Local density of states in three-dimensional photonic crystals: Calculation and enhancement effects

Abstract

The modulus of electric field eigenvector in photonic crystals (PCs), $|{\mathbf{E}}_{n}(\mathbf{k},\mathbf{r})|,$ is confirmed to be variant for a set of k points in a k star, namely, $|{\mathbf{E}}_{n}(\ensuremath{\alpha}[\mathbf{k}],\mathbf{r})|\ensuremath{\ne}|{\mathbf{E}}_{n}(\mathbf{k},\mathbf{r})|,$ where ${\ensuremath{\alpha}}$ represents the operations of the lattice point group. Therefore, in the calculation of the local density of states (LDOS), all k points in the entire Brillioun zone should be included; otherwise, the results are generally incorrect. According to group theory, the transformation relation $|{\mathbf{E}}_{n}(\ensuremath{\alpha}[\mathbf{k}],\mathbf{r})|=|{\mathbf{E}}_{n}(\mathbf{k},{\ensuremath{\alpha}|t{}}^{\ensuremath{-}1}\mathbf{r})|$ (here t represents the translation operator of the crystal, including the glide operation) is rigorously proven. This transformation can greatly save computing time spent in the diagonalization procedure of the eigenmode equation. Based upon this transform, a correct expression of calculating the LDOS is presented. This expression of the LDOS satisfies symmetry of the lattice. The detailed comparisons with the previous results based upon $|{\mathbf{E}}_{n}(\ensuremath{\alpha}[\mathbf{k}],\mathbf{r})|=|{\mathbf{E}}_{n}(\mathbf{k},\mathbf{r})|$ are given. The enhancement effects of the LDOS in the PCs with a fcc structure and a diamond structure, compared to the vacuum case, are revealed. It is found that the enhancement effect of the LDOS in the diamond structure is much larger than that in the fcc structure.