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In this paper, we propose two different methods for time-domain finite-difference analysis of uniform temporally and spatially dispersive metasurfaces using their zero-thickness sheet representations using the Generalized Sheet Transition Conditions (GSTCs). Metasurfaces are described here using their effective surface susceptibilities which are assumed to exhibit Lorentzian temporal dispersion characteristics. For both methods, the spatial dispersion of the surface susceptibilities (i.e., their dependence on the angle of incidence) are represented using the extended GSTCs presented in [1]&#x2013;[3]. However, the first method takes advantage of a polynomial expansion of the angle-dependent surface susceptibilities in terms of the transverse wavevector to implement spatial derivatives of the electric and magnetic polarization as well as the average field on the surface, leading to a coupled set of field equations encompassing the entire surface. Limitations for this method are presented in terms of poor conditioning for a coupled system of equations and an inconvenient extension to the higher-order expansion of the susceptibility terms. The second method lifts these limitations by solving the spatial dispersion problem in the spatial frequency domain at every time step. Both methods are validated for custom Lorentzian models and two canonical physical cells while comparing their transmission and reflection coefficients with analytical results.

Time-Domain Analysis of Temporally and Spatially Dispersive Metasurfaces in GSTC-FDTD Frameworks