The ideas of employing the unique properties of metamaterials and plasmonic media for cloaking and invisibility applications have been recently suggested and investigated by several groups 1–5, and they may find numerous applications in physics and technology. While many of the recent designs of the cloaking structures are based on the transformation optics and exact formulas 6, the original concept proposed by Alù and Engheta 1 suggests to use simpler plasmonic and metamaterial coatings in order to reduce drastically the total scattering cross-section of an object by relying on a nonresonant scattering cancelation 7. This approach is based on the observation that, in a certain range of frequencies, the polarization vector in a plasmonic material may be antiparallel with respect to that in a dielectric, implying that a dipole moment of the opposite phase may be induced in a plasmonic shell placed around a dielectric or even conducting object.
This scattering-cancelation approach is based on the local negative polarizability of metamaterials, and its realization was recently demonstrated experimentally at microwave frequencies 8, where an array of metallic fins embedded in a high-permittivity fluid was used to create a metamaterial plasmonic shell capable of cloaking a dielectric cylinder, with 75% reduction of total scattering width. The recent extensions of this concept are based on the use of multilayered plasmonic shells 9, 10.
In this Letter, we develop further the applications of the multilayered plasmonic cloaks 9, 10 and study the electromagnetic scattering of small particles by employing the structure with a double-shell metamaterial coating (see Fig. 1) that provides simultaneously the shielding of the cloaked volume and substantial reduction of the scattering losses. We demonstrate, by employing realistic parameters that, by using a layer with an epsilon-near-zero material allows to reduce substantially the scattering losses leading to the effect of electric field super-localization when a large amount of energy is concentrated inside a small shell volume.
We consider an isotropic cloaking structure (see Fig. 1) composed of two layers of different materials, the simpler version of the multilayered plasmonic shell 9. The use of two layers of the coating materials allows to avoid the coupling between the object and an external shielding shell of the cloak and to suppress higher-order-mode scattering 10. By employing this structure, we can achieve two major goals: (i) to realize a screening of a closed spatial region from an external electromagnetic field, so that an object placed inside this region does not affect the external field (i.e. it becomes “invisible”), and (ii) to cover the system “cloak-and-object” by an additional layer reducing the overall scattering losses.
In our structure, a double-layer shell surrounds a cloaking region of the radius a with the dielectric permittivity
the inner coating layer (a screening shell) of the larger radius b has the permittivity and the external coating layer (a scattering compensating shell) has the permittivity and external radius R. The structure is placed into a surrounding medium characterized by the permittivity
First, this problem can be treated by the well-known Mie theory to the case of a multilayered spherical object 9. The main result is given by the exact solutions expressed through the Mie coefficients of order n,
for both TE and TM polarizations 11; this solution defines the total scattering by the cloaked system, and it can be analyzed to achieve the scattering suppression in different frequencies 9. However, to achieve the full suppression of the dipole moment of the shielding shell, the resonant excitation of volumetric plasmons in the screening layer is required. The amazing property of the double-shell design is the super-localization effect of the field in the screening layer, with the degree of super-localization determined by the layer thickness.
The scattering by a double-shell structure can be analyzed much easier in the quasi-static approximation when the electric field inside the structure is described by the potential such that the electric field is defined as follows, Solving this equation, we find the potential distribution in all domains. The field inside the cloak has a simple form, where is the polar angle. This field vanishes when and this condition corresponds to the screening of the object by a layer with zero permittivity 10. Outside the cloak, i.e. in the region the solution can be presented in the form
where is the intensity of the external field, and P is the total dipole moment of the structure,
where The dipole moment vanishes when and this condition defines a re-lation between the shell thicknesses,
Next, we employ the theory outlined above for a design of a cloaking structure with realistic parameters and then simulate it numerically with the help of the commercial software CST Microwave Studio 2011. We compare the results with the scattering of the electromagnetic plane waves by a metal sphere, whose dimension equals to a half wavelength. As expected, such a metal sphere perturbs the wavefront substantially and forms a shadow (see Fig. 2(a)).
In order to realize the desired cloaking structure, we select the compensating layer equal to The value of permittivity of the scattering compensating layer is This value corresponds to a common isotropic dielectric medium, and it defines suitable size of the whole structure and, from Eq. (2), the inner radius of the compensating layer, In the numerical simulations, we include losses for both the compensating and screening layers following and Frequency dispersion of the screening layer is described by the Drude model,
Importantly, the operating frequency of the shielding layer coincides with the plasma frequency for the screening layer, since the real part of permittivity of the screening layer vanishes at this point.
Numerical results for the phase of the electromagnetic wave interacting with the cloaking structure are shown in Fig. 3(a), where a complete restoration of the wavefront behind the cloaked area is observed. The amplitude distribution of the electromagnetic field (Fig. 2(b)) and the energy flux density (Poynting vector) (Fig. 3(b)) clearly show uniformity of the electric field component outside the compensating layer. It indicates the absence of absorption, scattering, and reflection from the structure. Furthermore, we also observe a uniform distribution of the energy flux density outside the system, the local field enhancement in the screening layer, and the absence of the field in the inner (shielding) region when condition (2) is satisfied (see Fig. 4).
For the further analysis of the performance of the cloaking structure, we compare the electric field intensity dependence on frequency in the inner region and in the free space At the plasma frequency, the field amplitude is minimal at the inner region and maximum outside the structure. When we deviate from the plasma frequency, the electromagnetic wave scattering grows and the ability to hide an object reduces because the electromagnetic field penetrates into the inner region of the cloak (see Fig. 4). This effect is caused by the deviation from zero of the permittivity of the screening layer described by the Drude model.
One of the important qualitative characteristics of the cloaking structure is the reduction of scattering in the presence of the cloak. The analytical scattering cross-sectional width is determined as
where are the Mie amplitudes for the TM modes. The scattering width, SCS, is the scattered power divided by the power flux density (Poynting vector) of the incoming wave.
An effective reduction of the average volume scattering in comparison with the scattering of an ideally conducting sphere with the size equal to the size of the screening region is shown in Fig. 5, where we observe that the scattering of the cloak is minimal at the plasma frequency, and it is substantially less than the scattering of a perfectly conducting sphere (dashed curve).
In conclusion, by employing realistic parameters and materials, we have demonstrated numerically that a novel type of multilayer cloak, consisting of a dielectric layer and a layer of an epsilon-near-zero material, can suppress substantially the scattering from a sphere and at the same time shield its interiors. The physical principles of operation of this cloaking structure allow to hide any object with the size comparable to the wavelength of radiation in any frequency range, with various parameters.