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 $A^{e}$ in the interface of different substrates. The significance of the proposed theory is that it can readily be used to transfer the solution of $A^{e}$ to that of its complementary metallic elements $A^{c}$ . First, the Babinet principle is established for $A^{e}$ and its complementary magnetic elements $A^{m}$ according to the uniqueness theorem of boundary conditions, and $A^{m}$ is replaced with the corresponding metallic elements $A^{c}$ based on the dual principle. Then, the waves associated with $A^{e}$ and $A^{c}$ 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 $A^{e}$ and $A^{c}$ . Next, the tangential transmission matrices of $A^{c}$ 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 $A^{e}$ 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.