Correlations of the phase gradients of the light wave propagating in a turbulent medium in the regime of weak scintillations

TL;DR

The paper addresses how atmospheric turbulence affects the phase of a monochromatic beam in the weak scintillation regime, focusing on the off-diagonal phase-gradient correlation $Q_{xy}$ that is insensitive to the outer scale $L_0$. It uses direct numerical simulations with a chain of phase screens and a parabolic diffraction model to compute the pair correlations of the log-envelope $\eta=\ln(\Psi/\Psi_0)$ and their gradients, comparing against analytic expressions from Ref. $KLS25$ and related work. Findings show that the zeroth-order expression for $Q_{\alpha\beta}$ agrees with simulations for $\sigma_R^2$ up to about 0.5, and that $Q_{xy}$ exhibits a maximum at $r^\ast \approx 1.66 (\sigma_R^2)^{3/5} r_0$ independent of $L_0$, confirming uniform perturbation theory up to this regime. The results have practical implications for adaptive optics and turbulence diagnostics, suggesting that off-diagonal phase-gradient statistics can be used to estimate the Fried parameter $r_0$ and validate perturbative models, with planned experimental validation and extensions to non-Kolmogorov turbulence.

Abstract

We investigate numerically correlation functions of the phase of light waves that propagate through turbulent media. Special attention is paid to the off-diagonal component of the correlation function of the phase gradients which is insensitive to the outer scale of turbulence. The results of our simulations are in a good agreement with the analytical expressions obtained in Ref. \cite{KLS25}. Thus, we numerically confirm expectations based on uniformity of the perturbation theory for the logarithm of the envelope.

Figures (4)

• Figure 1: The pair correlation function (\ref{['pairnum']}) as a function of $r/r_0$ for $\sigma_R^2=0.5$, $r_0=3.75$ cm, $L_0=20$ m. The solid line is drawn using the analytic expression (\ref{['BKLS24']}) and the filled squares represent the numerical data extracted from the simulations.
• Figure 2: The product $-r_0^2 Q_{xy}$ as a function of $r/r_0$ taken at $x=y=r/\sqrt2$ for $\sigma_R^2=0.14$, $r_0$=5.69 cm, $L_0$=20 m, where $Q_{xy}$ is the non-diagonal component of the pair correlation function of the phase gradients. The solid line is drawn using the analytic expression given in Ref. KLS25 and the filled squares represent the numerical data extracted from the simulations.
• Figure 3: The product $-r_0^2 Q_{xy}$ as a function of $r/r_0$ taken at $x=y=r/\sqrt2$ for $\sigma_R^2=0.5$, $r_0$=3.75 cm, $L_0$=20 m, where $Q_{xy}$ is the non-diagonal component of the pair correlation function of the phase gradients. The solid line is drawn using the analytic expression given in Ref. KLS25 and the filled squares represent the numerical data extracted from the simulations.
• Figure 4: A dependence of $\sigma_R^2$ on $r^\ast/r_0$, where $r^\ast$ is the separation at which the maximum of the off-diagonal component of the correlation function of the phase gradients $Q_{xy}$ is observed. The squares represent analytical predictions, the triangles are results of the numerical simulations and the solid line is determined by $\sigma_R^2=0.43(r^\ast/r_0)^{5/3}$.