Ray-based propagation of the cross-spectral density

We explore the propagation of the cross-spectral density for scalar and electromagnetic fields based on generalized radiances that are exactly conserved along rays. Two formulas are derived: The first uses all rays to calculate the cross-spectral density exactly, while the second uses only the subset of those rays that pass through a single spatial point to construct an infinite series expression for the cross-spectral density. The evaluation of the truncated series is examined numerically for a variety of fields of varying angular width and coherence and is found to exhibit better convergence to the cross-spectral density when the rays through the centroid between the two observation points are used, when the fields are less coherent, and when the fields are more paraxial. In generalizing the series formula, two new cross-spectral correlations associated with the flux and energy density are examined.