1. Introduction

Photonic crystals (PhCs) are materials that show a periodic arrangement of the dielectric constant on a length scale comparable to the wavelength of light. In the past decade, the study of PhCs has received considerable interests because it demonstrates the ability of PhCs to prevent the propagation of electromagnetic waves in a certain frequency range [1, 2]. Even without a complete photonic bandgap (PBG), PhCs can be used to manipulate the propagation of light. Although three-dimensional (3-D) PhCs suggest the most interesting ideas for novel applications, two dimensional (2-D) structures also find several unique applications [3], including the feedback mirror in laser diodes [4], the waveguides and communication fibers [5], the 2-D periodic structures in living animals [6] and so on. A number of methods for PhC fabrication have been developed. Among them the holographic technique [7] 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. To investigate the use of this technique in fabrication of periodic microstructures, we have made a series of studies on the interference of three and four noncoplanar beams (ITNB and IFNB) [8, 9].

Since the existence of PBGs is the most important property and the basis of many applications of PhCs, how to obtain a PBG for a given structure and make it as large as possible is a fundamental task of PBG engineering. A unique feature of the PhCs produced by holographic method is that the shape and size of the resultant element spots, referred to as “atoms” of the structure here, take the form of the equal-intensity surfaces of the interference field, and naturally they vary with the beam design and the choice of the threshold intensity, usually different from the regular shapes like a cylinder or a sphere reported before. Therefore a more specific PBG study of the PhCs of this kind considering their special “atom” shape and size becomes necessary. In this paper we will take 2-D triangular lattices as an example to investigate the PBG property of holographically recorded PhCs, and show that the holographic method gives us a new freedom for PBG engineering.

2. The PBG of two-dimensional triangular titania arrays fabricated by interference of four umbrellalike beams

In our last Letter [8], we have demonstrated a four umbrellalike beams interference technique with a diffractive beam splitter to fabricate 2-D triangular lattices. In this paper, we will further provide a theoretical analysis of the PBG and spectral properties of the 2-D lattices of this kind. As shown in Ref. [8], the intensity of interference field for this lattice can be expressed as

If the regions of high intensity in the interference field are defined as bright lattice and dark otherwise, we can obtain identical bright and dark lattices shown in Fig. 1(a)–(I) with a ₁=a(3^1/2/2, -1/2) and a ₂=a(0, 1) as their translation basis vectors. The corresponding primitive vectors in reciprocal space are b ₁=(2π/a)(2×3^1/2, 0), b ₂=(2π/a)(1/3^1/2, 1). The first Brillouin zone turns out to be a hexagon as shown in Fig. 1(a)–(II). Obviously the “atoms” in both the bright and dark lattices shown in Fig. 1(a)–(I) take the shape of approximately round-corner rectangle instead of circle, ellipse or square, and this shape slightly varies with intensity threshold. Due to the breakdown of symmetry laws, the k-lines Γ-M-K-Γ of 1/12 of the irreducible Brillouin zone are not sufficient to identify the minimum band gap for the non-symmetry “atom”. To calculate the photonic band structures, we consider k-points around the boundary of the irreducible Brillouin zone determined by five symmetry points Γ(0, 0, 0), M (1/(2×3^1/2), 1/2, 0), K (0, 2/3, 0), M′(-1/(2×3^1/2), 1/2, 0) and K′(1/3^1/2, 1/3, 0), in units of 2π/a. To confirm the existence of a band gap, we also compute the frequencies for k-points inside the Brillouin zone.

 

Fig. 1. (a) (I) 2-D triangular photonic lattice fabricated by four umbrellalike beam interference technique; (II) The first Brillouin zone with the symmetry points indicated.(b) The relation between threshold intensity and the filling ratio of dielectric material, where line (I) is for titania and line (II) is for GaAs.

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Using the photolithographic technique reported by Shishido etc. [10], 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. Based on this fact, we employ the block-iterative frequency-domain methods for Maxwell’s equations in a plane wave basis [11] to investigate the gap properties for 2-D triangular titania array. In these calculations, grid size is 32×32×1, mesh size is 7 and tolerance is 1×10^-7. If grid size is 128×128×1 and tolerance is 1×10¹⁰, the band frequencies differ from those calculated with the former parameters by a maximum less than 0.5%. Most bands differ by less than 0.2%. Thus, we believe that all results herein are accurate to within 0.5% of their true values. The chemical property of photoinitiator determines an intensity threshold. The polymerization occurs only when the intensity is over the threshold. By controlling the exposure intensity, introducing a intensity threshold I_t (here I_t∊(0, 1] for Eq.(1)), and washing out the underexposed regions, the filling ratio (FR) of titania can be selected. Consequently the dielectric constant distribution ε(x,y) of resultant lattice should be 1 in the region I₀(x,y)<I_t and 4 in the region I₀(x,y)≥I_t.

2.1 The PBG for TE or TM polarization

The relation between I_t and FR of the titania is shown in Fig. 1(b)–(I). 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. For TM polarization, the calculated band diagram indicates that no gap occurs when titania is the host dielectric. For TE polarization, only one gap between E₁ and E₂ (where E_i represents the ith band for TE polarization, and M_i labels that for TM polarization) appears when I_t is or above approximately 0.5 (the corresponding titania FR is about 81.6%). The width of the TE gap (measured by gap to midgap ratio Δω/ω ₀) varies with the FR as indicated in Fig. 2(a). Using Brent's algorithm [12], we can find the favorable FR that maximizes the TE gap. Band diagrams for TE polarization is shown in Fig. 2(b) in this case. Frequency axes in all plots are calculated using normalized frequency units f _n=ωa/2πc, where ω is the frequency and c is the speed of light). It shows that the maximum 10.8% TE gap opens (frequency range is 0.376~0.419) when I_t is 0.888 (the corresponding FR is 60.4%).

2.2 Directional PBG

Although no full PBG can be expected on PhCs discussed here, quite significant directional PBG can be obtained. For example, we examined the TE and TM band diagrams in Γ-M direction and a broad band gap possessed by TE and TM simultaneously is observed. Fig. 3(b) and (c) give the directional band diagram of the optimized gap, where the optimized I_t is 0.941 (FR=56.3%) and the band gap range determined by E₁ and M₂ is from 0.329 to 0.382, which means 14.9% relative band size is obtained. For comparisons, we also give the corresponding transmission spectra (in Γ-M direction) calculated with non-orthogonal FDTD method [13] in Fig. 3(a) and (d). In these calculations, a Berenger type of perfectly matched layer (PML) [14] is implemented as absorbing layer, the total length of the scattering region is 4×(3)^1/2 a, and the electromagnetic fields are integrated for 16 blocks of 1024 time steps each. Obviously, quite significant transmittance dips are observed near 0.356 for both TE and TM polarizations. The gaps in the band structure are exactly corresponding to minima of the transmittance power.

 

Fig. 2. Calculated results for TE polarization. (a)Gap to midgap frequency ratio (Δω/ω ₀) as a function of titania FR; (b) Optimized photonic band gap structure with FR=60.4%.

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Fig. 3. Calculated titania PhC transmission spectra (a and d) and directional photonic band diagrams (b and c) for TE and TM polarizations, respectively.

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2.3 The PBG of inverted GaAs structure

The important remaining question is how to turn these directional PBGs into the full PBGs. Several methods have been suggested for obtaining full PBGs in 2-D situations. 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 [7] or by using a photoresist that has been exposed to the interference field as mask for the GaAs etching [15]. In the following we examined a structure by supposing the dielectric constant ratio can reach 13.6:1 (13.6 is the GaAs’s dielectric constant) as other authors used [16]. Then the dielectric constant ε(x,y) equals 13.6 in the area I0(x,y)<I(t) and 1 otherwise.

The FR of the GaAs in this case is shown in Fig. 1(b)–(II). Adopting a similar analysis on data for TE polarization, one finds two PBGs at different FRs. Fig. 4 shows the band diagram for TE wave at the optimized intensity thresholds of 0.999 and 0.210 (the corresponding FRs are 49.4% and 7.1%), respectively. The first gap (Fig. 4(a)) locates between E₃ and E₄ (0.429~0.486) and the second (Fig. 4(b)) appears between E₄ and E₅ (0.953~1.016). Then two band gaps of 12.5% and 6.4% are obtained. The two gaps disappear rapidly because of the serious degeneracy along Γ-M’ direction when the FR deviates far from the optimized filling fraction. On the other hand, many gaps appear for TM polarization and their number varies with the I_t. Fig. 5(a) shows the dependence of the PBG on the I_t, extracted from band structure calculations when I_t changes from 0.0 to 1.0. We plot in Fig. 5(b) the calculated sizes of the forbidden gaps of TM wave normalized to the midgap frequency for different I_t. The maximum relative gaps for M_1-2, M_2-3, M_4-5, M_5-6 and M_8-9 (where M_i-j represents the gap appearing between the ith and jth TM bands) are 43.3%, 5.3%, 9.9%, 6.7% and 8.8%; the central frequencies of these gaps are 0.399, 0.454, 0.664, 0.562 and 0.717; the intensity thresholds are 0.233, 0.486, 0.505, 0.840 and 0.823; and the corresponding FRs are 7.8%, 17.9%, 18.7%, 36.4% and 35.2%, respectively. These aforementioned results further indicate the rule of thumb for photonic band gap [2]: TM band gaps are favored in a lattice of isolated high-ε regions, and TE-gaps are favored in a connected lattice.

In Ref. [17], Villeneuve etc. have shown that single-atom triangular lattices of dielectric rods in air do not give rise to full PBGs even when the dielectric constant is up to 20.25, and asymmetry is introduced to lift the band degeneracies. While in our studies, the degeneracy of TE modes along Γ-M’ attenuates and the PBG of TE modes begins to appear between the third and fourth band (E_3-4) when the intensity threshold is more than about 0.8, and the center frequency of E_3-4 is around 0.573. The center frequency for the 5–6 band gap of TM modes M_5-6 is around 0.579 in the same case. Therefore, the two band gaps begin to overlap. The central frequencies of E_3-4 and M_5-6 move downward with the increase of intensity threshold, and the down-moving speed of TE polarization band is a bit faster than that of the TM polarization band, as shown in Fig. 5(a). The most interesting result for the single-atom triangular structure of GaAs rods in air which has not been reported before [16] is that a maximum 2.8% full PBG between M₅ and E₄ (from 0.521 to 0.536) shown in Fig. 6 appears when intensity threshold is 0.900 (the corresponding FR is 40.5%). Although the absolute PBG is not large, it has showed the important effect of “atom” shape on PBG properties.

 

Fig. 4. Calculated optimized photonic band structure for TE polarization with the dielectric constant ratio 13.6:1. (a) The FR of GaAs corresponding to the maximum E_3–4 is 49.4%; (b) The FR of GaAs corresponding to the maximum E_4–5 is 7.1%.

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Fig. 5. (a)Gap map for the inverted GaAs structure, where arrows indicate these maximum TM Gap-midgap ratios; (b)TM Gap-midgap ratios as a function of intensity threshold.

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Fig. 6. The photonic band structure in the optimized case that the intensity threshold is 0.900. The solid curves are for the TE polarization, and the dotted curves are for the TM.

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3. Larger bandgaps obtained by another recording geometry

To obtain some larger bandgaps, we also investigated another triangular lattice that can be formed by the recording geometry proposed in Ref. [9][18]. The intensity of interference field in that case 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 with a1=a(1/2,3^1/2/2) and a ₂=a(-1,0) as their translation basis vectors. Adopting a similar analysis, for the inverted GaAs structure, Fig. 7 shows the effect of the parameter I_t on the size and position of the band gaps. The gap map clearly indicates that up to four full PBGs appear during I_t changes from 1.6 to 2.5, and their number varies with the I_t. By using Brent’s algorithm in the calculation of band, the four PBGs, B_3–4 (where B_i-j presents the band gap locating between the ith and the jth bands), B_9–10, B_12–13 and B_22–23 (due to the overlap of E_1–2 and M_2–3, E_3–4 and M_6–7, E_5–6 and M_7–8, and E_9–10 and M_13–14), can be optimized. The maximum relative gaps for B_3–4, B_9–10, B_12–13 and B_22–23 are 18.7%, 8.2%, 6.7% and 2.6%, respectively. The most valuable result not indicated in Section 2 is that four sizeable full PBGs simultaneously appear when I_t is about 2.0 (see Fig.7). It has shown the important effect introduced by different incident wave design on PBG properties.

 

Fig. 7. Gap map of full PBG for the inverted GaAs structure

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4. Conclusions

In summary, using the block-iterative frequency-domain method and non-orthogonal FDTD method, we have provided a band gap analysis for two classes of 2-D triangular lattice. Through the above analysis, 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 results show that, for the 2-D triangular lattice, larger bandgaps can be obtained by proper recording geometry design. This rule can be further extended to the analysis on the other 2-D and 3-D lattices.