Figure 1: (a) The band structure of a two-dimensional photonic crystal composing of alumina cylinders arranged in a square lattice for the TM polarization. The radii of the cylinders are . The relative permittivity of the cylinders is 8.8. At the point, a triply degenerate state is formed, and the linear dispersion near the zone center is highlighted in (b). (c) The effective permittivity (black solid line) and permeability (red dash line) of this alumina photonic crystal. Note that at the Dirac-like point. (d)–(f) The field patterns of the three eigenmodes near the Dirac-like point with a very small k along X direction.