Extraction of the Electromagnetic Parameters of a Metamaterial Using the Nicolson–Ross–Weir Method: An Analysis Based on Global Analytic Functions and Riemann Surfaces

Abstract

The characterization of electromagnetic metamaterials (MMs) plays a fundamental role in their engineering processes. To this end, the Nicolson–Ross–Weir (NRW) method is intensively used to recover the effective parameters of MMs, even though this is affected by the branch ambiguity problem. In this paper, we face this issue in the context of global analytic functions and Riemann surfaces. This point of view allows us to rigorously demonstrate the mathematical foundations of an algorithmic approach for avoiding the branch ambiguity problem, in which the phase unwrapping method is merged with K-K relations for recovering the effective parameters of an MM. In addition, exploiting the intimate relationship between the K-K relations and the Hilbert transform, a simple variant of the above algorithm is presented.

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

“…Only recently, the link between phase unwrapping and the operation of analytic continuation of holomorphic functions has been demonstrated [22]. Following a previous couple of works [22,23], in this study, we aim to better clarify the connection between the phase unwrapping method and the analytic continuation framework and identify which way is best suited to reconstruct the refractive index of metamaterial structures. The paper is organized as follows: Section 2 provides the elements of theory needed to relate analytic continuation and phase unwrapping.…”

“…To understand how the relation ( 9) can be exploited in a phase-unwrapping scheme, we have to reconsider the NWR Equation ( 1) from a more abstract point of view. As discussed in [23], the set of all the analytic logarithms L(•) of ( 6) makes up a special mathematical object called a global analytic logarithm L. A particular domain of definition is related to L: the Riemann surface S(L) [23]. Its structure is composed of a numerable set of replies of Ċ, S p (L), overlayed on top of each other and sorted by the ascending index p, suspended above Ċ, and glued to each other along the slit extending from 0 to −∞, which each sheet owns [23], as shown in Figure 1.…”

“…As discussed in [23], the set of all the analytic logarithms L(•) of ( 6) makes up a special mathematical object called a global analytic logarithm L. A particular domain of definition is related to L: the Riemann surface S(L) [23]. Its structure is composed of a numerable set of replies of Ċ, S p (L), overlayed on top of each other and sorted by the ascending index p, suspended above Ċ, and glued to each other along the slit extending from 0 to −∞, which each sheet owns [23], as shown in Figure 1. If z ′ is a point of Ċ and z is the point belonging to S q (L), for the q-th sheet of S(L), which lies above the point z ′ of Ċ, the value taken by L on z, L z′ , is given by [23]:…”

“…Its structure is composed of a numerable set of replies of Ċ, S p (L), overlayed on top of each other and sorted by the ascending index p, suspended above Ċ, and glued to each other along the slit extending from 0 to −∞, which each sheet owns [23], as shown in Figure 1. If z ′ is a point of Ċ and z is the point belonging to S q (L), for the q-th sheet of S(L), which lies above the point z ′ of Ċ, the value taken by L on z, L z′ , is given by [23]:…”

“…Based on the above, if γ is the copy on S(L) of the path γ(t) lying in Ċ, we have from (11) that between the values taken by L(•) at z ′ ∈ γ(t) and the values assumed by L at z ∈ γ, the following relation must hold [23]:…”