Differential uncertainty analysis for evaluating the accuracy of S-parameter retrieval methods for electromagnetic properties of metamaterial slabs.

We apply a complete uncertainty analysis, not studied in the literature, to investigate the dependences of retrieved electromagnetic properties of two MM slabs (the first one with only split-ring resonators (SRRs) and the second with SRRs and a continuous wire) with single-band and dual-band resonating properties on the measured/simulated scattering parameters, the slab length, and the operating frequency. Such an analysis is necessary for the selection of a suitable retrieval method together with the correct examination of exotic properties of MM slabs especially in their resonance regions. For this analysis, a differential uncertainty model is developed to monitor minute changes in the dependent variables (electromagnetic properties of MM slabs) in functions of independent variables (scattering (S-) parameters, the slab length, and the operating frequency). Two complementary approaches (the analytical approach and the dispersion model approach) each with different strengths are utilized to retrieve the electromagnetic properties of various MM slabs, which are needed for the application of the uncertainty analysis. We note the following important results from our investigation. First, uncertainties in the retrieved electromagnetic properties of the analyzed MM slabs drastically increase when values of electromagnetic properties shrink to zero or near resonance regions where S-parameters exhibit rapid changes. Second, any low-loss or medium-loss inside the MM slabs due to an imperfect dielectric substrate or a finite conductivity of metals can decrease these uncertainties near resonance regions because these losses hinder abrupt changes in S-parameters. Finally, we note that precise information of especially the slab length and the operating frequency is a prerequisite for accurate analysis of exotic electromagnetic properties of MM slabs (especially multiband MM slabs) near resonance regions.