1. Introduction

In the last few years the study of photonic crystals (PHCs) has drawn considerable attention [1,2]. The major physical interests have always concentrated on the search for the dielectric modulated structures and material components that possess a sizeable photonic bandgap (PBG). Although three-dimensional (3-D) PHCs suggest the most interesting ideas for novel applications, two-dimensional (2-D) PHCs also can be used to design a wide range of passive optical devices, including the feedback mirror in laser diodes [3], wave guides and communication fibers [4], cavity resonators [5] and so on. Fabrication of PHCs and PBG engineering are among important tasks. The holographic technique [6–10], using multiple noncoplanar beam interference, has some advantages such as one-step recording in the fabrication of PHCs that work in the visible or near infrared range.

In our last letter [8], we demonstrated an interference technique of three noncoplanar beams (ITNB) to fabricate 2-D triangular lattices. The remaining question is what effect the intensity threshold has on the filling ratio (FR), the shape of “atoms” and then the PBG of the final structure, and how we should control the intensity threshold experimentally to optimize the PBG. In this letter, we will further provide a theoretical modeling of the PBG and spectral properties of the 2-D trigonal lattices proposed in Ref. [8]. Because it is easy to produce a PHC with a thickness larger than several dozen micrometers while the corresponding lattice constant is usually less than one micrometer [6], we can reasonably treat the PHCs as ideal 3-D structures with 2-D periodicity in PBG analysis.

2. The PBG of two-dimensional triangular titania arrays

As shown in Ref. [8], the intensity of interference field for this lattice can be expressed as

where a is the lattice constant. If the region of high intensity in the interference field is defined as bright lattices, we can obtain bright lattices shown in Fig. 1(a) with a ₁=a(1/2,3^1/2/2) and a ₂=a(-1,0) as their translation basis vectors. The corresponding primitive vectors in reciprocal space are b ₁=(2π/a)(0,2/3^1/2), b ₂=(2π/a)(-1,1/3^1/2). The first Brillouin zone (BZ) turns out to be a hexagon as shown in Fig. 1(a). If the dark lattice whose intensity is lower than the given intensity threshold can be washed out, the shape of the remaining bright “atoms” changes from approximately hexagon to approximately circle when the intensity threshold is given from large to small. Obviously, the symmetry of the scatters and that of the lattices are the same. To calculate the photonic dispersion curves, the k-points are traced along the boundary of the irreducible BZ determined by three symmetry points Γ=(0,0), M=(0,1/3^1/2) and K=(1/3,1/3^1/2), in units of 2π/a.

 

Fig. 1. (a) 2-D triangular photonic lattice fabricated by the interference technique of three noncoplanar beam and the first Brillouin zone with the symmetry points indicated; (b) and (c), dotted lines (I) present the relation between the intensity threshold and the FR of dielectric, and solid lines (II) present the derivative of curves (I), where (b) is for titania and (c) is for GaAs.

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Using the photolithography technique reported by Shishido etc. [11], 2-D titania triangular arrays can be directly fabricated by polymerization of a photosensitive titanium-containing monomer film. The effective dielectric constant of the titania array is found to be about 4. In the following, the block-iterative frequency-domain method for Maxwell’s equations in a plane wave basis [12] is employed to investigate the gap properties for the 2-D triangular titania arrays. In the calculations, grid size is 32×32×1, mesh size is 7 and tolerance is 1×10^-7. By controlling the exposure intensity, introducing an intensity threshold I_t (here I_tε (1.5, 6) for Eq. (1)), and washing out the underexposed regions, the filling ratio of titania can be selected. Consequently, the dielectric constant distribution ε(x, y) of result lattice should be 1 in the region I ₀(x, y)<I_t and 4 in the region I ₀(x, y)≥I_t. The relation between I_t and FR of the titania is shown in Fig. 1(b)-I. The derivative of curve-I is also given in Fig. 1(b)-II. It is clearly that the FR changes sharply when the intensity threshold is near 2. To optimize the structures so as to maximize the width of the PBG, the effect of different threshold and corresponding FR has been examined, and we present here some useful results.

2.1 The PBG for TE or TM polarization

In this paper, TM polarization corresponds to E field in the z direction (see Fig. 1(a)) and vice versa for TE. For TE polarization, numerous symmetry-induced degeneracies are observed at the symmetry points of the BZ. As a result, no gap is observed whatever the I_t is.

For TM polarization, we have studied the evolution of PBG’s as function of I_t. Fig. 2(a) shows the dependence of the PBG on the I_t, extracted from band structure calculations when I_t changes from 1.5 to 6.0. All band gaps appearing for the ten lower-energy bands have been plotted for all possible values of the I_t. The boundaries of shadows indicate the maxima and minima observed in the bands of the calculated band structure for various I_t. Each shadow indicates a PBG region. Frequency axes in all plots of this paper are calculated using normalized frequency units (f _n=ωa/2πc, where ω is the frequency and c is the speed of light). Clearly three gaps are observed for the ten lower-energy bands when I_t is or above approximately 2.0 (the corresponding FR is about 74.73%). They locate between M₁ and M₂, M₃ and M₄, and M₆ and M₇ (where M_i represents the ith band for TM polarization, and E_i below for TE polarization). The variation of the TM gap width with I_t (measured by gap to midgap ratio Δω/ω₀) is indicated in the inset of Fig. 2(a). Using Brent’s algorithm [13], we can find the favorable FR that maximizes the TM gaps. The maximum relative gaps for M_1–2, M_3–4, and M_6–7 (where M_i-j denotes the gap appearing between the ith and jth TM bands) are 22.19%, 12.74% and 4.285%; the central normalized frequencies of these gaps are 0.4614, 0.7120 and 0.9993; the intensity thresholds are 4.100, 3.140 and 3.050; and the corresponding FRs are 21.23%, 37.06% and 38.94%, respectively. Figure 2(b) displays the photonic band structure for I_t=3.0, which is the configuration yielding larger size of M_3–4 and M_6–7 gaps.

 

Fig. 2. (a) TM gap map for the 2-D triangular titania arrays; (b) TM photonic band structure for I_t=3.0.

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Fig. 3. (a) Gap map of directional PBG for 2-D titania arrays; (b) Gap map of full PBG for the inverted GaAs structure, where blue area is for TM polarization, red is for TE polarization and yellow area is for both.

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2.2 Directional PBG

Although no full PBG is expected for PHCs discussed here, several directional PBGs can be obtained. For example, we show in Fig. 3(a) the evolutions of PBG in Γ-M direction with respect to the intensity threshold I_t. Many gaps appear for TE or TM polarization. The essential result here is that numerous gaps existing in TM polarization (M_1–2, M_3–4, M_4–5 and M_10–11) eventually overlap TE gaps (E_1–2, E_2–3, E_4–5 and E_9–10), respectively, which leads to four directional PBGs (E₁-M₂, E₂-E₃ or E₂-M₄, E₄-M₅ and M_10–11). The optimized relative gaps calculated by Brent’s algorithm for these directional PBGs are 8.923%, 7.396%, 4.276% and 3.154%; the central frequencies of these gaps are 0.5194, 0.6774, 1.042 and 1.477; The intensity thresholds are 4.5881, 2.982, 4.961 and 3.942; and the corresponding FRs are 14.75%, 40.29%, 10.49% and 23.44%, respectively. Figures 4(a) and (c) illustrate the directional photonic band structures of the PHCs in the case of I_t=3.0 and 4.6, respectively. The former corresponds to a larger E_2–3 gap and the latter corresponds to larger E₁-M₂ and E₄-M₅ gaps. For comparisons, in Figs. 4(b) and (d) we also give the corresponding transmission spectra (in Γ-M direction) calculated with non-orthogonal FDTD method [14]. In these calculations, a Berenger type of perfectly matched layer (PML) [15] is implemented as absorbing layer, the total length of the scattering region is 10×3^1/2 a, and the electromagnetic fields are integrated for 16 blocks of 1024 time steps each. Obviously, the gaps in the band structure are exactly corresponding to minima of the transmittance power.

 

Fig. 4. The directional photonic band diagrams (a and c) and calculated transmission spectra (b and d) for TE (red line) and TM (blue line) polarizations respectively, where (a) and (b) are for I_t=3.0, and (c) and (d) are for I_t=4.6.

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3. The PBG of two-dimensional triangular inverted GaAs structure

The important remaining question is how to turn these aforementioned directional PBGs into the full PBGs. Several methods have been suggested for obtaining full PBGs in 2-D situations. For example, it has shown that large band gaps can be obtained by such as varying dielectric contrast ratio and filling factors, inserting a third component into the existing photonic crystals [16], reducing the structure symmetry [17], using non-circular rods [18], and subsequently by rotating the non-circular rods [19], rotating the lattice structure [20], using anisotropic dielectric materials [21], placing rods of various shapes on different lattice configurations such as square [22], triangular [21, 23], honeycomb [17] and so on. Each approach may have its advantages and shortcomings. For holographic technique, one can obtain an inverted crystal with higher refractive-index contrast by using some photoresist to fabricate template for in-filling other high refractive index materials or by using a photoresist that has been exposed to the interference field as mask for the GaAs etching [24]. In the following, we examine an inverted structure by supposing the dielectric contrast ratio reaches 13.6:1 (13.6 is the GaAs’s dielectric contrast) as other authors used [18, 25]. Then the dielectric constant ε(x, y) equals 13.6 in the area of I ₀(x, y)<I_t and 1 otherwise. Both the relation between I_t and FR of GaAs and its derivative are given in Fig. 1(c).

 

Fig. 5. Photonic band structures when the intensity threshold is 1.6 (a) and 2.0 (b) for TE (red line) and TM (blue line) polarizations, respectively.

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Our aim here is to modify the intensity threshold to obtain a full PBG. Figure 3(b) shows the effect of the parameter I_t on the size and position of the band gaps. The gap map clearly indicates that the gap for TE or TM polarization drops off sharply if I_t is less than about 1.6 or greater than about 2.5, this fact limits the overlap of TE and TM polarization gaps. On the other hand, up to four full PBGs appear during I_t changes from 1.6 to 2.5, and their number varies with the I_t. The PBG B_12–13 (where B_i-j presents the band gap locating between the ith and the jth bands) corresponding to the largest I_t range (from about 1.6 to 2.5) is due to the overlap of E_5–6 and M_7–8 gaps. In the following we will discuss the evolution of B_12–13 as an example. Figure 5(a) shows the result of band calculation for I_t=1.6. Band structure analysis indicates that degeneration at the symmetry points of the BZ closes the TE PBG in the case of lower intensity threshold. The bands will move downward with the increase of intensity threshold, and the down-moving speed of E₅ at the three symmetry points is a bit faster than that of E₆ when I_t changes from 1.6 to 1.7. Then the full PBG B_12–13 determined by E₅(Γ) and E₆(Γ) gradually appears (where E₅(Γ) presents the symmetry point Γ of E₅). During I_t increases from 1.7 to 1.9, the down-moving speed of E₅(Γ) is a bit faster at the beginning stage and a bit slower at the ending stage than that of E₆(Γ), and a maximum 6.684% full PBG calculated by Brent’s algorithm appears from 0.9653 to 1.032 when I_t=1.762 (corresponding FR is 10.78%). The down-moving speed of M₇(M) (or M₇(K)) is slower than E₅(Γ) when I_t changes from 1.9 to 2.0, and the B_12–13 translates into being bounded on the lower side by the TM polarization gap boundary M₇(M) (or M₇(K)) and on its upper side by the TE polarization gap boundary E₆(Γ). Adopting a similar analysis for other I_t and other three PBGs, B_3–4, B_9–10 and B_22–23 (due to the overlap of E_1–2 and M_2–3, E_3–4 and M_6–7, and E_9–10 and M_13–14), one can find their evolution at different intensity threshold. By using Brent’s algorithm in the calculation of band structure again, B_3–4, B_9–10 and B_22–23 can also be optimized. The maximum relative gaps for B_3–4, B_9–10 and B_22–23 are 18.71%, 8.235% and 2.552%; the central frequencies of these gaps are 0.4039, 0.7098 and 0.9985; the intensity thresholds are 2.000, 1.944 and 2.000; and the corresponding FRs are 25.27%, 20.65% and 25.27%, respectively. The most valuable result that has not been reported before [18] is that the four sizeable full PBGs simultaneously appear when I_t is about 2.0 (see Fig. 5(b)). It has shown the important effect of “atom” shape introduced by holography on PBG properties.

4. Conclusions

In conclusion, using the block-iterative frequency domain method and the non-orthogonal FDTD method, we have shown the existence of PBGs for a new class of 2-D triangular lattice, which can be fabricated by an interference technique of three noncoplanar beams proposed by us [8]. A specific problem in holographic lithography, the effect of threshold selection on the FR, the “atom” shape and then the PBG properties, is comprehensively investigated. Our calculations of band structures as a function of I_t indicate that PBGs of 2-D triangular titania arrays open only for TM polarization, and four broad directional PBGs along Γ-M can be obtained. Furthermore, for a higher dielectric contrast of 13.6:1, up to four sizeable full PBGs can be produced with an 18.15% relative gap-width of the maximum full PBG. These results have shown the potential of the holographic method in controlling the “atom” shape of the PHC and then improving their PBG properties, and are useful in the fabrication of 2-D PHCs with the technique of ITNB.