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

I. V. Kolokolov, V. V. Lebedev

TL;DR

This work develops a perturbative diagrammatic framework for phase correlations of light propagating in a turbulent medium by reformulating the parabolic wave equation in terms of the log-envelope, introducing the nonlinear equation for $\eta=\ln(\Psi/\Psi_0)$, and computing one-loop corrections to the phase correlation functions and Green’s function. The analysis shows that the perturbation theory for phase correlations is controlled by the small Rytov variance $\sigma_R^2$ and remains uniform in observation distance, with additional small factors arising at large separations relative to the Fresnel scale. Large-scale fluctuations are treated via a symmetry-based shift that cancels leading contributions to equal-$z$ correlators but can affect the high-$k$ behavior of the Green’s function, necessitating resummation. The results clarify the regime of validity for perturbative approaches to atmospheric-phase distortions and inform interpretation of Shack-Hartmann measurements and adaptive optics calibrations.

Abstract

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 its asymptotic behaviors. Some non-perturbative conclusions are made using the symmetry properties of the equation. These results allow us to conclude that the applicability condition for the perturbation theory is the smallness of the Rytov dispersion, $σ_R^2$, and this condition holds uniformly over the distances between observation points.

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

TL;DR

This work develops a perturbative diagrammatic framework for phase correlations of light propagating in a turbulent medium by reformulating the parabolic wave equation in terms of the log-envelope, introducing the nonlinear equation for , and computing one-loop corrections to the phase correlation functions and Green’s function. The analysis shows that the perturbation theory for phase correlations is controlled by the small Rytov variance and remains uniform in observation distance, with additional small factors arising at large separations relative to the Fresnel scale. Large-scale fluctuations are treated via a symmetry-based shift that cancels leading contributions to equal- correlators but can affect the high- behavior of the Green’s function, necessitating resummation. The results clarify the regime of validity for perturbative approaches to atmospheric-phase distortions and inform interpretation of Shack-Hartmann measurements and adaptive optics calibrations.

Abstract

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 its asymptotic behaviors. Some non-perturbative conclusions are made using the symmetry properties of the equation. 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.
Paper Structure (15 sections, 112 equations, 6 figures)

This paper contains 15 sections, 112 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main relations
  3. Logarithm of the envelope
  4. Linear approximation for $\eta$
  5. Perturbation theory
  6. Feynman diagrams
  7. First-order corrections: analytics
  8. Triple vertex
  9. Role of large-scale fluctuations of $\nu$
  10. Conclusion
  11. Zeroth approximation for the perturbed plane wave
  12. Correction $G_1$ to the Green's Function
  13. First corrections $\varPhi_1$, $\varXi_1$ for the initial plane wave
  14. Large separations
  15. Small separations in $\varXi_1$.

Figures (6)

  • Figure 1: Diagrammatic representation for $\varPhi_0,\varXi_0,\varPhi^\star_0$ in accordance with Eqs. (\ref{['respo1']},\ref{['respo2']}).
  • Figure 2: Diagrammatic representation of a first-order (one-loop) correction to the Green's function $G$.
  • Figure 3: Diagrammatic representation of a first-order (one-loop) correction to the pair correlation function $\varPhi$.
  • Figure 4: Diagrammatic representation of a first-order (one-loop) correction to the pair correlation function $\varXi$.
  • Figure 5: Diagrammatic representation of a first-order (one-loop) correction to the triple vertex function.
  • ...and 1 more figures