Generalization of the optical theorem to a singular source in the presence of a half-space

The scattering of the field of an electric dipole by a local penetrable body in the presence of a transparent half-space is considered. A relation is obtained that is similar to the optical theorem in the case of an incident plane wave. With this generalization, the fluorescence enhancement factor or the efficiency of an optical antenna can be calculated without computing the absorbed energy, which considerably reduces the computational costs.