Splitting algorithms for paraxial and Itô-Schrödinger models of wave propagation in random media

TL;DR

This work develops a full discretization and splitting-based framework for simulating wave-beam propagation in random media under the paraxial and Itô-Schrödinger models. It establishes rigorous convergence rates, showing first-order mean-square accuracy in the axial step $\Delta z$ and second-order accuracy in moments for centered splitting, with super-algebraic spatial discretization under smooth inputs. The analysis combines closed-form moment equations for the Itô-Schrödinger model with a Duhamel expansion for the paraxial regime, and is complemented by supplementary material proving convergence in distribution. Numerical experiments corroborate the theoretical rates and demonstrate phase-screen-like behavior and speckle formation, highlighting the method’s robustness even when $\theta \ll \Delta z$. Overall, the paper provides a principled, high-accuracy approach to simulating propagation in random media with practical implications for optical beam modeling in turbulence and related stochastic-wave contexts.

Abstract

This paper introduces a full discretization procedure to solve wave beam propagation in random media modeled by a paraxial wave equation or an Itô-Schrödinger stochastic partial differential equation. This method bears similarities with the phase screen method used routinely to solve such problems. The main axis of propagation is discretized by a centered splitting scheme with step $Δz$ while the transverse variables are treated by a spectral method after appropriate spatial truncation. The originality of our approach is its theoretical validity even when the typical wavelength $θ$ of the propagating signal satisfies $θ\llΔz$. More precisely, we obtain a convergence of order $Δz$ in mean-square sense while the errors on statistical moments are of order $(Δz)^2$ as expected for standard centered splitting schemes. This is a surprising result as splitting schemes typically do not converge when $Δz$ is not the smallest scale of the problem. The analysis is based on equations satisfied by statistical moments in the Itô-Schrödinger case and on integral (Duhamel) expansions for the paraxial model. Several numerical simulations illustrate and confirm the theoretical findings.

Figures (5)

• Figure 1: Error plots from first order scheme for the paraxial and Itô-Schrödinger models
• Figure 2: Rate of convergence with the second order splitting scheme under the Itô-Schrödinger model
• Figure 3: Statistical averages from the first-order simulations for the paraxial approximation and Itô-Schrödinger models. The left and right panels display the real part of wavefield mean value and its intensity, respectively.
• Figure 4: Plots showing evolution of the beam profile and cross-section of the reference solution at $z=1$ (absolute value) for different $\sigma$. The first panel corresponds to the beam evolution for $\sigma=0.125$ while the second panel plots the absolute value of the signal at $z=Z$. The third and fourth panels correspond to the same plots for $\sigma=1$. The last panel plots the scintillation index at the beam's center.
• Figure 5: Receiver reading at $z=1$ for increasing $\sigma$.

Theorems & Definitions (27)

• Remark 1.1
• Theorem 2.1: Path-wise estimates
• Theorem 2.2: Moment estimates
• Remark 2.3
• Corollary 2.4
• Remark 2.5
• Lemma 2.6
• proof
• Lemma 3.1
• proof