Local and nonlocal homogenization of wave propagation in time-varying media
Christian Döding, Barbara Verfürth
TL;DR
This paper develops a formal two-scale homogenization framework for Maxwell-based wave propagation in time-varying metamaterials with a rapidly oscillating time-periodic permittivity ε_η(t) = ε(t/η). By decomposing fields into macroscopic and microscopic components and expanding in η, the authors derive leading-order local effective equations with harmonic-time averaged permittivity ε_hom = (∫_0^1 ε^{-1} dτ)^{-1}, and higher-order corrections that become nonlocal, captured by a negative correction coefficient ε_cor and higher derivatives such as Δ^2 u_0. The work treats both electric and magnetic cases, yielding distinct but analogous effective laws, and shows how temporal interface conditions influence initial data for homogenized equations. Recasting the results in the Maxwell framework reveals both local and nonlocal constitutive relations, with numerical experiments validating the predicted convergence orders and illustrating the practical accuracy of the homogenized models for optics, elasticity, and acoustics in time-modulated media. The findings extend prior results, unify electric/magnetic treatments, and provide a rigorous basis for simulating wave propagation in temporally engineered materials.
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.