Homogenization of Periodic Structures Using the Multimodal Transfer Matrix Method

This work presents a method for obtaining the constitutive parameters of periodic structures from the computation of their dispersion relation and average fields. The method uses the scattering parameters (S-parameters) of multiple Bloch modes of a single unit cell. The corresponding multimodal scattering matrix is obtained with a suitable general-purpose electromagnetic software. Further post-processing of this scattering matrix is then carried out, which allows for the computation of the dispersion relation of structures with realistic finite conductivity or made of lossy dielectrics, as well as the calculation of the attenuation constant and the retrieval of the impedance, permittivity, and permeability. The proposed method is applied to homogenize some systems of interest: an artificial electric plasma built with wires, a free-space matched left-handed metamaterial based on two laterally shifted split ring resonators, a high-permittivity artificial dielectric based on densely arranged square metal patches, and a
$\mu $
-near-zero metamaterial made of metallic cubes embedded in a dielectric. The retrieved material parameters are found to accurately describe the scattering of finite samples of the corresponding homogenized structures. This research is limited to orthorhombic unit cells smaller than half the free-space wavelength to avoid diffracted beams. Besides, since only one propagation direction is considered, only the transverse components of constitutive parameters are retrieved.