Design of open devices based on multi-folded transformation optics

Open devices with homogeneous material parameters are proposed and designed based on multi-folded transformation optics, including open cloak device, open field concentrator and open field amplifying device. In comparison with the previous transformation devices, the proposed open devices possess open windows with compact and embedded structures, providing a flexible approach for remote control or upgrade. The open cloaking devices can hide arbitrarily shaped/sized object in the core region, making it disappeared in visually for the outside viewers, while the open field concentrator can enhance or store EM energy in the core region, and the open field amplifying device can magnify the scattering field of a small object, generating an bigger illusory image with differential material parameter and size. The effectiveness and correctness of the proposed devices are validated by the numerical results obtained based on the commercial finite element software COMSOL Multiphysics. Such scheme is believed to find potential applications in remote controlling with impressive new functions.


Introduction
AS an ingenious mathematical approach, transformation optics (TO) [1][2][3][4][5][6] provides a powerful and convenient way to control the electromagnetic (EM) fields arbitrarily, arousing much attentions in the past two decades. Based on this methodology, novel devices with varied functions and applications have been proposed, designed and experimented with an unprecedented prosperity, including invisible cloaks [7][8][9][10][11][12][13], concentrators [14][15][16][17][18], illusory devices [19][20][21][22][23][24][25][26][27][28] and antennas [29][30][31][32][33][34][35] etc. The most striking device among them is invisible cloak, where some properly designed materials are utilized to direct the EM waves propagates smoothly around an arbitrarily shaped object, making the entire device (including the coated object) become invisible for the outside viewers. However, since the EM waves are guided by the coated materials, no wave penetrate into the hidden region, making it impossible for the coated object to interact with the outside world. Furthermore, the coated object cannot move freely in such an enclosed device, making it difficult to meet the needs of replacement or upgrade. In order to overcome these drawbacks, an external cloak that hide an object at a certain distance was proposed by Lai et al [12], where complementary medium and 'anit-object' was used to cancel the scattering field generated by the pre-defined hidden object. Additionally, by adding another specific anti-object into the complementary medium, the external cloak acts as an illusion device [19] that makes one object looks like another. However, such an extraordinary device lacks of flexibility and it is merely invisible for an object with specific shape, size and location, i.e., any variation of these factors may greatly deteriorate the stealth effect of the device. It is worth seeking a way to hide an object with arbitrarily shapes or sizes as well as material exchange with the outside world.
As another attractive TO based device, field concentrator can increase and store the EM energies in the core region of the device, and it may find potential applications in solar cells or EM sensors. Since the external EM fields around the concentrator is undisturbed, the entire device acts as an ideal invisibility cloak for an outside viewer. Meanwhile, the concentrator also acts as an illusion device that can render a small object located at the core material to look like another bigger object [14,16]. Due to this magnifying effect, even minimal divergence or gaps in the core region will result in a deterioration of the desired performance. Furthermore, it is hard to check or predict the fabricating quality of the core materials when it was fully coated by a properly designed material, let alone replacing or upgrading the internal core material. Therefore, it makes sense to design a concentrator or amplifying device with an open structure, meeting remotely controllable, reciprocal and upgradeable demands of the modern society.
In this paper, based on the multi-folded transformation optics, novel open devices with homogeneous parameters are proposed and designed. The open property allows the hidden object coated by the proposed devices to interact with the outside world and provides a way of remote control or upgade. This open property is generated by compactly embedded segments instead of isolated components, differentiating our study from previous work and providing a robust application in a motion circumstance. Three examples are provided to validate the effectiveness of the proposed open devices. First, an open cloak device is introduced to hide an arbitrarily shaped object which can keep stable or move freely in the hidden region. The results show that the invisibility of the proposed open cloak keeps well, and it can hide objects with arbitrary shapes or positions. A more significant feature is that the structure of the open cloak is compact and integral, which means that it is more stable and robust, and more suitable to be used in a motion circumstance. Second, an open field concentrator that has most area of the core region opened to the outside world is presented. The EM waves are increased perfectly in the core region, indicating that the concentrator has a field enhancement effect. The simulation results show that this enhancement effect is independent of the incident wave direction or the stimulus source. Finally, by embedding a dielectric object into the core region of the proposed open filed concentrator, an open amplifying device is proposed and designed, from which a small dielectric object located at the core region is amplified to a larger image object visually with a different material property in the background medium (air). Full-wave finite element simulations validate the expected behaviors of our proposed devices. It is believed that the proposed opening devices have prospectively potential applications in antennas and propagation fields, including antenna/radar stealth, target camouflage or illusion, novel electromagnetic field sensor or device designing and fabrication etc. Furthermore, it provides a flexible way for remotely controlling the electromagnetic wave.

Open device design
According to TO theory, the permittivity and permeability tensors in the virtual space and the physical space are governed by: where L is the Jacobian transformation matrix between the local distorted coordinates in the virtual space and the Cartesian coordinates in the physical space, L det is the determinant of L. Figure 1 demonstrates the schematic diagram of our proposed open devices. All of these devices are obtained by two transformation steps. Firstly, a stretching or compressing transformation method is utilized to transform a virtual space into a conventional device with an enclosed structure, including invisible cloak, concentrator, and amplifying device etc. Subsequently, by employing the multi-folded transformation optics, these conventional devices are further folded and compressed into open devices with a compact and integral structure while maintaining identical performance.
To start with the open cloak device illustrated in figure 1(a), a tiny square PEC (should be small enough for invisibility) is located at the center of the virtual space. The virtual space is divided into eight triangular regions labeled as 1, 2, 3, ,8,  as the top column shown in figure 1(a). In the step 1, these triangular regions are transformed into eight different regions labeled as ¢ ¢ ¢ ¢ 1 , 2 , 3 , , 8  respectively, and the original tiny square PEC is mapped into a bigger one, as the middle column shown in figure 1(a). Therefore, a conventional homogeneous cloak is obtained from the step 1.
In the step 2, we choose two polygons B E A G B (green, bottom panel of figure 1(a)) in the physical space respectively. As a result, the conventional cloak is transformed into an invisible device which has an open window that allows for material and information interaction with the outside world. We named this invisible device as an 'open-cloak' device which can hide an arbitrarily shaped object in the area that is bordered by the inner boundaries of the device. Similarly, two steps are needed to design an open field concentrator depicted in figure 1(b). In the first step, a square ring in the virtual space is divided into eight triangular regions labeled as 1, 2, 3, ,8,  as shown in the top panel of figure 1(b). These regions are transformed into regions labeled as ¢ ¢ ¢ ¢ 1 , 2 , 3 , , 8  respectively, as shown in the middle panel of figure 1(b). Furthermore, the center region bordered by A B C D 1   In this letter, linear coordinate transformation is employed to achieve transformation mediums with homogeneous material property. Both the virtual space and the transformation space are divided into several triangle regions. The transformation equation between triangles in the physical space and its image in the virtual space is defined as: where e and f are the non-homogeneous term of linear equation (2), and L is the Jacobian matrix governed by the following formula: )indicate the before and after transformation coordinates respectively, and 1, 2, 3 represent the vertex order of the triangles. Since the constitutive parameter of the transformation medium is determined by the Jacobian matrix L only, it is not required to calculate the non-homogeneous term of linear equation (2).
Thus, the required material parameters of each triangles of the proposed open devices are obtained by substituting corresponding vertex coordinates into equations (3) and (1).
For an open-cloak, the coordinates of the vertexes are A 0, 0.04 ,     figure 1(a)), the material parameters of regions ¢ 3 and ¢ 4 must be multiplied, because these compressed regions are obtained from the conventional cloak.
Similarly, for an open concentrator, the coordinates of the vertexes are A 0, 0.04 ,

Numerical simulations and discussion
The commercial finite element software (COMSOL) is adopted to validate the effectiveness and accurateness of the proposed open devices. Simulations are carried out under a transverse electric (TE) plane wave or a cylindrical wave irradiation with a frequency of 10 GHz in this letter. Let's start with the validation of the proposed open cloak. In figure 2, a unit plane wave normally incident from left to right to investigate the electric field (E z ) distribution around a PEC object or a cloak device. From figures 2(a) and (b), it is observed that the EM waves are smoothly guided by the properly designed materials and the wave fronts are restored well for both the conventional cloak ( figure 2(a)) and the proposed open cloak ( figure 2(b)). However, when without the cloak device, the scattering field of the PEC object is strong, as shown in figure 2 as shown in figure 3(a). In this case, scattering field is observed, especially in the forward direction. In figure 3(b), the same oval shaped dielectric object is coated by the proposed open-cloak device and subjected to the same TE plane wave. Obviously, the scattering field generated by the object is greatly reduced when the object is covered by the cloak device. Thus, the oval-shaped object become invisible for the outside world. Furthermore, no electric field penetrates into the core region in this case since the inner boundaries colored by the red lines in figure 1(a) is set as PEC. In the second case, a cup-shaped object with e m = = 3.9, 1 o o is instead of the oval-shaped dielectric object, as shown in figures 3(d) and (e). Similarly, scattering field is observed when the cup-shaped object is directly exposed to the free space (air), as demonstrated in figure 3(d). However, the scattering field is minimized when the object is wrapped by the open-cloak device, and the object become invisible, as illustrated in figure 3(e). All these two cases mentioned above discussed the capability of the open cloak to hide arbitrary dielectric object, regardless of its shape, size or location.
In the third case, we replace the dielectric object with a magnetic medium. Figures 3(g) and ( 3(h)). Furthermore, far field differential RCS is calculated for all cases mentioned above, as shown in figures 3(c), (f) and (i), where blue-colored lines indicate the RCS of the bare objects exposed in the air, while the red-colored lines represent RCS with the open-cloaking devices. It is clear that the scattering field is well suppressed when the object is covered by the proposed open device. All discussions above validate the capability of the proposed open cloak device to hide any object regardless of its shape, size or location. In a short word, the invisibility is independent of objects, shapes and positions.
Comparing to a traditional cloak device [1,3,[7][8][9][10], the proposed open cloak provides the capability of material or information convertible with the outer world as well as an external cloak [12]. However, it should be noted that the invisibility of an external cloak is greatly depended on a pre-defined position, shape and size of a hidden object while the proposed device is independent of them, which provides a feasible approach to hide a moving object. Although the concept of open cloak has been proposed by Han et al [36][37][38] in 2010, the inhomogeneity and anisotropy of material parameters was a big challenge for fabrication. In contrast, the homogeneous open cloak proposed here will relax the implementation difficulty of the device. Furthermore, different from the remote device proposed by Zheng et al [39], the open cloak device developed here is composed of a compact, embedded and continuously structure which have more robustness to avoid the field perturbation caused by position offset or impedance mismatch that may exists in an isolated structure devices, even in a motion circumstance.
Next, we focus on the validation of the EM field concentration of the proposed open concentrator. Figure 4 illustrates the distribution of the electric field and the total energy density of the developed novel concentrator. In figures 4(a)-(c), a TE plane wave is irradiating on the concentrator along the x-direction, the y-direction and with an oblique incidence angle of p 4, / respectively. It is observed that the EM waves are perfectly focused into the core region of the open concentrator, validating that the field concentrator can be made open (at least partially open) to the outside world while keep the overall performance unchanged. Furthermore, the simulation results indicate that the field concentration is independent of the incidence direction of the electric filed and the device is prominently invisible to the outer world. In figure 4(d), a line source with unit power located at (−0.06 m, −0.06 m) is irradiating on the proposed open concentrator. It is observed that the field concentration effect is also effective under the irradiation of a line source, confirming that this concentration feature is independent of the stimulus source.  1 is directly exposed to the irradiation of a TE plane wave. Comparing figure 5(a) with figure 5(b), it is observed that the electric field distribution of them are almost identical, confirming the effectiveness of the proposed amplifying device to amplify an object that is located at the core region of the device. In  Finally, a brief discussion on future experiment of such open devices are taken. There is no denying the fact that the highly anisotropic material combined with negative or near-zero values is still a big challenge to fabricate in transient mode. One possible approach is use metamaterial structure with split ring resonators (SRRs) and metal rods [7,[39][40][41]. Generally, these highly off-diagonal anisotropy values in the x-y plane should be transformed to a diagonal tensor for both permittivity and permeability in the u-v plane (a / where e , xx e , xy e yy andm zz are the components of the off-diagonal parameter tensor in x-y plane, e , u e v and m w is the components of the diagonal parameter tensor in the u-v plane. The required mediums can be obtained by carefully tune the geometrical sizes of the metamaterial units and the permittivity of the substrate, which can specifically refer to [39]. However, most of the resonant based implementation have narrow bandwidth and lossy, which sacrifices the performance of the device. Another potential approach is use periodical L-C transmission line network, where relevant capacitors and inductors are used to equivalently realize the required permittivity and permeability mediums [42,43]. Still, L-C network only valid in the low frequency band (usually below the C-band) where the unit size is far less than working wavelength (l 20 / ) and the parasitic effects of lumped components are negligible. Furthermore, the using of graphene to dynamically tune the permittivity by means of chemical doping or gate voltage [44] is a prospective approach to achieve TO-based device, but it is still challenged to tune an anisotropic medium. Fortunately, for a direct current (DC) mode, a remote function cloak was successfully experimented by Chen et. al in [45], where negative resistor network with active elements is used to implement. Therefore, the device proposal here may be find experimental verification soon at DC frequency in future, which may find potential applications in medical and geologic research.  the open devices proposed here have a much compact, stable and robust structure than that of the previously remote devices. The open cloak has the capability to hide objects with arbitrary shapes, sizes and positions. The open concentrator enhances the EM energy in a core medium which is opened to the outer world. The open amplifying device magnifies the scattering field of an object that is embedded in the core material and renders the object to look like another object with a larger size and other parameters. The simulation results validate the effectiveness of the proposed devices. Although it still a challenge to realize such open devices at high EM band, we believed that the scheme presented here could be extended to other EM devices design and promoted the potential applications in remote control of microwave or optical engineering.