Extension of effective permittivity for metallic elements on the substrates interface

Previously, the influence of substrate interface on the metallic element is always described by the specific effective permittivity. Here, the metallic element on the substrate interface can be converted into that in a homogenized equivalent substrate with almost arbitrary permittivity. Specifically, according to the definition of the permittivity, the substrate interface with a small thickness can be regarded as a homogenized equivalent substrate. Based on the tangential components of the electrical fields, it can be derived that the equivalent permittivity is strongly related to the thickness of the equivalent substrate. Finally, the simulation results demonstrate that the electromagnetic response of the metallic element placed in the homogenized equivalent substrate keeps unchanged as that on the substrate interface. Thus, the proposed method not only provides a generalized means to describe the electromagnetic influence of the substrate interface on the metallic element, but also greatly simplifies the analytical design of multi-layered metasurface.