Mode symmetries required for creating photonic Dirac cones in the Brillouin-zone center

Abstract

Summary form only given. The linear dispersion relation around certain points in the Brillouin zone of optical periodic structures, which is often called (photonic) Dirac cones, has been attracting considerable interest during the last five years [1], since it can realize interesting optical phenomena such as unidirectional propagation of surface modes, optical simulation of Zitterbewegung, pseudo-diffusive transmission of optical waves, scatter-free waveguides, and lenses of arbitrary shapes. In particular, Huang et al. [2] clarified that isotropic Dirac cones in the Brillouin-zone center can be created by accidental degeneracy of two modes for two-dimensional square and triangular dielectric photonic crystals of C4v and C6v symmetries. On the other hand, we showed for metamaterials characterized by well-defined electromagnetic resonant states localized in their unit structures that the combination of A1 and E modes of the square lattice of C4v symmetry and the combination of A1g and T1u modes of the simple cubic lattice of Oh symmetry create isotropic Dirac cones [3], while the combination of E1 and E2 modes of the triangular lattice of C6v symmetry leads to isotropic double Dirac cones [4].Quite recently, we clarified that the presence or absence of Dirac cones and/or double Dirac cones in the Brillouin-zone center is solely determined by the symmetry of the two modes [5, 6]. We formulated a degenerate perturbation theory for the vector electromagnetic field and applied it to the problem of the creation of Dirac cones in the Brillouin-zone center by accidental degeneracy for periodic structures of C2v, C4v, C6v, and Oh symmetries. We derived a necessary condition by which we could easily select candidates for mode combinations that enabled the creation of the Dirac cone. We further analyzed the structure of a matrix that determined the first-order correction (linear in k) to eigen frequencies by examining its transformation by symmetry operations. Thus, we could obtain the analytical solution of dispersion curves in the vicinity of the zone center and could judge the presence of the Dirac cone. The results of analytical calculations agreed with the numerical photonic-band calculation. Figure 1 gives such examples that realize isotropic double Dirac cones, where the dispersion curves of simple cubic photonic crystals of dielectric spheres are shown for two cases of accidental degeneracy. In this presentation, we extend our calculation to photonic crystals and metamaterials of different spatial symmetries and examine the method to realize Dirac cones in optical frequencies. In particular, we discuss two cases: deformed cubic lattice and two-dimensional square lattice.