Babinet Principle for Anisotropic Metasurface with Different Substrates under Obliquely Incident Plane Wave

In this paper, we propose a theory by extending the Babinet principle to periodic subwavelength anisotropic metallic crevice elements Ae in the interface of different substrates. The significance of the proposed theory is that it can readily be used to transfer the solution of Ae to that of its complementary metallic elements Ac. First, the Babinet principle is established for Ae and its complementary magnetic elements Am according to the uniqueness theorem of boundary conditions, and Am is replaced with the corresponding metallic elements Ac based on the dual principle. Then, the waves associated with Ae and Ac are considered as plane waves since their period is smaller than the operation wavelength. Thereby, the Babinet principle is reexpressed as mathematical relations between tangential transmission matrices of Ae and Ac. Next, the tangential transmission matrices of Ac are solved using the corresponding circuit model, where the effective relative permeability is introduced to describe the inductive influence of the dual substrate. Finally, the tangential transmission matrices of Ae are derived, and the proposed theory is verified by two examples. The theoretical results agree well with the simulated ones at normal incidence (for various substrates) and over a large range of incidence angles (up to 75° and 60° for the two examples, respectively). The measured results support the simulation and the proposed theory.