1. Introduction

There is currently a growing interest in the theoretical and practical possibility of cloaking objects from the observation by electromagnetic fields [1–9]. The basic idea of the cloaking structure proposed by J. Pendry [1], is to use anisotropic transform medium whose permittivity and permeability are obtained from a homogeneous isotropic medium, by transformations of coordinates. The idea was successfully confirmed both by full-wave simulations [5, 6] and by experimental demonstration at microwave frequencies [7]. However, the design uses artificially structured metamaterial with inclusions of subwavelength metallic split-ring resonators (SRRs), and cannot be easily implemented for an optical cloak, which is certainly of particular interest. Recently, W. Cai et. al. have proposed an optical cloaking device for transverse magnetic (TM) polarization [8], but their design still requires metamaterial with anisotropic distribution of the permittivity, which is realized by subwavelength inclusions of metal wires in the radial direction embedded in a dielectric material.

Anisotropic materials with desired permittivity properties can be produced by a layered structure of thin, alternating dielectric layers (or metal and dielectric layers). Such planar systems have been proposed to demonstrate the subwavelength imaging [10], and to build photonic funnels for sub-diffraction light compression and propagation [11]. Recently, “optical hyperlens” made of cylindrical structure of anisotropic medium has been proposed, which has the capability of far-field imaging with resolution below the diffraction limit [12, 13]. Such anisotropic “hyperlens” has also been realized experimentally by concentric layered structures consisting of alternating layers of metal and dielectric [14, 15].

In this paper, we present the approach of realizing the electromagnetic cloaking by concentric layered structures instead of using the metamaterial with subwavelength structured inclusions. We show that by properly designing the realization of the anisotropic distribution of the permittivity required for the cloak through layered structure of homogeneous isotropic materials, the low-reflection and power-flow bending properties of the electromagnetic cloaking structure could be obtained.

2. Realizing anisotropic cylinder by layered structure of homogeneous isotropic materials

For the sake of simplicity, we restrict the problem to a two dimensional (2D) case. The proposed electromagnetic cloaking structure described here is based on the concentric layered structures consisting of alternating layers A and B of different homogeneous isotropic dielectric materials, as depicted in Fig. 1. When the layers are sufficiently thin compared with the wavelength, we can treat the whole layered structure as a single anisotropic medium with the dielectric permittivity as [10–12]

where, ε _A, ε _B are the permittivities of the layer A and layer B, respectively, ε_θ, ε_r are the angular and radial components of the effective anisotropic permittivity tensor, respectively, and η is the thicknesses ratio of the two layers:

Treating the layered structure as an effective anisotropic medium is based on the effective medium approximation, which requires that the thickness of each layer is much less than the wavelength and the number of the layers is large enough. When considering the finite thickness of a practical layered structure, it has been demonstrated in a planar case that the effective-medium approximation becomes more appropriate as the layers are made thinner [10]. Here we give the electromagnetic analysis on the cylindrical case in the following.

Consider the electromagnetic wave scattering for an infinite conducting cylinder shelled either with a concentric layered structure (Fig. 1(a)) or with an equivalent anisotropic medium (Fig. 1(b)). A plane wave with TM polarization is assumed to impinge along the x direction upon the shelled cylinder. The axial components of the incident and scattered magnetic field vector H outside the cylinder may be expressed as [16]

where, J_n, H_n ⁽²⁾ are, respectively, the Bessel function and the Hankel function of the second kind, k ₀ is the wave number of the outside medium. The magnetic field inside the mth dielectric layer (designated by the subscript m=1 to 2N) of the layered structure is expressed by

where, k_m is the wave number in the mth dielectric layer. In Eq. (3) and (4), a time dependence of the form e^jωt is assumed for the electromagnetic field quantities but is suppressed throughout. A_mn, B_mn, and C_n are arbitrary constants that can be determined by enforcing the boundary continuity condition at r=a, r=b, and at the interfaces between each dielectric layer. The electric fields in each medium are determined through

For a TM incident wave from a magnetic line source (with magnetic current of I_m, and located at r=r ₀), the incident magnetic field is described as

The scattering electromagnetic fields can be analyzed similar to that of the plane wave case.

 

Fig. 1. TM wave incident on an infinite conducting cylinder (in yellow) shelled with (a) concentric layers structure with alternating layers of dielectric A and B, (b) equivalent anisotropic cylindrical medium with radius-dependent, anisotropic material parameters. Both of the shells have inner and outer radius of a and b, respectively.

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When the conducting cylinder is shelled with the equivalent anisotropic medium (Fig. 1(b)), the electromagnetic fields scattering could be calculated similarly by application of Sommerfeld’s bundle of rays field representation via a polarization dependent coordinate transformation [17]. The electromagnetic wave scattered by the cylinders could be verified by calculating the far-field scattering pattern which is proportional to

First, we calculate the scattering fields by the conducting cylinder shelled with 2N layers of alternating dielectric A and B with same thickness (η=1) as depicted in Fig. 1(a). We choose ε_A=7.46, ε_B=0.54, a=λ, b=2λ, where λ is the wavelength of the incident electromagnetic wave. The far-field scattering pattern is calculated and plotted in Fig. 2 for the shells composed of different numbers of alternating dielectric A and B (N=5, and 20). The scattering pattern is also plotted in Fig. 2 for the conducting cylinder without any shell for comparison. Next, we calculate the scattering fields by the same conducting cylinder shelled with the equivalent anisotropic medium of same thickness, which has a 2D anisotropic permittivity of ε_r=1.0, and ε_θ=4.0, determined by Eq. (1). The far-field scattering pattern is plotted and compared in Fig. 2. It shows clearly that as the layers are made thinner by increasing the number of layers from 10 to 40, the layered structure shell has nearly the same scattering property as the equivalent anisotropic medium shell. This means that we are able to use a concentric layered structure to realize a cylindrical shell of a 2D rotationally invariant anisotropic medium. The thinner the layers, the better this layered structure approaches the scattering performance of the corresponding effective anisotropic medium.

 

Fig. 2. Comparison of the far-field scattering patterns for a bare conducting cylinder, the conducting cylinder shelled with a concentric layered structure of different number of layers, and that shelled with an anisotropic cylindrical medium. All values have been normalized to the scattering pattern of the bare conducting cylinder at θ=0.

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3. Electromagnetic cloaking properties of layered structures

As proposed in Ref. [5–7], cloaking a central cylindrical region of radius a by a concentric cylindrical shell of radius b requires a cloaking shell with the following radius-dependent, anisotropic relative permittivity and permeability:

Here we follow the TM wave illumination as considered in [8]. In this case only µ_z, ε_θ, and ε_r are of interest and must satisfy the requirement of Eq. (8). Similar to the consideration in [6, 7], to make only one component spatially inhomogeneous and also to eliminate any infinite values in Eq. (8), we can choose a reduced set of medium parameters as

This allows us to completely remove the need for magnetic response of the material, which is especially important for making cloak at optical frequency. The shell with the reduced set of parameters provides the same wave trajectory inside the cloaking medium, but it will induce some unfavorable reflection at the outer boundary due to the impedance mismatch.

Now we consider the realization of the above radius-dependent, anisotropic shell through layered structures of homogeneous isotropic materials. As verified in the previous section, an anisotropic cylindrical shell (with ε_θ, ε_r as the angular and radial permittivity components) could be mimicked by concentric layered structure consisting of alternating layers of two different dielectrics (with ε _A, ε _B as the permittivity), and the relation of the material parameters between them could be determined by Eq. (1), if only the thickness of each layer is much less compared with the wavelength. For an anisotropic shell with radius-dependent ε_r, we could still use a layered structure to mimic it by enforcing the permittivity of the alternating layers to vary with the radius. If the permittivity of the neighboring layers is changing gradually, we will show in the following examples that it is a good approximation to follow Eq. (1) to design the permittivity of the alternating dielectrics in the layered structure.

We consider the following design to explore the realizability of the cloaking through layered structures. For an example, we assume a cylindrical cloaking shell with inner radius a=λ and outer radius b=2λ. Similar to other practical realization of the invisibility cloaks [5–7], we first consider a stepwise homogeneous N-layer approximation of the ideal continuous parameters required by Eq. (9), and the continuous radius-dependent, anisotropic medium (as shown in Fig. 1(b)) could be represented approximately by N discrete layers of homogeneous anisotropic medium. Then we mimic each homogeneous anisotropic layer by an alternating layers of isotropic dielectric A and B (as shown in Fig. 1(a)), and the permittivities of the two layers are designed by Eq. (1) from the corresponding anisotropic layer. Fig. 3 depicts the material parameters of the proposed cloak of alternating layers of isotropic dielectric A and B, each with N=20 layers and of equal thickness (η=1). The relative permittivity of dielectric A increases gradually with the radius of each cylindrical layer from 0.005 to 0.536, while the relative permittivity of dielectric B decreases gradually from 8.00 to 7.46.

To demonstrate the performance of the designed electromagnetic cloak, we assume a TM plane wave incidence along the x direction, and calculate the electromagnetic wave scattered from a conducting cylinder (with radius a=λ) shelled with the proposed cloak using the semi-analytical method described in the previous section. The calculated results of magnetic-field distribution around the cloaked cylinder together with the power flow lines are illustrated in Fig. 4. As shown in Fig. 4(a), with the cloaking structure the wave fronts are deflected and guided around the cloaked region, and return to the original propagation direction with very small perturbation. When removing the cloaking structure (Fig. 4(b)), the waves are severely scattered by the object resulting in a remarkable backward reflection and an evident shadow cast behind the conducting cylinder.

 

Fig. 3. The relative permittivity components required for an ideal reduced set of parameter (εr(r), εθ(r)), and that for the corresponding layered structure with alternating dielectric A and B (εA(m), εB(m)). The inset describes the anisotropic shell divided into stepwise homogeneous N-layer (with permittivity as εr(m), εθ(m)), and the mimic of each layer by alternating layers of dielectric A and B (totally 2N layers).

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Due to the reduced set of parameters given by Eq. (9), there is an unavoidable but low level of reflection observed in Fig. 4(a), which is caused by the impedance mismatch at the outer boundary. The power reflection is about 11% estimated from the ratio of the inner and outer radii, R _ab=a/b=0.5 [8]. This agrees with the backward reflection deduced from the standing wave observed in Fig. 4(a). Decreasing R _ab or using a thicker cloak can further reduce the backward reflection. For example, if we choose a=λ and b=3λ, we have obtained less scattering from the cloak shell with a reduced power reflection of about 4%.

To show the quantitative behavior of the EM scattering reduction, we have calculated the far-field scattering patterns for the conducting cylinder with (blue line) and without the layered cloaking structure (red line) in Fig. 4(c). The far-field scattering pattern is calculated through Eq. (7) for all radial directions and normalized by the maximum value in the non-cloaked case. The strong lobes in the forward and backward directions for the conducting cylinder without any cloak in Fig. 4(c) correspond to the EM wave shadow and large backward reflection as illustrated in Fig. 4(b). When the cylinder is cloaked with the layered structure, the far-field scattering pattern is reduced in almost all radial directions (except at the directions for the two scattering nulls of the conducting cylinder). The forward scattering has been reduced for 12 dB and the backward scattering for 7 dB, resulting in much reduced EM wave shadow after the object as well as much reduced backward reflection.

 

Fig. 4. The calculated magnetic-field distribution around the conducting cylinder (a) with a cloak of concentric layered structure (movie, 3.67 MB) [Media 1], and (b) without cloak (movie, 3.67 MB) [Media 2], (c) the far-field scattering pattern. Power-flow lines (in black) in (a) show the smooth deviation of electromagnetic power around the cloaked object. The white circles outline the cloak.

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The above calculations have clearly demonstrated the capability of reducing the scattering from the conducting object inside the cloak and the power-flow bending properties of the proposed electromagnetic cloak. Such a concentric layered structure could get rid of the requirement of radius-dependent, anisotropically distributed relative permittivity usually realized by metamaterials with subwavelength structured inclusions. The most concerned issue of the proposed layered structure is the requirement of the alternating dielectrics with gradually increased or decreased permittivity, which is not easy to realize and to control from a practical point of view. An alternative approach is to construct a concentric layered structure that the permittivity of the alternating dielectrics is fixed, while the thickness ratio of the two layers varies to approximately satisfy the Eq. (9), since it will be easier to control the thickness of the dielectric layers.

For an example, a layered structure with N=10 has been designed with a=λ and b=2λ. The relative permittivity of the alternating layers is assumed to be ε_A=0.01, ε_B=4, and the thickness ratio η of the two layers varies from 2.3 to 130 according to Eq. (9). To verify the performance of the cloak, we assume a TM incident wave from a line source and calculate the electromagnetic wave scattered from a conducting cylinder (with radius a=λ) shelled with the proposed cloak. The magnetic-field distribution is plotted in Fig. 5 (a). Although there is imperfectness in the scattered fields caused by the reduced set of parameters and small approximation made in the design of the permittivity and the thickness ratio, Fig. 5(a) clearly demonstrates the low-scattering, shadow-reducing and power-flow bending properties of the electromagnetic cloaking structure. The electromagnetic wave scattering is also compared in Fig. 5 (b) for the non-shelled conducting cylinder irradiated by the same line source.

 

Fig. 5. The magnetic-field distribution around the conducting cylinder, (a) with a cloak of layered structure, and (b) without cloak, for a TM incident wave from a line source. The white circles outline the cloak.

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Materials with permittivity close to zero required for dielectric A in this design could be available at infrared and visible range. For examples, the permittivity of the noble metals and polar dielectrics follows Drude or Drude-Lorenz dispersion models [18, 19], and the real part of their permittivity effectively goes to zero at their plasma frequency, which, usually lies in the terahertz regime for polar dielectrics and some semiconductors [20] and in the visible and ultraviolet for noble metals [18–21]. Composite materials realized by properly embedding metallic nanoparticles and nanowires in a dielectric medium could also have effective permittivity near zero in a wide range of frequencies up to the visible [22, 23]. The operation frequency band of the proposed cloak should be limited by the material dispersion; especially the dispersion near the frequency the permittivity crosses the zero. Although the design works only for narrow frequency band at this time, we believe it might still have great potential interest.

4. Conclusions

We have proposed an electromagnetic wave cloak by realizing the radius-dependent, anisotropic material through layered structures of homogeneous isotropic materials. The performance of the cylindrical cloak has been demonstrated through rigorous calculation of the electromagnetic wave scattering, which reveals the low-scattering and power-flow bending nature of the properly designed cloaking structure. This approach is comparable with the metamaterial-based cloak [8] in terms of EM field distribution, the power flow, as well as the forward and backward EM scattering reductions. Our proposal has no requirement of any anisotropy or inhomogeneity of the material constitutive parameters which usually need metamaterials with structured inclusions to realize. Such cloak is possible to be realized by thin layers of normal materials or composites, therefore may lead to a simpler path to an experimental demonstration of electromagnetic cloaking, especially in the optical range.