Distributed spherical harmonics expansions are a powerful approach for the efficient solution of inverse source problems. They allow for a relatively accurate representation of the geometric support of the source distribution without the overhead of handling a mesh and mesh-specific basis functions representing the commonly utilized surface current densities. Standard spherical harmonics expansions distributed over a Huygens surface around the source domain are, however, redundant. Therefore, we discuss ways towards the reduction of this redundancy, where we are in particular interested in the construction of spherical harmonics with predominant radiation into the actual solution domain. Several sets of such spherical harmonics are presented and their utilization with measured and synthetic near-field data is demonstrated.

Inverse source solutions for field transformations and diagnostics are mostly working with surface current densities on appropriately defined Huygens surfaces around the test object. This gives excellent modeling flexibility, but the handling of the discretized surface current representation requires also substantial computational effort. Distributed spherical harmonics expansions of low order, for example based on a solution space partitioning as used in the multilevel fast multipole method, have somewhat reduced modeling flexibility, but they can save considerable computational effort, and they are, thus, an excellent choice of expansion functions for many practical inverse source problems. We discuss different forms of distributed spherical harmonics expansions comprising purely scalar spherical modes and vector spherical modes. Moreover, we discuss how the different surface source expansions consisting of electric and magnetic surface currents with Love condition or without, or consisting of directive Huygens radiators can be related to corresponding distributed 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.