Upscaling of Helmholtz Equation Originating in Transmission through Metallic Gratings in Metamaterials

We investigate the transmission properties of a metallic layer with narrow slits. We consider (time-harmonic) Maxwell's equations in the H-parallel case with a fixed incident wavelength. We denote η > 0 as the typical size of the complex structure and obtain the effective equations by letting η → 0. For metallic permittivities with negative real part, plasmonic waves can be excited on the surfaces of the slits. For the waves to be in resonance with the height of the metallic layer, the corresponding results can be perfect transmission through the layer.