Statistical regimes of electromagnetic wave propagation in randomly time-varying media

TL;DR

This work addresses how electromagnetic waves propagate in randomly time-varying media and how temporal disorder reshapes energy transport. The authors apply the invariant imbedding method to derive exact moment equations for reflectance, transmittance, and energy in a spatially uniform medium with random time dependence of the permittivity, considering delta-correlated Gaussian and piecewise-constant disorder. For unidirectional input, the energy statistics traverse three regimes—gamma-like at early times, negative exponential at intermediate times, and a quasi-log-normal long-time scaling—whereas symmetric bidirectional input yields true log-normal statistics at all times; long-time growth follows $\langle U^n\rangle = \frac{n!}{(2n-1)!!}\exp\left[n(n+1)\frac{g\omega_0 t}{4}\right]$ (or unity prefactor for symmetric input). The findings establish a unified framework for statistical wave dynamics in time-modulated systems and offer design principles for dynamically tunable photonic devices, with robustness across disorder models and a momentum-conservation constraint linking statistics to initial conditions.

Abstract

Wave propagation in time-varying media enables unique control of energy transport by breaking energy conservation through temporal modulation. Among the resulting phenomena, temporal disorder-random fluctuations in material parameters-can suppress propagation and induce localization, analogous to Anderson localization. However, the statistical nature of this process remains incompletely understood. We present a comprehensive analytical and numerical study of electromagnetic wave propagation in spatially uniform media with randomly time-varying permittivity. Using the invariant imbedding method, we derive exact moment equations and identify three distinct statistical regimes for initially unidirectional input: gamma-distributed energy at early times, negative exponential statistics at intermediate times, and a quasi-log-normal distribution at long times, distinct from the true log-normal. In contrast, symmetric bidirectional input yields genuine log-normal statistics across all time scales. These findings are validated using two complementary disorder models--delta-correlated Gaussian noise and piecewise-constant fluctuations--demonstrating that the observed statistics are robust and governed by input symmetry. Momentum conservation constrains the long-time behavior, linking the statistical outcome to the initial conditions. Our results establish a unified framework for understanding statistical wave dynamics in time-modulated systems and offer guiding principles for the design of dynamically tunable photonic and electromagnetic devices.

Figures (9)

• Figure 1: Log-log plot of the temporal evolution of the reflectance moments $\langle R \rangle$, $\langle R^2 \rangle$, $\langle R^3 \rangle$, and $\langle R^4 \rangle$ for Model 1 with $g = 0.003$. Numerical results from Eq. (\ref{['eq:z']}) are shown as black solid lines. The orange dashed lines represent the short-time behavior predicted by Eq. (\ref{['eq:short']}), while the green dashed lines correspond to the intermediate-time behavior described by Eq. (\ref{['eq:exp']}). For $n\ge 2$, a clear crossover from the initial regime governed by Eq. (\ref{['eq:short']}) to the regime of Eq. (\ref{['eq:exp']}) occurs near $\omega_0t\approx 1$.
• Figure 2: Temporal growth of energy moments for $g = 0.003$. The evolution of $\langle U \rangle$, $\langle U^2 \rangle$, $\langle U^3 \rangle$, and $\langle U^4 \rangle$ is shown. Solid lines represent numerical results from Eq. (\ref{['eq:z']}), and dashed lines show the analytical prediction from Eq. (\ref{['eq:u1']}). Excellent agreement is observed in the long-time regime $\omega_0 t \gtrsim g^{-1}$.
• Figure 3: Temporal variation of $U_n$ for $g = 0.003$. Here, $U_n = \langle U^n \rangle \exp\left[-n(n+1)g\omega_0 t / 4\right]$. In the long-time regime, numerical results show excellent agreement with the analytical predictions from Eq. (\ref{['eq:u1']}).
• Figure 4: Temporal evolution of $F_3$ for $g = 0.003$. $F_3 \equiv \langle R^3 \rangle \langle R \rangle^3 / \langle R^2 \rangle^3$ evolves from an initial value of 5/9 (gamma distribution), passes through 0.75 (negative exponential distribution), and approaches 1.35, consistent with the quasi-log-normal form given in Eq. (\ref{['eq:u1']}).
• Figure 5: Scaled moments $U_n / [1 + R(0)]^n$ as functions of $R(0) = |{w}|^2$ for $n = 1$–5 at $\omega_0 t = 25000$. $S(0) = |{v}|^2$ is fixed at 1.