Constitutive Parameters of a Metamaterial Slab Retrieved by the Phase Unwrapping Method
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Constitutive Parameters of a Metamaterial Slab Retrieved by the Phase Unwrapping Method J. J. Barroso & U. C. Hasar
Received: 29 October 2011 / Accepted: 27 December 2011 / Published online: 8 January 2012 # Springer Science+Business Media, LLC 2012
Abstract The constitutive parameters of a metamaterial slab showing negative refractive index over the 9-12 GHz frequency range are unambiguously retrieved from complex scattering parameters upon using the phase unwrapping method. Such a technique relies on properly adding to the phase of the complex transmission coefficient a stepwise function constructed from critical frequencies at which the phase of the transmission coefficient reaches ∓π. Keywords Metamaterials . Constitutive parameters . Transmission/reflection measurement methods . Scattering parameters . Phase ambiguity . Transmission coefficient
1 Introduction In metamaterials both the size of the composite particles and the period of lattices are made smaller than the wavelength at the particle resonance. So a homogenized description applies for the description of the properties of metamaterials in terms of conventional material parameters [1–3]. In this context, the use of constitutive parameters provides a useful way to understand the exotic propagation characteristics of metamaterials without the need to consider the complex local-field structure around the metamaterial elements. Among the complex permittivity (ε) and complex permeability (μ) retrieval methods [4– 11], the transmission/ reflection (T/R) technique as envisaged by Nicolson and Ross (NR) [4] is a non-iterative method that retrieves the complex permittivity and permeability of homogeneous materials from measurements of complex scattering (S-) parameters. Elegant J. J. Barroso (*) Plasma Laboratory, National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil e-mail: [email protected] U. C. Hasar Department of Electrical and Electronic Engineering, Ataturk University, Erzurum, Turkey e-mail: [email protected] U. C. Hasar Department of Electrical and Computer Engineering, Binghamton University, New York, NY 13902, USA
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J Infrared Milli Terahz Waves (2012) 33:237–244
and compact, the explicit NR procedure confronts nevertheless a problem that arises from the presence of complex exponential propagation factor in the transmission coefficient which, while generating multiple branches, can lead to ambiguities in determining the retrieved values for ε and μ. In a variety of approaches, this problem has been treated in several reports [7–12], yet failing in some cases as will be discussed later. In the present paper, on the other hand, this phase-ambiguity problem is solved by using the phase unwrapping method. Alternatively, based on the fact that the real part (attenuation factor) in the complex propagation factor is independent of the phase ambiguity, while only the determination of the imaginary part (phase factor) depends on the phase change, other works [13] have enforced causality to calculate through t
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