Design of an isotropic all-dielectric metasurface on a substrate

A mathematical statement of the problem of one-wavelength antireflective coating based on a metasurface is formulated. The constraints on geometric parameters of the structure are found. A penalty function is proposed to ensure the applicability of the physical model and to provide the uniqueness of the desired minimum. As an example, the problem of antireflective metasurface composed of PbTe spheres located on a germanium substrate is considered. The reflectance spectrum of such a structure is found in the range 8-12 um and is compared with the spectrum of the metasurface without a substrate and with the spectrum of an uncovered substrate.