Spatiotemporal Tracking of Persistent, Localized Speckles in Turbulent Atmospheric Propagation

TL;DR

This work investigates the spatiotemporal evolution of individual speckles formed during turbulent atmospheric propagation, revealing that speckles can be spatially localized and temporally persistent even after global coherence decays. By tracking speckles with an intensity-threshold algorithm and defining observables such as $\mathcal{C}(z)$, $\tilde{r}(z)$, $\tilde{r}_j(z)$, and $\Delta z_j$, the study links localization and persistence to turbulence strength via the inner scale $\ell_0$, beam width $w_0$, and refractive-index structure parameter $C_n^2$. The results show that fragmentation and decorrelation occur earlier for larger beams, while a subset of speckles remain confined and long-lived, akin to partial Anderson localization, especially in an intermediate regime of diffraction and turbulence. These findings offer a framework for discrete, object-level analysis of light in turbulent media and suggest that speckle statistics could aid in estimating time-averaged $\langle C_n^2\rangle$ for improved long-range sensing and imaging systems.

Abstract

Light propagation through turbulence produces speckles, whose ensemble behavior is typically characterized by snapshot intensity statistics. Here, we track the spatiotemporal evolution of individual speckles and quantify fragmentation, localization, and persistence under different diffraction and turbulence scales. Beam fragmentation coincides with complete spatial decorrelation defined by the magnitude-squared coherence. Fragmentation occurs closer to the source for larger beams, which indicates that smaller beams are more robust to decoherence. Subsequently, speckles are both spatially localized and persistent over distances significantly longer than their associated Rayleigh length. The combination of localization and persistence impacts the statistics of light relevant to their long-distance signaling and sensing.

Figures (4)

• Figure 1: Speckle simulation and tracking. Top row: spatial beam profiles from the source to the receiver plane with representative turbulence profiles. The mode field radius $\tilde{r}(z)$ (gray), source plane speckle (magenta), longest lived non-initial speckle (green), and other detected speckles (blue) are tracked based on their nearest neighbor conditions. Bottom row: magnified views (red box) of observation points (1-3). Detected structures have a local peak intensity greater than $\eta I_{\rm max}$.
• Figure 2: Propagation of Gaussian profiles under (A) weak turbulence ($\sigma_R^2 = 0.558$), (B) moderately-strong turbulence ($\sigma_R^2 = 5.58$), and (C) strong turbulence ($\sigma_R^2 = 55.8$). The transverse profiles (a1-a4, b1-b4, c1-c4) show the width evolution of the MFR (gray) and speckles (magenta, lime, blue) for the same realizations shown in the trajectory plots on the left located at $z = 0, 975, 1950, 3000\ \text{m}$. Conditions: $w_0 = 1\ \text{cm}$ and $\ell_0 = 5\ \text{mm}$.
• Figure 3: Transition to speckle fragmentation. (A) Spatial coherence $\mathcal{C}(z)$ (left axis) for varied beam widths $w_0$ and inner scales $\ell_0$ under strong turbulence $\left(C_n^2=10^{-13}\,\mathrm{m}^{-2/3}\right)$. The decorrelation distances $L_{\rm c}$ (vertical lines) loosely correspond with the focusing regime of the scintillation index $\sigma_I^2$ (right axis, green), whose analytic relation only depends on $\ell_0$. (B-E) Trajectory plots of representative realizations corresponding to (A). The tracks are projected with dashed lines in the $y-z$ plane. The shaded planes (dashed borders, $z = L_{\rm c}$) mark the transition to speckle fragmentation.
• Figure 4: Evolution of the spot-normalized MFR $\tilde{r}(z)/w_0$ of the global beam (gray) and tracked speckles (magenta, lime, blue) under (A) weak ($\sigma_R^2 = 0.558$), (B) moderately-strong ($\sigma_R^2 = 5.58$), and (C) strong ($\sigma_R^2 = 55.8$) turbulence. The ensemble averages (dashed), standard deviation (shaded dark), and the maxima/minima (shaded light) across all 500 runs show increasing confinement. The inset histograms of the spot-normalized Rayleigh lengths $\Delta z_j/ z_{0,j}$ show the transition of persistence statistics from a bimodal to a long-tailed distribution.