1. Introduction

Recently, extensive attention has been drawn to achieving the invisibility of objects. Alù et al. [1, 2] reported an idea that small reflections can be realized through plasmonic and metamaterial coatings for objects of carefully matched size and shape. Pendry et al. [3] proposed a far more general approach which is based on the spatial coordinate transformation. Comparing with Alù’s theory, it can be applied to the problems of any shape, dimension under any wavelength condition, which has been successfully verified by full-wave simulations [4]. Subsequent studies on invisibility cloaks, such as rounded cylinders [3, 4], elliptical cylinders [5, 6], eccentric elliptical cylinders [7], and even arbitrary shapes [8], were conducted according to the coordinate transformation theory. Furthermore, the first practical realization of a cylindrical cloak has been attained at microwave band in Ref. [9]. However, the design was constructed by artificial anisotropic metamaterial with inclusions of subwavelength metallic split-ring resonators (SRRs), whose permeability is sensitive to the frequency and polarization of the incident wave [10]. Feng et al. supplied a method of realizing the electromagnetic cloaking by concentric layered structures instead of using the metamaterial with subwavelength structured inclusions. Despite that the material of each layer is homogenous and isotropic, the whole structure cannot be considered as an isotropic one because of the local angular component of the permittivity. So the polarization of the incident electromagnetic wave is confined to the transverse magnetic (TM) mode [11].

On the other hand, almost all of the cloaks in above works adopt mediums with ultra-low refractive index, which is usually realized in the frequency ranges with high loss, such as at resonance or plasma areas of the metamaterials. This will induce an inevitable absorption in the cloaks.

In this paper, we propose an annular cloak with concentric multilayered structures, which is indeed homogenous and isotropic. A special distribution of the radial refractive index, obtained according to the geometrical optics theory, is applied in order to reduce scattering and shadow when electromagnetic wave passed through. And full-wave simulations on such a cylindrical design are performed to demonstrate the cloaking ability to any polarized incident wave from different direction. The refractive index of the material used here is relatively high comparing with those former designs [3, 4, 5, 6, 7, 11 and 13].

2. Cloaking method

According to the coordinate transformation theory, the original space is squeezed from a volume into a corresponding space with a spherical or cylindrical concealment in it. Since Maxwell’s equations are form-invariant during the transformations, the property of the cloaking material can be interpreted by the coordinate transformation. Due to the anisotropically compressed original space, medium properties of the cloaking material obtained by this theory are inevitably anisotropic [3, 12]. On the other hand, ray tracing results show that parts of the rays in the cloaking material need to be inflected sharply to flow around the concealed area [3]. Thus anisotropy is required to meet the constraint of the large ray bending. Despite that some reduced sets of medium [11, 13] parameters have been used, the spatially anisotropic components still exist, which results in the limit of the polarization of the incident wave and the practical realization.

In condition of little backward scattering and shadow, we realize the invisibility by an annular graded-index shell with a special refractive index distribution, which makes most of the incident rays pass though the shell in their original direction according to Snell’s Law. As shown in Fig. 1, we divide the domain of the incident wave into three parts: the outer area (I), the inner area (II) and the near-axis area (III). A perfect cloak implies that the incident ray can be focused as required or made avoid objects and flow around them like a fluid, returning undisturbed to their original trajectories. Therefore, the rays in Area I are not necessary to bend very much. Rays in Area II deflect a little and then come back in the cloak. However, those near-axis rays in Area III need a sharp turn to go around the central region, which requires a rapid change of the refractive index. Contrarily, for the outer rays, the refractive index should be closed to that of the background so that rays can pass through the cloak smoothly. Here we settle down this contradiction at the expense of the scattering loss of the rays in Area III, which need a sharp turn. Considering that this area is rather small, the effects of these back scattering wave can be neglected. By contraries, the scattering of rays in Area II cannot afford to ignore for Area II is much larger than Area III. Moreover, it is difficult to make these rays bend and then return without the help of the anisotropy of the medium parameters [3]. Therefore, in our design, these rays (ray ii in Fig. 1) are conducted forward in a large deflection angle so that they cannot be scattered backward, but interfere with the incident rays outside the cloak. Consequently, there will be a shadow behind the cloak because of the loss in Area I and II. Fortunately, the loss could be partly compensated by wave (ray i and iv in Fig. 1) from Area I, which is weakly focused and dispersed in Area I and II symmetrically and smoothly.

 

Fig. 1. Refraction of rays in different areas and the two consisted lenses.

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Overall, part of the inner rays (Area II) is bent in a large deflection. Near-axis rays (Area III) are lost by scattering. Those rays passed through, however, can be dispersed as symmetrically as possible, so that the field distribution on the other side of the cloak comes back. The penalty of such a design is the scattering of the near-axis rays and the aberrant superposition area arising from the interaction between waves from Area II and the outside waves.

3. The cloak design

To achieve the effect described above, we combined a diverging lens with a luneberg lens (Fig. 1). The diverging lens, which is a convex lens with refractive index less than one, composed the inner part of the cloak to refract the rays in Area II. A hollow luneberg lens with an external focal point at infinity wrapped the diverging lens. The radius-dependent refractive index is determined by Eq. (1) [14]:

where r ₁ is the internal focus and r is the radius of each layer. The two terms have been normalized by the radius of the lens. Here the effective internal focus would be much longer than its original focus r ₁ due to the cavity in the Luneberg lens. Hence, the outer rays can be focused weakly and fill up the shadow caused by the loss of rays in Area I and II. Additionally, being focused by luneberg lens outside, part of the inner rays (ray iii in Fig. 1) from the diverging lens could also compensate the shadow. The permittivity chosen here is a scalar (permeability µ=1), yet the reduced sets of the permittivity mentioned above were tensors [9, 11, 12 13]. Thus, it ensures that the permittivity or refractive index is only radius-dependent and the total cloak exhibits isotropic property. The final refractive index distribution along the radius is computed and illustrated in Fig. 2(a). The refractive index curves of two kinds of lens are connected by a short interpolation line so that the total distribution is continuous.

 

Fig. 2. (a) The refractive index distribution of the cloaking shell along the radius and multilayered approximation (N=40) to this continuous distribution; (b) the model of the 2D cylindrical multilayered cloaking structure with an inner diameter a=λ, outer diameter b=2λ, each color contains four layers.

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We considered the realization of the graded-index isotropic shell with the specialized distribution through multilayered approximation. As the fabrication and measurement requirements of a two-dimensional (2D) cloak are much simpler than those of a three-dimensional (3D) one, we took a 2D cylindrical cloak as an example. We assumed a cylindrical cloaking shell with inner radius a=λ, and outer radius b=2λ. The distribution of the refractive index shown in Fig. 2(a) can be approximately represented by N discrete layers medium. If the thickness of each layer is much less than the wavelength while the total number of the layers is sufficiently large, the shell can be treated as a homogenous material. Fig. 2(b) shows the stepwise 40-layer approximation model of the ideal continuous parameters supplied by Fig. 2(a). Some composite materials, with refractive index below 1 in certain frequency region [15, 16], are considered to achieve this specialized refractive index distribution.

4. Simulations and results

The multilayered medium simulations of cloaking structures were performed by means of HFSS, which is based on finite element method [17], to explore the effect and feasibility of the cloaking function. Fig. 3 shows the computational domain in which a transverse-magnetic (TM) polarized time-harmonic uniform plane wave is incident on a PEC shell of diameter a=λ wrapped by the cloak as specified above with outer diameter b=2λ. The shell ensures that no radiation can get into the concealed volume [4], nor can any radiation get out.

Consequently, an object of any shape or material can be placed in the interior of the thin shell (provided it fits) without influencing the fields anywhere in the domain. A copper cylinder at the inner radius of the cloak was chosen as the concealed object.

 

Fig. 3. The computational domain for the 2D cylindrical cloaking structure: the boundary condition around the domain is radiation; the top and bottom faces are PMC.

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Figure 4 shows the resulting simulated magnetic-field distribution and Poynting vectors for the copper rod with and without the cloak in the case of a TM plane wave incidence. As shown in Fig. 4(a), in the cloaking medium, wave fronts are deflected, guided around the copper cylinder, and then well reconstructed behind the cloak with minimal scattering. Outside the cloaking shell, interference patterns are formed by the interaction between the rays from Area II and the outside wave, which causes two feint shadows behind the cloak. In other parts of the domain, the plane wave is almost unaltered. The Poynting vectors shown in Fig. 4(c) demonstrate traces of rays in different areas. Rays near the axis touch the PEC shell, scattered and depleted strongly. Rays in Area II shoot out of the shell from the two sides. The Poynting vectors in other parts are bent smoothly, guided around the cloaked region, and focused slightly to restore the original propagation. After removing the cloaking structure (Fig. 4(b, d)), the waves are severely scattered by the object, resulting in a remarkable backward reflection and an evident shadow cast behind the conducting cylinder.

 

Fig. 4. Distribution of magnetic-field around the cloaked object (a) with a multilayered cloak, (b) without the cloak and distribution of Poynting vectors around the cloaked object (c) with a multilayered cloak and (d) without the cloak. The black circles outline the cloak.

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Incident waves from different sources were simulated to demonstrate the cloaking performance. Fig. 5(a) depicts the magnetic-field distribution under the source of elliptically polarized plane wave. Fig.5(c) illustrates the cloaking performance in the TE field stimulated by a line source. The electromagnetic wave scattering is also compared in Fig. 5(b) and (d) for the non-shelled conducting cylinder irradiated by the elliptical polarized wave and the line source respectively. In both cases, the concealment can be achieved by the cloak designed above. Results shown by Fig. 5(a) and (c) conform that the proposed cloaking principle is valid for both TE and TM incident waves, and therefore it can be extended to any incident wave by discomposing it as a sum of a TE and a TM component. Hence the limitation of the incident wave polarization is rather small. Fig. 5(c) also illustrates that the cloaking performance is feasible to incident wave from any direction.

 

Fig. 5. The calculated magnetic-field distribution in the vicinity of the conducting cylinder (a) with a multilayered cloak, (b) without cloak for elliptically polarized incident wave; and the electronic-field distribution (c) with a cloak of concentric layered structure, (d) without the cloak for a line source. The black circles outline the cloak.

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5. Conclusion

A cloak of homogenous and isotropic material has been presented based on the geometrical optics theory. It can be achieved by an isotropic shell with a special distribution of radius-dependent refractive index, which is realized by multilayered structures of homogeneous isotropic materials. The effectiveness of the cloaking performance has been verified by full-wave simulations in different cases. For practical applications, it is hoped that the cloaking material is available in the nature. Though the invisibility is imperfect because of the scattering of the near-axis rays and the focusing approximations of the luneberg lens, the simplicity of the material needed and the insensitivity to polarization do provide a feasible way to realize a practical cloak in the region from microwave to optical frequency.