1. Introduction

Control of electromagnetic wave with metamaterials is of great topical interest, and is fuelled by rapid progress in electromagnetic cloaks [1–6]. Recent proposals for electromagnetic cloaking techniques include plasmonic cloaking due to scattering cancellation [7,8], transformation based cloaking [1,2], active cloaking [9], broadband exterior cloaking [10], transmission-line based cloaking [11], cloaking due to anomalous resonance [12,13], and so on. The scattering cancellation technique can be achieved for example by cancelling radiation from the induced dipole moments of the scatter by introducing another object, in which dipole moments of the opposite direction are induced. Transformation based cloaking techniques [14] rely on the transformation of coordinates, e.g., a point in the electromagnetic space is transformed into a special volume in the physical space, thus leading to the creation of the volume where electromagnetic fields do not exist, but are instead guided around this volume. The active cloaking uses sensors and active sources near the surface of the region, and could operate over broad bandwidths. Broadband exterior cloaking is based on three or more active devices. The devices, while not radiating significantly, create a “quiet zone” between the devices where the wave amplitude is small. Objects placed within this region are virtually invisible. Transmission-line technique [14] is based on the use of volumetric structures composed of two-dimensional or three-dimensional transmission-line networks. In these structures, the electromagnetic fields propagate inside transmission lines, thus leaving the volume between these lines effectively cloaked. Cloaking by anomalous resonance enables dielectric bodies of finite size to be perfectly cloaked by certain cylindrical arrangements of materials of positive and negative permittivities known as superlenses, but apparently not larger objects [15].

More recently, Lai et al [16] proposed a new recipe for an invisibility cloak, which is based on complementary media, composed of a dielectric core and an “antiobject” embedded inside a negative index shell. It can cloak an object with a prespecified shape and size within a certain distance outside the shell. In the foregoing investigations, however, the numerical simulations and parameter designs are devoted to circularly cylindrical invisibility cloak, which are cloaks with rotational symmetry. Toward the practical and flexible realizations of the electromagnetic cloaks, we present the general material parameters for the cylindrical complementary medium-assisted cloak with arbitrary cross sections, and validate them by numerical simulation. We show that the material parameters developed in this paper can be also specialized to the complementary medium-assisted cloak with regular shapes, such as circular, elliptical and square, which represents an important progress towards the realization of the cloak with arbitrary cross sections, based on complementary medium. Meanwhile, we compared the performance of the external cloaks based on linear and non-linear transformation, and some interesting phenomena are found.

2. Theoretical model

Combining the concepts of complementary media and transformation optics, material parameters for the 2D cloak with arbitrary geometries are derived. The schematic diagram of the space transformation is shown in Fig. 1 , where three cylinders enclosed by contours $aR(\theta )$, $bR(\theta )$and $cR(\theta )$divide the space into three regions, i.e., the core material layer ($r^{″}<aR(\theta )$), the complementary layer ($aR(\theta )<r^{\prime }<bR(\theta )$) and the outer air layer ($bR(\theta )<r<cR(\theta )$). Here, $R(\theta )$ is an arbitrary continuous function with period $2\pi$ [17]. According to the coordinate transformation method, the permittivity ${\epsilon }^{i^{\prime }{j}^{\prime }}$ and permeability ${\mu }^{i^{\prime }{j}^{\prime }}$ tensors of the transformation media can be written as [18,19]

 

Fig. 1 (a) The system composed of air layer ($bR(\theta )<r<cR(\theta )$), the complementary media layer ($aR(\theta )<r^{\prime }<bR(\theta )$) and the core material layer ($r^{″}<aR(\theta )$) that is optically equal to a large circle of air ($r<cR(\theta )$). (b) A scheme to cloak an object of ${\epsilon }_o$, ${\mu }_o$ by placing the “antiobject” of ${\epsilon }_o^{\prime }$, ${\mu }_o^{\prime }$ in the complementary media layer.

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The complementary media is obtained by the coordinate transformation of folding the air layer into the complementary layer with the linear coordinate transformation of

And then, the Jacobian transformation matrix and its determinant can be obtained as

Substituting (3) and (4) into (1), we can obtained the permittivity and permeability tensors for the complementary layer as

And then, we can obtain the Jacobian transformation matrix and its determinant as

3. Simulation results and discussion

First we demonstrate the scheme shown in Fig. 1(a), i.e., the air layer ($bR(\theta )<r<cR(\theta )$) and the complementary media layer ($aR(\theta )<r^{\prime }<bR(\theta )$) that is optically equal to a large circle of air ($r<cR(\theta )$). Geometry parameters used in the simulation is chosen as$R(\theta )\text{=cos(4}\theta \text{)+sin(2}\theta \text{)+3}$,$a=0.1$, $b=0.5$ and $c=0.9$. We consider the case of cylindrical wave irradiation, of which the wavelength is $\lambda =1$unit. Figure 2 shows the electric field distribution in the vicinity of the transformation region composed of a core material and a complementary medial layer. The line source with a current of 1A/m is located at (−3, −3). The absence of scattered waves clearly verifies the invisibility of the whole system.

 

Fig. 2 The electric field (Ez) distributions in the vicinity of the core material ($r^{″}<aR(\theta )$) and the complementary layer ($aR(\theta )<r^{\prime }<bR(\theta )$) under cylindrical wave irradiation.

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Next we demonstrate the scheme shown in Fig. 1(b), i.e., the cloaking of an object by placing its “antiobject” in the complementary layer. The dielectric object with radius r = 0.3λ, parameters${\epsilon }_o=2,{\mu }_o=1$, is centered at (−1.5, −1.5), as shown in Fig. 3(a) , which also shows its scattering pattern under cylindrical wave irradiation. In order to make it invisible, we include an “antiobject” with parameters${\epsilon }_o^{\prime }=2{\epsilon }^{i^{\prime }{j}^{\prime }}$and${\mu }_o^{\prime }={\mu }^{i^{\prime }{j}^{\prime }}$into the complementary layer. The image of the “antiobject” is obtained according to the linear transformation of Eq. (2). The calculated electric field shown in Fig. 3(b) clearly demonstrates the “external” cloaking effect. It is worth noting that the object to be cloaked is placed outside the cloaking shell, and the cloaking effect comes from its “antiobject” embedded in the complementary media. In the space closely adjacent to the cloaked object, the cloaking effect doesn’t exist, and the pressure fields are very strong due to the surface mode resonance induced by the multiple scattering of acoustic wave between object and the cloak device. We emphasize that there is no shape or size constraint on the object to be cloaked, as long as it fits into the region bounded by $r=bR(\theta )$and $r=cR(\theta )$. In Fig. 3(c)-3(f), we show the cloaking scheme of the two curved shell. In this case, the image of the “antiobject” is also a curved shell according Eq. (2). Figure 3(c) is the scattering pattern of the dielectric shell of${\epsilon }_o=2$, ${\mu }_o=1$, which is fitted into the region bounded between $0.52R(\theta )<r<0.58R(\theta )$. In Fig. 3(d) the dielectric shell is hidden by the cloak with an “antiobject” located between the contours of $r^{\prime }=0.42R(\theta )$ and $r^{\prime }=0.48R(\theta )$in the complementary layer. The cylindrical wave pattern in Fig. 3(d) manifests the clocking effects. Next we consider the circular shell with parameters${\epsilon }_o=-1$,${\mu }_o=1$. The scattering pattern for such a shell shown in Fig. 4(e) is similar to that of metal shell. In such a case, the “antiobject” of the shell with parameters, ${\epsilon }_o^{\prime }={\mathrm{-\epsilon }}^{i^{\prime }{j}^{\prime }},{\mu }_o^{\prime }={\mu }^{i^{\prime }{j}^{\prime }}$is located in the complementary layer between the contours of $r^{\prime }=0.42R(\theta )$and$r^{\prime }=0.48R(\theta )$. The electric field distribution in the vicinity of the cloak is shown in Fig. 3(f). Again, the cylindrical wave pattern manifests the cloaking effect.

 

Fig. 3 Electric field distributions under cylindrical wave irradiation. (a) The circular dielectric object with$r=0.3$is centered at (−1.5, −1.5). (b) The object in (a) is hidden by the cloak with arbitrary shape. (c) The circular dielectric shell of ${\epsilon }_o=2$, ${\mu }_o=1$. (d) The shell in (c) is hidden by the cloak with embedded “antiobject” shell of ${\epsilon }_o^{\prime }=2{\epsilon }^{i^{\prime }{j}^{\prime }}$,${\mu }_o^{\prime }={\mu }^{i^{\prime }{j}^{\prime }}$. (e) The shell with parameters of${\epsilon }_o=-1$, ${\mu }_o=1$. (f) The shell in (e) is hidden by the cloak with embedded “antiobject” shell of${\epsilon }_o^{\prime }=-{\epsilon }^{i^{\prime }{j}^{\prime }}$, ${\mu }_o^{\prime }={\mu }^{i^{\prime }{j}^{\prime }}$.

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Fig. 4 Electric field distribution in the vicinity of the cloak under cylindrical wave irradiation. The line source is located at (3, 0), (−3, 0), (0, 3) and (0, −3) for (a), (b), (c) and (d).

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To investigate the interaction of the cloak with electromagnetic wave from different orientations, the current line source is located at four different positions in the computational domain, and the electric field distributions are simulated, as shown in Fig. 4. It can be clearly seen that the cylindrical waves are restored to the original wave fronts when passing through the cloak, and the circular dielectric object is perfectly hidden independent on the orientation of the incident electromagnetic wave.

Figure 5 shows the electric field distribution in the computation domain under TE wave irradiation. The incident TE wave with λ = 1 unit is from left to right. In Fig. 5(a), the absence of scattered waves clearly verifies the invisibility of the system composed of core material and the complementary layer. Figure 6(b) shows the cloaking of the dielectric circular object. Figure 6(c) and (d) shows the cloaking of the shell with ${\epsilon }_o=2$and${\epsilon }_o=-1$, respectively. Although the incident TE waves are distorted in the transformation region, they restore their original wave fronts when passing through, and the cloaking effect is independent on the type of the exciting source.

 

Fig. 5 Electric field distribution in the computation domain under TE wave irradiation. (a) The electric field distribution in the vicinity of the system composed of core material and the complementary layer. (b) The cloaking of the dielectric circular dielectric object. (c) The cloaking of the shell of ${\epsilon }_o=2$, ${\mu }_o=1$. (d) The cloaking of the shell of ${\epsilon }_o=-1$, ${\mu }_o=1$.

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Fig. 6 Electric field distribution in the computation domain of the cloak with loss tangent of 10⁻⁴ (a), 0.01 (b), and 0.1 (c).

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Since artificial metamaterials are always lossy in real applications, it does make sense to investigate the effects of loss on the performance of the cloak. Electric field distributions of the cloaks with electric and magnetic-loss tangents ($tg\delta$) of 10⁻⁴, 0.01, and 0.1 are displayed in panels (a), (b) and (c) of Fig. 6. In the case of $tg\delta =$10⁻⁴ and 0.01, the performance of the cloak is basically undisturbed, as shown in Fig. 6(a) and (b). In such cases, the effects of loss can be ignored. But when the loss tangent of the metamaterials is 0.1 or more than that, it deteriorates the performance of the cloak mainly in the transformation region and the forward-scattering region of the near field, as shown in Fig. 6(c).

According to the procedure illustrated in Ref [17], material parameters for the complementary medium-assisted cloaks with circular, elliptical and square cross sections can be obtained from Eqs. (5) and (8). The electric field distributions in the computation domain under cylindrical wave irradiation are simulated as shown in Fig. 7 , which clearly demonstrate the generality of the material parameters developed in this paper for designing the complementary medium-assisted cloaks with arbitrary geometries. Besides, under TE plane wave irradiation, the cloaking effect can also be observed, as shown in Fig. 8 . The simulation results for circular cloak are in good agreement with Ref [16], which further confirms the effectiveness and the generality of the material parameters we developed.

 

Fig. 7 Electric field distributions for the complementary medium-assisted cloak with circular (a), elliptical (b), and square (c) cross sections under cylindrical wave irradiation.

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Fig. 8 Similar to Fig. 7, but for TE plane wave irradiation.

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Similarly, the external cloak based on nonlinear transformation is studied. Here, we consider two kinds of nonlinear transformation as shown below.

The material parameters can be obtained according to the procedure illustrated in section 2, and such results are not included herein for brevity. Next, we will discuss the characteristic of the external cloak based on linear and nonlinear transformation. Taking the case of a circular cloak as a special example, the simulation results are shown in Fig. 9 , where panels (a) and (b) show the simulation results under linear transformation of Eq. (2), (c) and (d) show the cases under nonlinear transformation of Eq. (9), while (e) and (f) show the cases under nonlinear transformation of Eq. (10). From Fig. 9(a), (c) and (e), we can observe that to cloak the same dielectric object with radius$r=0.2\lambda$, centered at (−1.25, 0), the shape of the “antiobject” is quite different for linear and nonlinear transformations. Therefore, the image of the “antiobject” mapped into the complementary media layer is not only dependent on the contour of the cloak but also dependent on the coordination transformation, which is an interesting feature of the complementary medium-assisted external cloak. Meanwhile, from Fig. 9(b), (d) and (f), we can find that for the external cloaks in the same size and with the same “antiobject” of $r=0.2\lambda$ centered at (−0.75, 0), an object with much larger geometry size can be cloaked based on nonlinear transformation of Eq. (10), as shown in Fig. 9(f). It is worth noting that in Eq. (10), we just choose $m=4$in our simulation; increasing the value of m will enlarge the outer air layer in the transformation region, as a consequence, a much larger object can be cloaked for the given cloak size. It means that the nonlinear transformation has some advantages over the linear transformation for miniaturizing the size of the external cloak.

 

Fig. 9 Comparison of the characteristic of the external cloak based on linear and nonlinear transformation. (a) and (b) are corresponding to the linear transformation of Eq. (2). (c) and (d) are corresponding to the nonlinear transformation of Eq. (9). (e) and (f) are corresponding to the nonlinear transformation of Eq. (10). In panels (a), (c) and (e), the dielectric object with radius$r=0.2\lambda$, is centered at (−1.25, 0). In (b), (d) and (f), the “antiobject” with radius$r=0.15\lambda$, is centered at (−0.75, 0).

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4. Conclusion

Material parameters for the complementary medium-assisted cloak with arbitrary geometries are derived. The cloak is composed of the core material and the modified complementary layer embedded with the “antiobject”, and it can make the object outside its domain invisible. We have investigated the influence of electric and magnetic-loss of the metamaterials on the performance of the device. Results show that for loss tangent less than 0.01, the performance of the cloak is basically undisturbed; increasing the loss tangent will disturb the forward-scattering region of the near field. This work has greatly improved the designing flexibility of the complementary medium-assisted cloak, since material parameters for the cloak with arbitrary geometries can be easily obtained for the given contour equations. We show that the material parameters can be also specialized to the 2D cloaks with regular shapes, such as circular, elliptical and square, which represents an important progress towards the realization of arbitrary shaped complementary medium-assisted cloak. Moreover, we find that the shape and size of the “antiobject” is dependent on the contour of the cloak and the coordination transformation. Interestingly, the object with much larger size can be hidden by the cloak based on non-linear transformation, which shows some advantages in open up an avenue for miniaturization in future cloak design.