Circuit modeling of Metasurfaces with Printed Elements

Metasurfaces are the 2D version of metamaterials technology and are designed as planar periodic structures with unit cell size of less than $0.1 \lambda$. The metamaterials can be characterized by the homogenization method as surfaces with effective tensor electric ($3 \times 3$ matrix) and magnetic ($3 \times 3$ matrix) susceptibilities. In case of metasurfaces with printed elements and arbitrary geometry (our case study) the metasurface can be represented by a tensor shunt admittance ($2 \times 2$ matrix) at the interface between two transmission lines representing the two media on the two sides of the metasurface. The tensor shunt admittance can be extracted based on the boundary conditions at the location of the metasurface elements. As such the tensor shunt admittance depends on the metasurface geometry and the media on the two sides of the metasurface. In the literature, we can find two fundamental methods to extract the tensor shunt admittance. In the first method, the extraction is based on the analysis of the unit cell scattering parameters, while in the second method the extraction is based on the analysis of the sampled near field on top of the unit cell metasurface element. The scattering parameters analysis is performed for the dominant Floquet mode in the metasurface unit cell using two orthogonal polarization measurements or simulations. The circuit modeling holds for a relatively narrow incident angular sector around the normal to the metasurface plane. For larger incident angles on the metasurface the shunt admittance model fails due to diffraction and spatial dispersion effects from the metasurface elements. To improve the model performance beyond 50° incident angle there are two options. The $1^{\text {st }}$ option is to recalibrate the model based on measurements or simulations taken at 50° and compute a new tensor shunt admittance based on the data measured at 50°. The $2^{\text {nd }}$ option is to modify the model of the shunt admittance and a more complex circuit modeling is required to represent the metasurface positioned at the interface between two different media. The circuit modeling of the metasurface accelerates the design cycle of a multilayer metasurface structure like radomes and flat lenses instead of the analysis based on a volumetric problem with an effective refractive index.