If a field g (x ,z) satisfies the diffusion equation ∂<sup>2</sup>g /∂x<sup>2</sup>+2<i>j k</i> (∂g /∂z)=0, then its ambiguity function <i>X</i>(x ,v ,z)=∫<sup>∞</sup>g (η+x/2,z)g <sup>*</sup> (η-x/2,z)e <sup>-jrη</sup><i>d</i>η satisfies the wave equation <i>v</i><sup>2</sup>(∂<sup>2</sup><i>X</i>/∂<i>x</i><sup>2</sup>)-<i>k</i><sup>2</sup>(∂<sup>2</sup><i>X</i>/∂<i>z</i><sup>2</sup>)=0. A theory of Fresnel diffraction and Fourier optics results, involving merely coordinate transformations of the independent variables of the aperture ambiguity function. As an application, a simple expression for the width of the diffracted beam is derived in terms of certain moments of the amplitude of the incident wave. The analysis is extended to signals crossing a layer of a random medium. At the exit plane, the field is partially coherent and it spreads as it propagates. The broadening of beam width due to the loss in coherence is related to the statistical properties of the layer.
Athanasios Papoulis, "Ambiguity function in Fourier optics," J. Opt. Soc. Am. 64, 779-788 (1974)