Merging Effective Medium Concepts of Spatial and Temporal Media: Opening new avenues for manipulating wave-matter interaction in 4D

While wave-matter interactions can be tailored in space via spatial inhomogeneities in material parameters, temporal and spacetime media are becoming increasingly popular as they may enable the full control of electromagnetic (EM) waves in four dimensions (4D). Here, expanding our previous work on the effective medium concept of temporal media, we develop more general effective medium theories that combine spatial multilayered media with temporal multistepped changes of the permittivity. This work represents a combination of spatial and temporal inhomogeneities in a single structure. As the temporal modulation is applied within certain spatial multilayers (not all the layers), our approach may relax the need for temporally modulating the
whole
medium where the wave travels while still achieving frequency conversion (a key feature of temporal multistepped media). The theoretical formulation and closed-form expressions for the effective permittivity of such spacetime effective media are presented. The proposed structure is studied via numerical simulations, demonstrating the possibility of designing spacetime effective media using a combination of temporally and spatially modulated materials. The increased degrees of freedom provided by our approach may open new possibilities for manipulating wave-matter interaction in 4D.