Inverse Source Solutions with Huygens-Surface Conforming Distributed Directive Spherical Harmonics Expansions

Time-harmonic inverse source solutions are commonly working with electric and magnetic surface current densities defined and discretized on appropriately chosen Huygens surfaces. An efficient meshless alternative are spherical harmonics expansions of low order distributed along the chosen Huygens surface, which still possess pretty good spatial localization properties. Similar to expansions with electric and magnetic surface current densities, distributed expansions with Cartesian spherical harmonics or standard vector spherical harmonics are redundant when they are placed on Huygens surfaces. This redundancy is reduced by allowing only spherical harmonics, which can be excited by surface current density distributions in a specific plane, and by forming directive spherical harmonics. This results in a considerable reduction of the necessary number of unknowns for representing the sources and improved conditioning of the discretized integral equations. Inspired by the Huygens radiator concept, appropriate directive harmonics on the basis of Cartesian and vector spherical harmonics are derived, implemented, and compared in terms of their performances. The validity of the presented techniques is confirmed via inverse source solutions for synthetic and real measurement data.