Nonconducting electromagnetic media with rotational invariance: Transition operators and Greens's functions

For an accurate description of multiple scattering of electromagnetic waves, such as in random dielectrics or dielectric lattices, the full off-shell single scatterer transition (T-) operator is required. Here we study this quantity for the case of a nonconducting medium with rotational invariant permeabilities. We start with the scattering problem where rotational invariance is not yet assumed and find a class of mutually different T-operators which all have the same on-shell restriction. Next we extend the usual method of expressing scattered fields for rotational invariant systems in terms of the solutions of two scalar wave equations to the resolvent (Green's function) associated with the vector wave equation. We find that it can be expressed in terms of the resolvents of two scalar operators. Finally we turn to the Mie case (a dielectric sphere in vacuum) for which we obtain explicit expressions for the corresponding Green's functions and the general off-shell T-matrix elements.