Comparison between the linear and nonlinear homogenization of graphene and silicon

In this paper, we use a versatile homogenization approach to model the linear and nonlinear optical response of two metasurfaces: a plasmonic metasurface consisting of a square array of graphene cruciform patches and a dielectric metasurface consisting of a rectangular array of photonic crystal (PhC) cavities in a silicon PhC slab waveguide. The former metasurface is resonant at wavelengths that are much larger than the graphene elements of the metasurface, whereas the resonance wavelengths of the latter one are comparable to the size of its resonant components. By computing and comparing the effective permittivities and nonlinear susceptibilities of the two metasurfaces, we infer some general principles regarding the conditions under which homogenization methods of metallic and dielectric metasurfaces are valid. In particular, we show that in the case of the graphene metasurface the homogenization method describes very well both its linear and nonlinear optical properties, whereas in the case of the silicon PhC metasurface the homogenization method is less accurate, especially near the optical resonances