Rotational-translational addition theorems for spheroidal vector wave functions

Abstract

Rotational-translational addition theorems for spherical and spheroidal vector wave functions are established. These theorems concern the vector wave functions M a {M^a} and N a {N^a} (with a = r , x , y , z a = r, x, y, z ) which can be obtained and used to treat various electromagnetic problems such as multiple scattering of a plane wave from prolate spheroids (with arbitrary spacings and orientations of their axes of symmetry) or radiation from thin-wire antennas. For sake of completeness, rotational-translational addition theorems for the vector wave function L are also established. This work is a natural extension of previous studies concerning simpler transformations of coordinate systems, such as rotation or translation. The two cases r ≥ d r \ge d and r ≤ d r \le d are distinguished, where d d is the distance between the centers of the spheroids.