Critical frequency of metasurfaces on dielectric half-space

In this letter, the critical frequency of the periodic metasurface in a dielectric half-space is studied. First, according to the mode expansion theory, the wave of the periodic metallic elements can be decomposed into different eigen-modes. Then, as the frequency increases, partial high-order modes are excited by the periodic metallic elements that may propagate into the substrates as spics. Finally, the critical frequency is analytically derived for a homogenized surface without spics. This study shows that the reflected and transmitted spics can co-exist, but there is a distinct difference between their corresponding critical frequencies. Thus, the actual critical frequency of homogenized metasurface in a dielectric half-space is not only related to the periodicity and incident angle, but also to the substrates.