Optical cross-purity

Cross-spectral purity, a concept introduced by Mandel in 1961, refers to an optical field having identical spectra at two points whose superposition yields the same spectrum. Here, we generalize this concept after discretizing the modal structure of the field, so that cross-purity can then apply to any optical degree-of-freedom (DoF). Making use of two binary DoFs (polarization and two spatial modes), the separability of the associated 4×4 coherence matrix is the condition for cross-purity. We find that the rank of the coherence matrix (the number of non-zero eigenvalues) plays a surprising yet crucial role in determining optical cross-purity: whereas rank-1 and rank-2 fields are <i>always</i> cross-pure, rank-3 fields in contrast are <i>never</i> cross-pure, and only a subset of rank-4 fields are cross-pure.