A novel homogeneous illusion device with arbitrary polygonal cross section that acts as either invisible cloak or shifting medium has been proposed and designed based on coordinate transformation method. The material parameters of the device are derived and the effectiveness is verified by full-wave simulation. Results show that whether the illusion device acts as invisible cloak or shifting medium depends on a value of shifting distance which is about 2a (is circum-radius of the outer polygon).When the shifting distance is larger than 2a, the illusion device acts as an invisible cloak, and otherwise it acts as shifting medium. The stealth effect of all kinds of illusion devices are investigated, including identical-size mapped polygonal devices or non-identical size mapped polygonal devices. The results show that the device is a novel interactive cloak and is different from pre-proposed cloaks. The shifting properties of the device are validated by two examples, including moving the target object virtually and generating illusionary image of a line source. The material parameters of the device are homogeneous, which makes it more practicable in reality. It is hoped that our works may open an avenue for designing novel invisible cloaks, and are helpful for speeding up the potential applications of the illusion devices, such as aircraft or military equipment stealth, target objects camouflage or protection.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Based on the form invariance of the equation, coordinate transformation theory achieves the arbitrary control of field distribution by changing the equivalent physical parameter distribution of materials in the physical space, and provides a powerful and convenient way for the flexible design of metamaterial devices [1,2]. Following this theory, all kinds of metamaterial electromagnetic devices with broad application prospects and novel functions have sprung up like mushrooms. In , Chen et al. made a review on transformation optics approaching, and investigated its potential to create functional devices such as cloaks, concentrators, rotators and illusion optical devices. Mei et al.  further investigated the transformation based devices and its experiments implementing, mainly focus at radar illusion device, microwave bend structure, flattened Luneburg lens, planar focusing lens antenna, broadband carpet cloak. In , Sun et al. introduced the applications, extensions, new branches and recent developments of the transformation optics approaching and corresponding transformation based devices. Several outstanding works related to this theme have been reported in [6–8].
Recently, illusion devices based on coordinate transformation have aroused much attention among scientists and scholars. Unlike the invisible cloak, the illusive device uses a reasonable parameter designed to make the concealed object exhibit a completely different field mode from its own, which makes the detectors outside cannot obtain the real information of the object being concealed by illusionary devices. Based on the concept of complementary media, Yang et al.  proposed a superscatterer device which can enhance the electromagnetic wave scattering cross section of an object to make it look like a bigger one. Lai et al.  reported an illusion optical device that can make an arbitrary object appear like some other object. From then on, illusion devices with all kinds of functionalities have been proposed, including super absorbers , shrinking devices [12,13], illusionary source radiation [14,15], and have been extended from electromagnetic field to other multi-physical field, such as acoustics , electrostatics , and thermal dynamics . As a branch of illusion medium, shifting medium has been investigated widely recent years. Due et al.  reported a novel super-resolution imaging devices by using shifting medium. Mei et al.  proposed an illusion device exhibiting shifted or transformed image of the target simultaneously. Zang et al.  designed a rotating medium which can be extended to optical illusions to break the diffraction limit and generate overlapping illusion.
Inspired by the illusion device designing method, we report a novel homogeneous illusion device in this paper, which exhibits cloaking or shifting properties depending on the value of shifting distance. Firstly, by properly chosen the shifting distance, arbitrary polygonal cross section invisible cloak or shifting illusionary devices are obtained. The stealth effects of identical-sized mapping and non-identical-sized mapping devices are investigated and analyzed, which show that both of them have perfect invisibilities. It is worth noting that the stealth effect is different from pre-proposed cloaks, including Pendry’s cloak , external cloaks [22, 23], and reciprocal cloak [24, 25]. For one thing, though the hidden object is also concealed by the cloaking device, it can perceive information from outside. For another, the cloak designed here possesses double negative materials in the left half part, which acts like the complementary medium of the external cloak or reciprocal cloak, but the right half part is positive. It may open an avenue for designing novel interactive cloak, and easily extend to other physical fields such as acoustics and thermodynamics. Furthermore, by smoothly reducing the shifting distance, the stealth effect disappears when the boundaries in original and physical space are nearly closed to or adjacent to each other, which makes the device acted as a shifting illusion medium. Different from the previous cloak and illusion medium [19-20, 26–28], the shifting distance between invisible cloak and illusion medium lies in about.To the best of our knowledge, this is the firstly report of such interesting phenomenon. Simulation results confirm that all material parameters vary linearly with the shifting distance when d2a, and the influence of shifting distance increase is offset by material parameters. We believe such devices can effectively increase the deceptiveness of the protected targets, which has potential application prospects in military camouflage and other fields, for example, to hide aircraft or weapons, or generate illusionary information or light source, or protecting objects by shifting a distance from its real location.
2. Theoretical model
According to coordinate transformation theory, under a space transformation from the original coordinate to a new coordinate, the permittivity and the permeability in the physical space are given by
Figure 1 is a schematic diagram for the construction of the 2D N-sided polygonal cross section shifting-illusion device. The purpose of the transformation is to shift the polygon C [gray colored region in Fig. 1(a)] to a new polygon B [black dashed region in Fig. 1(a)]. The center of the polygon C and B is at origin and (-d, 0), respectively, where d is the shifting distance. The light blue colored region and gray region consist the physical space, and the entire region bordered by polygon A, polygon B and the red dashed lines compose the virtual space. The virtual space can be divided into n quadrilaterals, such as,,…,,…,.For each quadrilateral, it is divided into two triangles. For example, the quadrilateral is divided into triangle and triangle, as shown in Fig. 1(b). Mapping boundary onto while keeping the boundary fixed, makes the virtual space be transformed into physical space. The entire transformation process can be seen as a projection of transforming polygon B which is originally outside polygon A into the interior of polygon A, and it makes the medium obtained through this transformation method have the bi-functional effects of both stealth and shifting illusion.
Noted that the N-sided polygonal and C share the same center at (0, 0), and polygon B is shifted along x axis by a distance of d and then centered at (-d, 0), the general expression of the ith vertex of polygon A, B and C in Fig. 1 can be defined as:
For the outer triangles, the corresponding coordinate transformation is expressed as:
Equations (3) consists of 6 unknown parameters which need 6 equations to solve out, and the three vertexes of each sub triangle in pre transformation and post transformation can easily fulfill the requirement and help to solve the equations. Take the outer triangle in physical space as an example, it is mapped from the triangle, then by substituting the corresponding vertexes coordinate of these two triangles into Eqs. (3), we can easily obtain the following 6 equations:
Equations (4) can be rewritten as a matrix expression as follows:
Matrix operations yield a simple formulation:
The Jacobian matrix of Eq. (3) is given by:
The determinant of Jacobian matrix is. By substituting and into Eqs. (1), the constitutive parameters of the outer triangle area is obtained as follows:
For the internal triangle, the corresponding coordinate transformation can be written as follows:
By substituting the vertex coordinates of pre-transformation triangle and post-transformation triangle into Eqs. (9) and rewritten as the matrix forms, the following equations can be easily obtained:
The Jacobian matrix and its determinant of transformation obtained from Eqs. (9) are shown as follows:
Substituting Eq. (10) into (1) to obtain the constitutive parameters as follows:
In the next section, we carry out full-wave simulations using the commercial finite element solver COMSOL Multiphysics to verify the correctness and effectiveness of the derived Eqs. (8) and (12), and show the properties of the novel illusion device developed here.
3. Simulation results and discussion
In this paper, we report that the shifting distance determines the characteristics of the illusion device, that is, the device acts as a cloak when the distance is larger than (is circum-radius of the outer polygon),while it acts as a shifting or illusionary medium when the distance is less than . Specific interpretations are discussed in the following sections.
3.1 Stealth effect
First of all, we demonstrate the N-sided illusion devices acts as an invisible cloak by properly chosen the shifting distance. In the simulation, 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 to be 5GHz under TE polarization or cylindrical irradiation. The total calculation domain is chosen as , and the geometry parameters of the polygons are chosen as,,. The material parameter of the aircraft shaped object is chosen as.
Figure 2 demonstrates electric filed distribution in the vicinity of the illusion device. The TE plane wave is incident from left to right, from bottom to top, and oblique incident with an angle of for Figs. 2(a), 2(b) and 2(c), respectively. The absence of scattered waves clearly verifies the invisibility of the whole system, which is independent on the incident wave irradiation. Simultaneously, it seems that the outside electric filed can penetrate into the core region, makes the system is information exchangeable with the outside. It is worth mentioning that this property is also different from our previously reported reciprocal cloak or external cloak [23, 24]. We will discuss the details later.
Next, we will demonstrate that such device has the capability to steal target, as shown in Fig. 3. Firstly, an aircraft shaped object is surrounded by a polygon device, and is placed inside the illusion device, as shown in Fig. 3(a). The shifting distance is chosen as, which ensuing that the polygon B in original space is totally located in the calculation domain.
It is observed that the near fields of the illusion system exhibit no scattering field, which confirms the stealth effect of such illusion device. Figure 3(d) makes a contrast with Fig. 3(a), where the aircraft shaped object centered at (0, 0) is directly irradiated by the TE plane wave. The strong scattering field around the object without the illusion device further confirms the invisibility of the device proposed here. Furthermore, the stealth effect of the device is investigated under plane wave irradiation with different incident direction. Figures 3(b) and 3(c) illustrate the near electric field distribution of the illusion device under vertical incident wave and oblique incident wave with an angle of, respectively. Figures 3(e) and 3(f) demonstrate the corresponding field distribution when without the device. Both the vertical and oblique incident TE plane wave validates the effectiveness of the stealth effect, and confirms that the invisibilities are independent of the incident wave direction. It is worth mentioning that the invisibility of the proposed device is independent of the position, material or the shape of the embedded object, i.e., any object with arbitrary shape or material parameter embedded in the proposed device is invisible for the outside world. Besides, it should be noted that the stealth effect under the illumination of cylindrical incident waves is also effective. It is not shown here for brevity.
To further investigate the invisibility properties of the device, the normalized far electric field is calculated, as shown in Fig. 4. The red colored line demonstrates the normalized far field distribution of the illusion devices under the TE plane wave incident from vertical direction [as shown in Fig. 3(b)], while the blue colored line indicates the far field distribution when without the device [as shown in Fig. 3(e)]. It is observed that the scattering field of the aircraft shaped object is reduced to below −30dB with the device, and it is −15dB lower than the field without the device. The largest decline of far field occurs at and when coated with the device. The obvious declining of the normalized field further verifies the effectiveness of the illusion device.
Next, we demonstrate that the stealth effect can also be observed in other arbitrary N-sided polygonal cross section illusion devices. By fixing the shifting distance at , we can easily obtain the parameters of those illusion devices from Eqs. (8) and (12).We take 5-sided, 6-sided and 20-sided polygonal cross section illusion devices as examples in this paper, and calculate the near and far electric field, as shown in Fig. 5, and an aircraft shaped object has been put inside of the illusion devices. Figures 5 (a) to 5(c) show the electric field distribution in the vicinity of the devices under a TE plane wave incident from vertical direction. We can observe that no scattering occurred through the whole illusion system, which validate the effectiveness of the stealth. We also calculate the normalized far field of these illusion devices to verify the invisibilities of them, as shown in Fig. 5(d). We observe that the far field gain declines at least −10dB with the illusion devices, comparing with the bare object without the device. We also observe that the field gain declines more when increasing the edges of the polygon. The obviously decline of the far filed gain agrees well with the near filed, and confirms the accuracy and effectiveness of the designed illusion device.
It should be noted that the shifting distance is fixed at in the above simulations. In fact, the invisibility is still effective when the distance is larger than, as shown in Fig. 6. The shifting distance is chosen as, and for Figs. 6 (a) and 6(b) respectively. The left quadrangle in both Figs. 6(a) and 6(b) indicates the original location of the polygon B, which centered at (−0.2m, 0) and (−0.24m, 0) respectively. It can be clearly seen that the stealth effect is more obvious when the shifting distance is larger than.
It is worth mentioning that all aforementioned discussion is based on a common thing that original polygon B in original space is kept identical size to polygon C in the transformation procedure. However, the analysis model in section 2 is also effective when polygon B and C have different sizes, and the material parameters can be easily derived from Eqs. (8) and (12). Next, we give two examples to demonstrate the stealth effect of the illusion devices with different sizes of polygon B and C. The shifting distance is chosen as .The geometrical parameters of polygons are chosen as,,, and ,, for Figs. 7(a) and 7(b) respectively. An aircraft shaped object is put inside the illusion device. Both the near electric field and normalized far electric field is shown in Fig. 5 to illustrate the scattering properties of the illusion devices. The white dashed quadrilaterals in Figs. 7(a) and 7(b) indicate the original coordinates of polygon B. It is observed that no scattering occurs when the object is concealed by the illusion device, as shown in Figs. 7(a) and 7(b). From Fig. 7(c), we observe that the normalized far field of the aircraft shaped object is obviously declined when it is surrounded by the illusion device. The strength drops obviously when the size of polygon B is small, as the blue colored line illustrated in Fig. 7(c). Though the far field of the illusion device with large sized polygon B [as the blue colored line shown in Fig. 7(c)] is not obviously lower as that of small sized polygon B [as the blue colored line shown in Fig. 7(c)], it is still much lower than that of the bare object without the device [as the black colored line shown in Fig. 7(c)]. Both the absence of the scattering field in the vicinity of the device and the obviously drop of the far filed confirms the effectiveness of stealth effect of the devices.
Next, we briefly discuss the material parameters distribution. According to Eqs. (8) and (10) in section 2, we can draw a brief conclusion that the parameters are homogeneous since it only depends on several constants. In this section, we demonstrate the parameters distribution for both circumstances of identical or non-identical size of polygon B and C, as shown in Fig. 8. In the simulation, the geometric parameters are chosen as, , for the identical sized illusion device, and, , for the non-identical sized ones. The shifting distance is fixed at for both of them. It is observed that material parameters are homogeneous but anisotropic in the sub-triangles throughout the cloak structure for both identical or non-identical size of polygon B and C. However, there exists difference between these two circumstances. When it comes to identical size of polygon B and C, two sub-triangles in each quadrant have the identical constitutive parameters, which make their distribution look like continuous, as shown in Figs. 8(a) - 8(d). But the material parameters of the two triangles in each quadrant are different from each other when the polygon B and C have different sizes, as shown in Figs. 8(e) - 8(h).We also observe that the left half part of the devices have double negative parameters (i.e., both the permeability and permittivity are negative), but the right half part of the device have almost positive material parameters, i.e.,,, and.This phenomenon may be interpreted from the perspective of transformation procedure, as shown in Figs. 1(a) and 1(b). The left half part of the physical space is based on folded transformation, and usually obtains negative parameters. But the right part of the physical space is based on compressed transformation, and generally has positive parameters. Furthermore, the folded transformation is used to cancel the original space while the compressed transformation is used to restore space. It seems that the continuous boundary and symmetric structure makes the device acts as an invisible cloak when the shifting distance is large ().
As far as we know, these novel properties haven’t been reported previously, and are different from reciprocal cloak or external cloak [22, 24]. Though complementary medium (usually negative parameters) and core medium (usually positive parameters) are also used in reciprocal cloak and external cloak, they are usually composed of a continuous and closed region respectively, and rarely seen in half closed structure as presented here. It may open an avenue for designing novel interactive cloak, and may have potential applications in military stealth or disguising. The material parameters derived here are homogeneous, anisotropic and non-singularities, which makes it more practicable to realize. The double negative materials may be implemented by using metamaterial structures such as split ring resonators (SRR) and wire arrays [29–31], and positive materials may be realized by layered effective mediums [32–34].
3.2 Illusion effect
From above analysis, the stealth effect of the illusion device is effective when the shifting distance is chosen as 0.16m. Does the stealth property is still effective when the shifting distance is changed?
In this section, we report that such stealth effect is disappeared and act as a shifting illusion device when the shifting distance is shortened. It is worth mentioning that the material parameter equations of the illusion medium have similar forms as Eqs. (8) and (12), but the values are depend on the shifting distance d. Furthermore, the material parameters of the illusion device proposed here are homogeneous and negative, which need well designed metamaterials to realize. Two examples show the potential applications of predesigned illusion devices.
Firstly, we give out an example to utilize the shifting properties of such an illusion device to camouflage target objects. For giving a clear and convincing illustration for the performance of illusion device, an aircraft shaped object with parameters of is considered in simulation, as shown in Fig. 9. The geometric parameter of the device are chosen as ,,.Figs. 9(a) and 9(c) show the electric field distributions in the vicinity of aircraft shaped object with and without the illusion device when the shifting distance is chosen as . It can be observed that outside the external boundary of the illusion device, the field pattern in Fig. 9(c) is similar to that in Fig. 9(a). That is to say, the illusion device makes the electromagnetic image of the aircraft shaped object centered at (−0.1m, 0) appears exactly the same as that of the one centered at (0, 0) with the predesigned material parameters. Figures 9(b) and 9(d) are similar to Figs. 9(a) and 9(c), but have another shifting distance of. It is much obvious that the scattering pattern in Fig. 9(d) is exactly identical with that in Fig. 9(b), which confirms that the electromagnetic image of the aircraft shaped object centered at (−0.06m, 0) appear exactly the same as that of the another object centered at (0,0). Both the examples shown in Fig. 9 validate the shifting properties of the illusion device. Such examples indicate that the illusion device may have potential applications in military field to camouflage target objects, such as camouflage aircrafts or weapons.
In another example, we demonstrate how to make an optical source shifted visually to make confusion, as shown in Fig. 10. A line source with a current intensity of 10−2A/m is located at the center of the illusion device, as shown in Figs. 10(a) and 10(b).The material parameters of the illusion device are predesigned according to Eqs. (8) and (12) with shifting distance of 0.06m and 0.1m, respectively. So it hallucinates people that the line source is located at the other places, such as (−0.06m, 0) and (−0.1m, 0) as shown in Figs. 10(c) and 10(d). This property may have potential applications in radar or light source protection.
As the above analysis, when the shifting distance is larger than, it acts as a cloak, and otherwise it acts as shifting medium. This phenomenon is very interesting. In order to exploring the mechanism behind this phenomenon, we plot the relationship between the material parameter and shifting distance, as shown in Fig. 11. From Figs. 11(a)-11(h), it is found that both the inner triangle and outer triangle have identical material parameter for corresponding areas. Singularities appear at ( is the circum-radius of inner polygon C), as shown in Fig. 11(b), 11(c), 11(f) and 11(g). It is caused by the singular matrix A and B in Eqs. (6) and(10) when or.When the shifting distance increases to about, the linear relationship between material parameters and variable distance d appears, and the influence of shifting distance increase is offset by the material parameters. Experimental verification will be carried out latter.
In summary, a homogeneous novel illusion device with arbitrary polygonal cross section that acts as either invisible cloak or shifting medium has been proposed and designed based on coordinate transformation method. The stealth or illusion effect of the device depends on the shifting distance, i.e. when the shifting distance is larger than, the device acts as an invisible cloak, and otherwise it acts as shifting medium. Stealth effects are observed from both circumstances of original polygon B and transformed polygon C have identical size or non-identical size, and are independent of the incident wave orientation. Different from aforementioned cloaks, the illusionary cloak is a reciprocal device, which possesses half of positive and negative materials. We believe it may open an avenue for designing novel interactive cloak, and may have potential applications in military stealth or disguising. The shifting phenomenon occur when the shifting distance is smaller than, and it may have potential applications in camouflage target objects, or generate illusionary image to protect target objects. Experiment results will be shown in our later work.
National Natural Science Foundation of China (Grant Nos. 61461052, 11564044, 61863035).
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