Calculating photonic Green’s 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 or other photonic systems. The method is based on an extension of the finite-difference time-domain (FDTD) method, originally proposed by Yee [IEEE Trans. Antennas Propag. 14, 302 (1966)], also known as the order-$N$ method [Chan, Yu, and Ho, Phys. Rev. 51, 16 635 (1995)] which has recently become a popular way of calculating photonic band structures. We give a transparent derivation of the order-$N$ method which, in turn, enables us to give a simple yet rigorous derivation of the criterion for numerical stability as well as statements of charge and energy conservation which are exact even on the discrete lattice. We implement this using a general, nonorthogonal coordinate system without incurring the computational overheads normally associated with nonorthogonal FDTD. We present results for local densities of states calculated using this method for a number of systems. First, we consider a simple one-dimensional dielectric multilayer, identifying the suppression in the state density caused by the photonic band gap and then observing the effect of introducing a defect layer into the periodic structure. Second, we tackle a more realistic example by treating a defect in a crystal of dielectric spheres on a diamond lattice. This could have application to the design of superefficient laser devices utilizing defects in photonic crystals as laser cavities.