In recent years, considerable research has been carried out relative to the electromagnetic (EM) propagation and refraction characteristics in metamaterials with emphasis on the origins of negative refractive index. Negative refractive index may be introduced in metamaterials via different methods; one such is the condition whereby the Poynting vector of the EM wave is in opposition to the group velocity in the material. Alternatively, negative refractive index also occurs when the group and phase velocities in the medium are in opposition. The latter phenomenon has been extensively investigated in the literature, including recent work involving chiral metamaterials with material dispersion up to the first order. This paper examines the possible emergence of negative refractive index in dispersive chiral metamaterials with material dispersion up to the second order. The motivation is to determine if using second- as opposed to first-order dispersion may lead to more practical negative index behavior. A spectral approach combined with a slowly time-varying phasor analysis is applied, leading to the analytic derivation of EM phase and group velocities, and the resulting phase and group velocities and the corresponding phase and group indices are evaluated by selecting somewhat arbitrary dispersive parameters. The results indicate the emergence of negative index (via negative phase indices along with positive group indices, as reported in the literature) or negative index material (NIM) behavior over information bandwidths in the low RF range. The second-order results are not significantly better than those for first-order results based on the theoretical analysis; however, greater parametric flexibility exists for the second-order system leading to the higher likelihood of achieving NIM over practical frequency bands. The velocities and indices computed using the Lorentzian and Condon models yield an NIM bandwidth around 200−400 Mrad/sec, about 2 orders of magnitude higher than that for the parametric approach; more importantly, NIM is found not to occur in the first order when using practical models.
In recent work, electromagnetic propagation velocities for plane waves in dispersive metamaterials were calculated assuming frequency dispersion up to the second order. The three velocities were expressed in terms of dispersive coefficients under certain simplifying constraints. Frequency domains were found to exist around resonances where group and phase velocities are in opposition, implying possible negative index behavior. In this paper, we incorporate in the derived equations physical models (including Debye, Lorentz and Condon) for material dispersion in permittivity, permeability and chirality in order to further examine the consequences of second-order dispersion leading to negative index for practical cases, and also evaluate the resulting phase and group indices.
Negative refractive index arises typically in metamaterials via multiple routes. One such avenue is the condition where the Poynting vector of the electromagnetic wave is in opposition to the group velocity in the material. An earlier work along this route in a chiral material led to the well-known result of requiring very large (non-realizable) chirality. Thereafter, a combination of chirality together with first-order dispersion was examined using plane wave electromagnetic analysis. To arrive at the conclusions in that approach, the three wave velocities (energy, group and phase) were derived under first-order dispersion in permittivity, permeability and chirality. Negative index in this approach was established under the condition of contra-propagating group and phase velocities. Regions of negative index were found analytically by assuming standard dispersive models (such as Condon). In this paper, we will re-visit the negative index problem under higher-order dispersion. In addition, we will re-examine the plane wave propagation model under parametric dispersion where each material parameter (ε, μ, κ) is dispersively expanded up to the second order in frequency. Such a physical effect may be traced to group velocity dispersion (GVD) in the material. Field solutions are then obtained under the GVD effect, and extended to the evaluation of the energy, phase and group velocities.
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