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

TL;DR

This paper provides a comprehensive analysis of electromagnetic wave propagation in media with randomly time-varying permittivity, revealing distinct statistical regimes and the influence of input symmetry on wave energy distribution.

Contribution

It introduces exact moment equations and identifies three statistical regimes, offering a unified framework for understanding wave statistics in time-modulated media.

Findings

Identifies three statistical regimes: gamma, negative exponential, and quasi-log-normal.

Symmetric input yields genuine log-normal statistics across all times.

Results are robust across different disorder models and linked 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.