Photonic band structure calculations of two-dimensional Archimedean tiling patterns

We present a study of photonic band structures of two-dimensional Archimedean tiling patterns. The tilings we have investigated are $(4.8^2)$, $\left(6^3\right)$, (4.6.12), and $(3^2.4.3.4)$, which have been discovered computationally and experimentally in self-assembled microphase separation of $ABC$ star block terpolymer systems. Using plane-wave method, we have calculated eigenvalue equations for various combinations of dielectric contrast on the complex patterns. We demonstrate the existence of complete photonic band gaps in the (4.6.12) structure. Furthermore, we find that complete photonic bands readily open in the $(3^2.4.3.4)$ structures in the same way as in dodecagonal quasicrystals. Complex tilings open up a way to construct photonic crystals.

Figure 1

Archimedean tilings: A set of integers $(n_1.n_2.n_3.\cdots )$ denotes a tiling of a vertex type in the way that $n_1$-gon, $n_2$-gon, and $n_3$-gon, …, meet consecutively on each vertex. Superscripts are employed to abbreviate when possible. There exist only 11 types of tiling by regular polygons, where all vertices are of the same type.

Figure 2

(Color online) MC simulation results: Phase diagram as functions of arm-length ratio of $ABC$ star terpolymer system with symmetric interactions between three components. Morphologies in the central region are five cylindrical structures: $\left({4.8}^2\right)$, $\left(6^3\right)$, $(3^2.4.3.4)$, dodecagonal quasicrystal, and (4.6.12). Three-dimensional structures are perforated layer (PL), $\text{lamella}+\text{cylinder}$ $(\mathrm{L}+\mathrm{C})$, single diamond (Diamond), columnar piled disk (CPD), lamella-in-sphere (L-in-S), and $\text{lamella}+\text{sphere}$ $(\mathrm{L}+\mathrm{S})$. See details in Ref. 2.

Figure 3

(Color online) MC simulation result of $(3^2.4.3.4)$ by $A9B7C14$ star-shaped terpolymers in a box with $25\times 48\times 48$. Small domains represent $B$, large ones are $C$, and $A$ is transparent. (a) Top view. The skeleton $(3^2.4.3.4)$ net is superimposed. (b) Cylindrical structure is shown. Both pictures represent the volume rendering of averaged densities over ${10}^3$ MC steps at $\beta =0.071$.

Figure 4

(Color online) Minimal fundamental triangles of (a) $\left(6^3\right)$, (b) (4.6.12), and (c) $\left({4.8}^2\right)$ Archimedean tilings, which belong to the single junction class. Limiting cases (d)–(f) and the resulting tiling (g) (3.6.3.6), (h) (3.4.6.4), and (i) $\left(4^4\right)$, respectively. Other limiting cases (j) $\left(3^6\right)$, (k) (3.6.3.6), (l) $\left(4^4\right)$, and (m) $\left({3.12}^2\right)$. (n) St. Andrew’s cross having curved polygons from the minimal triangle of $\left({4.8}^2\right)$.

Figure 5

Area ratios of photonic crystals: (a) $A:B:C=1:1:1$ in the $\left(6^3\right)$ structure, (b) $A:B:C=2:1:2$ in the $\left({4.8}^2\right)$ structure, and (c) $A:B:C=1:1:2$ in the (4.6.12) structure. These ratios are used in photonic band calculations.

Figure 6

Unit cell of the $(3^2.4.3.4)$ structure. The area ratio calculated in this paper is $A:B:C=9:7:14$ or 9:7:16. The two-dimensional space group is $p4gm$.

Figure 7

Plots of the Fourier series expansions of $1∕\epsilon \left(\mathbf{x}\right)$ by means of (a) $l=11$ (529 waves) and (b) $l=30$ (3721 waves) for $(3^2.4.3.4)$ with dielectric constants ${\epsilon }_A=13$, ${\epsilon }_B=13$, and ${\epsilon }_C=1$, where $l$ is the maximum integer of $\mid l_1\mid$ or $\mid l_2\mid$ in Eq. (4). Shaded regions indicate $1∕\epsilon <0.13$.

Figure 8

Band gap as a function of $1∕l$ for $(3^2.4.3.4)$ with dielectric constants ${\epsilon }_A=13$, ${\epsilon }_B=13$, and ${\epsilon }_C=1$, where $l$ is the maximum integer of $\mid l_1\mid$ or $\mid l_2\mid$ in Eq. (4). Upper points correspond to the bottom of the air band, and lower points correspond to the top of the dielectric band. From right to left, these points correspond to $l=11$, 16, 22, 25, 30, and 37, namely, 529, 1089, 2025, 2601, 3721, and 5625 plane waves, respectively. These lines converge to 0.504 and 0.481, indicating that the band gap remains.

Figure 9

Photonic band structure for $\left(6^3\right)$ with dielectric constants ${\epsilon }_A=13$, ${\epsilon }_B=1$, and ${\epsilon }_C=1$: $E$ polarization (solid line) and $H$ polarization (broken line).

Figure 10

Photonic band structure for $\left(6^3\right)$ with dielectric constants ${\epsilon }_A=1$, ${\epsilon }_B=13$, and ${\epsilon }_C=13$: $E$ polarization (solid line) and $H$ polarization (broken line).

Figure 11

Photonic band structure for $\left({4.8}^2\right)$ with dielectric constants ${\epsilon }_A=13$, ${\epsilon }_B=1$, and ${\epsilon }_C=1$: $E$ polarization (solid line) and $H$ polarization (broken line).

Figure 12

Photonic band structure for $\left({4.8}^2\right)$ with dielectric constants ${\epsilon }_A=1$, ${\epsilon }_B=13$, and ${\epsilon }_C=13$: $E$ polarization (solid line) and $H$ polarization (broken line).

Figure 13

(Color online) Photonic band structure for (4.6.12) with dielectric constants ${\epsilon }_A=13$, ${\epsilon }_B=1$, and ${\epsilon }_C=1$: $E$ polarization (solid line) and $H$ polarization (broken line). The gap-midgap ratios $(\Delta \omega ∕\omega )$ are $9.3\times {10}^{-2}$ (bottom) and $3.0\times {10}^{-2}$ (top).

Figure 14

Photonic band structure for (4.6.12) with dielectric constants ${\epsilon }_A=13$, ${\epsilon }_B=13$, and ${\epsilon }_C=1$: $E$ polarization (solid line) and $H$ polarization (broken line).

Figure 15

(Color online) Photonic band structure for $(3^2.4.3.4)$ with dielectric constants ${\epsilon }_A=13$, ${\epsilon }_B=1$, and ${\epsilon }_C=1$: $E$ polarization (solid line) and $H$ polarization (broken line). The gap-midgap ratios $(\Delta \omega ∕\omega )$ are $2.2\times {10}^{-2}$ (bottom) and $3.1\times {10}^{-2}$ (top).

Figure 16

(Color online) Photonic band structure for $(3^2.4.3.4)$ with dielectric constants ${\epsilon }_A=1$, ${\epsilon }_B=1$, and ${\epsilon }_C=13$: $E$ polarization (solid line) and $H$ polarization (broken line). The gap-midgap ratio $(\Delta \omega ∕\omega )$ is $5.1\times {10}^{-2}$.

Figure 17

(Color online) Photonic band structure for $(3^2.4.3.4)$ with dielectric constants ${\epsilon }_A=13$, ${\epsilon }_B=13$, and ${\epsilon }_C=1$: $E$ polarization (solid line) and $H$ polarization (broken line). The gap-midgap ratio $(\Delta \omega ∕\omega )$ is $5.4\times {10}^{-2}$.

Figure 18

Electric field of near $M$-point $E$ polarization mode in the dielectric band for $(3^2.4.3.4)$ with dielectric constants ${\epsilon }_A=13$, ${\epsilon }_B=13$, and ${\epsilon }_C=1$. Different shades indicate the plus and minus of the field. The field mainly concentrates into $A$ regions, leading to the reduction of energy.

Figure 19

Electric field of near $M$-point $E$ polarization mode in the air band for $(3^2.4.3.4)$ with dielectric constants ${\epsilon }_A=13$, ${\epsilon }_B=13$, and ${\epsilon }_C=1$. Different shades indicate the plus and minus of the field. The field mainly concentrates into $B$ regions; however, it invades $C$ air cylinders more and thus increases the displacement energy.

Figure 20

The largest gap-midgap ratio against dielectric fluctuations: $E$ polarization of dielectric cylinders (solid circle) and $H$ polarization of air cylinders (open circle) are plotted for $\left(6^3\right)$, $\left({4.8}^2\right)$, (4.6.12), and $(3^2.4.3.4)$. Double circles correspond to the cases of complete photonic band gaps. The least-squares fit is estimated from solid or open data points; the slope is 0.19 or 0.46, respectively.