Electromagnetic characterization of substrated metasurfaces

In this paper we present a macroscopic model of a metasurface-optically dense grids of resonant scatterers located on a refracting interface. Similar models were previously built for the case when the scatterers are non-resonant electric dipoles and for the case when there is no substrate. In the present case scatterers are resonant, can have magnetic response, and for substrates with large refraction index acquire also substrate-induced bianisotropic properties. Representing the homogenized response of a metasurface to incident electric and magnetic fields through surface susceptibilities we derive the reflection and transmission formulas for semi-infinite and finite-thickness substrates. We develop a robust algorithm of the retrieval of the characteristic surface parameters from the reflection and transmission coefficients measured or exactly calculated for three different incidence angles. With numerical examples we show that the retrieved susceptibilities of metasurfaces are independent on the incidence angle and can be really called characteristic parameters.