S-theorem (on regularization): Green´s function-induced distributed elementary sources — Second kind

Standard singular dyadic Green´s functions (DGFs) in computational electromagnetics are responses to idealized dipoles - Dirac´s delta functions. The latter are generalized symbolic functions defined as the limit of a sequence of (η-)parametrized functions. Any member of the sequence, with non-vanishing η, is a function in ordinary sense having finite (non-zero) or infinite support. Utilization of such distributed source functions, rather than symbolic distributions, renormalizes singularities automatically and results in regularized DGFs. In this work a novel physics-inspired distributed elementary source function has been constructed for the first time. Maxwell´s equations in general media can be split into two complementary systems of partial differential equations: diagonalized-and supplementary forms, D- form and S- form, respectively. Given a boundary value problem, the D-form with respect to a distinguished direction in space allows directly determining field components transversal to the distinguished direction. The remaining two field components (parallel to the distinguished direction) can be determined a posteriori from the transversal components by employing the S-form. Using the S-form, a novel distributed elementary source has been constructed leading to self-consistently regularizing DGFs. The results have been firmly established by providing the complete proof of a theorem.