Complete photonic band gaps in supercell photonic crystals

We develop a class of supercell photonic crystals supporting complete photonic band gaps based on breaking spatial symmetries of the underlying primitive photonic crystal. One member of this class based on a two-dimensional honeycomb structure supports a complete band gap for an index-contrast ratio as low as $n_{\mathrm{high}}/n_{\mathrm{low}}=2.1$, the lowest index contrast known to support a complete band gap for a two-dimensional (2D) photonic crystal. This same design principle is used to develop the first photonic crystal slab that supports a band gap for dual-polarization visible light. The complete band gaps found in such supercell photonic crystals do not necessarily monotonically increase as the index contrast in the system is increased.

Figure 1

[(a), (c), (e)] Schematics of 2D photonic crystals formed of a network of a high dielectric material with $n=2.4$ (black region) in a background of air, $n_{\mathrm{air}}=1$. The unit cells for the corresponding band structures in panels (b), (d), and (f) are shown as green diamonds or hexagons. Parameters which characterize these crystals are indicated in purple. The structures in panels (a) and (c) are completely specified by the line thickness, $t$, while the structure in panel (e) is completely specified by three parameters, the thick line thickness, $t_1$, the thin line thickness, $t_2$, and the shortest distance from a thick line to the center of its corresponding thick-bordered hexagon, $r$. $l$ denotes the distance between two vertices in the primitive honeycomb lattice. [(b), (d), (f)] Band structures around the border of the irreducible Brillouin zone for the photonic crystals shown in panels (a), (c), and (e), respectively, in which the TE bands are shown in red and the TM bands are shown in blue. In panel (b), the second TM band is shown as cyan (light gray), while the corresponding folded bands in the supercell Brillouin zone in panels (d) and (f) are also shown in the same color. In panels (b) and (d), $t/l=0.3636$, while in panel (f), $t_1/l=0.4149$, $t_2/l=0.0816$, and $r/l=0.8145$, and a complete band gap is found with width $\mathrm{\Delta }\omega /\overline{\omega }=8.6\%$ between the eighth and ninth bands (yellow rectangle). Band structures were calculated using MIT photonic bands (MPB) [45].

Figure 2

Plot of the optimized complete bandgap width as a function of the index contrast, $n_{\mathrm{high}}/n_{\mathrm{low}}$ for six different 2D photonic crystals, the supercell network honeycomb lattice discussed in Fig. 1 (red squares), the decorated square lattice designed in Ref. [30] (green diamonds), the traditional inverse triangular lattice (magenta downward-pointing triangles), the decorated network honeycomb lattice designed in Ref. [34] (blue circles), the traditional network square lattice (dark green upward-pointing triangles), and the supercell network square lattice discussed in Fig. 4 (yellow right-pointing triangles). In the schematics, the black regions correspond to the high index material, $n_{\mathrm{high}}$, while the white regions correspond to low index material, $n_{\mathrm{low}}$. Optimized complete bandgaps were calculated using MPB [45].

Figure 3

(a) Schematics of a 3D photonic crystal slab consisting of titanium dioxide hexagonal rods with $n=2.57$ at $\lambda =650$ nm mounted on a periodic substrate of silica, $n=1.45$, with air above, $n=1$. Each titanium dioxide hexagonal rod also contains a hexagonal air hole in its center, while the substrate rods are solid. This system can be completely parameterized using exactly the same choice of $t_1$, $t_2$, and $r$ from the 2D photonic crystal shown in Fig. 1, as well as the height of each rod, $h$, and the width of the substrate rods, $w$. For the structure in panel (a), $t_2=0$. (b) Band diagram for the photonic crystal slab shown in panel (a). The gray regions of the band diagram indicate the continuum of radiation modes which lies above the light line. As can be seen, a complete band gap with width $\mathrm{\Delta }\omega /\overline{\omega }=5.6\%$ (yellow rectangle) opens between the seventh and eighth bands for $t_1/l=0.7482$, $t_2/l=0$, $r/l=0.7405$, $h/l=3.750$, and $w/l=1.0392$, where $l$ is again the distance between two vertices in the underlying primitive honeycomb lattice. For $\overline{\lambda }=650$ nm tuned to the center of the band gap, the band gap would extend over the range 633–667 nm, with $t_1=78.9$, $r=78.0$, $h=394.9$, and $w=109.4$ nm. Likewise, for $\overline{\lambda }=1.55\mu \text{m}$, the band gap extends over 1.509–$1.591\mu \text{m}$, with $t_1=188$, $r=186$, $h=941$, and $w=261$ nm. Band structures were calculated using MPB [45].

Figure 4

[(a), (c), (e)] Schematics of 2D photonic crystals formed of a network of a high dielectric material with $n=3.2$ (black region) in a background of air, $n_{\mathrm{air}}=1$. The unit cell for the corresponding band structure in panels (b), (d), and (f) is shown as a green square. Parameters which characterize these crystals are indicated in purple. The structures in panels (a) and (c) are completely specified by the line thickness, $t$, while the structure in panel (e) is completely specified by three parameters, the thick-line thickness, $t_1$, the thin-line thickness, $t_2$, and the shortest distance from a thick line to the center of its corresponding thick-line bordered square, $r$. $l$ denotes the distance between two vertices in the primitive cell. [(b), (d), (f)] Band structures around the border of the irreducible Brillouin zone for the photonic crystals shown in panels (a), (c), and (e), respectively, in which the TE bands are shown in red and the TM bands are shown in blue. In panel (b), the second and third TM bands are shown as light blue (medium gray) and cyan (light gray), respectively, while the corresponding folded bands in the supercell Brillouin zone in panels (d) and (f) are also shown in the same colors. In panels (b) and (d), $t/l=0.32$, while in panel (f), $t_1/l=0.34$, $t_2/l=0.16$, and $r/l=0.605$, and a complete band gap is found with width $\mathrm{\Delta }\omega /\overline{\omega }=4.6\%$ between the 12th and 13th bands (yellow rectangle). Band structures were calculated using MPB [45].

Figure 5

(a) Schematic showing a coarse pixelized version of the supercell honeycomb photonic crystal. The unit cell is depicted by the green hexagon, the irreducible spatial region is the blue triangle, and those pixels which are independent degrees of freedom are shown as cyan diamonds. (b) Initial dielectric pattern chosen for the nonlinear optimization algorithm. There are 110 independent degrees of freedom in this structure. (c) Optimized band gap as a function of index contrast for the pixelized structures. The optimization process produced three different types of structures, examples of which are shown in panels (d)–(f), with panel (d) being the structure for $n_{\mathrm{high}}/n_{\mathrm{low}}=2.2$, (e) $n_{\mathrm{high}}/n_{\mathrm{low}}=2.5$, and (f) $n_{\mathrm{high}}/n_{\mathrm{low}}=3.3$.

Figure 6

(a) Numerically optimized complete photonic band gaps as a function of index contrast for five different simplified structures depicted in panels (b)–(e), and the structure from Fig. 1. Parameters of the structures are indicated in purple.