CALCULATING PHOTONIC GREENS-FUNCTIONS USING A NONORTHOGONAL FINITE-DIFFERENCE TIME-DOMAIN METHOD

In this paper we shall propose a simple scheme for calculating Green's functions for photons propagating in complex structured dielectrics o r other photonic systems. The method is based on an extension of the f inite-difference time-domain (FDTD) method, originally proposed by Yee [IEEE Trans. Antennas Propag. 14, 302 (1966)], also known as the orde r-hi method [Chan, Yu, and Ho, Phys. Rev. 51., 16 635 (1995)] which ha s recently become a popular way of calculating photonic band structure s. We give a transparent derivation of the order-hi method which, in t urn, enables us to give a simple yet rigorous derivation of the criter ion for numerical stability as well as statements of charge and energy conservation which are exact even on the discrete lattice. We impleme nt this using a general, nonorthogonal coordinate system without incur ring the computational overheads normally associated with nonorthogona l FDTD. We present results for local densities of states calculated us ing this method for a number of systems. First, we consider a simple o ne-dimensional dielectric multilayer, identifying the suppression in t he state density caused by the photonic band gap and then observing th e effect of introducing a defect layer into the periodic structure. Se cond, we tackle a more realistic example by treating a defect in a cry stal of dielectric spheres on a diamond lattice. This could have appli cation to the design of superefficient laser devices utilizing defects in photonic crystals as laser cavities.