Reciprocal space approach to effective constitutive parameters of periodic composites

We adopt a technique previously used for crystalline solids to calculate the effective permittivity of periodic nanostructured composites. Our technique, which is based on the reciprocal space representation of a mesoscopic permittivity tensor, allows one to accurately take into account the local field corrections to the effective permittivity of 2D and 3D metamaterials made of lossy anisotropic constituents. To demonstrate the feasibility and to assess the convergence and accuracy of computational procedure used, we consider two 2D geometries for which exact analytical solutions are known (checkerboard geometry and square geometry) and one geometry, for which an exact solution is known in an asymptotic limit (quasi-one-dimensional geometry). It has been shown that the accuracy of the method can be improved when using Keller'ss duality relation.