Local and nonlocal homogenization of wave propagation in time-varying media

TL;DR

This paper develops a mathematical framework for homogenizing wave propagation in time-varying media, revealing local and nonlocal effective behaviors through asymptotic analysis, applicable across various wave-based physical models.

Contribution

It introduces a two-scale asymptotic expansion method to derive effective equations, including nonlocal relations, for electromagnetic waves in temporal metamaterials.

Findings

Effective equations derived for wave propagation in time-varying media.

First-order corrections follow local material laws.

Second-order corrections exhibit nonlocal constitutive relations.

Abstract

Temporal metamaterials are artificially manufactured materials with time-dependent material properties that exhibit interesting phenomena when waves propagate through them. The propagation of electromagnetic waves in such time-varying dielectric media is governed by Maxwell's equations, which lead to wave equations with temporal highly oscillatory coefficients for the electric and magnetic fields. In this study, we analyze the effective behavior of electromagnetic fields in time-varying metamaterials using a formal two-scale asymptotic expansion. We provide a mathematical derivation of the effective equations for the leading-order homogenized solution, as well as for the first- and second-order corrections of the effective solution. While the effective solution and the first-order correction are governed by local material laws, we reveal a nonlocal constitutive relation for the second-order corrections. Special attention is also paid to temporal interface conditions through initial values of the homogenized equations. The results provide a mathematically justified framework for the effective description of wave-type equations of time-varying media, applicable to models in optics, elasticity, and acoustics.