Local density of states and modes of circular photonic crystal cavities

Because the penetration depth of light in the band gap is of the order of one lattice constant of a photonic crystal, structures such as circular photonic crystals (CPC) confine light. We show that an extension of the CPC concept to holes in any high-index material yields confinement only in the case that the hole size is varied jointly. Applying the Bloch-Floquet theorem in such a rotationally symmetric case leads to a decomposition of the Green’s tensor. By using the photonic local density of states (LDOS) components in a CPC, the eigenfrequencies of localized cavity modes can be efficiently found.

Figure 1

Gap-rate equicontour plot showing the pair values of hole radius versus distance for a hexagonal structure of air holes in silicon $(n=3.4)$. $H_z$ polarized light with ${\lambda }_0=1.55\mu \mathrm{m}$ assumed. The inset shows stop-band frequencies vs relative radius.

Figure 2

Two-dimensional circular photonic crystal with seven-fold rotational symmetry consisting of three rings. Center positions of two neighboring cylinders in the same ring are labeled by ${\mathit{r}}_{l,j}$ and ${\mathit{r}}_{l,j+1}$.

Figure 3

(Color) (a) Complete LDOS and its $K=0$ component at the point with coordinates $x=1.0a$ and $y=0.2a$. Distribution of the electric fields: (b) $K=0$ mode, (c) $K=1$, (d) $K=2$, (e) $K=3$.

Figure 4

(Color) (a) Complete LDOS calculated from ${\mathit{G}}^{\mathit{e}}$ and the $K=2$ component of the LDOS corre- sponding to ${\mathit{G}}^{\mathit{h}}$. (b) Comparison of $K=2$ components related to ${\mathit{G}}^{\mathit{e}}$ and ${\mathit{G}}^{\mathit{h}}$. The first one shows a slowly oscillating background. The curves are calculated for the point with coordinates $x=0.70a$ and $y=0.25a$.

Figure 5

(Color) Distribution of the magnetic fields: (a) Our sample structure with $K=0$ mode; (b) $K=1$; (c) $K=2$; (d) $K=3$; (e) $K=4$; (f) $K=8$ mode representing a well-localized whispering-gallery mode. The field distribution of degenerated modes is affected by the position of the source.