Operator approach to effective medium theory to overcome a breakdown of Maxwell Garnett approximation

We elaborate on an operator approach to effective medium theory for homogenization of the periodic multilayered structures composed of nonmagnetic isotropic materials, which is based on equating the spatial evolution operators for the original structure and its effective alternative. We show that the zeroth-, first-, and second-order approximations of the operator effective medium theory correspond to electric dipoles, chirality, and magnetic dipoles plus electric quadrupoles, respectively. We discover that the spatially dispersive bianisotropic effective medium obtained in the second-order approximation perfectly replaces a multilayered composite and does not suffer from the effective medium approximation breakdown that happened near the critical angle of total internal reflection found previously in the conventional effective medium theory. We establish the criterion of the validity of the conventional effective medium theory depending on the ratio of unit-cell length to the wavelength, the number of unit cells, and the angle of incidence. The operator approach to effective medium theory is applicable for periodic and nonperiodic layered systems, being a fruitful tool in the fields of metamaterials and subwavelength nanophotonics.