Propagation of the spectral correlation function in a homogeneous medium

Abstract

When two infinitely extensive electromagnetic waves of wavelengths λ1, and λ2propagate in the Fresnel region of a homogeneous medium, the spectral correlation function, 〈 A ( x ′ , y ′ ; λ 1 ) A * ( x , y ; λ 2 ) 〉 , gradually changes with increasing propagation distance. An exact expression is developed which describes these changes. The changes are exhibited in terms of both modulus and phase. In the special case where the two wavelengths are identical, the spectral correlation function reduces to the autocorrelation function. In this case, the function, as expected, is found to conserve during propagation. The paper considers two spectrally well-separated plane waves from the visible part of the spectrum at 450 and 650 nm. Numerical evaluations are given for a typical spectral correlation function that might arise from these waves in conditions of good seeing. It is shown that this spectral correlation function is capable of propagating with a high degree of conservation over distances of the order of the atmospheric depth. It is inferred from this that the spectral correlation function that arises when visible waves pass through the atmosphere is determined only by the accumulated effect of the local atmospheric turbulence; the propagation distance itself makes no significant contribution to the final value attained by the function. This result simplifies the task of describing the statistical properties of polychromatic images formed by large astronomical telescopes. It also offers insights and tools for dealing with the problems associated with laser beam propagation through the atmosphere.