Space-group theory of photonic bands

The wide-range application of photonic crystals and metamaterials benefits from the enormous design space of three-dimensional sub-wavelength structures. In this work, 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 non-zero frequencies~($M'$). 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 work provides theoretical guidelines for the choice of spatial symmetries in photonics design.