Analysis of multimode point-defect cavities in three-dimensional photonic crystals using group theory in frequency and time domains

We present analysis of a multimode point-defect cavity in a three-dimensional (3D) photonic crystal belonging to the point group, utilizing group-theoretical classifications in both the plane-wave expansion (PWE) and 3D finite-difference time-domain (FDTD) methods. In the PWE method, application of the projection operator to the electromagnetic fields is proposed to classify the point-defect modes into a basis of the irreducible representation of the point group. This is a simple computational process. A group-theory formulation is developed for Maxwell's equations in the time domain, involving the electric and magnetic current densities. It is demonstrated that a temporal analysis can be made independently about each basis vector of the irreducible representation of the symmetry group and this fact establishes Bloch's theorem in the time domain. In the FDTD method, the number of the point-defect modes dealt with in each calculation becomes small so that analysis of the multimode point-defect cavity is much simplified. In practice, we investigate various properties of multimode point-defect cavities belonging to the point group C_2h, including resonant frequency, field distribution, quality factor, light-extraction efficiency, radiation pattern, and polarization.