On the mapping of Dirac-like cone dispersion in dielectric photonic crystals to an effective zero-index medium

Two-dimensional (2D) dielectric photonic crystals with accidentally induced Dirac-like points (DLPs) at the Brillouin zone center behave as an effective zero-index medium under certain conditions. We investigated different 2D square lattice dielectric PCs, such as dielectric rods of square and circular cross sections and core–shell rods in a square lattice, and cataloged the conditions under which these PCs can mimic an effective zero-index medium. We found that PCs with DLPs formed by dipole and quadrupole moments cannot be mapped to an effective zero-index metamaterial (ZIM) even if the DLP is at low frequency, indicating the applicability of effective medium theory. The effective ZIM can only be realized in PCs with DLPs if all three of the following conditions are fulfilled simultaneously: (i) the DLP is formed by monopole and dipole moments, (ii) the DLP is at low enough frequency, and (iii) there is only one propagating band in the extended states above the DLP frequency. The shape of the iso-frequency contours does not play a significant role toward the mapping of the DLP to an effective ZIM.