A free-space solution is found for the field correlation tensors as a Green’s function integral over the correlation tensor for the source currents. This solution is used to formulate a vector theory of partial coherence in an angular spectrum representation. The theory is used to analyze the angular distribution of radiation into the far field for several special cases of source polarization and coherence. It is found that the radiation pattern from unpolarized, quasihomogeneous sources depends on the coherence properties of the source in the same manner as described in scalar theory. However, for sources that are not unpolarized this is not generally the case.
William H. Carter, "Properties of electromagnetic radiation from a partially correlated current distribution," J. Opt. Soc. Am. 70, 1067-1074 (1980)