Explicit Conversion Formulas Between Spherical Wave Expansions With Scalar and Vector Expansion Coefficients

Abstract

Explicit formulas for converting spherical wave expansions with vector basis functions and scalar expansion coefficients into spherical wave expansions with scalar basis functions and vector expansion coefficients and vice versa are presented. The formulas are given in terms of Wigner-3- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$j$ </tex-math></inline-formula> -symbols. The conversion formulas are derived by spherical harmonics expansions of products of two spherical harmonics and by levering on recurrence relations of the associated Legendre functions. The expansions with vector coefficients are redundant and explicit formulas are given for the linear combination of vector coefficients of which the expanded fields cancel identically to zero. The redundancy in the expansion can be used to find compact expansions with a minimum expansion order. For every vector spherical wave function, two expansions using scalar spherical harmonics are presented: on the one hand, a radial-component free expansion and on the other hand, a minimum-order expansion. The correctness of all expansions is verified with a simple computer code up to a mode order of 65. The agreement of the expanded near and far fields in both types of expansions is better than 14 digits of accuracy for most cases.