Two-step homogenization of spatiotemporal metasurfaces using an eigenmode-based approach

Abstract

Metamaterials are a fascinating class of photonic materials since they allow us to control optical responses (largely) at will. Besides being an intellectual challenge, adding time variations into spatial metamaterials increases the degrees of freedom to tune their effective response, which motivates their exploration. However, to exploit such materials in the future design of functional devices, we may wish to treat them at the effective level to avoid considering all the mesoscopic details. To permit such effective treatment, we describe here an eigenmode-based approach to homogenize spatiotemporal metamaterials composed of a periodic arrangement of scatterers made from a time-varying material. Practically, we consider the periodic arrangement of spheres within one layer. In our two-step homogenization scheme, we first temporally homogenize that metasurface using the eigenmodes of the bulk time-varying material. Following this, we perform spatial homogenization by inverting the Fresnel coefficients of a slab made from a stationary material. These steps effectively describe the optical response of the spatiotemporal metasurface as a homogeneous slab. We validate our results by comparing the optical observables, i.e., reflectivity and transmissivity, of the metasurface with those of the homogenized slab, and we assess the limitations of the homogenization.