## Abstract

A novel external cloak with homogeneous properties is proposed and designed based on the coordinate transformation. By contrast with the reported external cloaks, the homogeneous properties make the cloak designed here applicable to practical fabrication. Both symmetric and asymmetric structured polygonal external cloaks are investigated here, which provides a new perspective to designing arbitrary transformation based devices, breaking through the limitation of previously reported cloak designs with symmetric structures and inhomogeneous material parameters. It represents important progress toward the practical fabrication of the metamaterial-assisted invisible external cloak.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In the past decade, invisible cloaks based on coordinate transformation have aroused much attention for its potential applications in various fields of human production and life, such as military equipment stealth, radiation shielding, wave-guiding or filtering. Generally speaking, the cloaks can be divided into two categories based on the information interactive properties: blind cloaks and interactive cloaks. Blind cloaks such as Pendry’s cloak [1], carpet cloak [2,3] or other arbitrary shaped cloaks [4–8] have one thing in common that the hidden object was impossible to interact with the outside since the electromagnetic field cannot penetrate into the concealed region. In order to overcome the drawbacks of the blind cloak, several information interactive cloaks have been proposed by scientists or scholars, such as open cloaks [9–11], external cloaks [12–14] or distributed outer cloaks [15,16]. Furthermore, in 2011, our group proposed a novel reciprocal cloak [17], of which the object hidden inside is not blind and can exchange information with the outside world.

It should be noted that most cloaks in early age usually have highly inhomogeneous properties, which hinder the practical fabrication of the devices [4–8, 12–14]. Removing inhomogeneous property of the cloak is significant for practical use. In [18], a one-directional diamond cloak with homogeneous parameter was firstly studied which is equivalent to a carpet cloak when the hidden object is sitting on a conducting ground plane. Later, several electromagnetic cloaks with homogeneous material parameters have been reported [19–22]. On the basis of the combination of complementary media and the embedded optical transformation, homogeneous external cloak has been proposed [23–25], which can hide an object within a certain distance outside the cloak. In our early works [26], we have proposed a diamond shaped reciprocal invisible cloak with homogeneous metamaterials. All these aforementioned transformation mediums with homogeneous material parameter are obtained by the utilization of the linear transformation approach [27–33]. However, it should be noted that most of these homogeneous external cloaks are symmetrical and lack of versatility. In [34], Han et al. reported a common method to obtain arbitrarily shaped blind cloak with homogeneous materials by employing liner transformation and twofold spatial compression method. Recently, Rajput et al. [35] extended this approach to design arbitrary shaped reciprocal cloak, which can hide the target object in the core area of the device. Nevertheless, investigation on arbitrarily shaped external cloak with homogeneous properties has not reported yet as far as we know.

In this paper, inspired by Han’s work [34], we utilize the linear transformation method to design arbitrary external cloak with homogeneous parameters. We first divide the N-sided polygon into N triangles. For each triangle, we re-divide it into three regions: cloaked region, complementary region and core region. And then, by utilizing folded and compressed transformations respectively in complementary region and core material region, arbitrary shaped external cloak with homogeneous parameters is obtained. In this external cloak, the complementary medium layer with embedded anti-object is used to cancel the scattering of the object outside the cloak, and the core material layer is used to restore the canceled space to keep impedance matching of free space. The invisibility of the proposed generalized approach has been validated by the numerical simulation for both the symmetric and asymmetric geometries. In contrast with the traditional external cloak device [12–14], the material parameters of the designed device are homogeneous, reducing the difficulty of practical application. Furthermore, the designed approach reported here is effective for both symmetric and asymmetric shaped transformation based devices, which break the limitation of traditional device design with symmetrical structures. It also provides a general method to design novel devices with versatility and homogeneous. We believe such external cloaks could find potential applications in military stealth, radiation shielding, or other communication systems.

This paper is organized as follows: Section 2 presents the theoretical analysis of the proposed novel external cloak. Simulation results and discussions are described in Section 3. Conclusions are drawn in Section 4.

## 2. Theoretical analysis

According to transformation optics, under a space transformation from the original coordinate $(x,y,z)$ to a new coordinate $[x^{\prime}(x,y,z),y^{\prime}(x,y,z),z^{\prime}(x,y,z)]$, the permittivity and the permeability in the transformed space are given by:

Before derivation of the constitutive tensors, we firstly decompose the whole original space into N triangles having one of the vertices at origin as shown in Fig. 1(a). Each triangular region in the original space is divided into three contoured regions: cloaked region, complementary region and core material region, respectively, as shown in Fig. 1(b). The cloaked region usually consists of background material and stealth object. The essence of the external cloak design is the parameter derivation of the complementary medium region and the core material region.

In order to obtain the material parameters of the complementary regions and core regions, we utilize folded mapping and compressed mapping method in these two regions respectively. Firstly, we employ the folded transformation to obtain the material parameters by mapping trapezium ${a_i}{a_{i + 1}}{b_{i + 1}}{b_i}$ onto trapezium ${c_i}{c_{i + 1}}{b_{i + 1}}{b_i}$. In the second step, we use compressed transformation method to obtain the constitutive tensors of the core material region by mapping triangle ${a_i}{a_{i + 1}}o$ into the triangle ${c_i}{c_{i + 1}}o$.

#### 2.1 Material parameters of complementary region

Figure 2 is a schematic diagram for the folded transformation procedure. Firstly, region ${a_i}{a_{i + 1}}{b_{i + 1}}{b_i}$ in the original space is divided into two sub-triangles: triangle ${a_i}{a_{i + 1}}{b_{i + 1}}$ and triangle ${a_i}{b_i}{b_{i + 1}}$, as shown in Fig. 2(a). Then, triangle ${a_i}{b_i}{b_{i + 1}}$ is folded into the outer triangle ${c_i}{b_i}{b_{i + 1}}$, and triangle ${a_i}{a_{i + 1}}{b_{i + 1}}$ is folded into inner triangle ${c_i}{c_{i + 1}}{b_{i + 1}}$, as shown in Fig. 2(b) and Fig. 2(c) respectively. The motive of the whole transformation is to fold the trapezium ${a_i}{a_{i + 1}}{b_{i + 1}}{b_i}$ in original space into trapezium ${c_i}{c_{i + 1}}{b_{i + 1}}{b_i}$, and generate an empty space ${a_i}{a_{i + 1}}{b_{i + 1}}{b_i}$ in physical space, which constituted the complementary region and cloaked region, respectively. The trapezium region ${c_i}{c_{i + 1}}{b_{i + 1}}{b_i}$ is utilized to cancel the trapezium region ${a_i}{a_{i + 1}}{b_{i + 1}}{b_i}$ in original space, and the generated cloaking region is used to hide the stealth object.

It should be noted that the N-sided polygon A, B and C in Fig. 1 share the same center at origin (0,0). The general expression of the $ith$ vertex of polygon A, B and C can be defined as:

For the outer triangles [as the blue colored triangle shows in Fig. 2(c)], the corresponding coordinate transformation is expressed as:

The Jacobian matrix of Eqs. (3) is given by:

The Jacobian matrix and its determinant of transformation can be obtained from Eq. (9) as follows:

#### 2.2 Material parameters of the core region

Figure 3 demonstrates the schematic diagram of the compressed transformation procedure. Firstly, the triangle region ${a_i}{a_{i + 1}}o$ in the original space (as the green colored region shows in Fig. 3(a)) is compressed into an intermediate triangle ${c_i}{a_{i + 1}}o$ (as the blue colored region shows in Fig. 3(b)). Then, the intermediate triangle ${c_i}{a_{i + 1}}o$ is further compressed into the triangle ${c_i}{c_{i + 1}}o$ (as the yellow colored region shows in Fig. 3(c)).

The target of the whole compressive procedure is mapping the triangle ${a_i}{a_{i + 1}}o$ in the original space into triangle ${c_i}{c_{i + 1}}o$ in the physical space, and restoring the space canceled by the aforementioned folding transformation, making the whole physical space maintain impedance matching of the free space.

Two steps are needed to derive the constitutive tensor of the core region:

Firstly, supposing that the corresponding coordinate transformation of the intermediate compressing procedure can be expressed as:

The Jacobian matrix of the intermediate transformation can be obtained from Eq. (9) as follows:

The Jacobian matrix of the transformation from Eq. (16) is obtained as:

From Eqs. (12) and (20), it can be observed that the material parameters are homogeneous since they depend on several constants only

## 3. Results and discussion

In this section, we carry out full-wave simulations using the commercial finite element solver COMSOL Multi-physics to verify the correctness of the derived Eqs. (8), (12) and (20), and demonstrate the invisibility of the novel cloak designed here.

For the convenience of description, we give out the simulation circumstance firstly. The whole computational domain is surrounded by a perfectly matched layer that absorbs the outward traveling waves. The frequency of the incident wave is set as 10 GHz under TE wave irradiation. The total calculation domain is chosen as $0.3m \times 0.3m$, and the geometry parameters of the polygons are chosen as $a = 0.075m$, $b = 0.05m$, $c = 0.03m$. The material parameters of the hidden object are chosen as ${\varepsilon _0} = - 1,{\mu _0} = 1$ for all simulations, and will be omitted in the following description for simplicity.

Figure 4 demonstrates the electric field (${E_z}$) distribution in the vicinity of the core material region($r \le c$) and the complementary region ($c \le r \le b$) with 3-sided, 4-sided, 5-sided and 6-sided regular polygonal external cloaks. The absence of scattered waves clearly verifies the invisibility of the whole system.

Figure 5 illustrates the material distribution of 4-sided regular polygonal external cloak. In order to clearly distinguish the material parameters of proposed cloak from others, we recited the results in our early work [36]. Figs. 5(a)–(d) are the material distributions of the N-sided polygonal external cloak reported in [36]. It is clear that the material distribution is inhomogeneous and anisotropic, which hindered the practical application of the device in [36]. However, by employing the linear transformation method proposed here, the parameters become homogeneous (as shown in Figs. 5(e)–5(h)), reducing the complexity in practical realization. It is worth noting that the material parameters of the anti-object are also homogeneous and nonsingular. Furthermore, both the permittivity and permeability of the cloak designed here are relatively small, making it easy to fabricate by modern microwave and meta-material engineering technologies. The large values of permeability can be realized by using ferrite sheets, while the double negative material in the complementary region can be realized by using split ring resonators (SRRs) and wire structures [37], where split ring resonators (SRRs) provides the negative permeability and wire structures contribute to the negative permeability. Since the core region of regular polygonal external cloak proposed here is homogeneous and isotropic, we omit to draw the divided geometric lines of the core region for cloak devices in the following simulations.

Figure 6 demonstrates the electric field (Ez) distribution in the vicinity of the bare dielectric objects without and with the proposed cloak. Figure 6(a) displays the scattering pattern of a dielectric shell (${\varepsilon ^{\prime}_o} = 2$, ${\mu ^{\prime}_o} = 1$) fitted into the region bounded between $0.055m \le r \le 0.065m$ under TE plane wave irradiation. In order to make the dielectric object invisible, we embedded an “anti-object” with parameters ${\varepsilon _0} = {\varepsilon ^{\prime}_o} \ast {\varepsilon ^{i^{\prime}j^{\prime}}},{\mu _0} = {\mu ^{\prime}_o} \ast {\mu ^{i^{\prime}j^{\prime}}}$ (${\varepsilon ^{\prime}_o}$ and ${\mu ^{\prime}_0}$ are the permittivity and permeability of the hidden object) into the complementary layer, as shown in Fig. 6(d). The image of the “anti-object” is obtained according to the linear transformation of Eqs. (3) and (9). It is worth noting that the object to be cloaked is placed in the cloaked region of the device, and the cloaking effect comes from its “anti-object” 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 electromagnetic wave between hidden 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 $r = a$ and $r = b$. In Figs. 6(b) and 6(c), a dielectric ball (${\varepsilon ^{\prime}_o} = - 1$, ${\mu ^{\prime}_o} = 1$) with radius of 0.0094 m is located at (-0.0506 m,0), and two dielectric quadrangles (${\varepsilon ^{\prime}_{o1}} = - 1$, ${\mu ^{\prime}_{o1}} = 1$ and ${\varepsilon ^{\prime}_{o2}} = 2$, ${\mu ^{\prime}_{o2}} = 1$) are fitted into the region bounded between $0.055m \le r \le 0.065m$. Figures 6(e) and 6(f) are corresponding external cloaking device with anti-object (${\varepsilon _{01}} = {\varepsilon ^{\prime}_{o1}} \ast {\varepsilon ^{i^{\prime}j^{\prime}}},{\mu _{01}} = {\mu ^{\prime}_{o1}} \ast {\mu ^{i^{\prime}j^{\prime}}}$ and ${\varepsilon _{02}} = {\varepsilon ^{\prime}_{o2}} \ast {\varepsilon ^{i^{\prime}j^{\prime}}},{\mu _{02}} = {\mu ^{\prime}_{o2}} \ast {\mu ^{i^{\prime}j^{\prime}}}$) embedded into the complementary regions. It can be found that the scattering pattern is strong when the dielectric objects are directly irradiated by the TE plane wave. However, when added with the cloak device with corresponding anti-object, all of them become invisible, since there are no scattering fields outside the devices. It is worth mentioning that the stealth effect is also effective under the illumination of transverse-magnetic (TM) mode. In this circumstance, the material parameters are easily obtained by replacing ${\mu _{xx}}$, ${\mu _{xy}}$, ${\mu _{yx}}$, ${\mu _{yy}}$ and ${\varepsilon _{zz}}$ components with ${\varepsilon _{xx}}$, ${\varepsilon _{xy}}$, ${\varepsilon _{yx}}$, ${\varepsilon _{yy}}$ and ${\mu _{zz}}$ components. Simulation result is not shown here for brevity.

Although both the cloaks reported in [27] and here are homogeneous and anisotropic, they have differences. For one thing, the concealed object is covered by the cloaked medium for the former, but it is outside the shielding medium for the latter. For another, electromagnetic field cannot penetrate into the core region, making the former one acts as a blind one. However, it is information exchangeable for the latter one since the field can interact with the object. Furthermore, the presented device in this paper has extended the external cloak reported in [12] to arbitrarily polygonal shapes and removed the inhomogeneity of the device.

For brevity and clarity, we take 4-sided external cloak shown in Fig. 6(d) as an example and give out the material parameters of each regions, as shown in Table 1. It can be seen that both the complementary regions and anti-object have homogeneous and double negative anisotropic values which need double negative (DNG) meta-materials such as split ring resonators (SRRs) and wire structures to realize.

To further investigate the invisibilities of the device, the normalized far field is calculated, as shown in Fig. 7. The blue colored line in Fig. 7(a) demonstrates the normalized far field distribution of the annular dielectric object, and the red colored line indicates the normalized far field distribution when it is covered by a 4-sided polygonal cloak. The corresponding near field distribution is shown in Figs. 7(a) and 7(d) respectively. It is found that the normalized far field is decreased when the dielectric object is covered with the cloak. The maximum far field intensity of the bare dielectric object is 15 dB, but it falls to -14 dB when covered with the cloak. Similarly, blue colored lines in Figs. 7(b) and 7(c) demonstrate the far field distributions of a bare dielectric ball and two dielectric quadrangles, respectively, and the red colored lines represent the corresponding far filed when they are hidden by the cloak. It can be observed that the maximum far field intensity reaches -5 dB and 9 dB for the bare dielectric ball and dielectric quadrangles respectively. However, when covered with the cloaking devices, it decreases to -16 dB and -18 dB respectively. The obvious decreasing of the far field intensity vividly verifies the invisibilities of the external cloak.

It should be noted that though the proposed cloak can hide an object outsides from the device, it is limited to only predefining object with predefined location and position. To analyze the performance of the stealth effect with different locations of the concealed object, we calculated the normalized far field distributions of them, as shown in Fig. 8. Firstly, we fixing y = 0 and varying the x coordinate values of the dielectric ball from x=−0.0513 m to x=−0.0499. Since the anti-object of the external cloak is obtained by fixing the dielectric ball with radius as r = 0.0094 m located at x=−0.0506 m, y = 0, any disturbance of the location can affect the performance of the device, as the blue or black colored lines shown in Fig. 8(a). Secondly, we fixing x = 0.0506 m and varying the y coordinate values of the dielectric ball from y=−0.023 m to y = 0.02 m, as shown in Fig. 8(b). Similarly, the disturbance of the location greatly weaken the performance of the proposed device. Thus, the proposed device should be carefully designed to achieve a splendid stealth effect.

Although aforementioned discussions are focused on the regularly symmetric structure, it should be noted that the material parameter distribution described by Eqs. (8), (12) and (20) are effective for any arbitrarily shaped devices, including symmetric or asymmetric geometries. Two sceneries are given out here: 20-sided regular polygonal external cloak and asymmetric arbitrary polygonal shaped external cloak. The 20-sided polygonal cloak displays how to approximate a cylindrical external cloak, and asymmetric arbitrarily shaped polygon demonstrates the versatility of the proposed method in cloak design. Coordinates of the asymmetric arbitrarily shaped polygonal external cloak are chosen as ${a_1}(0.046m,0),\, {a_2}(0,0.044m),\,{a_3}( - 0.038m,0),\,{a_4}(0, - 0.044m),\,{b_2}(0,0.022m)$, ${b_3}( - 0.03m,0),\,{b_4}(0, - 0.034m),\,{c_1}(0.018m,0),\,{c_2}(0,0.01),\,{c_3}( - 0.014m,0),\,{c_4}(0, - 0.02m).$ Figure 9 illustrates the near electric field distribution of the dielectric object without or with the cloaking device. Figures 9(a) and 9(b) show the scattering pattern of two dielectric shell segments which are fitted into the region bounded between $0.055m \le r \le 0.065m$. In order to hide these objects, corresponding anti-objects with parameters of ${\varepsilon _0} = - {\varepsilon ^{i^{\prime}j^{\prime}}},{\mu _0} = {\mu ^{i^{\prime}j^{\prime}}}$ are embedded into the complementary region according to the linear transformation Eq. (3) and Eq. (9), as shown in Figs. 9(c) and 9(d), respectively.

It is observed that the scattering pattern disappears when the dielectric shell segments are covered with the proposed cloak. The absence of the scattering field validates the invisibility of the device.

Figure 10 demonstrates the normalized far field distribution of the dielectric object without and with the cloak corresponded to Fig. 9. In Fig. 10(a), the blue and red colored lines indicate the normalized far field of the dielectric object without and with the cloak, corresponded near field distributions are shown in Figs. 9(a) and 9(c), respectively. It can be clearly seen that the maximum far filed intensity reaches 8 dB for the case of bare annular dielectric segment; however, it decreases to -20 dB when covered with the cloak. Similarly, blue and red colored lines in Fig. 10(b) indicate the far field distribution of the dielectric object without and with the cloak whose near field distribution are show in Figs. 9(b) and 9(d), respectively. The maximum far field intensity of the bare dielectric quadrangle in Fig. 10(b) is 8 dB, but it is declined to -11 dB when covered with the cloak. The decrease of the far field agrees well with the near field distribution, and confirms the invisibility of the external cloak. Furthermore, the effectiveness of the invisibility confirms the versatility of the method proposed here, providing an approach for both symmetric and asymmetric structure cloak design.

## 4. Conclusion

In summary, a novel external cloak with homogeneous materials parameters is proposed and designed based on the coordinate transformation method. Full-wave simulation demonstrates the perfect invisible properties of the external cloaks. The homogeneous of the material parameters makes the cloak designed here applicable to practical fabrication. Furthermore, the invisibility of both the symmetric and asymmetric arbitrarily polygonal shaped cloak is investigated, which provide a general approach to design arbitrary shaped external cloak. It is expected that our works are helpful for fabricating the external cloak and speeding up its application in the field of aircraft stealth, radar stealth, or other military equipment stealth.

## Funding

National Natural Science Foundation of China (NSFC) (11564044, 61461052, 61863035).

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