Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices

Metamaterials comprising lattices of small resonant scatterers and (optionally) infinite wires are considered. The crystal planes of these lattices contain magnetic dipoles and electric dipoles (or wires). The set of so-called Bloch material parameters is discussed. A class of lattices for which these parameters describe the transfer matrix of an individual monolayer is considered. These lattices are called as Bloch lattices. It is shown that for Bloch lattices and only for them the Bloch material parameters can be directly extracted from the S-parameters of a finite-thickness metamaterial slab. Material parameters retrieved in this way in the previous literature are either Bloch material parameters or senseless (not invariant with respect of the number of monolayers in the composite slab). Explicit examples of Bloch and non-Bloch lattices are presented. Some intermediate results (two definitions of the Bloch impedance, frequency bounds of the Lorentz–Lorenz formula for the local material parameters, etc.) having practical and theoretical importance are discussed.