Percolation effects in the checkerboard Babinet series of metamaterial structures

The proposed checkerboard Babinet series consists of pairs of Babinet complementary metamaterial structures, with the checkerboard pattern being the self-complementary structure. In this work, formulas are derived for the effective dielectric function and transmittance of all structures in the series, and it is shown that the evolution of the structures from holes to islands is a topological analog of the percolation problem, and that the highly singular checkerboard pattern represents the percolation threshold. The results are in agreement with the percolation theory of Bergman and Imry.