Predictive design of parallel-plate metamaterial scattering-cancellation cloaks beyond the long-wavelength, perfect-conductor limit: the virtual-interface shift, electrically large objects, and loss-aware optimization

Silveirinha, Alù, and Engheta (Phys. Rev. E 75, 036603 (2007)) showed that a metamaterial of metallic parallel-plate (or radial-plate) implants in a dielectric host can emulate the epsilon-negative cover required for scattering-cancellation cloaking of a dielectric cylinder at microwaves, provided the interface fringe fields are compensated by thin dielectric gaps of thickness equal to a virtual-interface displacement $\delta\approx0.1a$ ( $a$ being the plate spacing). The paper itself acknowledged that this design rests on a long-wavelength, perfect-conductor, electrically-small-object idealization. The open problems are: a precise characterization of the frequency-, angle-, and material-dependent virtual-interface shift $\delta(k_0,k_z)$ beyond first order; the extension of the two-parameter cover concept to electrically large objects whose scattering involves many multipole orders; and a systematic, loss- and manufacturability-aware design procedure for the large spacer permittivities the recipes demand. The extension to plasmonic implants at infrared and optical frequencies was published by the same authors in 2008 and is prior progress rather than part of the open problem; the three items above constitute the open problem.

Background

Scattering-cancellation (plasmonic) cloaking was proposed by Alù and Engheta in 2005: an object whose total scattering cross section (the total power scattered in all directions, normalized by the incident power density; per unit length, the 'scattering width' $Q$ , for two-dimensional problems) is dominated by its electric dipole can be made nearly invisible by surrounding it with a thin cover of epsilon-negative (ENG) material, i.e., a medium whose relative permittivity is negative. The induced dipole of the cover then cancels that of the object, suppressing the dominant scattering order. Natural ENG ('plasmonic') materials exist at optical and infrared frequencies, but not generally at microwaves, which motivated the 2007 source paper: to synthesize the required ENG response at microwaves with a metamaterial, an artificial composite of subwavelength metallic inclusions in a dielectric host.

The specific platform is a two-dimensional parallel-plate metamaterial: metallic plates of spacing $a$ , infinite along one direction and embedded in a dielectric of relative permittivity $\varepsilon_\mathrm{diel}$ , illuminated by an $E$ -polarized wave (electric field parallel to the plates) with free-space wavenumber $k_0=\omega/c$ and longitudinal wave-vector component $k_z$ along the plates. Below the cutoff of the parallel-plate waveguide mode, each inter-plate channel behaves as an effective plasma: the standard homogenization formula gives $\varepsilon_\mathrm{eff}/\varepsilon_0=\varepsilon_\mathrm{diel}-((\pi/a)^2-k_z^2)/k_0^2$ , which is negative in a designed band, so the composite emulates a homogeneous ENG slab. This homogenization is only approximate: at a physical truncation of the plates, the fundamental Floquet harmonic (a Bloch-wave component of the periodic structure) of the metamaterial does not match the uniform free-space harmonic, so higher-order evanescent diffraction modes are excited as fringe fields localized at the interface. Using an exact Wiener–Hopf solution (a classical factorization technique for scattering by semi-infinite periodic structures), Silveirinha, Alù, and Engheta showed that the exact reflection coefficient equals the homogenized one times a phase factor $\mathrm{e}^{2\Gamma_0\delta}$ , with $\Gamma_0=\sqrt{k_z^2-k_0^2}$ and a 'virtual interface' displacement $\delta$ inside the metamaterial. To first order and for long wavelengths ( $k_0a\ll\pi$ ), $\delta\approx0.1a$ ; hence a practical design rule: a slab that must emulate an ideal ENG slab of thickness $d$ should have a metallic region of thickness $d-2\delta$ with dielectric gaps of thickness $\delta$ on both faces. Applied to a cylindrical cover of $N$ radial plates in a dielectric spacer (whose spacing $a(r)=2\pi r/N$ makes the effective plasma frequency radial, $\varepsilon_\mathrm{eff}(r)/\varepsilon_0\approx\varepsilon_\mathrm{diel}-((N/2)^2-l^2)/(k_0r)^2$ for azimuthal harmonic $l$ ), the rule recovers the analytically predicted scattering reduction of a dielectric cylinder — but only for electrically small cylinders whose scattering is dominated by one harmonic, only in the long-wavelength regime, and only for perfectly conducting, lossless implants. The recipes also require large spacer permittivities when the implant count $N$ is large (e.g., $\varepsilon_\mathrm{diel}=35.4$ for $N=12$ , dropping to $14.1$ for $N=8$ ), raising loss and manufacturability questions. These limitations, stated in the paper's own concluding discussion, motivate the open problem; the term 'spatial dispersion' (wave-vector-dependent effective parameters) and the general program of nonlocal homogenization — correcting effective-medium descriptions for such interface and lattice effects — name the modern research area into which the $\delta$ -characterization question falls.

Problem Statement

Turn the parallel-plate (radial-plate) metamaterial scattering-cancellation cover of Silveirinha, Alù, and Engheta (2007) from a long-wavelength, perfect-conductor, electrically-small design into a predictive and realizable design methodology. Concretely, as acknowledged in the source paper's concluding discussion, the open problem comprises the three items below:

Characterize the virtual-interface displacement $\delta(k_0,k_z)$ — and through it the required gap thicknesses and effective permittivity — beyond the first-order long-wavelength estimate $\delta\approx0.1a$ : determine its full dependence on frequency ( $k_0$ ), incidence angle/wave-vector component ( $k_z$ ), spacer permittivity $\varepsilon_\mathrm{diel}$ , and the finite conductivity, frequency dispersion, and losses of the metallic implants and the high-index spacer, so that the homogenized design is predictive over the range of frequencies, incidence angles, and spacer permittivities for which the parallel-plate channels support an effective ENG response, rather than only for $k_0a\ll\pi$ and perfectly conducting plates.
Extend the scattering-reduction technique to electrically large objects, for which many multipole (azimuthal-harmonic) orders contribute to the scattered field: identify cover geometries and parameterizations with sufficient degrees of freedom to cover or cancel the several dominant higher-order multipoles, and quantify the achievable total-scattering-width reduction and its frequency band.
Establish a systematic design/optimization procedure that accounts for realistic material constraints — lossy high-permittivity spacers, dispersive and lossy implants, and manufacturability of many radial plates — including the trade-off that large implant counts require very large spacer permittivities, and quantify how these constraints, together with the causality and passivity of the constituent materials, bound the achievable scattering reduction and bandwidth.

The extension to plasmonic implants at infrared/optical frequencies, which the source paper reported as underway, was published by the same authors in 2008 (Phys. Rev. B 78, 075107) and is prior progress rather than part of the open problem; items 1–3, possibly addressed within the parallel-plate platform or by a directly comparable ENG-emulating implementation, constitute the open problem.

References

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