Abstract
To retrieve complex-valued effective permittivity and permeability of electromagnetic metamaterials (EMMs) based on resonant effect from scattering parameters using a complex logarithmic function is not inevitable. When complex values are expressed in terms of magnitude and phase, an infinite number of permissible phase angles is permissible due to the multi-valued property of complex logarithmic functions. Special attention needs to be paid to ensure continuity of the effective permittivity and permeability of lossy metamaterials as frequency sweeps. In this paper, an automated phase correction (APC) algorithm is proposed to properly trace and compensate phase angles of the complex logarithmic function which may experience abrupt phase jumps near the resonant frequency region of the concerned EMMs, and hence the continuity of the effective optical properties of lossy metamaterials is ensured. The algorithm is then verified to extract effective optical properties from the simulated scattering parameters of the four different types of metamaterial media: a cut-wire cell array, a split ring resonator (SRR) cell array, an electric-LC (E-LC) resonator cell array, and a combined SRR and wire cell array respectively. The results demonstrate that the proposed algorithm is highly accurate and effective.
Highlights
INTRODUCTIONAs a nascent artificial material, electromagnetic metamaterials (EMMs) based on resonant effect exhibit some exotic electromagnetic characteristics including negative permittivity, permeability or refractive index in certain frequency range that can be rarely found in natural materials.[1,2] In general, the EMMs can be classified into three categories: negative electric permittivity medium often made of split ring resonators (SRR) array;[2] negative magnetic permeability medium composed of cut wire array[1] or electric-LC resonator (E-LC) array;[3] and left-handed materials (LHM) made of combined SRR and thin metal wire array[4,5,6] which exhibits simultaneous negative permittivity and permeability
The electromagnetic metamaterials (EMMs) can be classified into three categories: negative electric permittivity medium often made of split ring resonators (SRR) array;[2] negative magnetic permeability medium composed of cut wire array[1] or electric-LC resonator (E-LC) array;[3] and left-handed materials (LHM) made of combined SRR and thin metal wire array[4,5,6] which exhibits simultaneous negative permittivity and permeability
One is called scattering parameters method (SPM) proposed by Smith et al.,[12] in which the impedance z and refractive index n of a metamaterial slab is first deduced from its transmission and reflection coefficients, and both electric permittivity and magnetic permeability are calculated through ε = n/z and μ = nz
Summary
As a nascent artificial material, electromagnetic metamaterials (EMMs) based on resonant effect exhibit some exotic electromagnetic characteristics including negative permittivity, permeability or refractive index in certain frequency range that can be rarely found in natural materials.[1,2] In general, the EMMs can be classified into three categories: negative electric permittivity medium often made of split ring resonators (SRR) array;[2] negative magnetic permeability medium composed of cut wire array[1] or electric-LC resonator (E-LC) array;[3] and left-handed materials (LHM) made of combined SRR and thin metal wire array[4,5,6] which exhibits simultaneous negative permittivity and permeability. One is called scattering parameters method (SPM) proposed by Smith et al.,[12] in which the impedance z and refractive index n of a metamaterial slab is first deduced from its transmission and reflection coefficients, and both electric permittivity and magnetic permeability are calculated through ε = n/z and μ = nz. An automated phase correction (APC) algorithm is proposed to circumvent mathematical ambiguity of the complex-valued optical properties by the use of phase tracing and compensation, and continuity of the phase of the optical properties as a function of frequency can be ensured for lossy EMMs. In Section II, basic theoretical retrieval formulas of electric permittivity and magnetic permeability from scattering parameters via impedance and refractive index are discussed.
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