Rotational-translational addition theorems for spheroidal vector wave functions

Abstract

Abstract. Rotational-translational addition theorems for spherical and spheroidal vector wave functions are established. These theorems concern the vector wave functions Mfl and Na (with 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 and r < d are distinguished, where d is the distance between the centers of the spheroids.

Rotational-translational addition theorems for scalar spheroidal wave functions

“…It is interesting to note that, contrary to translational addition theorems [5][6][7] for spheroidal scalar wave functions, in the present rotational addition theorem there is never any change of the index i from one side of (17) to the other.…”

“…','7, due to Dalmas and Deleuil [6, Sec. 2;7], These theorems have found application in the problem of electromagnetic scattering of a plane wave from a pair of perfectly conducting prolate spheroids whose major axes are in parallel alignment [6,8,9]. Moreover, very thin conducting spheroids can be used to model thin-wire dipole antennas, and translational addition…”

“…Such theorems are necessary in the rigorous analysis of radiation and scattering by spheroids with arbitrary spacings and orientations. ','7, due to Dalmas and Deleuil [6, Sec. 2;7], These theorems have found application in the problem of electromagnetic scattering of a plane wave from a pair of perfectly conducting prolate spheroids whose major axes are in parallel alignment [6,8,9].…”

“…The alternative formulation, where a is given by the radius vector r, has resulted in the translational addition theorems for and = k 'v X M ','7, due to Dalmas and Deleuil [6,Sec. 2;7], These theorems have found application in the problem of electromagnetic scattering of a plane wave from a pair of perfectly conducting prolate spheroids whose major axes are in parallel alignment [6,8,9]. Moreover, very thin conducting spheroids can be used to model thin-wire dipole antennas, and translational addition 738 R. H. MACPHIE, J. DALMAS, R. DELEUIL theorems have recently been used to deduce the mutual admittance of spheroidal dipole antennas in parallel configuration [10].…”

Scattering of electromagnetic waves by a system of two dielectric spheroids of arbitrary orientation

“…In this communication, we obtain an analytic solution for the general case of scattering by two dielectric spheroids of arbitrary orientation, from which the formulation corresponding to conducting spheroids can be derived by particularization. As in the case of two perfectly conducting spheroids of arbitrary orientation, the solution given here has also been obtained on the basis of rotational-translational addition theorems for vector spheroidal wave functions derived by the authors [9], [10], and independently in [11]. As mentioned in [12], the importance of obtaining such a solution lies in the fact that it can be used along with the exact solution for two perfectly conducting spheroids of arbitrary orientation, as benchmarks, to be considered in development of a database for validating numerical codes and also in cross validating numerical and experimental results.…”