote by haW i , bW iji=1 , 3 a set of primitive and reciprocal vectors, respectively, of the desired Bravais lattice. On the other hand, criterion siid is far more W j combirestrictive. Indeed, only a very limited set of hC , K i nations generate intensity patterns with sufficiently high symmetry. Suppose that the desired holographic structure has a cubic symmetry ssc, bcc, or fccd. Figure 1 shows several reciprocal lattice vectors of a simple cubic lattice, grouped by length. In order of increasing length, the first three sets of lattice vectors in the cubic reciprocal space are • Bs = h±bW si j of length 2p / ap , reaching the centers of the cube faces sred onlined, • Bb = h±bW bi j ø hbW bi − bW bj jiÞj of length 2pÎ2 / ap , reaching the centers of the cube edges, sgreen onlined, and • Bf = h±bW fi j ø h±sbW f1 + bW f2 + bW f3dj of length 2pÎ3 / ap , reaching the cube corners. sblue onlined i and j run from 1 to 3 and bW si , bW bi , and bW fi denote the three primitive vectors of the sc, bcc, and fcc reciprocal lattices, respectively. W vectors from one of these sets generates a Choosing K holographic structure with a specific cubic symmetry. The first order reciprocal lattice vectors generate an sc spatial lattice, the second order vectors a bcc lattice, and the third order vectors a fcc lattice. The fourth set of vectors snot shown in Fig. 1d again generates an sc lattice, with a lattice FIG. 1. sColor onlined The reciprocal space of a simple cubic lattice. The first sredd, second sgreend, and third sblued order neighbors are located at the centers of the cube faces, the centers of the cube edges, and the cube corners, respectively. W b1, W b2, and W b3 denote the primitive simple cubic reciprocal lattice vectors. constant of uau / 2 in this case. Similarly, higher order sets generate intensity patterns that can be described as one of the fundamental lattices ssc, fcc, or bccd with increasingly small lattice constants. We therefore restrict our choice for the set of target vectors to the first three, illustrated sets. We define the set T of “target vectors” as a subset T , K W ,K W ,K W ,K W ,K W ,K W j corresponding to nonzero terms ; hK 1 2 3 12 13 23 in Eq. s3d sor, equivalently, nonzero coefficients in Cd. Accordingly, we define CT , C as the subset of nonzero coefficients of C. The requirement of high symmetry can be achieved by choosing T simply to be a subset of one of the vector sets B*. Since DIsrWd is insensitive to the sign of the target vectors in T, an upper bound on the size of the sets T and CT is 3 in the case of Bs, 6 in the case of Bb, and 4 in the case of Bf. A. Face-centered cubic structure We first illustrate the choice of C coefficients using the fcc Bravais lattice. Since there are only four distinct directions in the Bf set sblue in Fig. 1d, the target vector set can be denoted W as T = hTW 1 , TW 2 , TW 3 , TW 4j. One possible choice for the three K i vectors is the following: W = 2p s1,1,1d, K 1 ap W = 2p s0,2,0d, K 2 ap W = 2p s− 1,1,1d. K 3 ap s7d W ,K W ,K W ,K W j and C This leads to the choice T = hK 1 3 12 23 = h* , 0 , * , * , 0 , * j, where * indicates a nonzero real coeffi- 046605-4 PHYSICAL REVIEW E 71, 046605 s2005d PHOTONIC BAND GAP TEMPLATING USING OPTICAL… cient. We denote the nonzero target coefficients as CT = ht1 , t2 , t3 , t4j so that Eq. s3d becomes 4 DIsrWd = ti cossTW i · Wrd. o i=1 s8d Noting that the photonic band structure is invariant to the space group of the crystal, we choose CT and find a translation vector, dW , such that DIsrWd is invariant under changes of d the form Wr → Sj,dW srWd ; Odj srWd + dW , where we have defined Od = hOdj j j=1 , 48 as the set of fcc point group operations. Such an intensity pattern satisfies the requirement for the smallest d possible IBZ. Operating on DIsrWd with Sj,dW yields d S W 4 j,d DIsrWd→ 4 o ti cosfTW i · Sj,ddW srWdg = o ti cosfOrj sTW id · Wr + jig, i=1 i=1 s9d where Or = hOrj j j=1 , 48 is the set of point group operations of the fcc reciprocal space, and hji = TW i · dW ji=1 , 4 is a set of four phase factors. Since the target vector set T is a subset of K, in general the ji are not independent, and are a subset of W · dW . In order hg1 , g2 , g3 , g1 − g2 , g1 − g3 , g2 − g3j, where gi = K i to simplify the problem, we consider only the case for which ji P h0 , pj, which is consistent with the condition ji P hg1 , g2 , g3 , g1 − g2 , g1 − g3 , g2 − g3j if one assumes gi P h0 , pj. For our particular choice of target vectors, it can be verified that Orj sTW id P h±TW 1 , ± TW 2 , ± TW 3 , ± TW 4j. Since the sign of TW i leaves DIsrWd unchanged, it follows that, for our restricted values of ji, the point group operations Odj simply permute the coefficients htij as they appear in Eq. s8d, in the form d S W 4 j,d 4 DIsrWd = o ti cossTW i · Wrd→ o SksidtP jsidcossTW i · Wrd, i=1 i=1 where P j is a permutation of the indices h1,2,3,4j completely determined by Od and Sk is a sign vector defined as ig1 ig2 ig3 Sk ; he ,e ,e ,e ig1 ig2 − e ,e ig1 ig3 − e ,e ig2 FIG. 2. The iso-intensity surfaces of the fcc structure generated by Eq. s11d. sad An opal-like architecture appears for h = + 1. sbd A diamond-like architecture appears for h = −1. In both cases, the light intensity is mapped to the color bar. phases are independent, such as in the case hg1 , g2 , g1 − g3j. According to s10d, we can without loss of generality always choose the first three components from Sk to be 11. The fcc intensity pattern can then be written as DIfccsrWd = cossbW f1 · Wrd + cossbW f2 · Wrd + cossbW f3 · Wrd ig3 − e j, + h cosfsbW f1 + bW f2 + bW f3d · Wrg, 3 where k = 1 , 8 corresponds to one of the 2 possible choices for the hg1 , g2 , g3j set. Invariance of DIsrWd can therefore be determined by requiring utiu = 1 for all i and using dW to compensate for any permutation-induced sign changes. The eight Sk vectors are given as columns in the following table: S1 S2 S3 S4 S5 S6 S7 S8 g1 g2 g3 g1 − g2 g1 − g3 g2 − g3 1 + − + − + − + − + + − − + + − − + + + + − − − − + − − + + − − + + − + − − + − + + + − − − − + + 2 s10d The intensity pattern contains four components of Sk for some k, with the condition that three of the corresponding s11d where h = ± 1 and hbW fi j are the primitive vectors of the fcc reciprocal space. The choice h = + 1 leads to an intensity pattern resembling an fcc lattice of spheroid “atoms”, as illustrated in Fig. 2sad. This indicates that a fundamental band gap does not open. sIt is possible for a higher-order gap to open between bands 5 and 6, characteristic of disconnected F4̄3m structures f64g, which are fcc lattices of slightly nonspherical atoms.d On the other hand, the choice h = −1 produces an intensity pattern, shown in Fig. 2sbd, exhibiting a strong resemblance to a diamond network structure and corresponding to the double diamond sDd surface f60g. This is the diamond-like structure studied in previous theoretical reports f56,58,59g. Accordingly, our calculations, presented in the next section, show that a PBG as large as 25% opens between the second and third bands in a structure whose solid component has a dielectric constant of 11.9 sSid. It is 046605-5 PHYSICAL REVIEW E 71, 046605 s2005d TIMOTHY Y. M. CHAN, OVIDIU TOADER, AND SAJEEV JOHN interesting to note that a single sign in the intensity pattern differentiates between the fcc lattice of spheroids and the diamond structure and therefore determines the existence of a fundamental photonic band gap. In fact, if one extends to noninversion symmetric structures sSec. III A 2d, taking h = 1 and adding a p / 2 phase to the corresponding cosine term, a structure which retains both a fundamental gap and a gap between bands 5 and 6 can be achieved ssee Fig. 5d. In the optical interference lithography technique, this sign can be controlled directly though the choice of laser beam amplitude and polarization pairs. 1. Diamond structure Now that we have obtained the desired intensity pattern, Eq. s11d with h = −1, it is straightforward to determine holographic beam parameters which produce this pattern. First, it W of the incident is necessary to determine the wave vectors G i laser beams, according to Eqs. s1d and s7d. The following vectors of length Î5p / ap satisfy our requirements: W = p s0,− 2,− 1d, G 0 ap W = p s2,0,1d, G 1 ap W = p s0,2,− 1d, G 2 ap W = p s− 2,0,1d. G 3 ap W + eibi sinsw dRW , eW i = cosswidU i i i with wi describing the relative amplitudes and bi the relative phase between the two linear polarization vectors. For linear polarizations, the relative phase is zero and hence a linear polarization vector can be expressed as W + sinsw dRW , eW i = cosswidU i i i hE0,E1,E2,E3j = h1.29,1.52,1.34,1.08j, hw0, w1, w2, w3j = h34.8°,66.6°,40.8°,14.6°j, It is then necessary to determine the polarization vectors which satisfy s13d hf1, f2, f3, f12, f13, f23j = h0,0,0,0,0,0j s14d and fsee Eqs. s3d and s4dg. In order to describe the polarization W , it is convenient to introduce two muvector for a given G i W , and “right”, RW , to tually perpendicular unit vectors “up”, U i i W , RW , G W , d such that complete an orthogonal triad sU i i i W =G W 3 ẑ/uG W 3 ẑu, R i i i W = RW 3 G W /uG W u, U i i i i W 3 RW = G W /uG W u, U i i i i hb0, b1, b2, b3j = h90.0°,− 90.0°,90.0°,− 90.0°j. s18d These polarizations achieve a value of 6.93 for the intensity background I0, and the total intensity reaches a maximum of 12.6. It should be noted that this particular numerical solution is not unique f56,58,59g. The beam configuration represented by this solution is shown in Fig. 3. The plane spanned W and G W sG W -G W d is perpendicular to the plane spanned by G 0 2 0 2 W W W W W and G W equals by G1 and G3 sG1-G3d. The angle between G 0 2 W and G W and has the value 126.9°. The the angle between G 1 3 polarization ellipses are shown next to the corresponding beams, depicting the path traced by the polarization vector as one looks down the wave vector. The dashed lines passing through the polarization ellipses lie in the respective planes defined above. The linear polarization parameters s15d where sx̂ , ŷ , ẑd define unit vectors in a specific laboratory W i ẑ, then by convention we choose coordinate frame. sIf G i RW i = x̂.d An elliptical polarization vector can then be expressed as the sum of two linear polarization vectors in the up and right directions: s17d where in this case wi can be interpreted as a polarization angle. Beam polarizations and intensities which produce the desired pattern are derived by finding parameters that minimize the intensity background I0, while satisfying the desired values for the c and f coefficients. If one applies an overall scale to the c coefficients, then the minimal value of I0, as well as the four beam intensities, will be scaled accordingly. The minimization is performed using a differential evolution method ssee, e.g., f65gd, which finds a global minimum for I0. In this method, one generates a randomly distributed set of N-dimensional vectors in the N-dimensional parameter space sin which the constraints on c and f are satisfiedd. Each vector is “mated” with a random combination of the other vectors to create a trial vector. If the trial vector improves the value of I0, then it replaces the original vector. The mating step is iterated until the minimum is found. By minimizing I0 while setting hc1 , c2 , c3 , c12 , c13 , c23j = h1 , 0 , 1 , 1 , 0 , −1j and hcossf1d , cossf2d , cossf3d , cossf12d , cossf13d , cossf23dj = h1 , 1 , 1 , 1 , 1 , 1j, one arrives at the following elliptical polarization parameters: s12d hc1,c2,c3,c12,c13,c23j = h1,0,1,1,0,− 1j s16d hE0,E1,E2,E3j = h2.08,0.93,1.29,2.26j, hw0, w1, w2, w3j = h− 29.3°,6.5°,− 46.9°,− 100.7°j s19d also satisfy Eqs. s13d and s14d while minimizing I0. However, as noted in f59g, the ratio Imax / I0 in this case is 17.6/ 11.97, which is smaller than the one obtained above 046605-6 PHYSICAL REVIEW E 71, 046605 s2005d PHOTONIC BAND GAP TEMPLATING USING OPTICAL… FIG. 3. The diagram of a beam configuration which generates the fcc sdiamondd intensity pattern using elliptically polarized W and G W is perpendicular to the beams. The plane spanned by G 0 2 W W plane spanned by G1 and G3. The ellipses at the end of each wave vector describe the polarization ellipse of the corresponding beam, W , with the dashed line spassing through the as one looks down G i W -G W or G W -G W plane. polarization ellipsed corresponding to the G 0 2 1 3 W W The R and U pairs for each beam, defined in Eq. s15d, are shown next to the corresponding polarization ellipse. with elliptical polarizations. This linear polarization beam configuration is shown in Fig. 4. The polarization angles are shown as one looks down the wave vector, relative to the W -G W or G W -G W plane. G 0 2 1 3 2. Noninversion symmetric structure By extending to noninversion symmetric structures it is possible to achieve a connected F4̄3m scorresponding to the space groupd structure. This structure has a two-atom basis where, unlike the diamond lattice, the two atoms are disproportionate in size. It has been shown that connected F4̄3m structures display PBGs both between the second and third bands, and between the fifth and sixth bands f64g. The corresponding intensity pattern can be written as FIG. 4. The diagram of a beam configuration which generates the fcc sdiamondd intensity pattern using linearly polarized beams. W and G W is perpendicular to the plane The plane spanned by G 0 2 W and G W . The polarization angles as one looks down spanned by G 1 3 the wave vector are shown next to the corresponding beams, with the dashed line passing through the polarization circle correspondW -G W or G W -G W plane. The RW and U W pairs for each beam, ing to the G 0 2 1 3 defined in Eq. s15d, are shown next to the corresponding polarization angle. achieved with z = 0 f64gd. The photonic band structure obtained in this case is shown in Fig. 6. The desired intensity pattern can be obtained by adjusting the polarization vectors and amplitudes of the laser beams used to generate the diamond structure above, while retainW . In this case, the laser beam ampliing the wave vectors G i tude and polarization pairs must obey DIsrWd = cossbW f1 · Wrd + cossbW f2 · Wrd + cossbW f3 · Wrd + cosfsbW f1 + bW f2 + bW f3d · Wr + zg, s20d with z = p / 2. Note that the fcc lattice of spheres and the diamond structure are achieved with z = 0 and z = p, respectively. Figure 5 shows the iso-surface of this intensity pattern which generates a structure with a solid volume fraction of <23%, where high intensity regions correspond to the solid parts of the structure. When silicon is used to synthesize the structure in an air background, a 12.5% PBG emerges between bands 2 and 3 scharacteristic of the diamond structure achieved with z = pd, and a nearly 5% PBG emerges between bands 5 and 6 scharacteristic of the FCC lattice of spheroids FIG. 5. The iso-intensity surface generated by Eq. s20d, with z = p / 2 at a solid volume fraction of <23%, with the field intensity mapped to the color bar. When the high intensity regions map to silicon and the low intensity regions to air, this structure displays a 12.5% PBG between bands 2 and 3, and a nearly 5% PBG between bands 5 and 6. 046605-7 PHYSICAL REVIEW E 71, 046605 s2005d TIMOTHY Y. M. CHAN, OVIDIU TOADER, AND SAJEEV JOHN FIG. 6. The photonic band structure diagram for the noninversion symmetric fcc structure shown in Fig. 5 with an 11.9:1 dielectric contrast. The positions of the high symmetry points are shown in the inset. hc1,c2,c3,c12,c13,c23j = h1,0,1,1,0,1j s21d hf1, f2, f3, f12, f13, f23j = h0,0,0,0,0, p/2j. s22d and Since the structure lacks inversion symmetry, elliptical polarizations must be used. One set of elliptical polarization parameters which minimize I0 with the constraints imposed by Eqs. s21d and s22d is hE0,E1,E2,E3j = h1.36,1.36,1.36,1.36j, hw0, w1, w2, w3j = h37.7°,37.7°,37.8°,37.8°j, hb0, b1, b2, b3j = h28.6°,208.6°,28.6°208.6°j. s23d These parameters achieve a value of 7.39 for the intensity background I0 and a maximum total intensity of 14.78. Figure 7 shows this beam configuration. The dashed lines W passing through the polarization ellipses lie in either the G 0 W W W -G2 or G1-G3 plane. All four ellipses are tilted by an angle of approximately 53.2° from the corresponding plane. In all cases, the major axis is 4.1 times the length of the minor axis. B. Body-centered cubic structure Next, we consider a holographic structure based on the bcc Bravais lattice. There are six distinct directions in the Green set of vectors in Fig. 1, corresponding to the Bb set. In order to obtain a nontrivial pattern which possesses the full symmetry of the bcc Bravais lattice, one needs to use all six of these directions. Unfortunately, none of the eight possible combinations for C generates a structure with a full photonic band gap. It is possible to relax the symmetry conditions by removing the inversion symmetry, which results in the gy- FIG. 7. The diagram of a beam configuration which generates W the noninversion symmetric fcc structure. The plane spanned by G 0 W W W and G2 is perpendicular to the plane spanned by G1 and G3. The ellipses at the end of each wave vector describe the polarization W , with the ellipse of the corresponding beam, as one looks down G i dashed line spassing through the polarization ellipsed corresponding W -G W or G W -G W plane. In each case, the major axis of the to the G 0 2 1 3 polarization ellipse srepresented by the dotted line passing through the polarization ellipsed is tilted by approximately 53.2° with reW pairs for each beam, spect to the corresponding plane. The RW and U defined in Eq. s15d, are shown next to the corresponding polarization ellipse. 046605-8 PHYSICAL REVIEW E 71, 046605 s2005d PHOTONIC BAND GAP TEMPLATING USING OPTICAL… W = p s− 1,− 1,− 1d, G 0 ap W = p s− 1,1,1d, G 1 ap W = p s1,1,− 1d, G 2 ap W = p s1,− 1,1d. G 3 ap FIG. 8. The optimized bcc architecture generated by Eq. s25d with a solid volume filling fraction of <22%. When the solid regions sinterior of depicted iso-intensity surfacesd consist of silicon and the background is air, a PBG of 21% is obtained. roid sGd surface described in f60g. Instead, we relax our symmetry conditions by choosing only four of the six target directions, giving a target vector set T = hTW 1 , TW 2 , TW 3 , TW 4j, which leads to a larger IBZ and an intensity pattern of lower W vectors is symmetry. One possible choice for the three K i W = 2p s0,1,1d, K 1 ap W = 2p s1,1,0d, K 2 ap W = 2p s1,0,1d. K 3 ap s24d W ,K W ,K W ,K W j, and C This leads to the choice T = hK 1 3 12 23 = h* , 0 , * , * , 0 , * j. Using a similar derivation to the one presented in the fcc case but replacing the fcc point group operations with those of the bcc point group, one can write the bcc intensity pattern in terms of the vectors in T as DIbccsrWd = cossTW 1 · Wrd + cossTW 2 · Wrd + cossTW 3 · Wrd + h cossTW 4 · Wrd, The beam amplitudes and polarizations obey and hc1 , c2 , c3 , c12 , c13 , c23j = h1 , 0 , 1 , 1 , 0 , −1j hf1 , f2 , f3 , f12 , f13 , f23j = h0 , 0 , 0 , 0 , 0 , 0j. Elliptically polarized beam parameters which obey these constraints and minimize I0 are hE0,E1,E2,E3j = h1.19,1.19,1.19,1.19j, hw0, w1, w2, w3j = h30.0°,30.0°,30.0°,30.0°j, hb0, b1, b2, b3j = h90.0°,− 90.0°,90.0°,− 90.0°j. s27d It is interesting to note that, with this configuration, the bcc intensity pattern can be created using four laser beams of equal intensities. With this configuration, the intensity background I0 is 5.66, with the total intensity ranging from 0 to 2I0. The beam configuration represented by this solution is W and G W sG W -G W d is shown in Fig. 9. The plane spanned by G 0 2 0 2 W and G W sG W -G W d. perpendicular to the plane spanned by G 1 3 1 3 W W W The angle between G0 and G2 equals the angle between G 1 W and has the value 109.5°. The polarization ellipses and G 3 are shown next to the corresponding beams, depicting the path traced by the polarization vector as one looks down the wave vector. The dashed lines passing through the polarization ellipses lie in the respective planes defined above. Linearly polarized beams can be used to generate the bcc structure, but, as in the case of the diamond structure, the minimum value of I0 is larger than in the case of elliptical polarizations. Here, the value of I0 achieved is 10.17 and the maximum total intensity is 15.83: hE0,E1,E2,E3j = h1.11,1.98,1.97,1.09j, s25d where h = ± 1. The choice h = + 1 produces an intensity pattern resembling a bcc lattice of spheres, which produces no fundamental gap. However, the choice h = −1 generates a bcc structure whose basis is similar to that of the diamond structure. Figure 8 shows the optimized structure when h = −1, which produces a PBG of 21% for a silicon structure in an air background. Holographic beam parameters which produce this intensity pattern are determined straightforwardly as before. The W vectarget vectors in T can be generated by the following G Î tors of length 3p / ap : s26d hw0, w1, w2, w3j = h− 6.1°,21.9°,107.7°,39.7°j. s28d This linear polarization beam configuration is shown in Fig. 10. Again, the polarization angles are shown as one W -G W or G W looks down the wave vector, relative to the G 0 2 1 W -G3 plane. C. Simple cubic Finally, we consider a holographic structure based on the sc Bravais lattice, which is the simplest of the cubic structures. There are only three distinct directions in the set Bs 046605-9 PHYSICAL REVIEW E 71, 046605 s2005d TIMOTHY Y. M. CHAN, OVIDIU TOADER, AND SAJEEV JOHN FIG. 11. The optimized sc architecture generated by Eq. s29d with a solid volume filling fraction of <24%. When the solid regions scorresponding to regions with a red hued consist of silicon and the background is air, a PBG of 10.5% is obtained. FIG. 9. The diagram of a beam configuration which generates the bcc intensity pattern using elliptically polarized beams. The W and G W is perpendicular to the plane spanned plane spanned by G 0 2 W W by G1 and G3. The ellipses at the end of each wave vector describe the polarization ellipse of the corresponding beam, as one looks W , with the dashed line passing through the polarization eldown G i W -G W or G W -G W plane. The RW and U W pairs lipse corresponding to the G 0 2 1 3 for each beam, defined in Eq. s15d, are shown next to the corresponding polarization ellipse. sred set onlined of vectors in Fig. 1 and consequently only three vectors in the target vector set, i.e., T = hTW 1 , TW 2 , TW 3j. In W = bW s, so that the C this case, we make the simple choice TW i = K i i coefficients are given by C = h* , * , * , 0 , 0 , 0j, where * denotes a nonzero real coefficient. Again, without loss of generality the first three nonzero coefficients can be chosen to be 1; in this case, this applies to all nonzero coefficients. The intensity pattern is thus given by DIsrWd = cossTW 1 · Wrd + cossTW 2 · Wrd + cossTW 3 · Wrd. s29d This pattern has a simple cubic Bravais lattice and is invariant to all symmetry operations of the simple cubic point group, thereby satisfying our design requirements. The intensity iso-surface of the optimized Si sc structure is shown in Fig. 11. A full photonic band gap of 10.5% opens between bands 5 and 6 for this optimized structure, which has a solid volume filling fraction of <24%. The target vectors in T can be generated by the following W G vectors of length Î3p / ap : W = p s− 1,− 1,− 1d, G 0 ap W = p s1,− 1,− 1d, G 1 ap W = p s− 1,1,− 1d, G 2 ap W = p s− 1,− 1,1d, G 3 ap FIG. 10. The diagram of a beam configuration which generates the bcc intensity pattern using linearly polarized beams. The plane W and G W is perpendicular to the plane spanned by G W spanned by G 0 2 1 W and G3. The polarization angles as one looks down the wave vector are shown next to the corresponding beams, with the dashed line W -G W or G W -G W plane. The RW and U W pairs for corresponding to the G 0 2 1 3 each beam, defined in Eq. s15d, are shown next to the corresponding polarization vector. s30d and beam polarizations which obey hc1 , c2 , c3 , c12 , c13 , c23j = h1 , 1 , 1 , 0 , 0 , 0j and hf1 , f2 , f3 , f12 , f13 , f23j = h0 , 0 , 0 , 0 , 0 , 0j and minimize I0 are 046605-10 hE0,E1,E2,E3j = h1.79,1.02,1.02,0.88j hw0, w1, w2, w3j = h50.1°,− 7.8°,− 67.8°,52.2°j, PHYSICAL REVIEW E 71, 046605 s2005d PHOTONIC BAND GAP TEMPLATING USING OPTICAL… FIG. 12. The diagram of a beam configuration which generates the sc intensity pattern using elliptically polarized beams. The W ,G W , and beams are set up in an “umbrella” configuration, where G 1 2 W W G3 each make an angle of 70.5° with G0 and are spread evenly around the azimuth. Beam 0 is elliptically polarized with a polarization ellipse whose major axis is rotated 15.3° from the projection W onto the plane normal to G W . The other three beams are of G 1 0 linearly polarized, with polarization vectors making identical angles W -G W planes. of 52.2° with the G 0 i hb0, b1, b2, b3j = h275.8°,0°,0°,0°,j. FIG. 13. The diagram of a beam configuration which generates the sc intensity pattern using linearly polarized beams. Each of the W ,G W , and G W make an angle of 70.5° with G W and wave vectors G 1 2 3 0 are spread evenly around the azimuth. The polarization angle of W onto the beam 0 is measured with respect to the projection of G 1 W , whereas the other polarization angles are meaplane normal to G 0 W -G W plane, for i = 1 , … , 3, with the wave sured with respect to the G i 0 vector pointing into the page. s31d In this case, an “umbrella” configuration of the beams is used W , G W , and G W ssee Fig. 12d. Each of the wave vectors G 1 2 3 W makes an angle of 70.5° with G0 and is spread evenly around the azimuth. Only beam 0 is elliptically polarized; the major axis of the polarization ellipse sdepicted as the dashed line passing through the polarization ellipsed makes an angle of W onto the plane normal to G W . 15.33° with the projection of G 1 0 The other three beams are linearly polarized, with polarizaW tion vectors making identical angles of 52.24° with the G 0 W planes. The background intensity I is 6.07, and the total -G i 0 intensity ranges from 0.07 to 12.07. If one constrains all four polarizations to be linear, the parameters hE0,E1,E2,E3j = h1.92,0.64,1.28,1.28j, hw0, w1, w2, w3j = h− 15.5°,67.8°,7.7°,− 52.2°j s32d yield a minimum value of 7.35 for I0, with the maximum total intensity reaching 13.35. The linear beam configuration is shown in Fig. 13. D S D S D S D 2p 2p 2p ,− , , TW 2 = ap ap cp 2p 2p 2p , ,− , TW 3 = ap ap cp 2p 2p 2p , , . TW 4 = ap ap cp cp /ap = utansjdu/Î2. The diamond sfccd and bcc structures described in Sec. III can be considered as special cases of a larger class of photonic crystals. The varying intensity pattern associated with these structures can be written as DIsrWd = cossTW 1 · Wrd + cossTW 2 · Wrd + cossTW 3 · Wrd − cossTW 4 · Wrd, s33d s34d The general structure can be modeled in a tetragonal unit cell whose aspect ratio is given by the quantity cp / ap . By varying the aspect ratio, one can scale the structure along the z direction. The aspect ratio can be written in terms of the acute angle between any one of the target vectors and the z axis, j, by IV. GENERALIZED TETRAGONAL STRUCTURES where the target vectors, TW i, are given by S 2p 2p 2p TW 1 = − , , , ap ap cp s35d The diamond structure is achieved when j = arctansÎ2d = 54.7° scorresponding to cp / ap = 1d and the bcc structure results when j = 45° scorresponding to cp / ap = 1 / Î2d. The desired target vectors can be obtained by changing the angle between wave vectors, which we denote by n, in the counter-propagating scheme used for the fcc and bcc structures described in Sec. III ssee, e.g., Fig. 3d. For target vectors characterized by a given j, an angle n given by 046605-11 PHYSICAL REVIEW E 71, 046605 s2005d TIMOTHY Y. M. CHAN, OVIDIU TOADER, AND SAJEEV JOHN as j varies, for structures with a dielectric contrast of 11.9 to 1, corresponding to silicon in air. The solid volume fraction in all cases is 22.1%. The maximum PBG occurs at j = 54.74°, which is the diamond structure. Figure 15 shows the photonic band structure for a structure with j = 50°, between the bcc and fcc cases. A full photonic band gap of 23% opens for this structure. V. ROBUSTNESS OF HOLOGRAPHIC PHOTONIC CRYSTALS FIG. 14. The relative size of the full photonic band gap as a function of the angle, j, between the target vectors, TW i, and the z axis. The structure has a solid volume fraction of 22.1%. The solid has a dielectric constant of 11.9, corresponding to silicon. The bcc structure corresponds to j = 45° and the diamond sfccd structure corresponds to j = 54.74°. SÎ n = 2 arccos cossjd 2 − cos2sjd D s36d is sufficient. As shown in Figs. 3 and 9, angles between wave vectors of 126.9° and 109.5° satisfy the conditions for fcc and bcc structures, respectively. Beam polarizations which satisfy the constraints hc1 , c2 , c3 , c12 , c13 , c23j = h1 , 0 , 1 , 1 , 0 , −1j and hcossf1d , cossf2d , cossf3d , cossf12d , cossf13d , cossf23dj = h1 , 1 , 1 , 1 , 1 , 1j can then be found using the minimization procedure described previously. Figure 14 shows the size of the relative photonic band gap We now turn our attention to the properties of the inversion symmetric holographic photonic crystals obtained from the intensity patterns derived in the previous section. For a given intensity pattern, it is first necessary to determine the optimal value of the threshold Ithr in the simplified shape function, Eq. s5d. The choice of Ithr corresponds to choosing a particular iso-intensity surface, and therefore the solid volume filling fraction of the resulting crystal. The one-to-one mapping from the Ithr to the volume fraction is displayed in Fig. 16 for the fcc, bcc, and sc structures. The functional dependence of the volume fraction smeasured in %d on Ithr is not strictly linear, but can be taken to be so for the volume fractions of interest. The linear part of the mapping can be written as f f = − 21Ithr + 50, f b = − 21Ithr + 50, f s = − 29Ithr + 50, s37d where f f , f b, and f s are the volume fractions of the fcc, bcc, and sc structures, respectively. Since the fcc and bcc structures are both specific examples of the generalized structure described in Sec. IV, the mapping is identical for the two FIG. 15. The photonic band structure of the generalized structure with j = 50°, at a solid volume fraction of 22.1% and a dielectric contrast of 11.9 to 1, corresponding to silicon. The high symmetry points and the irreducible Brillouin zone are shown in the inset. This exhibits a PBG of 23%. 046605-12 PHYSICAL REVIEW E 71, 046605 s2005d PHOTONIC BAND GAP TEMPLATING USING OPTICAL… FIG. 16. The volume solid versus threshold used in the shape function Eq. s5d, for fcc, bcc, and sc structures. The fcc and bcc curves overlap each other because both structures are members of the generalized HLPC structures described in Sec. IV. The solid material is found in the regions where DIsrWd . Ithr. structures. One may note that this mapping is symmetric around a volume fraction of 50%. In fact, it can be shown that, for the three cubic structures we have obtained, taking Ithr → −Ithr produces an equivalent but inverted structure ssolid and air regions are interchangedd, to within a translation of the origin. In terms of a physical sexperimentally expt expt → 2I0 − Ithr . measuredd intensity, this corresponds to Ithr This implies that the desired structure can be achieved equivalently by both single and double inversion processes simply by adjusting the threshold value accordingly. Similarly, this means that one can compensate for the distinction between positive and negative photo-resist materials by changing the holographic exposure time or overall intensity. Consider the shape functions Sd of the direct structure and Si of the inverted structure, given by SdsrW,Ithrd = Q„DIsrWd − Ithr…, SisrW,Ithrd = Q„Ithr − DIsrWd…. s38d We endeavour to find rW = Wr8−rW such that SdsrW , Ithrd = SisrW8 , −Ithrd. This implies that DIsrW8d = −DIsrWd. This condition is met provided that W · rW = s2n + 1dp K i i W · rW = s2n + 1dp , K ij ij s39d where ni and nij are integers, for all i such that ci Þ 0 and all i, j such that cij Þ 0. Recall that ci and cij are the members of the set C defined in Eq. s6d. Equation s39d can only be satisfied if, when cij is nonzero, either ci or c j is zero. The format of the set C of the three inversion symmetric structures discussed above satisfies this condition. It can be shown that the corresponding translation vectors, rW , are rW f = sa / 2 , a / 2 , a / 2d for the fcc structure, rW b = sa / 2 , a / 2 , 0d, for the bcc structure, and rW s = sa / 2 , a / 2 , a / / 2d, for the sc structure. FIG. 17. The relative size of the full photonic band gap as a function of the solid volume fraction, for the fcc, bcc, and sc structures. The solid has a dielectric constant of 11.9, corresponding to silicon. The optimized fcc structure is <21.5% solid and generates a relative gap of 25%. The optimized bcc structure is <22% solid and generates a relative gap of 21%. The optimized sc structure is <24% solid and generates a relative gap of 10.5%. Figure 17 displays the relative size of the full photonic band gap as a function of the solid volume fraction for the fcc, bcc, and sc structures composed of a material with a dielectric constant of 11.9 sSid in an air background. The optimized fcc structure occurs at a solid volume fraction <21.5% and has a relative fundamental gap sbetween the second and third bandsd of 25%. The optimized bcc structure, which is <22% solid, also has a fundamental gap, which is 21% of the gap center frequency in this case. Finally, for a solid volume fraction of <24%, a full photonic band gap of 10.5% opens between bands 5 and 6 for the optimized sc structure. The photonic band structure diagrams for the three optimized structures are shown in Fig. 18. Figure 19 shows the dependence of the relative photonic band gap on the index of refraction of the solid material for the three optimized structures. For the fcc structure, a full photonic band gap appears when the refractive index is larger than about 1.97. The relative width increases with the index of refraction until saturation at about 38%. The refractive index threshold for the bcc structure is approximately 2.24 and the relative PBG saturates at 34%. The index threshold for the sc structure is 2.77 and the relative PBG saturates at 16.5%. The global intensity contrast, defined as maxsu2DIsrWdud / I0, is of important practical concern. The larger the contrast, the more effective the developing process to create the polymer photonic crystal template. An alternative metric for measuring the global intensity contrast can be found in f58g. The intensity contrast depends both on the shape being created and on the particular beam configuration used to create that shape. Table I shows the global intensity contrast for each of the beam configurations for the inversion symmetric structures discussed in Sec. III. For each structure, the use of elliptically polarized beams improves the contrast over the case when only linearly polarized beams are used. The elliptical beam configuration for the bcc structure achieves a 046605-13 PHYSICAL REVIEW E 71, 046605 s2005d TIMOTHY Y. M. CHAN, OVIDIU TOADER, AND SAJEEV JOHN FIG. 19. The relative size of the full photonic band gap as a function of the index of refraction of the solid material, for the optimized fcc, bcc, and sc structures. The background is assumed to be air. The threshold index of refraction for the emergence of a photonic band gap is 1.97 for the fcc structure, 2.24 for the bcc structure, and 2.77 for the sc structure. The inset shows a magnified view of the curve, near the threshold indices of refraction for the fcc and bcc structures. ing” of the interface between “exposed” and “unexposed” polymer even though the iso-intensity contours are perfectly smooth. Therefore, it is useful to maximize the gradient of the intensity pattern at precisely the threshold intensity isosurface. This minimizes the spatial extent of “exposure roughening”. In order to compare the intensity contrasts for the fcc, bcc, and sc structures, we define an average intensity contrast factor for an iso-intensity surface at Ithr, CavgsIthrd as CavgsIthrd = E W ¹W fDIsrWdg · dA ]V sI0 + 2Ithrd E , s40d W nW · dA ]V where V corresponds to the region which becomes the interior of the resulting structure and ]V corresponds to its surface si.e., the iso-intensity surfaced. In other words, the con- FIG. 18. Photonic band structure diagrams for the optimized sad fcc, sbd bcc, and scd sc holographic photonic crystals structures characterized by an 11.9:1 dielectric contrast. The positions of the high symmetry points together with the corresponding irreducible Brillouin zones are shown in the insets. “perfect” contrast ratio of 1, meaning that a value of zero total intensity is found in the interference pattern. When comparing the exposure efficiency of different interference patterns, it is not sufficient to consider only the global intensity contrast. Disorder in the PBG template may arise from polymer inhomogeneities that lead to small, random variations in the exposure threshold from point to point in the bulk photo-resist. This, in turn, may cause “roughen- TABLE I. Global intensity contrast in the interference patterns of the various beam configurations. The global intensity contrast is defined as the ratio of the amplitude of the varying part of the intensity, maxsu2DIsrWdud, and the background intensity I0. The use of elliptical polarizations improves the global intensity contrast for each of the three structures. Structure Polarizations fcc elliptical linear elliptical linear elliptical linear bcc sc 046605-14 I0 maxsu2DIsrWdud Contrast 4Î3 11.97 4Î2 10.17 6.06 7.35 4Î2 4Î2 4Î2 4Î2 6 6 0.816 0.472 1 0.556 0.989 0.816 PHYSICAL REVIEW E 71, 046605 s2005d PHOTONIC BAND GAP TEMPLATING USING OPTICAL… FIG. 20. The plot of the average intensity contrast factor, Cavg, as a function of the volume fraction of solid for the fcc, bcc, and sc structures, created using elliptically polarized beams which produce minimal background intensities. The solid part of the photonic crystal corresponds to regions where the intensity exceeds the threshold value. trast factor gives the average value of the intensity gradient along the iso-intensity surface, normalized by the isointensity value. Normalization by the iso-intensity value means that the contrast factor is not symmetric with respect to a solid volume fraction of 50%. Using Gauss’ theorem and taking the area of the iso-intensity surface as A, the contrast factor can be rewritten as 1 Cavg = sI0 + 2IthrdA E W2 ¹ fDIsrWdgdV. s41d V Figure 20 shows a plot of the average intensity contrast factor, Cavg, as a function of the volume fraction of solid for the fcc, bcc, and sc structures. The contrast factor is calculated by discretizing the conventional scubicd unit cell in 2563 boxes and sampling the appropriate intensity pattern. The area of the iso-intensity surface in the cubic cell is calculated by approximating the surface with a triangular mesh. The volume integral is calculated by taking E V ¹W 2fDIsrWdgdV → o ¹W 2fDIsrWdgv , DIsrWdøIthr W 2fDIsrWdg is where v is the volume of the small voxels and ¹ calculated analytically from the corresponding intensity pattern. The solid part of the photonic crystal corresponds to regions where the intensity exceeds the threshold value. This could be achieved using negative photo-resists and double inversion. Near the respective optimal solid volume fractions s<22%d, the contrast factors for the fcc and bcc structures are approximately equal and larger than the sc contrast factor. However, near 78% solid volume fraction swhich corresponds to the inverted structured, the bcc contrast factor becomes larger than the fcc contrast factor. This is relevant if a single inversion process is used in conjunction with a nega- FIG. 21. The plot of the minimum intensity contrast Cmin W fDIsrWdg / sI + 2I du as a function of the solid volume = minDIsrWd=Ithru¹ 0 thr fraction for the fcc, bcc, and sc intensity patterns which produce minimal background intensities using elliptically polarized beams. tive photo-resist, or if double inversion is used with a positive photo-resist. In addition to considering average intensity gradient, we examine the minimum intensity gradient for various isointensity surfaces by finding the figure of merit f14g: Cmin = min u¹W fDIsrWdg/sI0 + 2Ithrdu. DIsrWd=Ithr s42d This is calculated analytically for every point on the triangular mesh used to approximate a given iso-intensity surface. The regions where contrast is minimum along the threshold iso-intensity surface represent the weak points of the photonic crystal template. These regions are most susceptible to disorder in the holographic process. Shown in Fig. 21, Cmin gives a measure of the worst case for each isointensity surface. In this case, the fcc interference pattern has the best characteristics near the optimized volume fraction of 22%, whereas the fcc and bcc patterns have similar characteristics near 78% volume fraction. It should be noted that the sc interference pattern appears to achieve the worst case near both the optimized direct and inverted volume fractions. Finally, we consider the sensitivity of the holographic photonic crystals to imprecision in the laser beam parameters. We do this by considering the trajectories of the photonic band edges as beam polarizations and amplitudes are perturbed from the optimal configurations. Figure 22 shows the photonic band edges of the fcc, bcc, and sc structures as a single beam amplitude is perturbed from the optimal value. Only the worst smost sensitived case out of the four possible beam perturbations is shown for each of the structures. The fcc structure shows the most robustness against amplitude perturbations, allowing the single beam amplitude to range from 70% to 250% of the optimal value before the gap closes, with the bcc structure showing similar characteristics. On the other hand, a 10% perturbation in the single beam amplitude closes the gap in the case of the sc structure. Next, we consider perturbations to the polarization vectors in the 046605-15 PHYSICAL REVIEW E 71, 046605 s2005d TIMOTHY Y. M. CHAN, OVIDIU TOADER, AND SAJEEV JOHN FIG. 22. The photonic band edges of the optimized fcc, bcc, and sc structures as functions of perturbations in the holographic beam amplitudes, DEi. The structure is assumed to consist of a material of dielectric constant 11.9 in an air background. FIG. 24. The photonic band edges of the optimized fcc, bcc, and sc structures as functions of perturbations in the angle b fsee Eq. s16dg. The structure is assumed to consist of a material of dielectric constant 11.9 in an air background. optimal elliptically polarized beam configurations used to create the structures described in this paper. Recall Eq. s16d where we write an elliptical polarization vector as the sum of W + eib sinswdRW . two linear polarizations such that eW = cosswdU Figure 23 shows the photonic band edges of the three structures when the quantity tanswd, which describes the relative amplitude of the two linear components, is perturbed from the optimal value. The dependence of the photonic band edges on perturbations of b, the relative phase between the linear polarization components, is shown in Fig. 24. For both graphs, only the worst case out of the four possible perturbations for each structure is displayed. For these elliptically polarized beam configurations, it is noted that the photonic band gap of the bcc structure is most robust against beam parameter perturbations, while the sc structure displays the least robustness. Finally, we consider perturbations to a single polarization angle, w, for the optimal linearly polarized beam configurations in Fig. 25, by changing the polar- ization angle of a single beam. Figure 25 shows the photonic band edges of the inversion symmetric fcc, bcc, and sc structures as the polarization angle of the most sensitive beam is perturbed. In the worst case, the photonic band gap of the bcc structure remains open after an 8° change in a single polarization angle, whereas for the fcc structure, a 6° perturbation closes the gap. Again, the sc structure is least robust. FIG. 23. The photonic band edges of the optimized fcc, bcc, and sc structures as functions of perturbations in the ratio tanswd fsee Eq. s16dg. The structure is assumed to consist of a material of dielectric constant 11.9 in an air background. VI. CONCLUSION In summary, we have derived beam configurations for creating fcc, bcc, and sc photonic crystals using holographic lithography through simple, intuitive symmetry considerations. The guiding principle in this procedure is that a large PBG can be obtained by choosing a structure with the smallest possible irreducible Brillouin zone. The fcc structure, which resembles the diamond network structure, exhibits a FIG. 25. The photonic band edges of the optimized fcc, bcc, and sc structures as functions of perturbations in single linear polarization angles w. The structure is assumed to consist of a material of dielectric constant 11.9 in an air background. 046605-16 PHYSICAL REVIEW E 71, 046605 s2005d PHOTONIC BAND GAP TEMPLATING USING OPTICAL… full photonic band gap of 25% when made with materials with a dielectric contrast of 11.9:1. Our results also reveal that an optimized geometrical structure within the unit cell enables the bcc lattice to exhibit a PBG comparable to that of the widely studied diamond lattice. Here the photonic band gap is 21% for materials with a dielectric contrast of 11.9:1. Finally, we have found a simple cubic holographic architecture which exhibits a PBG of roughly 11% in the case of Si:air materials. This sc PBG material can be achieved using a simple “umbrella configuration” for the interfering laser beams. In each case, we find that an equivalent but inverted structure can be achieved simply by changing the threshold intensity. This implies that there need be no distinction between direct structures and structures made by inversion techniques. The beam configurations which we describe are optimal with respect to maximizing the intensity contrast in the respective laser interference patterns. The fcc and bcc structures exhibit similar contrast near the optimal filling fractions, while the sc structure displays worse contrast. For all three structures, the use of elliptically polarized beams enables better intensity contrast than can be achieved by the use of linearly polarized beams alone. We have also studied the sensitivity of our holographic structures to perturbations of the amplitudes and polarization angles of the holographic laser beams. The fcc and bcc structures each continue to exhibit full photonic band gaps even as a single linear polarization angle is adjusted by 6° or a single beam amplitude is adjusted between 70% and 240% of the optimal value. 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