On the eigenfunction expansion of electromagnetic dyadic Green's functions

Abstract

A relatively simple approach is described for developing the complete eigenfunction expansion of time-harmonic electric ( \bar{E} ) and magnetic ( \bar{H} ) fields within exterior or interior regions containing an arbitrarily oriented electric current point source. In particular, these results yield directly the complete eigenfunction expansion of the electric and magnetic dyadic Green's functions \bar\bar{G}_{e} and \bar\bar{G}_{m} that are associated with \bar{E} and \bar{H} , respectively. This expansion of \bar\bar{G}_{e} and \bar\bar{G}_{m} contains only the solenoidal type eigenfunctions. In addition, the expansion of \bar\bar{G}_{e} also contains an explicit dyadic delta function term which is required for making that expansion complete at the source point. The explicit dyadic delta function term in \bar\bar{G}_{e} is found readily from a simple condition governing the behavior of the eigenfunction expansion at the source point, provided one views that condition in the light of distribution theory. These general expressions for the eigenfunction expansion of \bar\bar{G}_{e} and \bar\bar{G}_{m} reduce properly to those obtained previously for special geometries by Tai.