  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, also known as the
Order-N method, which has recently become a popular way of calculating photonic
band structures. We give a new, 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, non-orthogonal co-ordinate system without incurring the
computational overheads normally associated with non-orthogonal FDTD.
  We present results for local densities of states calculated using this method
for a number of systems. Firstly, 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. Secondly, 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 super-efficient laser devices
utilising defects in photonic crystals as laser cavities.
