A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization

The scattering-parameter extraction method of metamaterial homogenization is reviewed to show that the only ambiguity is that related to the choice of the branch of the complex logarithmic function (or the complex inverse cosine function). It is shown that the method has no ambiguity for the sign of the wavenumber and intrinsic impedance. While the method indeed yields two signs for the intrinsic impedance and thus the wavenumber, the signs are dependent. Moreover, both sign combinations lead to the same permittivity and permeability, and are thus permissible. This observation is in distinct contrast to a number of statements in the literature where the correct sign of the intrinsic impedance and wavenumber resulting from the scattering-parameter method is chosen by imposing additional physical requirements, such as passivity. The scattering-parameter method is reviewed through an investigation of a uniform plane wave normally incident on a planar slab in free space. The severity of the branch ambiguity is illustrated through simulations of a known metamaterial realization. Several approaches for proper branch selection are reviewed, and the suitability to metamaterial samples is discussed.