Ambiguity function in Fourier optics*

TLDR

The ambiguity function of a field satisfying the diffusion equation obeys a wave equation, and this framework underpins a Fresnel diffraction theory in Fourier optics via coordinate transformations of the aperture ambiguity function. The authors derive a simple expression for the diffracted beam width in terms of amplitude moments and extend the analysis to signals traversing a random medium layer. The beam width broadening caused by loss of coherence is linked to the statistical properties of the random layer.

Abstract

If a field g(x,z) satisfies the diffusion equation ∂2g/∂x2 + 2j k(∂g/∂z) = 0, then its ambiguity function χ(x,ν,z)=∫-∞∞g(η+x/2,z)g*(η-x/2,z)e-jνηd η satisfies the wave equation ν2(∂2χ/∂x2) − k2(∂2χ/∂z2) = 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.