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

## TL;DR

This paper explores how electromagnetic waves behave in materials with random time changes, revealing distinct statistical patterns based on input symmetry.

## Contribution

The study identifies three new statistical regimes of wave propagation in time-varying media and links them to input symmetry and momentum conservation.

## Key findings

- Gamma-distributed energy occurs at early times in unidirectional input.
- Symmetric bidirectional input leads to log-normal statistics across all time scales.
- Momentum conservation connects statistical outcomes 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.