Complete transverse representation of a correlation singularity of a partially coherent field

An understanding of correlation singularities is fundamentally important in imaging science. Until now spatial coherence studies have examined a two-dimensional projection of the four-dimensional correlation function, finding so-called correlation vortices or correlation ring dislocations in a given transverse plane. Here we describe the properties and symmetries of the full four-dimensional correlation function. The general solution is found to be a hyperbola in two reduced dimensions. For perfect coherence this reduces to crossed straight lines, whereas in the incoherent limit it reduces to parallel lines. These results elucidate a number of previous experimental and theoretical observations regarding correlation singularities and suggest other behaviors of such singularities.