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Complex Gaussianity of long-distance random wave processes

Guillaume Bal, Anjali Nair

TL;DR

The paper provides a rigorous derivation showing that, under scintillation (weak-coupling) scaling for the Itô-Schrödinger paraxial model, a deterministic incident beam evolves into fully developed speckle described by a circularly symmetric complex Gaussian field. It achieves this by analyzing all moments through phase-compensated dual-variable equations, proving convergence of finite-dimensional distributions to Gaussian limits in both kinetic and diffusive regimes, and establishing that irradiance becomes exponentially distributed with unit scintillation index. A diffusion-type equation governs the mean irradiance in the diffusive regime, yielding anomalous beam spreading with transverse variance growing as $z^{3/2}$. The results justify the common speckle-statistics assumptions, quantify self-averaging upon regional averaging, and extend Gaussian convergence from Fourier-domain descriptions to physical-space variables, with broad applicability to wide incident beams and potential extensions to partial coherence. Overall, the work provides a comprehensive, rigorous framework linking microscopic random fluctuations to macroscopic Gaussian speckle behavior in long-range beam propagation.

Abstract

Interference of randomly scattered classical waves naturally leads to familiar speckle patterns, where the wave intensity follows an exponential distribution while the wave field itself is described by a circularly symmetric complex normal distribution. In the Itô-Schrödinger paraxial model of wave beam propagation, we demonstrate how a deterministic incident beam transitions to such a fully developed speckle pattern over long distances in the so-called scintillation (weak-coupling) regime.

Complex Gaussianity of long-distance random wave processes

TL;DR

The paper provides a rigorous derivation showing that, under scintillation (weak-coupling) scaling for the Itô-Schrödinger paraxial model, a deterministic incident beam evolves into fully developed speckle described by a circularly symmetric complex Gaussian field. It achieves this by analyzing all moments through phase-compensated dual-variable equations, proving convergence of finite-dimensional distributions to Gaussian limits in both kinetic and diffusive regimes, and establishing that irradiance becomes exponentially distributed with unit scintillation index. A diffusion-type equation governs the mean irradiance in the diffusive regime, yielding anomalous beam spreading with transverse variance growing as . The results justify the common speckle-statistics assumptions, quantify self-averaging upon regional averaging, and extend Gaussian convergence from Fourier-domain descriptions to physical-space variables, with broad applicability to wide incident beams and potential extensions to partial coherence. Overall, the work provides a comprehensive, rigorous framework linking microscopic random fluctuations to macroscopic Gaussian speckle behavior in long-range beam propagation.

Abstract

Interference of randomly scattered classical waves naturally leads to familiar speckle patterns, where the wave intensity follows an exponential distribution while the wave field itself is described by a circularly symmetric complex normal distribution. In the Itô-Schrödinger paraxial model of wave beam propagation, we demonstrate how a deterministic incident beam transitions to such a fully developed speckle pattern over long distances in the so-called scintillation (weak-coupling) regime.
Paper Structure (24 sections, 22 theorems, 248 equations)

This paper contains 24 sections, 22 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. Random wave process and main results
  3. Itô-Schrödinger wave process.
  4. Rescaled process in scintillation regime.
  5. Spatially rescaled random vector.
  6. Main results.
  7. Main steps of the derivation.
  8. Limiting expressions for first and second statistical moments
  9. First moment
  10. Second moment
  11. Kinetic regime $\eta(\varepsilon)=1$.
  12. Diffusive regime with $\eta^{-1}=\ln\ln\varepsilon^{-1}$.
  13. Higher order moments and the Gaussian summation rule
  14. Moments of compensated wave field and integro-differential equations
  15. Approximation of higher moments in dual variables
  16. ...and 9 more sections

Key Result

Theorem 2.2

The mean zero random vector $\Phi^\varepsilon-\mathbb{E}[\Phi^\varepsilon]\boldsymbol{\Rightarrow}\tilde{\Phi}$ in distribution as $\varepsilon\to 0$, where $\tilde{\Phi}=(\tilde{\phi}_1,\ldots,\tilde{\phi}_N)$ is a circularly symmetric Gaussian random vector characterized by Here, $\widetilde{M}_{1,1}$ is the limiting centered second moment given by eqn:plane_wave_mod-eqn:mu_2_plane_wave_super_p

Theorems & Definitions (42)

  • Theorem 2.2: Kinetic regime
  • Remark 2.3: Kinetic regime for smooth incident beams
  • Theorem 2.4: Diffusive regime
  • Corollary 2.5: Scintillation in the diffusive regime
  • Corollary 2.6: Self-averaging in the diffusive regime
  • Theorem 2.7: Stochastic continuity and tightness
  • Corollary 2.8: Convergence of processes
  • Lemma 3.1: First and second moment limits
  • Lemma 4.1
  • proof
  • ...and 32 more