1. Introduction

Control of electromagnetic wave with metamaterials is of great topical interest, and is fuelled by rapid progress in electromagnetic cloaks [1–6]. The approach to achieve invisibility of an object generally includes optical transformation [1,2], scattering cancellation [7], and transmission-line technique [8,9]. A review of these works has been made by Alitalo and Tretyakov [10]. Among various approaches, the optical transformation method plays an important role in the design of metamaterial devices [11–15]. Through a specific transformation, Pendry et al. [1] firstly proposed the cylindrical and spherical cloak that is capable of guiding electromagnetic wave around the cloaking region without scattering. Later, this theoretical prediction was confirmed by full-wave simulations [3] and realized in experiments at microwave and optical frequencies [4,5] with help of the metamaterials. Inspired by these pioneering works, many further investigations on cloak have been conducted, including elliptical-cylindrical cloak [16], arbitrary shaped cloak [17], line-transformed cloak [18], area-transformed cloak [19], and cloaks with a twin cavity [20]. The above cloaks can conceal an object by steering waves around an enclosed domain so that any object located inside the domain is hidden from observation. However, the objects hidden by these cloaks are “blind”, since no outside electromagnetic waves can reach into the cloaking region. It would thus be desirable to design new invisibility cloaks in which the hidden object can see the outside world.

Recently, Lai et al. [21] proposed a new recipe for invisibility cloak based on the concepts of complementary media and optical transformation. This cloak is composed of a dielectric core and an “anti-object” embedded inside a negative index shell, and it can cloak an object with a pre-specified shape and size outside the cloaking shell. Later, the generalized material parameter equations for the external cloak with arbitrary cross section were developed by our group [22]. In the foregoing investigations, however, the practical fabrication of the external cloak has so far not been achievable, for all components of material parameters are the functions of radius. Towards the practical and flexible realizations of the external cloak, a simplified cylindrical electromagnetic external cloak is designed based on optical transformation method. The radial and tangential (transverse i.e. $r$and $\phi$direction) material parameters of the designed external cloak are constants, and only axial ($z$direction) parameter is a function of radius. Thus, it is possible for the external cloak to be constructed with two-dimensional (2D) metamaterials. Full wave simulations based on the finite element method verified the performance of the designed cloak. The simplicity of the material parameters would move the 2D metamaterial external cloak a step further towards the practical realization.

2. Method and simulation model

The schematic diagram for the design of the external cloak is shown in Fig. 1 , where three circles with radius of$R_1$,$R_2$,$R_3$divide the original space into three regions, i.e., $r\text{'}<R_1$, $R_1<r\text{'}<R_2$and$R_2<r\text{'}<R_3$. To design the cloak, we first fold the region$R_2<r\text{'}<R_3$intoregion$R_1<r<R_2$, and then compress the region$R_1<r\text{'}<R_3$into region$r<R_1$. Here r and r’ represent the radius of the transformed space and the original space. According to the coordinate transformation method and the form invariance of the Maxwell’s equations, the relative permittivity and permeability in the transformed space can be given as [23]:

 

Fig. 1 (a) A system composed of the core material layer ($r<R_1$) and the complementary media layer ($R_1<r<R_2$) is optically equal to a large circle of air ($r\text{'}<R_3$). (b) A scheme to cloak an object with parameters of $\epsilon$,$\mu$by placing the “anti-object” with parameters of $\epsilon \text{'}$,$\mu \text{'}$ in the complementary media layer.

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As for the material parameters in the core material layer ($r\in (0,R_1)$), it can be obtained through the linear coordinate transformation of$f_{cor}(r)=r{R}_3/R_1$. Then according to Eq. (1), the material parameters can be derived as:

Suppose a blue object with permittivity$\epsilon$and permeability$\mu$is located in the outer air layer. In order to make it invisible, we need to add a red “anti-object” with parameters$\epsilon \text{'}=\epsilon {\epsilon }_{com}$and $\mu \text{'}=\mu {\mu }_{com}$into the complementary media layer, as shown in Fig. 1(b). It should be noted that the “anti-object” is mapped into complementary media layer according to Eq. (3). The cloak is composed of the modified complementary layer embedded with the “anti-object” and a core material. Its working principle can be summarized as follows: First, the scattering of the object and surrounding space is optically cancelled by the modified complementary media layer embedded with the “anti-object”. Then, the optical path in the cancelled space is restored by the dielectric core material. In the next section, we will make full-wave simulation based on the finite element software COMSOL Multiphysics to demonstrate the designed material parameters and the performance of the external cloak.

3. Simulation results and discussions

In the simulation, we only consider the case of TE polarization, of which a plane wave or a cylindrical wave is impinging onto the cloak from a specific direction. As for TM polarization case, the simulation can be done in the same way. It is not included herein for brevity. The whole computational domain is surrounded by a perfectly matched layer that absorbs waves propagating outward from the bounded domain. For the 2D external cloak, the geometry parameters are chosen as$R_1=0.5m$,$R_2=\text{1m}$, and$R_3=\text{2m}$. Based on these geometry parameters, the material parameters for the core material layer and complementary media layer of the cloak can be easily calculated through Eqs. (4) and (5), and shown in Figs. 2(a) and 2(b), respectively. It is clear from Fig. 2(a) that all components of material parameters for the core material layer are homogeneous and greater than or equal to one. From Fig. 2(b), we can observe that only z component of material parameters for complementary media layer is space dependent. Although the values of material parameters are less than zero, they are accessible from the metamaterials. In what follows, the performance of the external cloak will be investigated from the following aspects.

 

Fig. 2 (a) Material parameters distribution for the core material layer of the external cloak. (b) Material parameters distribution for the complementary media layer of the external cloak.

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We first demonstrate the scheme shown in Fig. 1(a), i.e. a system composed of a core material layer ($r<0.5m$) and the complementary media layer ($0.5m<r<1m$) is optically equal to a large circle of air ($r\text{'}<2m$). Figure 3(a) shows the electric field distribution in the vicinity of this system when a plane wave with wavelength$\lambda =1$m propagates from left to right. The absence of scattered waves clearly verifies the invisibility of the whole system. Then, we demonstrate the scheme shown in Fig. 1(b), i.e. the cloaking of an object by placing its “anti-object” in the complementary layer. An circular dielectric object with radius r = 0.3m and parameters$\epsilon =1.5$,$\mu =1$is centered at (−1m, −1m), as shown in Fig. 3(b), which also show its scattering pattern under plane wave irradiation. In order to make it invisible, we include an “anti-object” with parameters of ${\epsilon }_z\text{'}=1.5{\epsilon }_{comz}\text{'}$and$\mu \text{'}={\mu }_{com}\text{'}$ into the complementary media layer, as shown in Fig. 3(c). As can be seen from Fig. 3(c), although the incident plane waves are distorted in the transformation region, they restore original wave fronts when passing through the cloak. Hence, the perfect cloaking effect is verified. It is worth noticing that the object to be cloaked is placed outside the cloaking shell, and the cloaking effect comes from its “anti-object” embedded in the complementary media layer. The white flecks in the figure represent overvalued fields which are caused by the surface mode resonance. In addition, the cloaking of a circular dielectric object with linearly changing permittivity$\epsilon =1-r/5$under cylindrical wave irradiation is also simulated, and shown in Fig. 3(d). A line source with a current of 1A/m is located at $(-\text{2}.\text{25m},-\text{2}.\text{25m})$to generate the cylindrical wave. In this case, the wavelength of cylindrical wave is set as$\lambda =0.5m$, thus the size of object to be cloaked is larger than the wavelength. It can be clearly seen that the wave fronts of the cylindrical wave are perfectly restored when the wave exits the cloak. That is to say, whether the object being cloaked is larger than the wavelength or not, the object can be perfectly cloaked when it fit into the region bounded by$R_2$and$R_3$, and its “anti-object” is located in the complementary layer. From above simulation results, we can conclude that arbitrary object with homogeneous or anisotropic material parameters can be cloaked when its anti-object is “custom-made” according to Eq. (3), and the performance of the cloak is independent on the excitation source and incident direction.

 

Fig. 3 Electric field ($E_Z$) distributions in the computational domain under plane wave ((a)-(c)) and cylindrical wave ((d)) irradiations. (a) The system is composed of core material layer ($0<r<0.5m$) and complementary media layer ($0.5m<r<1m$). (b) The circular dielectric object with radius $r=0.3\text{m}$and permittivity $\epsilon =1.5$is centered at $(-\text{1m},-\text{1m})$. (c) The object in (b) is hidden by the cloak composed of modified complementary media layer with an embedded “anti-object” and a core material. (d) Another circular dielectric object with linearly changing permittivity$\epsilon =1-r/5$is cloaked by a similar cloak of that in (c).

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Next, we will illustrate the cloaking scheme with another two examples. Figure 4(a) displays the scattering pattern of two circular dielectric objects. The one on the top-left has a linearly changing permittivity of $\epsilon =1-r/5$, while the permittivity of another one on the downside is$\epsilon =1.5$. In such a case, two “anti-objects” with material parameters of ${\epsilon }_z\text{'}=(1-r/5){\epsilon }_{comz}\text{'}$, $\mu \text{'}={\mu }_{com}\text{'}$and${\epsilon }_z\text{'}=1.5{\epsilon }_{comz}\text{'}$,$\mu \text{'}={\mu }_{com}\text{'}$are embedded at the corresponding “image” positions in the complementary media layer, respectively. The perfect recovered wave fronts of the impinging plane wave shown in Fig. 4(b) demonstrate the effectiveness of the external cloak for the two objects. Actually, more objects located outside the cloak can also be cloaked, and there is no geometrical or material constraint on the object to be cloaked, as long as it fits into the region bounded by$R_2$and$R_3$, and its “anti-object” is located in the complementary layer. In Fig. 4(c), we show the scattering pattern of a dielectric shell with anisotropic permeability of ${\mu }_r=0.3$ and${\mu }_{\phi }=-0.6$. The shell is bounded between the circles of $r=1.5m$and $r=1.8m$. Under these circumstances, the “anti-object” is a complementary “image” shell with ${\mu }_r\text{'}=0.3{\mu }_{comr}\text{'}$, ${\mu }_{\phi }\text{'}=-0.6{\mu }_{com\phi }\text{'}$, and ${\epsilon }_z\text{'}={\epsilon }_{comz}\text{'}$. The electric field distribution in the vicinity of the cloak is shown in Fig. 4(d). The perfect plane wave pattern manifests the excellent cloaking effect.

 

Fig. 4 (a) Scattering pattern of two circular dielectric objects with permittivity$\epsilon =1-r/5$on the top-left, and$\epsilon =1.5$on the downside. (b) The objects in (a) are cloaked by external the cloak. (c) Scattering pattern of a dielectric circular shell with anisotropic permeability of ${\mu }_r=0.3$ and${\mu }_{\phi }=-0.6$. (d) The circular shell in (c) is cloaked by the external cloak.

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Since metamaterials are always lossy in real applications, it is very necessary to study the effect of loss on the performance of the cloak. The electric field distributions in the vicinity of the external cloak with electric and magnetic loss tangents (tgδ) of 0.001, 0.01, 0.02, and 0.03 are displayed in panels (a), (b), (c) and (d) of Fig. 5 . Here, the permittivity of circular dielectric object to be cloaked is$\epsilon =1.5$. It can be seen in Figs. 5(a)–5(c) that the electric field distributions are basically undisturbed when loss tangents of 0.001, 0.01 and 0.02 are added into both permittivity and permeability of the material. But when the loss tangent of metamaterials is 0.03 or more than, it deteriorates the performance of the external cloak in the forward scattering region, as shown in Fig. 5(d). In order to give a more intuitive understanding of loss effect, electric field distributions along x axis of the external cloak with different loss tangents are shown in Fig. 6 . We can clearly observe that when loss tangents are less than or equal to 0.02, electric field distribution generally overlaps with the lossless case. Therefore, loss tangents of metamaterials being less than or equal to 0.02 is acceptable.

 

Fig. 5 The electric field$(E_Z)$distributions in the vicinity of the cloak with loss tangents of 0.001 (a), 0.01 (b), 0.02 (c) and 0.03 (d).

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Fig. 6 The electric field $(E_Z)$ distributions along x axis of the external cloak with different loss tangents.

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Finally, the robustness of external cloak with non-ideal parameters is studied. We multiply transverse material parameters for both the complementary media layer and core material by coefficient ${\delta }_1$, and axial material parameter by coefficient${\delta }_2$. It represents the slight changes in the transverse and axial parameters. When there is a slight change in the axial parameter, research on the robustness of the external cloak is carried out under three cases: (i) the transverse parameters are kept invariant, (ii) the impedance ($Z=\sqrt{{\mu }_{\phi }/{\epsilon }_z}$and$\sqrt{{\mu }_r/{\epsilon }_z}$) is kept invariant, (iii) the refractive index ($n=\sqrt{{\mu }_{\phi }{\epsilon }_z}$and$\sqrt{{\mu }_r{\epsilon }_z}$) is kept invariant. In the three cases, the corresponding changes in the transverse parameters can be easily obtained. In the simulation, the negative perturbation (${\delta }_2=0.9$) and positive perturbation (${\delta }_2=1.1$) in the axial parameter are taken into consideration, and the parameters of circular dielectric object to be cloaked is $\epsilon =1.5$and$\mu =1$. Figures 7(b) –7(d) and Figs. 8(b) –8(d) show the corresponding electric field distributions in the vicinity of the external cloak. Simulation results for the cloak without perturbation are also simulated for comparisons, as shown in Figs. 7(a) and 8(a). It is clear that whether the transverse parameters change or not, the performance of the external cloak will be affected by the positive or negative perturbation in the axial parameters. In practice, if we want to keep the cloaking effects of the external cloak while at the same time minimizing its scattering field, it is the best choice to keep the refractive index invariant.

 

Fig. 7 Electric field $(E_Z)$distributions in the vicinity of the external cloak without perturbation (a), with negative perturbation (${\delta }_2=0.9$) while transverse parameters are invariant (${\delta }_1=1$) (b); impedances are invariant (${\delta }_1=0.9$) (c), and refraction index is invariant (${\delta }_1=1/0.9$) (d).

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Fig. 8 Electric field $(E_Z)$ distributions in the vicinity of the external cloak without perturbation (a), with positive perturbation (${\delta }_2=1.1$) while transverse parameters are invariant (${\delta }_1=1$) (b); impedances are invariant (${\delta }_1=1.1$) (c); and refraction index is invariant (${\delta }_1=1/1.1$) (d).

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Far field intensities of the external cloak with negative and positive perturbation in axial parameter are shown in Figs. 9(a) and 9(b), respectively. It is seen that once the refraction index is kept invariant, negative or positive perturbation in axial parameter will have little influence on the performance of the external cloak. This is in good agreement with the conclusions obtained from the near field distribution shown in Figs. 7 and 8.

 

Fig. 9 Far field intensity of the external cloak with negative perturbation (a) and positive perturbation (b) in axial parameter.

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4. Conclusion

In conclusion, a cylindrical electromagnetic external cloak with simplified material parameters is designed by means of optical transformation method. We show that axial permittivity and permeability of the external cloak is dependent on the radius, while the transverse material parameters are constants. The cloaking effect is basically undisturbed when loss tangents of metamaterials is less than or equal to 0.02. Moreover, the external cloak is robust against the perturbation in the axial material parameters when the refractive index is kept invariant. We believe that this external cloak can be constructed with 2D metamaterials, and the design method can also be extended to other transformation devices.