Space Group Theory of Photonic Bands

The wide-range application of photonic crystals and metamaterials benefits from the enormous design space of three-dimensional subwavelength structures. In this Letter, we study the space group constraints on photonic dispersions for all 230 space groups with time-reversal symmetry. Our theory carefully treats the unique singular point of photonic bands at zero frequency and momentum, which distinguishes photonic bands from their electronic counterpart. The results are given in terms of minimal band connectivities at zero ($M$) and nonzero frequencies ($M^{\prime }$). Topological band degeneracies are guaranteed to be found in space groups that do not allow band gaps between the second and third photonic bands ($M>2$). Our Letter provides theoretical guidelines for the choice of spatial symmetries in photonics design.

Figure 1

Two types of band connectivities ($M$ and $M^{\prime }$) are distinguished by whether they are connected to $\omega =|\overrightarrow{k}|=0$.

Figure 2

Consequence of a nonsymmorphic symmetry on band connectivities. (a) $M=4$ for $\mathit{4}$ ($P{2}_1$) containing a twofold screw. (b) $M=2$ for $\mathit{7}$ ($Pc$) containing a glide. The numbers aside the vertical axes and the colors of the dispersions indicate the eigenvalue of the nonsymmorphic symmetry.

Figure 3

A possible band structure of space group $\mathit{23}$ ($I222$) along three different lines connecting $\mathrm{\Gamma }$ and $X$. The numbers aside the vertical axes indicate the eigenvalues ${\zeta }_{\alpha }$ of the $C_{2\alpha }$ rotation for $\alpha =x$, $y$, $z$. The three eigenvalues at the same point $X$ must satisfy ${\zeta }_x{\zeta }_y={\zeta }_z$, except at $\mathrm{\Gamma }$.

Figure 4

Examples of new photonic crystals with $M=2$ and $M>2$. (a) Space group $\mathit{131}$ ($P{4}_2/mmc$) with rod radius $0.15a$. There is a full band gap of 5% between the second and third bands. All bands along $A\text{−}Z$ are doubly degenerate. (b) Space group $\mathit{223}$ ($Pm\overline{3}n$) with rod radius $0.1a$. There is a sixfold degeneracy point at $R$. All bands along $R-X$ are doubly degenerate. Insets are the real-space structures of dielectric constant $\epsilon =13$ in the cubic unit cell of lattice constant $a$.