Impurity mode techniques applied to the study of light sources

The dispersion relation, density of states, wave-functions and single site impurity bound and resonant modes of a photonic crystal slab are computed. The photonic crystal slab is formed by embedding a hexagonal lattice array of dielectric cylinders in a background dielectric. Outside the photonic crystal slab is a uniform, isotropic dielectric medium. The solution of Maxwell's equations for the modes of a free-standing photonic crystal slab are obtained by treating a self-consistent matrix eigenvalue problem that is solved iteratively using a fast computer routine. The properties of single site impurities in photonic crystal slabs are determined using both Green's function methods and effective mass equations based on a Wannier function representation. The impurity dielectric constant supporting an impurity bound state is shown to be a multiple valued function of the frequency of the impurity mode. Asymptotic forms are obtained for the behaviour of impurity bound states at frequencies in the neighbourhood of the band edges. Resonant modes at pass band frequencies are computed for the impurities formed by replacing a single dielectric cylinder of the photonic crystal slab with background dielectric media. Comparison of the theoretical results is made with experiments on an electrically injected photonic crystal microcavity light source