Generalized Sheet Transition Condition FDTD Simulation of Metasurface

We propose an FDTD scheme based on Generalized Sheet Transition Conditions (GSTCs) for the simulation of polychromatic, nonlinear and space-time varying metasurfaces. This scheme consists in placing the metasurface at virtual nodal plane introduced between regular nodes of the staggered Yee grid and inserting fields determined by GSTCs in this plane in the standard FDTD algorithm. The resulting update equations are an elegant generalization of the standard FDTD equations. Indeed, in the limiting case of a null surface susceptibility ($\chi_\text{surf}=0$), they reduce to the latter, while in the next limiting case of a time-invariant metasurface $[\chi_\text{surf}
eq\chi_\text{surf}(t)]$, they split in two terms, one corresponding to the standard equations for a one-cell ($\Delta x$) thick slab with volume susceptibility ($\chi$), corresponding to a diluted approximation ($\chi=\chi_\text{surf}/(2\Delta x)$) of the zero-thickness target metasurface, and the other transforming this slab in a real (zero-thickness) metasurface. The proposed scheme is fully numerical and very easy to implement. Although it is explicitly derived for a monoisotropic metasurface, it may be straightforwardly extended to the bianisotropic case. Except for some particular case, it is not applicable to dispersive metasurfaces, for which an efficient Auxiliary Different Equation (ADE) extension of the scheme is currently being developed by the authors. The scheme is validated and illustrated by five representative examples.