Self-consistent Green’s function formalism for acoustic and light scattering, Part 1: Scalar notation

Abstract

A self-consistent Green’s function formalism in spherical coordinates, related to the scalar Helmholtz equation of acoustic and light scattering (if the latter is formulated in terms of Debye potentials, for example), is presented. The scatterer geometry is assumed to be nonspherical. The spherical geometry is covered as a limiting case. The four essential quantities of the formalism are the volume Green’s function, the surface Green’s function, the free-space Green’s function, and the interaction operator. First it is demonstrated that Huygens’ principle can be formulated purely in terms of Green functions. Based on this principle, Lippmann–Schwinger equations are derived for the volume Green’s function and the interaction operator, thus allowing iterations in terms of Born series. On the other hand, the appropriate definition of the interaction operator allows approximations of the volume and surface Green’s function by series expansions in terms of outgoing and incoming spherical wave functions, respectively, without any recourse to the physical boundary value problem under consideration. It turns out that the matrix elements of the interaction operator are strongly related to the so-called T-matrix introduced by P.C. Waterman [J. Acoust. Soc. Am. 45 (1969) 1417; Phys. Rev. D 3 (1971) 825], thus providing a better understanding of the latter. The advantage of the self-consistent Green’s function formalism is twofold. First, it serves as a consistent mathematical basis from which equivalences and differences between different solution schemes of scattering analysis, which have been developed so far, can be discussed. Second, the formalism clearly separate the methodological aspects of the solution schemes from the physical content of a certain boundary value problem.