Dynamical Control of Multilayer Spacetime Structures Using Extended Fourier Modal Method

We introduce two-dimensional space plus time (2D+1) structure and numerically investigate it using a developed multilayer simulation framework, for the first time. The new structure is consisting of crossed grating with time-varying permittivity which is inspired from 1D+1. In this regard, we extend Fourier Modal Method (FMM) in a general approach for spacetime multilayer states. Our proposed framework is fast, robust, and powerful compared to various finite difference methods. We use the scattering matrix technique to develop the proposed spacetime simulation method for multilayer structures using a non-uniform stack of layers. The method is perfectly suitable to investigate the spatiotemporal effects of surfaces/metasurfaces which covers both the transverse electric and magnetic (TE & TM) polarizations. The results show more freedom to control the optical outcomes of the multilayer considering two spatial periodicities in addition to the modulation frequency to reach nonreciprocity as one of the main consequences of the proposed structure. Moreover, we investigate the condition and limitation of breaking the Lorentz rule for spacetime structures. 2D+1 structure is more controllable than the 1D+1 due to its greater ability to adjust spatial manipulation in addition to temporal variations to reach nonreciprocity applications, digital coding, beam steering, etc.