Optical frequencies of spectra of photonic crystals

We develop an effective method for calculating optical frequencies of two-dimensional photonic crystals (PC), i.e. a periodic arrangement of infinite dielectric cylinders. These frequencies, ωN, correspond to the Bloch vector at the center of the Brillouin zone, k=0, and are labeled by the band index N=1,2,…. Using the plane-wave expansion method, we derive two eigenvalue equations for the optical frequencies of the E- and H-polarized modes. These equations are valid for any geometry of the unit cell and arbitrary cross-sectional form of the cylinders. We show that optical frequencies of the E- and H-modes satisfy two different sum rules. We apply our method to PC with high dielectric contrast between constituents and calculate numerically the dependence of the lowest optical modes on the filling fraction. These graphs show splitting of degenerated frequencies with increase of the filling fraction and may be helpful for design of structures with photonic band gaps.