# A Generalized Accurate Model for Complementary Periodic Sub-wavelength Metasurface Based on Babinet Principle

2020

In this article, the Babinet principle is applied for complementary periodic sub wavelength metallic elements $A^{e}$ and $A^{c}$ at the interface between different substrates. For the problem, the restriction of the Babinet principle is mainly resulted from the inhomogeneity of the different substrates or the anisotropy of the metallic elements. In order to break the restriction, a virtual substrate with an effective permittivity is inserted into the interface to cover $A^{e}$ and $A^{c}$ as an updated physical boundary condition. As the thickness of such a virtual effective substrate layer approaches zero, the updated model will not only remain the same as the original one for $A^{e}$ and $A^{c}$ but also can be solved by the existing Babinet principle with a feasible mathematical method. Finally, according to the network transmission theory and Babinet principle, the proposed theory gives the specific mathematical relationship between the tangential transmission matrices of $A^{e}$ and $A^{c}$ . Two examples are simulated to verify the theory, regardless of the substrate, metallic elements, and incidence angle. This method not only greatly simplifies the solutions of complementary metasurfaces but also shows higher precision than the previous work.

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