Design of a Minimized Complementary Illusion Cloak with Arbitrary Position

This paper has developed a complementary illusion cloak to hide the objects with arbitrary shapes and sizes. In the proposed cloak, a complementary cloak is first introduced by using the trapezoid coordinate transformation. Then an illusion device in terms of another trapezoid coordinate transformation is designed to cloak the object. With two coordinate transformations, the complementary illusion cloak device can be arbitrarily placed to hide the target object, and the sizes of the antiobject used to cloak the object are obviously compressed. Numerical results including target objects with different shapes are given to verify correctness and effectiveness of the proposed cloak.


Introduction
In recent years, transformation optics-based invisibility cloaks have attracted considerable interests.Making use of form-invariant coordinate transformation of Maxwell's equations, Pendry et al. [1] and Leonhardt [2] independently proposed the invisibility cloak and the concept of transformation media for electromagnetic (EM) waves.Lately, some key issues related to the coordinate transformation theory including ray tracing [3], numerical simulation [4], singularity treatment [5], and embedded coordinate transformations [6] have been addressed, and various invisibility cloaks have been proposed, for instance, elastic cloaking devices [7], conformally invisible cloaks [8], three-dimensional cloaks [9], and nonmagnetic optical cloak [10].Following this success, the coordinate transformation theory has resulted in the development of interesting devices including superscatterer [11], concentrators [12], and transparent metamaterial slabs [13].Besides, practical implementations of the invisibility cloaks have been discussed [14,15].However, one of disadvantages of the conventional invisibility cloaks is that the object hidden inside the cloaked domain is "blind," since no exterior electromagnetic waves can reach into the cloaked space.The concept of the complementary media [16,17] provides us with a new idea to construct an invisibility cloak which can hide the objects external to the cloak itself.The complementary media, most notably in the "perfect lens" [18], can optically "cancel" a certain volume of space at a certain frequency.By locating a complementary "image" of the object embedded in a negative index shell, which is called the "antiobject," an object as well as the surrounding space is optically canceled.Further, the correct optical path in the canceled space is restored by a dielectric core material, as shown in Figure 1.We call this kind of cloaks "open cloaks" and then the conventional cloaks are defined as "closed cloaks."Based on the complementary media, various open cloaks with different shapes [19], illusion cloaks [20,21], and anticloaks with arbitrary shapes [22] have been developed.However, because of the implementation of the optical transformation along one direction, the required antiobject in these open cloaks can only be compressed along the corresponding direction, while the dimensions of the antiobject along other directions keep unchanged.Hence, the miniaturization of the open cloak cannot be achieved.More importantly, the relative position between the transformation region of the open cloak and the target object is fixed when the optical transformation along one direction is employed.In this scenario, the position of the cloak device cannot be arbitrarily changed.
In this paper, we propose a trapezoid complementary illusion cloak to hide the objects with arbitrary shapes and  sizes.In the proposed illusion cloak, a coordinate transformation along one direction is first conducted to design a trapezoid complementary cloak.With the complementary cloak, a coordinate transformation along another direction is further implemented to generate a new illusion cloak device used to cloak the target objects.Considering the coordinate transformations along two different directions, the sizes of the required antiobjects are greatly reduced for a target object with arbitrary size and shape.Moreover for a given target object, we can arbitrarily design the location of the illusion complementary cloak to hide the target object by using two coordinate transformations.

Design of a Complementary Illusion Cloak
According to the coordinate transformation theory [1,2], when a space is transformed into another space of different shape and size, the permittivity tensor   and permeability tensor   in the transformed space or physical space   can be expressed according to the permittivity tensor  and permeability tensor  in the original space  as   = ΛΛ  /det(Λ) and   = ΛΛ  /det(Λ), where Λ is the Jacobian transformation tensor with components Λ  = / ( =   ,   ,   ;  = , , ) and det(Λ) is determinant of Λ.
Consider a two-dimensional complementary cloak, as shown in Figure 2(a), where region I acts as the restoring medium and regions II and III are used as a pair of complementary media.In the complementary cloak, the transformations from region III to region II and from the whole region including regions I, II, and III to region I are implemented.In order to easily carry out the corresponding transformations, we divide each region into three smaller regions, for example, the upper region, the middle region, and the lower region.Specifically, three subregions in each of three regions can be, respectively, marked as I upper , I middle , I lower , II upper , II middle , II lower , III upper , III middle , and III lower .Hence, the coordinate transformations between III and II are divided into three transformations including two triangle region transformations and one rectangle region transformation.The coordinate transformations between the whole region and region I can be similarly handled.It can be seen from Figure 2(a) that transformations from region III to region II and from the whole region to region I are the linear transformations along -axis.
(1) Specifically, as to the transformation from region III middle to region II middle , the points , , and  are mapped to points , , and , respectively, and thus we get  = − 3 / 1 ,  = 1, and  =  =  =  = 0.With the similar mapping, we can get the transformations from region III upper to region II upper and from region III lower to region II lower with  = − 3 / 1 ,  = 1, and other parameters of zero.If region III is air, the permittivity tensor and the permeability tensor in region II are diagonal matrices and can be expressed as follows: With the similar transformation process between the whole region and region I, the resultant material parameters of regions I upper , I middle , and I lower can be where From ( 2) and (3), we can clearly see that the cloak is made of a homogenous material, and the material parameters in entire complementary region and rectangular core region are just diagonally homogeneous, whereas only triangular core regions have anisotropic tensor properties.When there has an object with the permittivity of  and permeability of  in region III, it can be cloaked by placing an antiobject in region II, of which material parameters can be determined by   = ΛΛ  /det(Λ) and   = ΛΛ  /det(Λ).In addition, the transformation from the whole region to region I makes region I optically equivalent to the whole region.In this scenario, the electromagnetic response produced by an object in region I is the same as that caused by another object with different size and material parameters in the whole region, which is called the illusion.
It can be seen from the above procedure that in the design of the complementary cloak the coordinate transformation is only implemented along one direction, for example,  direction, and therefore the resultant whole transformation region and the required antiobject can be only compressed along  direction, which hinders the miniaturization of the whole device.Moreover, the use of only one coordinate transformation imposes restrictions on the relative position between the cloak and the target object.In order to solve these problems, two coordinate transformations along two different directions are proposed.Specifically, following above steps a coordinate transformation along a direction, for instance,  direction, is first implemented to achieve a trapezoid complementary cloak, in which region VI acts as the restoring medium, and regions V and IV are chosen as a pair of complementary media, as shown in Figure 2(b).And then a cloak device used to hide the object is illusively generated to become a new complementary cloak based on the coordinate transformation along another direction.In order to generate the illusion device used to hide the object, the illusion object related to the illusion device is required to locate in region VI.The shapes, locations, and material parameters of the illusion object can be exactly determined according to the coordinate transformation from the whole region including regions IV, V, and VI to region VI, which is similar to the coordinate transformation given by (1).Hence, an object in region III is cloaked by placing an International Journal of Antennas and Propagation  antiobject in region VI with two coordinate transformations along two different directions, as shown in Figure 3.With two coordinate transformations, the sizes of the antiobject are greatly reduced.Also we can see that the cloak used to hide target object is the complementary cloak generated by the illusion.Hence, unlike the conventional complementary cloak, the relative position between the whole device and the target object can be arbitrary in the proposed complementary illusion cloak.

Simulations and Discussions
A finite element solver (COMSOL Multiphysics) is used to verify and demonstrate the performance of the proposed complementary illusion cloak.Without loss of generality, TE  wave (the electric field perpendicular to the  direction) is considered so that the material parameters including    ,    ,    , and    are solved in the simulations.As the first example, consider a plane wave with the operating frequency of 900 MHz incident on a nonmagnetic triangle object outside a trapezoid complementary cloak, as shown in Figure 4.The background material is assumed to be air and the relative permittivity of a triangle object is 2. Geometry parameters corresponding to the cloak in Figure 2(a) are chosen as  1 = 0.4 m,  2 = 0.4 m,  3 = 0.2 m,  1 = 0.1 m, and  2 = 0.2 m, respectively, and thus the electric size of the cloak is 2.4 × 1.2 ( is the free-space wavelength).With the complementary cloak, we clearly see from Figure 4 that the triangle object is hidden, and meanwhile it can receive the incident electromagnetic wave so that the communication between the object and the outer space can be achieved.The size of the required antiobject is compressed along  direction due to the use of the coordinate transformation along  direction, while its size along  direction keeps unchanged.Hence the resultant cloak is in a large size.Moreover, it can be seen from Figure 4 that the complementary cloak can be only placed along axis to hide the target object due to the use of the coordinate transformation along -axis.
In order to simultaneously reduce the dimensions of the antiobject along  and  directions, a complementary illusion cloak is designed.Parameters associated with the cloak in Figure 2  can be obtained by using the complementary illusion cloak.Moreover, it can clearly be seen from Figures 5(b) and 5(c) that the sizes of the required antiobject are greatly reduced along both  and  directions.Moreover, with the use of two coordinate transformations, the complementary illusion cloak can be placed along -axis compared to the case of the above complementary cloak.In addition, Figure 6 shows the electric field distributions when the plane waves with the initial phases of 0 ∘ , 30 ∘ , 60 ∘ , and 90 ∘ are incident on the trapezoid complementary illusion cloak.According to Figure 6, good invisibility can be observed.
In the following, we consider a complementary illusion cloak with the following geometric parameters:  1 = 1 cm,  2 = 0.75 cm,  3 = 0.5 cm,  4 = 0.75 cm,  5 = 0.5 cm,  6 = 1 cm,  7 = 1.25 cm,  1 = 0.25 cm,  2 = 0.5 cm,  3 = 0.75 cm,  4 = 0.75 cm,  5 = 1 cm, and  6 = 1.25 cm. Figure 7 shows invisibility of the cubic nonmagnetic object with relative permittivity of 4, when the plane waves with the frequencies of 2 GHz, 2.5 GHz, and 3 GHz are incident from the left to the right on the complementary illusion cloak.In this example, the electric size of the cloak is 0.4 × 1.07.Figure 8 shows the good performance of the proposed cloak including small size, invisibility, and arbitrary position.

Conclusion
In this paper, a trapezoid complementary illusion invisibility cloak is designed according to the complementary cloak and illusion theory.A trapezoid complementary cloak is first achieved by a coordinate transformation along one direction.With the illusion theory, a new illusion cloak device

2 InternationalFigure 1 :
Figure 1: The external cloak based on the complementary media.

Figure 2 :
Figure 2: The transformation device: (a) the transformation along -axis and (b) the transformation along -axis.

Figure 3 :Figure 4 :
Figure 3: Schematic model to cloak a target object.

Figure 5 :
Figure 5: Electric field distributions for the trapezoid complementary illusion cloak: (a) without target object; (b) with target object for a normally incident wave; (c) with target object for an obliquely incident wave.
(b) are chosen as  4 = 0.2 m,  5 = 0.1 m,  6 = 0.5 m,  7 = 0.6 m,  3 = 0.3 m,  4 = 0.3 m,  5 = 0.4 m, and  6 = 0.5 m, respectively.A plane wave with the frequency of 300 MHz is incident along -axis, and therefore the electric size of the cloak is 0.8×0.8.With proposed complementary illusion cloak, the trapezoid complementary cloak in first example can be illusively generated.Figure 5(a) demonstrates the electric field distribution without the nonmagnetic target object.Figures 5(b) and 5(c) show the electric field distributions with the nonmagnetic target object for the normally and obliquely incident waves, respectively.Here the relative permittivity of the target object is 2. A good invisibility

Figure 7 :
Figure 7: Electric field distribution of the trapezoid complementary illusion cloak for the incident plane wave with different frequencies: (a) 2 GHz; (b) 2.5 GHz; (c) 3 GHz.

Figure 8 :
Figure 8: Electric field distribution of an object with an arbitrary boundary.