Determining the Effective Constitutive Parameters of Finite Periodic Structures: Photonic Crystals and Metamaterials

A novel approach to find the effective electric and magnetic parameters of finite periodic structures is proposed. The method uses the reflection coefficients at the interface between a homogenous half-space and the periodic structure of different thicknesses. The reflection data are then approximated by complex exponentials, from which one can deduce the wavenumber, and the effective electric and magnetic properties of the equivalent structure by a simple comparison to the geometrical series representation of the generalized reflection from a homogenous slab. Since the effective parameters are for the homogenous equivalent of the periodic structure, the results obtained are expected to be independent of the number of unit cells used in the longitudinal direction. Although the proposed method is quite versatile and applicable to any finite periodic structure, photonic crystals and metamaterials with metallic inclusions have been used to demonstrate the application of the method in this paper.