Analytic Continuation, Phase Unwrapping, and Retrieval of the Refractive Index of Metamaterials from S-Parameters

The heuristic homogenization approach is intensively employed to characterize electromagnetic metamaterials (MMs). The effective parameters are extracted within this framework using the Nicolson-Ross-Weir (NRW) method. Special attention must be devoted to handling this procedure because of the branch ambiguity issue affecting it, i.e., the lack of uniqueness in the evaluation of the effective refractive index neff rooted in the use of the multivalued complex logarithm to invert the Airy-Fresnel relation. Over the years, several techniques based on the phase-unwrapping approach have been introduced, but without any theoretical justification. In this paper, we aim to clarify the theoretical connection between the phase unwrapping method and the analytic continuation theory framework. Furthermore, three-phase-unwrapping approaches, which descend directly from the theory we discussed, are compared to identify which approach is best suited to reconstruct the complex refractive index of metamaterials when the NRW method is applicable.