Study on Light Propagation through Space-Time Random Media via Stochastic Partial Differential Equations

Abstract

In this letter, the theory of stochastic partial differential equations is applied to the propagation of light fields in space-time random media. By modeling the fluctuation of refractive index's square of the media as a random field, we demonstrate that the hyperbolic Anderson model is applicable to describing the propagation of light fields in such media. Additionally, several new quantitative characterizations of the stochastic properties that govern the light fields are derived. Furthermore, the validity of the theoretical framework and corresponding results is experimentally verified by analyzing the statistical properties of the propagated light fields after determining the spatial and temporal stochastic features of the random media. The results presented here provide a more accurate theoretical basis for better understanding random phenomena in emerging domains such as free-space optical communication, detection, and imaging in transparent random media. The study could also have practical guiding significance for experimental system design in these fields.

Figures (3)

• Figure 1: The schematic diagram depicts the two-point temperature difference measurement device and the large aperture coherent detection system used in this experiment.
• Figure 2: (a) Double‑logarithmic plot of the squared refractive index (obtained from outdoor measurements using the TempVue 20 Pt100 sensors) versus separation distance. Experimental data $*$ are compared with two piecewise linear fits (solid and dashed lines). (b1)-(b3) For a 2 mm receiving aperture, the ratios $\frac{\langle I^{q}\rangle}{\langle I \rangle^{q}} \vert_{\boldsymbol{x}_{0}}$ for $q = 4, 5, 6$ are plotted against $\frac{\langle I^{2}\rangle}{\langle I \rangle^{2}} \vert_{\boldsymbol{x}_{0}}$ on double‑logarithmic axes. The results confirm the validity of Eq.\ref{['w13']}.
• Figure 3: (a1)-(a2) For a 2 mm aperture, the 24‑hour Hurst index variations of $\mu(t_i, \boldsymbol{x}_0)$ and $\varphi(t_i)$ are strongly correlated (Pearson coefficient 0.905), whereas for a 300 mm aperture the correlation becomes insignificant (Pearson coefficient 0.341). (b) The sKL divergence between $Z_{R}(t)$ and $\mathcal{N}_{t} (0,1)$ is shown on a double‑logarithmic scale as the aperture increases from 2 mm to 300 mm. The dashed line indicates the theoretical upper bound attenuation $C_{2} R^{-\frac{\alpha}{2}}$ in the same coordinates. The results support Eq.\ref{['whurt']} and Eq.\ref{['w15']}.