Propagation of the spectral correlation function in a homogeneous medium

Abstract

When two infinitely extensive waves at λ1 and λ2 propagate in the Fresnel region of a homogeneous medium, the spectral correlation function 〈A(x′, y′; λ1)A*(x, y; λ2)〉 is not in general conserved. In this paper expressions that describe the behavior of this function during propagation are derived. Although the function itself is not conserved, the modulus of the volume contained under the function does conserve. In the limiting case for which λ1 = λ2, the spectral correlation function reduces to the autocorrelation function. As expected, conservation is found in this case. If λ1 and λ2 are closely separated, the spectral correlation function can propagate large distances without changing appreciably. A numerical example shows that the spectral correlation function typical of waves that have propagated through the atmosphere changes little even with propagation distances of the order of the atmospheric depth. The results contained in this paper are important to imaging in astronomy and laser-beam propagation through the atmosphere.