Determination of the Electrical Radius ka of a Spherical Scatterer from the Scattered Field

The inverse problem of scattering for a spherical vector scattering geometry is considered. The transverse scattered field components are related to the expansion coefficients of Hansen's vector wave expansion in terms of the scattered field matrix. This matrix needs to be inverted to obtain the unknown expansion coefficients which are required to recover the shape of the target in question. Since the particular properties of the spherical vector wave expansion may cause highly instable matrix inversion, an analytical, closed form solution of the determinant associated with the scattered field matrix was sought. For vector scattering geometries representing the mth degree multipole cases such closed form solutions for the associated determinant of truncated order 2N are derived, using a novel complementary series expansion for the employed forms of the associated Legendre's functions of the first kind. A novel determinate optimization procedure is presented which enables the specification of the optimal distribution of measurement aspect angles within any given finite measurement cone of the unit sphere of directions. The closed form solution for nonsymmetrical vector scattering geometries is presented in Appendix III only for the value N = 3 (m = 0 and 1) employing properties of quadratic forms as derived in Appendix II. It is then shown that the electrical radius ka of a perfectly conducting spherical scatterer can be directly recovered from a finite number of contiguous expansion coefficients similar to the cylindrical case presented in Boerner, Vandenberghe, and Hamid. Furthermore, relationships between contiguous expansion coefficients of both electric and magnetic type result, which are relevant to the general inverse problem since the scattered field can be uniquely expressed in terms of only one set of expansion coefficients associated with either the electric or magnetic vector wave functions.