Photonic band structures of two- and three-dimensional periodic metal or semiconductor arrays

The photonic band structures of a square lattice array of metal or semiconductor cylinders, and of an fcc array of metal or semiconductor spheres, are computed numerically. The frequency-dependent dielectric function of the metal or semiconductor is assumed to have the free-electron form \ensuremath{\epsilon}(\ensuremath{\omega})=1-(${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}^{2}$/${\mathrm{\ensuremath{\omega}}}^{2}$), where ${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}$ is the plasma frequency of the charge carriers. A plane-wave expansion is used to transform the vector electromagnetic wave equation into a matrix equation. The frequencies of the electromagnetic modes are found as the zeros of the determinant of the matrix. In general, the plane-wave method is found to be most effective for structures with filling fractions f\ensuremath{\le}0.1% but provides a semiquantitative description of the band structures of these systems for f>0.1%.