Alù and Engheta (Phys. Rev. Lett. 100, 113901, 2008) showed theoretically that a multilayered shell of plasmonic materials with suitably staggered plasma frequencies can cancel the dominant scattering multipole of a subwavelength dielectric nanoparticle simultaneously at two distinct optical wavelengths, and they acknowledged a set of unresolved issues that stand between this idealized two-frequency design and a general, realizable, experimentally verified multifrequency optical cloak: passivity-mandated resonant scattering peaks at intermediate frequencies between the cloaking windows, uncanceled higher-order multipoles as the electrical size of the cloaked system grows, the fabrication of many thin concentric nanoscale layers with tight tolerances, the realization of low-loss dispersive plasmonic materials meeting the prescribed permittivity trajectories, and the lack of any experimental verification of multifrequency invisibility for single nanoparticles and for collections of them. This problem asks for the generalization of multifrequency plasmonic scattering-cancellation cloaking from the demonstrated idealized dipole-regime design to realizable and experimentally demonstrable optical invisibility, together with the quantitative theory of the passive trade-offs that govern how many cloaking windows can be achieved and at what cost in intermediate-frequency scattering. Substantial partial progress exists (multifrequency graphene cloaks in the far infrared, comblike multilayer scattering engineering, active non-Foster broadband cloaking, Bode-Fano-type and quasistatic impossibility bounds for passive cloaking, and single-frequency near-infrared, single-frequency visible, and microwave experiments) but the core remains open: no quantitative theory of the multi-window passive trade-off for finite-size objects, and no experimental demonstration of multifrequency scattering-cancellation cloaking at optical frequencies.

Background

When an object is illuminated, it reradiates part of the incident power; the total scattering cross section quantifies this reradiated power, and for a sphere it is conventionally decomposed in Mie theory into contributions from multipole orders n=1,2,3,…n=1,2,3,\dots (dipole, quadrupole, etc.), where each order is represented by a Mie scattering coefficient whose vanishing suppresses that order. Two families of cloaking techniques exist. Transformation-optics cloaks reroute the wave around the object using spatially inhomogeneous, anisotropic, often resonant metamaterial parameter profiles; they are typically narrowband, bulky, and polarization-specific. Scattering-cancellation (plasmonic) cloaking instead covers the object with a thin shell of a plasmonic material, meaning a medium whose real permittivity is below that of the surrounding and can become negative, described at optical frequencies by a Drude model ε(ω)=ε0[1−ωp2/ω(ω+iγ)]\varepsilon(\omega)=\varepsilon_0[1-\omega_p^2/\omega(\omega+i\gamma)] with plasma frequency ωp\omega_p and loss rate γ\gamma. At frequencies below ωp\omega_p the shell's induced polarization is oppositely phased to the object's dominant multipole, so the shell's scattering destructively interferes with the object's, drastically reducing the total cross section; the mechanism is nonresonant and local. In 2008 Alù and Engheta proposed exploiting the frequency dispersion of such shells to cloak the same object at several distinct optical wavelengths simultaneously: in a multilayered shell, each plasmonic layer is assigned a different plasma frequency, so that a different layer provides the cancelling polarizability in each spectral window. For a dielectric sphere of radius 100 nm and permittivity 3ε03\varepsilon_0 they designed a two-layer cloak (inner shell 7.5 nm thick, outer shell 24 nm thick) whose Drude materials are required to have real permittivity 0.2ε00.2\varepsilon_0 at λ0=500\lambda_0=500 nm and λ0=625\lambda_0=625 nm respectively, satisfying the full-wave dipolar cancellation condition (vanishing real part of the Mie denominator, U1TM=0U_1^{\mathrm{TM}}=0) at both wavelengths; rigorous Mie calculations and independent numerical simulations gave total scattering-efficiency reductions of roughly 30-fold at 500 nm and 120-fold at 625 nm, together with near-field amplitude and phase restoration, for both loss levels considered (γ=10−4ωp\gamma=10^{-4}\omega_p and γ=10−2ωp\gamma=10^{-2}\omega_p). Crucially, the same calculations revealed a pronounced resonant scattering peak near λ0≈580\lambda_0\approx 580 nm, between the two cloaking windows, caused by the near-simultaneous vanishing of the imaginary parts of the dipolar and quadrupolar Mie denominators (an overlapping dipole-quadrupole resonance); the authors state that this inter-window peak is inherently dictated by the passivity (causality plus energy conservation) of the dispersive materials and is expected between any two cloaking frequencies, with realistic Ohmic loss substantially reducing but not eliminating it. A companion paper (New J. Phys. 10, 115036, 2008) extended the theory and explicitly analyzed the fundamental limitations dictated by passivity and causality. The concept of passivity is central: because passive media are causal, their permittivity satisfies Kramers-Kronig relations, so the dispersion that enables multifrequency operation is inseparable from resonant enhancement in the intervals between the design windows. The authors also identified the practical constraints on this program: each additional cloaking layer is a few to tens of nanometers thick, so the number of concentric layers is limited by nanofabrication; and the overall cloaked system must remain electrically small, since higher-order multipoles excited at larger electrical size are not canceled by the dipole-focused design. Related later techniques that interact with this trade-off but change the mechanism include mantle cloaking (a patterned metasurface shell represented by a frequency-dependent surface impedance), active non-Foster cloaking (metasurfaces containing negative capacitance, which is non-passive), and Bode-Fano-type bounds, the classical broadband-matching limits from circuit theory that constrain any passive system's reflectivity-bandwidth product and have been adapted to bound passive cloaking performance.

Problem Statement

The 2008 result of Alù and Engheta demonstrates, for an idealized design, that a two-layer plasmonic shell can cancel the dominant dipolar scattering of a subwavelength dielectric sphere at two prescribed optical wavelengths simultaneously, at the price of a passivity-related resonant scattering peak between the windows that Ohmic loss only partially suppresses. The open problem is to carry this multifrequency scattering-cancellation concept from that idealized dipole-regime demonstration to a general, realizable, and experimentally verified form, under one common objective: quantitative understanding of what passive multilayered plasmonic cloaks can and cannot achieve at multiple prescribed optical frequencies, and a working demonstration of it. Concretely, the interlocking questions left open by the source paper and not closed by subsequent literature are: (1) what are the fundamental passive-limit relations between the number, spectral spacing, and depth of the cloaking windows of a causal, passive multilayer cloak and the unavoidable resonant scattering in the frequency intervals between them, for objects whose electrical size is beyond the pure dipole regime — a generalization of the existing quasistatic and planar Bode-Fano-type bounds to the discrete multi-window, finite-size, higher-multipole setting; (2) how to design cloak layers that cancel several higher-order Mie multipoles simultaneously, since at larger electrical size the uncanceled higher-order harmonics dominate the residual scattering and each additional cloaking layer increases the outer radius and thus the electrical size; (3) which realistic low-loss dispersive plasmonic materials or metamaterial covers can provide the prescribed permittivity trajectories (for the original design, real permittivity near 0.2ε00.2\varepsilon_0 at the target visible wavelengths with loss γ≲10−2ωp\gamma\lesssim 10^{-2}\omega_p), and how fabrication tolerances and deviations from ideal concentric geometry of nanometer-thin multiple shells degrade the multifrequency performance; and (4) whether multifrequency invisibility — in both near field and far field, for single nanoparticles and for collections of them — can be experimentally verified at optical frequencies, given that existing experiments are either single-frequency (near-infrared or visible), at microwave frequencies, or computational only.

References

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