Homogenization of Fractal Metasurface Based on Extension of Babinet–Booker's Principle

In this letter, we develop a homogenization approach for a quasi-two-dimensional fractal-based metamaterial (fractal-based metasurface). The structure that we consider is a self-similar patterned metal screen based on the Hilbert curve. The screen can be placed above a metallic ground plane, be printed on a dielectric substrate, or it can be considered freestanding in air. In this study, we develop a theoretical model for the effective surface impedance of this structure by using the self-similarity of the fractal and the electromagnetic Babinet's principle. We numerically simulate the reflection of the plane waves incident on such a metasurface (with the substrate plane included). The properties of the Hilbert fractal metasurface are extracted from the data obtained in these simulations. Our aim is to find the effective parameters and construct an equivalent model for the fractal metasurface.