The Dynamics of the Transverse Optical Flux in Random Media

TL;DR

This work tackles how linearly polarized light evolves while propagating through correlated random media, focusing on the transverse energy and vortex dynamics as the paraxial approximation gives way to full vector Maxwell propagation. The medium is modeled with Gaussian-correlated index fluctuations defined by a correlation length σ and strength κ, and the vector Helmholtz equation is solved numerically using a modified Born series, with analyses of the ensemble-averaged transverse kinetic energy E_kin, vortex density ρ_v, and the incompressible kinetic-energy spectrum E_kin^i(k). Key contributions include (i) a paraxial expression for the evolution of E_kin and its breakdown in the vector regime, (ii) identification of three-stage vortex nucleation with a cubic-root growth for small σ and a kink for large σ, and (iii) demonstration of a driven steady state due to evanescent filtering and a spectral evolution from a paraxial form to an evanescent-filtered random field with a k^-3 tail; the results also reveal isotropization of the intensities in the asymptotic regime. Overall, the paper provides a quantitative framework bridging paraxial and nonparaxial propagation in random media, with implications for atmospheric scintillation, speckle statistics, and energy transfer in disordered photonic systems, and points to future work on multi-length-scale scatterers and nonlinear effects.

Abstract

We study the evolution of the kinetic energy (or gradient norm) of an incident linearly polarized monochromatic wave propagating in correlated random media. We explore the optical flux transverse to the mean Poynting flux at the paraxial-nonparaxial (vectorial) transition along with vortex counting. Here, by paraxial-nonparaxial transition we mean a gradual loss of validity of the paraxial approximation such that it is necessary to solve Maxwell-consistently employing the dyadic Green's function. The vortex number appears to increase approximately with a cubic root of the propagation distance for sufficiently small correlation length. Furthermore, a kink appears in nucleation rate at the position of maximum scintillation upon increasing correlation length. A driven steady state is reached due to the filtering of evanescent waves upon propagation. Finally, we present the spectrum of the incompressible kinetic energy and how it evolves from the paraxial case to that of a (nonparaxial) random field.

Figures (5)

• Figure 1: Propagation of an initially linearly polarized plane wave in a random medium. The planes depict the phase, the lines correspond to vortex lines [red (blue) topological charge $1$ ($-1$)], and the gray iso-surface depicts the intensity, showing speckle formation.
• Figure 2: (a) Comparison of numerical simulations (solid lines) and analytical predictions (dashed lines) for the evolution of the ensemble-averaged kinetic energy at the initial stage. $\Delta$ denotes the difference evaluated at $z = 50\lambda$. The color of the solid lines is given by $\mathcal{I}_y/\mathcal{I}_x$ and, therefore, indicates how vectorial the light becomes upon propagation. (b) The values of $\Delta$ for various $\sigma$ and $\kappa$, represented on a logarithmic scale. The gray lines represent isolines where $\kappa^2\lambda/\sigma^4$ is constant. The ensemble average was performed over ten realizations.
• Figure 3: (a) Evolution of vortex number per unit area $\rho_{\text{v}}$ for $\sigma = 0.5\lambda, \lambda, 1.5\lambda, 2\lambda$ and $\kappa = 0.25\sigma^{3/2}$, where the black solid line shows the trend for $0.5\lambda$. The dashed and dash-dotted horizontal lines are the estimations of $\rho_{\text{v}, \infty}$ obtained by applying two different filters to a random field in spatial Fourier domain. (b) Values of the asymptotic vortex density $\rho_{\text{v}, \infty}$ for various parameter sets of random media. In practice, use the vortex density at $z=280\lambda$ as asymptotic value. The points at the bottom row correspond to the four curves in panel (a) with the same coloring. Panel (c) shows the evolution of the scintillation index for the four curves in panel (a), the maximum of which (e.g., vertical dotted line for $\sigma=2\lambda$) coinciding with the first branching point and the position where vortex nucleation is inhibited. Panel (d) shows the ratio of intensities in $y$ and $x$ polarizations for the four curves in panel (a).
• Figure 4: (a) The vortex density for $\sigma = 0.5\lambda$ as a function of the normalized kinetic energy $E_{\rm kin}/\mathcal{I}_x$ is approximately linear for sufficiently large vortex density. (b) The increase in vortex density as a function of the propagation distance can be approximated by $\rho_\text{v} \lambda ^2\propto (z/\lambda-z_{\rm cr}/\lambda)^\beta$. We find $\beta=0.36, 0.34, 0.30, 0.27$ for $\kappa/\sigma^{3/2}=0.25, 0.3, 0.35, 0.4$. The purple line shows the propagation of the vortex density in the $x$ component for a source whose $x$ and $y$ components are both complex random fields (white noise) propagating in the random medium with $\kappa/\sigma^{3/2}=0.4$.
• Figure 5: (a) The incompressible kinetic energy as a function of the wave vector $k$ for a paraxial case (red) and a vectorial, nonparaxial case (blue). The dashed lines correspond to the theoretical models and the solid lines to the numerics. The black solid lines show the $k^{-3}$ ultraviolet asymptotics. (b) The dashed lines correspond to the theoretical calculations in panel (a), and the solid lines show the spectra upon propagation, showcasing how an initial plane wave moves from one limiting paraxial case to the random field.