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Semi-phenomenological effective permittivity approach to metallic periodic structures.

A detailed review of the theory of effective permittivity for one- and two-dimensional periodic structures shows its limited validity for metal-dielectric structures in the visible and near infra-red if the feature dimensions are comparable with the metal skin depth. We propose a phenomenological correction to the static formulae using a realistic assumption for the electric field behavior inside the metal features. This approach allows us to obtain analytical expressions for the effective permittivity in the case when the electric field is not sufficiently homogeneous within the unit cell of the gratings. A comparison with the numerical results of the Fourier modal method demonstrates the validity of the analytical formulae. Additional study is made on the impedance approximation at the outer boundaries of the periodical structure in order to propose analytical formulae for the reflection coefficient that permits better correspondence with the numerical results. The link between the values of effective permittivity and permeability defined as the ratios between the averaged fields, and the metamaterial permittivity and permeability is discussed.