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

TL;DR

This paper reviews the scattering parameter extraction method for metamaterial homogenization, clarifying that ambiguity arises only from the complex logarithm branch choice, not from the sign of wave parameters, and discusses proper branch selection methods.

Contribution

It clarifies the nature of ambiguity in the scattering parameter method and evaluates approaches for correct branch selection in metamaterial analysis.

Findings

Ambiguity is only due to complex logarithm branch choice.

Both sign combinations of impedance and wave number yield the same permittivity and permeability.

Proper branch selection methods are reviewed and their suitability discussed.

Abstract

The scattering parameter extraction method of metamaterial homogenization is reviewed to show that the only ambiguity is the one related to the choice of the branch of the complex logarithmic function (or the complex inverse cosine function), whereas it has no ambiguity for the sign of the wave number and intrinsic impedance. While the method indeed yields two signs of the intrinsic impedance, and thus the wave number, the signs are dependent, and 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 wave number, 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, and the severity of the branch ambiguity is illustrated through simulations of a known metamaterial realization. Several approaches for proper branch selection are reviewed and their suitability to metamaterial samples is discussed.