Retrieving the Effective Parameters of an Electromagnetic Metamaterial Using the Nicolson-Ross-Weir Method: An Analytic Continuation Problem Along the Path Determined by Scattering Parameters

Electromagnetic metamaterials (MMs) are composite structures that allow one to potentially develop unique and innovative microwave, millimetre wave, and optical devices due to their unusual physical properties. In this process, their electromagnetic characterization plays a fundamental role. Various procedures have been proposed to accomplish this task, but the Nicolson-Ross-Weir (NRW) method still appears to be the most commonly adopted one even though it is afflicted by the severe issue of branch ambiguity. In this paper, we have demonstrated that rigorously, as the branch ambiguity can be entirely overcome through the analytic continuation of a specific analytic logarithm element along the path determined in the complex plane by the scattering parameters of an MM under analysis. Furthermore, the underlying relationship between analytic continuation, phase unwrapping approach, implemented through a procedure devised by Oppenheim and Schafer for the homomorphic treatment of signals (hereafter named PUNWOS), and the Kronig-Kramers relation has been discussed and enlightened, demonstrating the full equivalence among the methods. To clarify this aspect, a couple of numerical examples is presented. The results discussed in this study open the possibility of employing the vast theoretical equipment developed in the phase unwrapping field to achieve the retrieval of MMs’ effective parameters when the NRW method is applicable.