Derivation of the transverse size of the mean speckle throughout propagation from the moments of the Wigner function

Partially coherent beams are characterized by the presence of speckle patterns. Modeling the evolution of speckle properties throughout propagation requires solving the beam propagation equation, potentially at the cost of resource-intensive calculations. Here, we derive analytical expressions for the transverse size of speckles by approximating the mean autocorrelation function of the electric field through Wigner function moments. As those moments follow particularly simple propagation laws, this approach allows us to describe the evolution of the mean speckle properties along the beam propagation with the use of a few scalar parameters, hence at a very low computational cost. We highlight the relevance of this approach by plotting the evolution of the speckle size in different examples, including Gaussian-Schell and multiple beam configurations. The formalism we introduce is general and applies to the case of multi-dimensional fields, which is of particular interest for studying the effect, on the speckle characteristics, of space-time couplings in dispersive systems.