Symmetry analysis of the localized modes associated with substitutional and interstitial defects in a two-dimensional triangular photonic crystal

Abstract

By using a finite-difference time-domain numerical method based on the numerical simulation of the excitation of an isolated eigenstate at a specific frequency by using an oscillating dipole embedded in a two-dimensional photonic crystal we have calculated both the eigenfrequencies and eigenfunctions of the localized defect modes induced by defect cylinders placed at the center or in interstitial positions within an otherwise perfect two-dimensional photonic crystal consisting of dielectric cylinders arrayed in a triangular lattice. In the case of a substitutional defect cylinder we have imposed boundary conditions appropriate to the irreducible representations of the ${C}_{6v}$ point group, and in addition to fully symmetric localized states of ${A}_{1}$ symmetry we have also found localized modes possessing ${A}_{2},$ ${B}_{1},$ and ${B}_{2}$ symmetry, and doubly degenerate modes of ${E}_{1}$ and ${E}_{2}$ symmetry. For defect rods placed in interstitial positions we have found localized modes that belong to the irreducible representations ${A}_{1},$ ${A}_{2},$ ${B}_{1},$ and ${B}_{2}$ of the ${C}_{2v}$ point group. We have also studied the effects of the geometrical and material parameters on the eigenfrequencies and eigenfunctions of the defect modes by varying the dielectric strength and/or radius of the defect rods. We have shown that the calculated eigenfrequencies obtained for a substitutional defect rod within the triangular lattice with lattice constant a whose radius ${r}_{d}$ is in the interval $0<{r}_{d}<0.5a$ are in good quantitative agreement with both the nondegenerate and degenerate modes obtained by the supercell method by Feng and Arakawa [Jpn. J. Appl. Phys. Part 2 36, 120(1997)].