Spatial correlation of phase-expansion coefficients for propagation through atmospheric turbulence

Spatial correlation functions of phase-expansion coefficients are derived for phase fluctuations of a wave that has propagated through a layer of atmospheric turbulence. First, a general expression is derived giving the correlation of the coefficients of phase-expansion functions orthogonal over an arbitrary circularly symmetric weighting function for an isotropic turbulence spectrum. Second, the correlations are evaluated analytically for a Gaussian weight and numerically for a uniform circular weight. Finally, some of the correlations for a layer are integrated over the Hufnagel 1974 structure constant model to obtain results applicable to ground-to-space propagation.