1. Introduction

Photonic bandgap (PBG) materials that facilitate the localization of light are of interest in optical communication and optical integration circuits. PBGs can be achieved in both photonic crystals (PCs) [1] and photonic quasicrystals (PQCs) [2,3]. Quasicrystals (QCs) are a special class of aperiodic crystals, which has long-range orientational order but lack translational symmetry. In comparison to conventional two-dimensional (2D) PCs, which are typically square or triangular lattices with an isotropic circular-shaped motif on the each lattice point [1], the PQCs can offer higher rotational symmetries, more isotropic Brillion zone (BZ), and hence potentially can open more uniform PBGs at a lower dielectric contrast [3]. While various PQCs have been fabricated experimentally [4,5], precise theoretical prediction on their photonic band structures remains challenging due to the non-periodic nature.

It has been suggested that the unique PBG behaviors in PQCs are dominated by short range environment [2]. Recently, a few 2D complex periodic structures have been constructed from small portions of QCs. In this case, the calculation work is not as sophisticated as that for PQCs, at the same time they can retain or approximate the desirable properties of PQCs [6,7]. While attractive, little has been studied on the effect of the selected QC portions and their arrangement on the PBGs at various dielectric contrast levels.

In order to understand the influence of local geometries and total structural symmetries on PBG, we construct fourteen complex periodic 2D PCs with different structural symmetries by arranging small portions of the square-triangular tiling 12-fold QC [3,8]. In comparison to other QCs, such as octagonal tiling [2], Penrose tiling [9] and regular Stampfli tiling with rhombi, squares and equilateral triangles as elements [10], the square-triangular dodecagonal tiling QC is simple and has the same nearest neighbor distance. Therefore, it is the focus of this study. The structural details of 2D patterns can be fine-tuned by varying lattice symmetry (hexagonal vs. square), QC portions [Fig. 1(a-c) ] and their arrangements. The PBGs of the corresponding 2D structures are calculated at two dielectric contrast levels [Si (ε = 13) and glass (ε = 2.1) rods in air]. We find that these 2D complex structures possess similar PBG properties in the Si PCs, whereas the PBG behaviors become much different in the glass PCs when varying the arrangements. In order to understand the basis that contributes to the difference in PBG behaviors in the two dielectric materials, we investigate the corresponding reciprocal lattices. All complex 2D patterns possess 12-fold-like features due to the analogue short-range environments. The similarity of the PBGs in Si PCs suggests that the PBGs at the high dielectric contrast level are dominated by the local geometry, which leads to the 12-fold-like characteristics in the reciprocal lattices. However, the role of the fine features in the reciprocal lattices, which are caused by the difference in arrangements of the basic elements, structural symmetries and coupling effect between local and lattice symmetries, becomes increasingly important as dielectric contrast is decreased. Further, we find that the lowest dielectric contrast required to open the transverse magnetic (TM) gap in the complex lattices with dielectric columns in air can approach as low as ε = 1.44, and the inverse structures with air-in-dielectric configuration may open a 2D complete gap in silicon nitride (ε = 4.1).

 

Fig. 1 (a-c) 12-fold QCs. The shadowed areas illustrate portions with different sizes. (d-f) The corresponding Fourier transformation of the lattice points shown in the shadowed areas. (g, h) Illustration of the same portion as shown in the red shadowed area in (a). (i, j) Illustration of the same portion as shown in the green shadowed area in (b). (g) and (h) or (i) and (j) are rotated by 90° respectively.

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2. Construction of complex 2D photonic structures from small portions of 12-fold quasicrystal and the corresponding structural symmetries

First, we studied the tiling elements from small portions of square-triangular tiling 12-fold QC with different sizes [Fig. 1 (a-c)]. Their Fourier transformations [Fig. 1 (d-f)] show that the 12-fold-like feature was maintained in all three portions in the high frequency region, which was attributed to the local geometry.

We then embedded such small portions in different lattices to construct fourteen complex periodic 2D photonic structures [Fig. 2 ] based on the equilateral triangular-square tiling, therefore, they maintain the same nearest neighbor distance as the 12-fold QC. Structures shown in Fig. 2(a-k) were constructed from the elements shown in Fig. 1(i) and (j), denoted as A and B, respectively. The elements were arranged in a hexagonal lattice [Fig. 2(a-g)] and a square lattice [Fig. 2(h-k)], respectively, and connected through “square-square” contact [Fig. 2(a)], “triangular-triangular” contact [Fig. 2(b)], and “square-triangular” contact [Fig. 2(c-g)]. In the latter, the ratios of A to B in the unit cells were varied as 2:1, 3:1, 1:2, 1:3 and 1:1, respectively. The 2D patterns shown in Fig. 2(a-f) belong to the plane group p6mm, whereas the total symmetry in Fig. 2(g) is reduced to p2mm.

 

Fig. 2 Complex periodic 2D structures constructed with small portions of 12-fold QCs showed in Fig. 1. The constructed rules are showed by blue and red veins or points. The grey shadowed areas indicate the unit cells. Inset: the corresponding plane groups.

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The structure in Fig. 2(h) was obtained by embedding the elements A (or B by rotating 90°) on a square lattice. However, its symmetry belongs to p2mm, and thus it has no 4-fold rotational symmetry. In Fig. 2(i), the element A (or B) was surrounded by the element B (or A) at the nearest neighbor positions, therefore, the ratio of A to B is 1:1 and the structure belongs to the plane group of p4gm. In Fig. 2(j), each A (or B) was surrounded by eight B’s (or A’s by rotating 90°) at the positions of the nearest and the second nearest neighbors. The ratio of A to B is 1:3 (or 3:1), and the total structural symmetry is p2mm. Structure in Fig. 2(k) contains the alternating arrays of A and B with the ratio of A to B = 1:1, and it belongs to the plane group p2mm.

The structures shown in Fig. 2(l-n) were constructed from different portions of the 12-fold QC. The one in Fig. 2(l) was built by arranging the elements shown in Fig. 1(a, g, h) in a square lattice, and its total symmetry is p4gm. The structures in Fig. 2(m) and (n) were constructed with the larger portion of 12-fold QC shown in Fig. 1(c). Their total symmetries belong to the plane group p2mm and p4gm, respectively.

3. PBGs of complex 2D photonic crystals at various dielectric contrast levels

The PBGs were calculated using MIT photonic bandgap (MPB) package [11] from the Si (ε = 13) or glass (ε = 2.1) rods in the air background with the 2D complex structures of infinite size in xy plane and an infinite height in z direction. Here, we focus on TM modes since isolated dielectric rods are conductive to TM gaps, whereas a connected lattice favors transverse electric (TE) gaps [1]. For each dielectric material, we optimized the filling fraction to achieve the maximum TM gap. Since the relative deviation from optimal filling fractions, ~12% for Si PCs and ~32% for glass PCs, was less than 2%, we performed PBG calculations at the optimal filling fraction of each dielectric material. The PBG results are summarized in Table 1 .

Table 1. TM bandgap properties in Si and glass photonic crystals with complex 2D structures.

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For 2D Si PCs investigated, the average mid-gap position for TM modes is 0.357 c₀/a, and the relative standard deviation (RSD) is extremely small, 0.368%. Here, a is the distance between the nearest neighbors, c₀ is the light velocity in vacuum, and c₀/a is the normalized frequency unit. The mean value of quality factor (QF = gap/mid-gap) of the TM gaps is 47.12% with RSD = 0.738%. All Si PCs possess similar mid-gap positions and QFs, suggesting that at a high dielectric contrast the PBG size and the mid-gap position may be determined by the local geometries, which are similar in all fourteen complex structures. Supporting this, the corresponding reciprocal lattices [Fig. 3(a-n) ] all display 12-fold-like features due to the analogue short-range arrangements. The results also imply that the complex periodic 2D structures could achieve more circular BZs than conventional 2D PCs.

 

Fig. 3 Reciprocal lattices of the complex 2D structures seen in Fig. 2(a-n), showing 12-fold-like features, in comparison with that of the square-triangular tiling 12-fold quasicrystal (o).

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Further, all fourteen structures can open TM gap with dielectric contrast lower than 2 (Table 1), and the lowest contrast of 1.44 can be achieved from structures shown in Fig. 2 (a). Specifically, all glass PCs possess TM gaps with an average gap position at 0.497 c₀/a and a small RSD, 0.742%. The latter is expected since the structures have the analogue 12-fold-like local geometries. However, the RSD of QF in glass PCs is 26.9%, much greater than that in Si PCs, 0.738%. In addition, the structures belonged to the plane group p6mm did not open the largest PBG. Instead, the champion TM gap with QF of 4.99% was achieved in the structure shown in Fig. 2(i) with a total symmetry p4gm. Interestingly, we found that the structure with symmetry p2mm [Fig. 2(g)] could open a relatively large TM gap (QF = 4.23%). It indicates that a high total symmetry may not be essential to open large PBGs in the complex structures.

Since the QC portions are superimposed on the triangular or square lattices, the local geometry, the lattice, and the coupling between them, could influence PBGs of the complex 2D PCs. Therefore, even using the same QC portion and the same lattice, the PBG quality could be different depending on the arrangement of elements. It is worth mentioning that additional lower TM gaps exist at 0.463 c₀/a with QF of 1.70% and at 0.461 c₀/a with QF of 1.60% in the glass PCs for structures shown in Fig. 2(a) and (e), respectively. The additional gaps can be attributed to the coupling between the Fourier transforms of the QC portions and the triangular reciprocal lattice.Moreover, we investigated the size of the QC portions to TM gap in the glass PCs. The structure [Fig. 2(l)] constructed from the smallest QC portion [Fig. (a), (g), (h)] can open up a large TM gap with QF = 4.44%. The employment of smaller portions to approximate local symmetry is attractive since it can decrease the size of unit cell, hence reducing computational work. However, structures in Fig. 2(m) and (n) were constructed from a larger QC portion [Fig. 1(c)] have completely different total symmetries, for example, p2mm for Fig. 2(m) and p4gm for Fig. 2(n), leading to different TM gap sizes: QF = 1.63% and 4.51%, respectively.

Zoorob et al. have shown that complete PBGs for both TE and TM polarizations can be achieved in the 12-fold PQCs (air-in-dielectric) at a very low dielectric contrast [3]. However, doubts remain how valid these claims are [12]. Here, we investigated the threshold to realize complete PBGs in the complex 2D PCs with air-in-dielectric configuration. The structures shown in Fig. 2(a) and (b) were chosen as the model structures since their primitive cells are relatively simple and contain fewer lattice points. Our calculation found that the lowest dielectric contrast to open a complete PBG was ε = 3.9 at an optimum air filling fraction, f _air = 56.4% from structure shown in Fig. 2(a), and ε = 4.1 at f _air = 44.3% from structure shown in Fig. 2(b), respectively. Hence, the complex structures may realize complete PBGs in silicon nitride, but not in glass.

4. Conclusion

We have investigated PBG properties of fourteen 12-fold-like dielectric rods-in-air structures with different structural configurations and QC portions at different dielectric levels. All structures show analogue TM PBGs at a high dielectric contrast (ε = 13) due to the similar local geometries. At a lower dielectric contrast (ε = 2.1), the arrangements of the basic elements, structural symmetries, and the coupling between the local and the lattice symmetries became noticeably more important, and a large deviation of TM gap size was observed. To open a TM PBG, the dielectric contrast could be as low as ε = 1.44, whereas the 2D complete PBG for both TE and TM polarizations might be achieved in the inverse structures with air-in-silicon nitride. The investigation of the reciprocal lattices has clearly shown that the complex 2D structures can create more circular BZs than conventional 2D PCs. Further, complex structures with low total rotational symmetries can open large TM PBGs in glass. Our study suggests that it will be important to design the local geometry in photonic crystals to manipulate the BZ and PBGs.