Perturbation theory for phase correlations of a light wave propagating in a turbulent medium

We theoretically investigate the correlation functions of the phase of a light wave propagating through a turbulent medium. We use an equation for the logarithm of a wave packet envelope, which includes a second-order nonlinear term. Based on this equation, we develop a diagrammatic technique to calculate corrections to the correlation function obtained in the linear approximation. We calculate the first corrections determined by one-loop diagrams and find their asymptotic behaviors. We found that at short distances between points, the first correction to the pair correlation function of phase gradients has the same scaling behavior as the zero approximation, and at long distances, the first approximation decays faster than the zeroth one. Therefore, there is no evidence to suggest that the correction will destroy the zeroth approximation. Some nonperturbative conclusions are made using the symmetry properties of the equation. We demonstrated that small details of an image received by an optical system through the atmosphere are smeared due to turbulent fluctuations, and we estimated the maximum size of these details in terms of the Rytov dispersion, σ_{R}^{2}. These results allow us to conclude that the applicability condition for the perturbation theory is the smallness of the Rytov dispersion, and this condition holds uniformly over the distances between observation points. The relative correction at distances smaller than or of the order of the radius of the first Fresnel zone is of the same order as the Rytov dispersion, while at larger distances, the relative correction is much smaller than the Rytov dispersion. Therefore, if the single parameter (the Rytov dispersion) is small, the zeroth approximation can be used for the correlation function of phase gradients at any distance.