Abstract
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.
Highlights
Electromagnetic metamaterials (MMs) are composites that exhibit unusual physical characteristics when exposed to the action of an external electromagnetic field [1], [2]
We rigorously demonstrated that the branch ambiguity problem affecting the NRW method can be fully overcome and, the complex refraction index Neff (ω) can be uniquely evaluated through the careful inversion of the exponential relationship between Neff (ω) and the scattering parameters of the MM at hand by the computation of a suitable right-inverse obtained through the analytic continuation of a specific analytic logarithm element along the path determined by the S-parameters in the complex plane
In this study, we rigorously demonstrated that the NWR branch ambiguity issue can be avoided by computing the proper right-inverse of the relationship (5) through the analytic continuation of a suitable analytic logarithm along the path determined in the complex plane by the scattering parameters of the MM under analysis
Summary
The associate editor coordinating the review of this manuscript and approving it for publication was Weiren Zhu. Free space propagation constant Effective thickness Complex logarithm Principal logarithm Real part operator Imaginary part operator An analytic function f (·)’s domain Complex plane Complex punctured plane f (·)’s range Inverse of f (·) Sub-domain of f (D). Versaci: Retrieving Effective Parameters of Electromagnetic Metamaterial Using NRW Method h(·) γ (·) γ (·). Right-inverse of f (·) Path in C Path in C Sub-domain of C Analytic logarithm Star-domain Analytic α-logarithm Natural logarithm Absolute value function α-argument function Complex number Set of arguments of z Set of the integer numbers Sub-domains of C Empty set Subordination operator Unique analytic logarithm
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