In a series of papers we have extended Rayleigh's technique (Lord Rayleigh, Philos. Mag. 34, 481 (1892)) from electrostatic to full electromagnetic problems, for singly, doubly and triply periodic systems. Rayleigh's method involves a set of lattice sums which consist of sums over terms with a function evaluated at each lattice point, and the evaluation of lattice sums is the most important and subtle part of this technique. The main reason is that the definition of lattice sums involves conditionally converging series over the direct lattice, and a direct evaluation is thus impractical if high accuracy is needed. The lattice sums involved in our method are represented in terms of absolutely converging series over the reciprocal lattice, and in contrast to the method used by Ewald, these series may be accelerated by succesive integrations to any order. By introducing the lattice sums, we obtain a representation of the Green's function in terms of a rapidly convergent Neumann series. Also, the representation in terms of absolutely converging series allows us to have some physical insight into the analytic properties of the lattice sums. For the coefficients in the multipole expansions of fields we have obtained a generalised Rayleigh identity.
The Extended Rayleigh Method (ERM) is capable of studying, numerically and analytically, problems in which the dielectric constant is piecewise constant and may take finite or infinite values. Due to the fact that there are no series expansions of the dielectric constant, the method may also be applied in cases when the dielectric constant takes on imaginary values. ERM is also consistent with an approach which has been used for static studies of composites, and is therefore suitable for studying the homogenization problem (in which an effective uniform refractive index is attributed to a composite material).