Resolution of Maxwell's equations by the spectral moments method. Global approach

Groupe de Dynamique des Phases Condensées, UMR 5581, Université Montpellier 2, 34095 Montpellier Cedex 05, France

Corresponding author: benoit@gdpc.univ-montp2.fr

Received: 9 April 2002
Revised: 20 December 2002
Accepted: 7 April 2003
Published online: 17 July 2003

Abstract

We present a systematic analysis, within the scope of electromagnetic theory, of the spectral moments method (SMM) and develop several ways to compute the linear response of any type of system. We show that the method can be used in diffraction studies regardless of the number, nature and form of the diffracting objects. Multiple diffraction is naturally taken into account in the computation. The method can thus be applied to determine propagation through any type of media. It is based on the computation of Green functions, solutions of discretized Maxwell's equations. Fourier transforms of Green functions are developed in continued fraction. Two approaches will be presented. In the first “global” approach, all space is discretized, the coefficients of continued fractions are computed directly from the dynamic matrix obtained by the discretization of Maxwell's equations and from sources and receivers. In the second “local” approach, only the diffracting system is discretized. This paper is devoted to the global approach. We study two important problems in electromagnetism, i.e. propagation of a plane wave through a heterogeneous layer and scattering of an isolated object. We present two computation techniques for plane wave propagation: one uses a small grid, is very rapid but the results are approximate; the other uses a large grid, is less rapid but the results are exact. We show that computing the reflectivity and/or transmissivity of photonic lattices is now a very simple problem. For scattering, we mainly report a series of tests on some canonical systems, such as cylinders or spheres, showing that SMM results are in very good agreement with the analytical results. Several types of absorbing boundary conditions are tested. We report results on backscattering cross-sections and the impulsional response of different one-, two- and three-dimensional systems.