Role of Short-Range Order and Hyperuniformity in the Formation of Band Gaps in Disordered Photonic Materials

We study photonic band gap formation in two-dimensional high-refractive-index disordered materials where the dielectric structure is derived from packing disks in real and reciprocal space. Numerical calculations of the photonic density of states demonstrate the presence of a band gap for all polarizations in both cases. We find that the band gap width is controlled by the increase in positional correlation inducing short-range order and hyperuniformity concurrently. Our findings suggest that the optimization of short-range order, in particular the tailoring of Bragg scattering at the isotropic Brillouin zone, are of key importance for designing disordered PBG materials.

Figure 1

Structural order in (a),(b),(e),(f) HD patterns at $\varphi =0.60$ and (c)–(f) SHU patterns at $\chi =0.50$. We compare (a),(c) typical point patterns in real space, (b),(d) structure factors $S(\mathbf{k})$, (e) radial distribution functions $g(r)$, and (f) angle-averaged structure factors $S(k)$. The two-dimensional plots in (b),(d) confirm that the distributions are isotropic. The inset in (e) shows that a correlation gap is present in HD and SHU packings for $r<a$. The inset in (f) demonstrates that long-wave length fluctuations $2\pi /k>2\pi /K$ are strongly suppressed (up to numerical noise) in the SHU pattern. All data is obtained by taking an average over 1000 independent patterns with $N=200$ points each. Vertical lines indicate the position of the band edges in reciprocal space for TM polarization. (g) Photonic properties of a dielectric structure composed of a network of silicon rods placed at the position of the points as shown in (a),(c). NDOS as a function of the normalized frequency $\nu a/c$ in TM polarization for the HD system at $\varphi =0.60$ and the SHU system at $\chi =0.50$. Here, $c$ is the speed of light in vacuum.

Figure 2

Decorated network structure derived from (a) a HD seed pattern with $N=200$, $\varphi =0.60$, and (c) a SHU seed pattern with $N=200$, $\chi =0.50$. (b) and (d) The average of the square amplitude of the corresponding Fourier transforms taken over 1000 network realizations.

Figure 3

Formation of PBGs for TM modes, TE modes and for both polarizations simultaneously. The normalized PBG width is plotted as a function of the parameters $\chi$ and $\varphi$ for (a) SHU and (b) HD seed patterns up to a maximal value of ${\chi }_{\max }=0.79$ and ${\varphi }_{\max }=0.91$. The shaded area marks the appearance of first signs of peaks in reciprocal space indicating the presence of quasi-long-range crystalline order. The dashed lines indicate the normalized PBG widths for otherwise identical structural designs when the points are arranged according to a hexagonal lattice seed pattern.

Figure 4

Robustness of the band gap for a HD structure at $\varphi =0.6$ (a) and a SHU structure at $\chi =0.5$ (b). The plot shows the averaged normalized PBG width for networks of dielectric walls, where a percentage $p$ of links is removed randomly. The shaded area represents the smallest and largest gap within the ensemble of ten configurations studied. The calculation was performed in TE polarization.