On the uniqueness of virtual substrate for metasurface in a dielectric half-space

In this paper, we study the uniqueness of a virtual substrate for periodic metallic elements in a dielectric half-space. When the periodic metallic elements are placed at the interface of different substrates, they can be regarded to be embedded into a virtual substrate whose thickness approaches zero. However, the process of the mathematical limit of the thickness seems to be independent of the choice of the virtual substrate. Thereby, it is necessary to verify whether the arbitrary virtual substrate holds for the case. It is theoretically verified that the permittivity of the virtual substrate should be unique in order to satisfy the physical boundary condition of the periodic metallic elements. The root of the phenomenon is that the mathematical limit gives the alternative means to approach the actual physical situation, but the actual physical situation determines the way how the mathematical limit approaches zero. Finally, for comparison, two different virtual substrates are designed to validate the theory, for alternative substrate, incidence angle, and metallic elements. Besides, the finding can also be used to simplify the analysis and design of the metasurface by converting the periodic metallic elements in a dielectric half-space to the same periodic metallic elements in a uniform substrate.