Asymptotic Analysis of Maxwell’s Equations to Study Shielding Through a Thin Periodic Perforated Metallic Layer

Maxwell’s equations are considered in a 3D domain separated by a thin periodic layer with transverse cylindrical holes. The periodicity and the thickness of the layer are δ≪1. The domain and the holes are non-conductive (e.g., air); that is, the imaginary part of their electric permittivity is zero, while the real part is strictly positive and denoted by ε1. The novelty of the work is in the regularization of the problem for the missing imaginary coefficient part and the convergence of the regularized problem. Then, it is rigorously handled by the periodic unfolding method. We assume that the imaginary part of the permittivity in the conductive layer is of order O(ε2/δ), where ε2 is a parameter. The asymptotic behavior of Maxwell’s equations for different cases of the pair (δ,ε2) leads in the homogenized limit to (i) partial shielding or impedance interface condition, (ii) complete shielding, and (iii) no shielding. This paper provides the optimal design for UV, VIS, IR, X-ray shielding grids and textiles, like UV-protective T-shirts or X-ray protective wear, on how to choose the appropriate metal, yarn, or wire thickness and distance between the yarns or wires, to shield fully or partially for the applied light wave.