Chapter 10 - Null-Field Method with Discrete Sources

Abstract

One of the fastest and most powerful numerical tools for computing nonspherical light scattering using spherical vector wave functions expansions is the null-field method. In the null-field method, the particle is replaced by a set of surface-current densities, so that in the exterior region the sources and fields are exactly the same as those existing in the original scattering problem. In recent years the discrete-sources method has become an effective means for solving a wide variety of boundary-value problems in scattering theory. Essentially, this method entails the use of a finite linear combination of fields of elementary sources to construct the solution. This chapter describes various formulations of the null-field method with discrete sources for the transmission boundary-value problem. The goal of the theoretical development is to derive a set of integral equations for the surface-current densities which guarantee the null-field condition inside the particle. This chapter describes the formulation of the transmission boundary-value problem in the classical setting of continuous and Holder-continuous functions. It then proves some fundamental lemmas which enable constructing complete systems of vector functions on the particle surface. Following this, the general null-field equations for the transmission boundary-value problems are described. The chapter also discusses the existence and the uniqueness of the solution, and shows the equivalence of these equations with a system of boundary-integral equations.. Finally, the numerical scheme of the null-field method with discrete sources and give some numerical results are presented.