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Complex Gaussianity and spatio-frequential memory effect of random wave processes

Guillaume Bal, Anjali Nair

Abstract

Wavefield speckle patterns are generated by interference of randomly scattered coherent light. In the weak-coupling regime of the Itô-Schrödinger paraxial model for long-distance wave propagation, we show the following multiscale character: a macroscopic envelope solves a deterministic diffusion equation while the local wavefield (the speckle) is described by a complex Gaussian process both in terms of spatial axial and lateral displacements as well as frequency and angular variations of the incident wavebeam. These results describe speckle patterns and corroborate chromato-spatial memory effects observed in laser light propagation through heterogeneous media.

Complex Gaussianity and spatio-frequential memory effect of random wave processes

Abstract

Wavefield speckle patterns are generated by interference of randomly scattered coherent light. In the weak-coupling regime of the Itô-Schrödinger paraxial model for long-distance wave propagation, we show the following multiscale character: a macroscopic envelope solves a deterministic diffusion equation while the local wavefield (the speckle) is described by a complex Gaussian process both in terms of spatial axial and lateral displacements as well as frequency and angular variations of the incident wavebeam. These results describe speckle patterns and corroborate chromato-spatial memory effects observed in laser light propagation through heterogeneous media.

Paper Structure

This paper contains 20 sections, 21 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Assumption on the medium.
  4. Convergence results.
  5. Speckle memory effect.
  6. Analysis of second-order moments
  7. Phase compensation
  8. Frequency correlations
  9. Two-point correlation function in diffusive regime
  10. Lateral shifts and tilts and proof of Theorem \ref{['thm:tilt_mem']}.
  11. Frequency and axial shifts and proof of Theorem \ref{['thm:chrom_ax_mem']}.
  12. Analysis and convergence of higher moments
  13. Phase compensation
  14. Multi-frequency moments at fixed distance
  15. Gaussian summation rule in all variables
  16. ...and 5 more sections

Key Result

Theorem 2.1

The random vector $\Upsilon^\varepsilon\Rightarrow\Upsilon$ in distribution as $\varepsilon\to 0$ where $\Upsilon$ is a complex Gaussian random vector with entries $\{\upsilon_j\}_{j=1}^N$ satisfying where $h_{j,l}=h_j-h_l$, $\tau_{j,l}=\tau_j-\tau_l$, $\Omega_{j,l}=\Omega_l-\Omega_j$, $\kappa_{j,l}=\kappa_j-\kappa_l$, and $m_{1,1}$ is given by eqn:m_11 below.

Theorems & Definitions (38)

  • Theorem 2.1: Convergence of finite dimensional distributions
  • Theorem 2.2: Tightness and stochastic continuity
  • Theorem 2.3: Convergence of processes
  • Theorem 2.4: Tilt memory effect
  • Theorem 2.5: Chromato-axial memory effect
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 28 more