Solving the helmholtz equation using multiply propagated fields

Computer models in electromagnetics are based primarily either on integral or on differential equations. The former arise from source integrals using some appropriate Green's function whereas the latter originate from the Maxwell curl equations. Although requiring volume rather than surface sampling even for spatially homogeneous problems, in contrast to integral-equation (IE) models, differential-equation (DE) models are geneally a better choice for problems involving spatial inhomogeneities. This is because such problems require volumetric sampling using either approach, but the DE model produces a sparse matrix rather than the full matrix of the IE formulation. In this paper we describe a new approach based on using multiply propagated fields for numerically solving the banded matrix that results from discretizing the Helmholtz equation. A computer-time savings of N^1/2 and N^2/3 for two-dimensional (2-D) and three-dimensional (3-D) problems, respectively, is made possible, where N is the total number of field samples or unknowns. For even moderate-size problems where 100 samples per linear dimension are used (N₂ = 10,000 and N₃ = 1,000,000), the time savings can be of the order of 100 and 10,000 respectively. Another advantage of this procedure, which we call Helmholtz equation multiple propagator (HEMP), is that the radiation or closure condition needed to terminate the spatial solution mesh for exterior problems can be enforced rigorously with essentially no additional computational cost. The method is illustrated for a 2-D problem by application to plane-wave scattering from an infinite, metal, circular cylinder. Results are presented for the mode amplitudes of the scattered field, the induced surface current, and the bistatic far field as obtained from HEMP, and shown to be in good agreement with the analytical results. Although limited here to the simplest possible application in order to establish its feasibility, the approach's advantage would be its applicability to 2-D and 3-D problems involving inhomogeneous, penetrable objects.