Effective parameters of resonant negative refractive index metamaterials: Interpretation and validity

We present a numerical study of resonant negative refractive index (NRI) metamaterials in which their effective parameters are calculated. For a periodic media consisting of split-ring resonators and metallic wires the effective refractive index is compared by means of two different methods. The first one is an inversion procedure in which the effective refractive index is calculated from the reflection and transmission coefficients of a finite structure and the second one consists its calculation from the phase velocity issued from the dispersion diagram. A significant difference between the two cases is highlighted in the frequency interval of interest (NRI regime) and for the finite media, counterintuitive observations are made. These anomalous features are observed in a frequency range in which there is a non-negligible contribution of the higher-order modes to propagation inside the periodic metamaterial. Hence in this particular frequency interval, the media cannot be described by an effective refractive index in a Fresnel sense. In other frequency regions, both methods converge and it will be shown that the metamaterial exhibits a negative refractive index.