Permeability and permittivity extraction issues for non-Foster and active metamaterials

Although a variety of methods have been proposed for the extraction of effective permittivity and permeability of metamaterials, certain underlying passivity constraints and assumptions would not be suitable for non-Foster metamaterials that incorporate active devices and power sources. Moreover, recent arguments suggest that a common passivity constraint in extraction methods does not resolve solution branch ambiguities. To address these issues, the fundamental principles of parameter extraction are revisited for the case of active materials where passivity cannot be assumed. The analysis follows along the lines of Nicolson-Ross-Weir approaches, where parameters are extracted from measured two-port S-parameters. It is shown that a convergence constraint for active materials requires that the magnitude of the product of the transmission coefficient and reflection coefficient must be less than unity. This allows metamaterials with gain, and simulation results are provided for a slab of active material that exhibits gain and satisfies the constraint.