Stacked Time-Varying Metasurfaces

Spatiotemporal metasurfaces offer unique opportunities for wave manipulation, however, their practical realization is often constrained by the requirement for in-plane spatial modulation, which necessitates a large number of time-varying elements. In this work, we introduce an alternative architecture based on a cascade of spatially uniform metasurfaces subjected to periodic temporal modulation. Although all metasurfaces share the same modulation frequency, their individual modulation functions are independently engineered to achieve a desired complex electromagnetic response. We develop a general theoretical framework for the design and optimization of such stacked metasurface systems, composed of dense arrays of cylindrical meta-atoms with time-varying plasma and/or collision frequencies. The effectiveness of the approach is demonstrated through the optimization of metasurface designs that enable magnet-free isolation at the fundamental frequency and a temporal analogue of circulators. Furthermore, we show that a metasurface stack can be implemented using only a few time-modulated elements embedded within a parallel-plate waveguide, opening new avenues for extremely compact, versatile, and scalable spatiotemporal platforms for next-generation photonic and microwave systems.