Space intensity distribution and projections of the cross-spectral density

It is shown that, in the paraxial approximation, the intensity distribution produced throughout the space by a partially coherent field depends only on certain projection integrals of the cross spectral density. These projections, in tum, can be recovered from knowledge of the spatial distribution of the optical intensity. This result can enlighten the recently evidentiated fact that two fields in different states of coherence can give rise to the same intensity everywhere. Some examples are discussed.