A closed-form conversion from spherical-wave- to complex-point-source-expansion

A simple closed-form expression for a complex point source (CPS) beam expansion of an arbitrary electromagnetic field is derived. The expansion process consists of two steps: first, a particular form of the equivalence principle is applied to a sphere enclosing the real sources, and a continuous equivalent electric current distribution is obtained in terms of spherical waves (SW); then, the continuous current is extended to complex space and its SW components are properly filtered and sampled to generate the discrete set of CPS. The final result is a compact finite series representation suitable for arbitrary radiated fields, and particularly efficient when the source is highly directional and/or the observation domain is limited to a given angular sector. The robustness of the process is demonstrated by showing its connection with the singular value decomposition of the radiation operator from a complex sphere.