Effective degree of coherence: A second look

In this paper, we show that the most general set of transformations of electromagnetic fields, for which overall (global) second-order coherence properties can reasonably be expected to remain unchanged, is the set of scaled unitary transformations. Building on our earlier results concerning coherence functionals that are invariant to scaled unitary transformations, we prove that the effective degree of coherence is the only such functional that is "additive" in the sense that it can be computed for linear combinations of fields from its values for pairwise sums of the constituent fields. Additionally, we highlight the fact that the invariance of the effective degree of coherence to scaled unitary transformations means that it has the same value when computed from most of the important representations of electromagnetic fields. We then go on to use the effective degree of coherence to provide a generalization of the scalar two-point degree-of-coherence function to a system consisting of two orthogonal Hilbert spaces. Interestingly, several commonly used measures of coherence and polarization turn out to be special cases of this generalization.