Dirac Points for Square Lattice Photonic Crystals

This work is concerned with Dirac points in dispersion surfaces associated with the wave equation where the material coefficient has the translational symmetries of a square lattice. It is proved in Keller et. al. (2018) that the spectrum of the Schrödinger operator over a square lattice does not have a Dirac point at the high symmetry point of the Brillouin zone, at least for the lower bands. In the high-contrast limit, we prove the existence of a Dirac point for the 2D Helmholtz operator over a square lattice away from the high symmetry point of the Brillouin zone, if the dielectric function of the periodic medium has reflection symmetry. In particular, the two-dimensional eigenspace at the Dirac point is spanned by eigenfunctions with opposite parity. Based on this observation, we propose a computational inverse design method using the local density-of-state method to generate a Dirac point at the desired Bloch wave vector and frequency. Numerous numerical results demonstrate the capability of our framework to generate a desired Dirac point robustly.