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

1. Problem Statement

The paper addresses the propagation of light waves through turbulent media (e.g., Earth's atmosphere), specifically focusing on the regime of strong scintillations.

• Context: In weak scintillation regimes, light beam distortions are small. However, over long distances or high turbulence levels (characterized by a large Rytov variance, $\sigma^2_R$), the beam breaks into individual speckles, leading to strong scintillations.
• Challenge: While the statistical properties of the light envelope ($\Psi$) in the strong scintillation regime are known to follow Gaussian statistics (due to the Central Limit Theorem), the analytical derivation of the pair correlation function of phase gradients ($Q_{\alpha\beta}$) has been difficult. Standard perturbation theories fail in this regime because the series for the envelope is non-uniform, whereas the series for phase gradients is uniform but complex.
• Goal: To analytically derive an expression for the phase gradient correlation function based on the Gaussian statistics of the envelope and to verify this via direct numerical simulations.

2. Methodology

The authors employed a dual approach combining rigorous analytical derivation and high-fidelity numerical simulation.

A. Analytical Derivation

• Assumption: In the regime of strong scintillations ($\sigma^2_R \gg 1$), the complex envelope $\Psi$ follows Gaussian statistics.
• Mathematical Framework:
  • The authors start with the parabolic wave equation governing the envelope $\Psi$ in a medium with refractive index fluctuations $\nu$.
  • They utilize the known pair correlation function of the envelope, $F(r) = \langle \Psi(r_1)\Psi^*(r_2) \rangle$, which is derived from the von Karman spectrum of turbulence.
  • Wick's Theorem: Since $\Psi$ is assumed Gaussian, they apply Wick's theorem to calculate the average of products of fields.
  • Integral Representation: They express the phase gradient correlation function $Q_{\alpha\beta} = \langle \partial_\alpha \phi(r_1) \partial_\beta \phi(r_2) \rangle$ using an integral representation involving an auxiliary exponential factor.
  • Result: By solving a system of equations for the "weighted" averages (incorporating the exponential factor), they derived an explicit analytical expression relating $Q_{\alpha\beta}$ directly to the envelope correlation function $F(r)$ and its derivatives:
    $$Q_{\alpha\beta} = \frac{1}{2} \partial_\alpha \partial_\beta \ln \left[ F \ln(1 - F^2) \right]$$

B. Numerical Simulations

• Setup: The team performed direct numerical simulations of a monochromatic plane wave ($\lambda = 0.55 \mu m$) propagating through a statistically homogeneous turbulent medium.
• Modeling Technique:
  • The continuous turbulent medium was modeled using the split-step Fourier method (or finite difference with splitting) involving a chain of random phase screens (100–200 screens).
  • The phase jumps on the screens were generated to mimic the Kolmogorov/von Karman spectrum of refractive index fluctuations.
  • Subharmonics Method: To accurately reproduce large-scale spatial behaviors without increasing the computational box size, they used a spectral method with added subharmonics.
• Parameters: Simulations were run for various Rytov variances ($\sigma^2_R$ ranging from 1.05 to 5.73) and Fried parameters ($r_0$).
• Data Processing: A Monte Carlo approach was used with 300–500 realizations to compute statistical averages of the phase gradients.

3. Key Contributions

1. Analytical Link: The paper successfully derives a closed-form analytical expression for the pair correlation function of phase gradients in the strong scintillation regime, expressing it entirely in terms of the envelope's pair correlation function. This bridges the gap between envelope statistics and phase gradient observables.
2. Validation of Gaussianity: The work confirms that while the envelope statistics are Gaussian, the resulting phase gradient correlations exhibit specific behaviors that deviate from standard perturbation theory predictions.
3. Identification of Regime Limits: The study rigorously defines the spatial limits ($r$) where the Gaussian approximation holds versus where it breaks down.

4. Key Results

• Agreement at Small Separations ($r < r_0$):
  • For distances $r$ smaller than the Fried length ($r_0$), the numerical simulation results for the off-diagonal component of the phase gradient correlation ($Q_{xy}$) show excellent agreement with the newly derived analytical expression (Eq. 29).
  • As $\sigma^2_R$ increases, the numerical profile of $Q_{xy}$ gradually converges from the zeroth-order perturbation theory result to the theoretical expression derived under Gaussian assumptions.
• Divergence at Large Separations ($r > r_0$):
  • For $r > r_0$, the numerical results deviate significantly from the Gaussian-based analytical prediction.
  • Explanation: The authors explain that while the envelope is Gaussian for small separations, the "irreducible" part of the fourth-order correlation function (non-Gaussian contributions) decays only as a power law at large distances, eventually dominating the reducible (Gaussian) part. Thus, the Gaussian assumption fails for $r \gtrsim r_0$.
• Behavior of $Q_{xy}$:
  • The off-diagonal component $Q_{xy}$ exhibits a minimum near the Fried parameter $r_0$.
  • At large separations, $Q_{xy}$ grows and approaches the zeroth-order perturbation value, consistent with regular perturbation theory applied to $\ln \Psi$.

5. Significance and Applications

• Adaptive Optics (AO): The phase gradient correlation function is directly measurable using Shack-Hartmann wavefront sensors. The ability to predict these correlations in the strong scintillation regime is crucial for designing and optimizing AO systems for long-range atmospheric propagation.
• Fried Parameter Estimation: The discovery that $Q_{xy}$ has a distinct minimum near $r_0$ in the strong scintillation regime offers a new, robust method for experimentally determining the Fried parameter ($r_0$) even when turbulence is severe.
• Theoretical Insight: The work clarifies the limitations of Gaussian approximations in turbulence. It demonstrates that while the envelope is Gaussian, the phase statistics are non-Gaussian at large scales, a nuance critical for high-precision optical modeling.
• Future Work: The authors plan to conduct experimental observations to validate these findings, noting that measuring phase correlations in strong scintillation is experimentally challenging but feasible.

In summary, this paper provides a critical theoretical and numerical framework for understanding light propagation in strong turbulence, offering a new analytical tool for phase gradient correlations and a potential new method for characterizing atmospheric turbulence.