1. Introduction

As electronic devices occupy an increasingly important position in modern warfare, how to make an object invisible from the enemy radar has become a hot research spot in recent years. In 2006, Pendry et al. proposed a new vision-electromagnetic invisible cloak, which is based on the optical transformation [1]. The coordinate transformation is used to squeeze space to control the electromagnetic waves propagation in metamaterials, so that the target in the cloak is invisible to the outside radiation [2]. The method for controlling electromagnetic fields is based on Maxwell’s equations form invariance principle in different spatial coordinates [3–5]. The theory of the electromagnetic cloak of stealth was first confirmed in the microwave regime by experiment [6]. In [7], the working principle was demonstrated in optical frequencies by using non-magnetic materials and tiny metal segments to constitute the invisibility cloak. The transformation approach has been extended for many electromagnetic applications [8–11]. Subsequently, a lot of analyses and research of electromagnetic cloaks have been done, such as full-wave simulations employing the finite element methods and the finite difference time domain methods, and the analysis of Fourier-Bessel functions [12, 13].

However, most of aforementioned examples verified the perfect effect of cloak of stealth, but did not consider embedded objects with anisotropic material inside the cloak. In [14], Chen et al. proposed a new concept-electromagnetic invisibility anti-cloak. An anti-cloak with a specific form of negative index anisotropic material is embedded inside the cloak, makes the external electromagnetic wave penetrate into the interior of the invisibility cloak and partially defeats the cloaking effect. This concept has been elaborated in [15–18]. In [15], the objects in the cloaked domain can receive the external energy via a “tunneling” between cloak and anti-cloak, and the waveform of electromagnetic waves is kept unchanged. In other words, the objects are still invisible for the outside.

In this paper, according to the anti-cloak theory, we derive the transformation formulae of the permittivity and permeability for designing the electromagnetic invisibility anti-cloak of two-dimensional arbitrary geometries. Different shapes of electromagnetic invisibility anti-cloaks are proposed to verify the correctness and effectiveness of the proposed formulae.

2. Invisibility anti-cloak for two-dimensional arbitrary geometries

Maxwell’s equations are form-invariant under the coordinate transformation. According to the coordinate transformation, a volume in the virtual space can be compressed into a shell surrounding the concealment volume in the real space [19]. This shell is the electromagnetic invisibility cloak which has the specific permittivity and permeability. The cloak can control the electromagnetic waves bypassing the concealed region and restore the waveform outside the cloak, so that the observer is insensible of the objects inside the cloak.

Assume the coordinate for the virtual space is $\text{(}u,v,w\text{)}$, and the coordinate for the real space is $\text{(}u\text{'},v\text{'},w\text{'}\text{)}$, so the coordinate transformation can be described by

Figure 1(a) shows the structure of a two-dimensional stealth cloak with an anti-cloak. Using a cylindrical coordinate system, the anti-cloak shell is embedded between the cloak shell and the masked object which permits the electromagnetic waves propagate into the cloak and the anti-cloak. The relative permittivity and permeability of the inner cylinder are ${\epsilon }_r$ and ${\mu }_r$, and the outside of the cloak is vacuum (${\epsilon }_r=1,{\mu }_r=1$).

 

Fig. 1 (a) Schematic diagram of stealth cloak with anti-cloak. (b) Coordinate transformation of the anti-cloak.

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To create a cloak shown in Fig. 1, we use a spatial transformation that compresses the cylindrical region with $0\le r\le R_3\left(\theta \right)$ into an annular region with $R_2\left(\theta \text{'}\right)\le r\text{'}\le R_3\left(\theta \text{'}\right)$,

3. Full-wave simulation and analysis

Firstly, a two-dimensional cylindrical anti-cloak is established to verify the above formulae. As shown in Fig. 2(b) , the cylinder is put along z-direction. The inner radius of the anti-cloak is $R_1(\theta \text{'})=a=100mm$, the outer radius is $R_2(\theta \text{'})=b=200mm$, and $R_3(\theta \text{'})=c=400mm$, the anti-cloak is filled with an inner cylinder (${\epsilon }_r=2,{\mu }_r=1$). In a two-dimensional space, the electromagnetic wave can be divided into a transverse electric (TE) polarization and a transverse magnetic (TM) polarization. If the source is a TE-polarized plane-wave spread vertical z direction, the effective electromagnetic parameters are ${{\epsilon }^{\prime }}_{zz}$, ${{\mu }^{\prime }}_{rr}$ and ${{\mu }^{\prime }}_{\theta \theta }$.

 

Fig. 2 Total electric field distribution near the anti-cloak under an incident TE plane-wave from left to right for (a) a dieletic cylindrical object (${\epsilon }_r=2,{\mu }_r=1$) with radius 100mm in a cloak; (b) a cylindrical object (${\epsilon }_r=2,{\mu }_r=1$) surrounded by an ideal anti-cloak embedded in the cloak; (c) a cylindrical object (${\epsilon }_r=2,{\mu }_r=1$) surrounded by a simplified anti-cloak embedded in the cloak.

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To demonstrate the electromagnetic properties of the anti-cloak, a commercial simulation tool based on the finite-element methods, Comsol Multiphysics [20] was used in this paper. A TE-polarized plane-wave illuminates from the left to the right, and the anti-cloak works at the frequency of 700MHz.

Figure 2(a) shows the total electric field when the cylindrical object with ${\epsilon }_r=2$and ${\mu }_r=1$ is enclosed by an annular cloak. It can be seen that the electromagnetic wave is smoothly bent around the cloaked region and the waveform is restored after the wave depart from the cloak. The object is invisible to the observer in this case. Simultaneously, the object cannot receive any incoming wave outside the cloak, and be “blindness” for the external situation. When a conformal anti-cloak with DNG materials is laid directly between the cloak and the object, it is capable of receiving the outer fields and restoring the plane waveform. Shown in Fig. 2(b) is an ideal anti-cloak with anisotropic parameters which derived from Eq. (7), the cloaking performance is perfect. Note that ${{\mu }^{\prime }}_{\theta \theta }$ is infinite at the inner boundary of the cloak and the outer boundary of the anti-cloak, which is difficult to realize in practice, so we can use the simplified parameters as shown in [21]. The cloak will have constants ${{\epsilon }^{\prime }}_{zz}=4$ and ${{\mu }^{\prime }}_{\theta \theta }=1$ with ${{\mu }^{\prime }}_{rr}$ varying radially from 0 to 0.25, while the anti-cloak have constants ${{\epsilon }^{\prime }}_{zz}=-2$ and ${{\mu }^{\prime }}_{\theta \theta }=-1$ with ${{\mu }^{\prime }}_{rr}$ varying radially from −1 to 0. The result shown in Fig. 2(c) indicates that the reduced cloaking material degrades cloaking performance but makes the anti-cloak easier to realize in practice with metamaterials.

Furthermore, several anti-cloaks with different special geometries, such as an elliptic anti-cloak with a plane curve obeying the equation in polar coordinate $R_2\text{(}\theta \text{)}=\text{b}\sqrt{\text{1}+\text{3}{\cos }^{\text{2}}\text{(}\theta \text{)}/\text{(3}{\sin }^{\text{2}}\text{(}\theta \text{)}+\text{1)}}$, a $\infty$-contour anti-cloak with a plane curve obeying the equation in polar coordinate $R_2\text{(}\theta \text{)}=\text{b}\sqrt{\text{1}+\text{3}{\cos }^{\text{2}}\text{(}\theta \text{)}}$, and an anti-cloak with arbitrary geometry expressed by the equation in polar coordinate $R_2\text{(}\theta \text{)}=\text{b(4}+\sin \text{(}\theta \text{)}+\sin \text{(3}\theta \text{)}+\text{1}\text{.5}\sin \text{(5}\theta \text{))}$ are established and simulated. They also obey the phenomenon that mentioned above. The full-wave simulation resuls are shown in Figs. 3(a) -3(c), respectively, which verify the effectiveness and correctness of the proposed anti-cloak design.

 

Fig. 3 Total electric fields distribution near the anti-cloak filled with dielectric (${\epsilon }_r=2,{\mu }_r=1$) under an incident TE plane-wave from left to right for (a) an elliptical anti-cloak; (b) a $\infty$-contour anti-cloak; (c) an arbitrary geometry anti-cloak.

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In addition, the formulae also can be applied to a polygonal anti-cloak. A hexagon anti-cload shown in Fig. 4(a) is divided into six regions, and each region has a different contour equation. Substituting the contour equation into the formula we get the permittivity and permeability of each region.

 

Fig. 4 Total electric field distribution near the anti-cloak filled with dielectric (${\epsilon }_r=2,{\mu }_r=1$) under an incident TE plane-wave from left to right for (a) a hexagon anti-cloak; (b) a discretization anti-cloak for unknown boundary object.

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As illustrated in Eq. (7), we observe that if the contour equation and electromagnetic parameters of an object are known, the parameters of the anti-cloak for this object can be deduced. However, at most time, the contour of a random object is difficult to be expressed with an equation. In this case, we can make the formulae discretization by sampling some points on the boundary of the object. Because the distances from the sampling points to the origin are easily measured, more sampling points can be used to restore the boundary. Moreover, the distances from the anti-cloak boundary to the origin are known for the reason that the anti-cloak is conformal with the stealth object. In order to verify this method is effective, we check the stealth effect of an anti-cloak with a random shape. The discretization permittivity and permeability can be determined with Eq. (7) by sampling the points every one degree. The result is shown in Fig. 4(b).

The fact that the electric fields exist in the cloaked area demonstrates that an observer in the anti-cloak can recieve the information from the outside, but remain invisibility for the outside. That is to say, an anti-cloak is equivalent to an invisibility cloak with a “window”, and through the “window”, information transmission between the outside and the inside can be implemented.

4. Conclusion

The design formulae for the electromagnetic invisibility anti-cloak of two-dimensional arbitrary shape objects are given in this paper. By setting the anti-cloak that is conformal with the stealth cloak and the masked object, we can get the uniform transformation for the cloak and anti-cloak of two-dimensional arbitrary geometries. In the region of the anti-cloak, the transformation media is anisotropic inhomogeneous, which can be realized by using electromagnetic metamaterials. Different two-dimensional anti-cloak models were given to verify the accuracy and effectiveness of the proposed design formulae. The anti-cloak plays an important role in the transmission of information from outside to inside of the cloak, meanwhile the object inside the cloak preserves invisibility to the outside. Relative to the invisibility cloak, the anti-cloak could be better to meet the practical application requirements.