  Analytic expressions of the spatial coherence of partially coherent fields
propagating in the Fresnel regime in all but the simplest of scenarios are
largely lacking and calculation of the Fresnel transform typically entails
tedious numerical integration. Here, we provide a closed-form approximation
formula for the case of a generalized source obtained by modulating the field
produced by a Gauss-Shell source model with a piecewise constant transmission
function, which may be used to model the field's interaction with objects and
apertures. The formula characterizes the coherence function in terms of the
coherence of the Gauss-Schell beam propagated in free space and a
multiplicative term capturing the interaction with the transmission function.
This approximation holds in the regime where the intensity width of the beam is
much larger than the coherence width under mild assumptions on the modulating
transmission function. The formula derived for generalized sources lays the
foundation for the study of the inverse problem of scene reconstruction from
coherence measurements.
