A dyadic Green's function representation is derived between two disjoint volumes with arbitrary external scattering. The result is a double-integral of transmit and receive plane waves multiplied with a dyadic kernel indicating their amplitude and polarization interactions. The kernel, which is named the angular spectrum of the system, is fully described in terms of vector spherical harmonics. It provides a solid mathematical and physical foundation to the scattering response matrix that hereto has only been conjectured and modeled in the literature.