Generalized Lorentz-Lorenz homogenization formulas for binary lattice metamaterials

Generalized Lorentz-Lorenz formulas are developed for the effective parameters of binary lattice metamaterials composed of a periodic arrangement of electric and/or magnetic inclusions. The proposed homogenization approach is based on a dual dipole approximation for the induced currents. The obtained formulas for the metamaterial effective electric and magnetic characteristics duly consider both electric and magnetic polarizabilities of the inclusions and completely describe the effects of frequency and spatial dispersion. Several numerical examples are provided to demonstrate the general applicability of the proposed formulas to different types of binary lattices and inclusions. It is shown that the proposed effective parameters have the capability of providing a physically sound and accurate description of wave propagation in the metamaterials in an extended range of frequencies in contrast to the equivalent parameters that can be defined in the absence of impressed sources and assuming a local anisotropic constitutive model, which hides inherent spatial dispersion effects and nonphysical features. To gain further insight into the metamaterial response and the physical meaningfulness of calculated effective parameters, the power flow of metamaterial supported modes is analyzed and its homogenized representation is compared to the complete description. A correspondence between the power flow due to the microscopic field and the effect of spatial dispersion in the homogenized parameters is established.