Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets

We describe a simple metamaterial homogenization procedure that calculates a limited set of constitutive parameters that are functions of both the frequency omega and the wave number k. This procedure is based on describing the inclusions of the metamaterial with current sheets excited by the local electromagnetic fields. It is shown that these constitutive parameters correctly predict several measurable properties of a metamaterial crystal including the dispersion relation, reflection from an interface, and emission from a simple directional metamaterial antenna.