On the relationship between the cross-spectral density and the Wigner distribution function

Abstract

It is shown that the angular spectrum of the cross-spectral density is a Wigner distribution function, which is equivalent to the marginal cross-spectral density of the field. The value of the Wigner distribution function at each position of the observation plane results from the contributions of aperture radiator pairs, around each position into the aperture, properly projected along the corresponding direction of propagation. The contributions are weighted by the complex degree of spatial coherence of the illuminating optical field, and their relative weights will be estimated by introducing a ‘visibility’ parameter. There are also forbidden directions, along which certain radiator pairs into the coherence patch cannot contribute depending on the phases of both the complex degree of spatial coherence and the aperture transmittance, but all radiator pairs provide their maximal contributions along orthogonal directions to the aperture plane.