A novel EM concentrator with open-concentrator region based on multi-folded transformation optics

Conventional concentrators with inhomogeneous coating materials that fully enclose the destined region pose great challenges for fabrication. In this paper, we propose to design an EM concentrator with homogeneous materials. Distinguished from conventional ones, the elaborately designed EM concentrator features a concentrator region that is open to the outer-world, which is achieved with multi-folded transformation optics method by compressing and folding the coating materials to create window(s). Based on this concept, we also investigate open-rotator and open rotational-concentrator devices, which could simultaneously rotate and store the EM waves in the central destined region. Due to the open nature of our proposed designs, we believe they will find potential applications in remote controlling with impressive new functionalities.

Start from concentrator 21 , Fabry Perot resonances were used to design and fabricate the concentrators 35 that achieved progressive attraction in novel energy devices. In addition, some unidirectional and angle-dependent based concentrators 29 have been investigated that work efficiently only in some specific incident wave-angles. Beyond this, rotators 20 are designed for the polarization and have wide applications in the antennas and wave-guiding that provides a reason of linearly polarized into a circularly polarized nature 18 . By taking the combine effects of concentrators and rotators, this phenomenon can be made in practice for Smart Grid applications 36 , large area wide band imaging 37 , commonly in laser LED-pumped devices 38 and also in micro solar cells arrays 39 . Specifically, parabolic concentrators provides practical impacts in radio broadcasting & motion picture recording 40 , while polarization based rotators manipulates effects in some add-drop filter systems 41 . Meanwhile, the TO-based EM concentrators and rotators yield straight path in the manufacturing and applications of metamaterials 18 .
Despite all these facts EM concentrators and rotators possess few discrepancies by representing some known reasons. As metamaterials provides very low practical outputs due to their low characteristics and narrow bandwidths 5,8,18 . Previous work concluded that in the designing of the TO based devices, the material parameters were set in inhomogeneous coordinate transformation which is hard to predict the realization phenomenon and difficult for fabrication 42,43 . Beside this, early EM concentrators and rotators show complete invisibility of the destined (concentration and/or rotator) region because of the designing method of these conventional devices that involves in the construction of enclosures the destined region by coating with properly designed materials [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] .

Results and Discussions
Conventional concentrators 13,21 are based on the radial mapping that store the EM energy in the smaller concentrator region. Meanwhile the concentrator device itself is also the example of invisibility as there is no scattering for the outside observers. Therefore, the expanding technique is applied in ref. 21 and the folding method is used in ref. 13 to design complementary media based concentrator. Thus, in previous works, the properly designed coating material is used to fully enclose the destined region and that coating material is composed by anisotropic and inhomogeneous materials that are difficult for practical implementations.
Distinguishing from the aforesaid methods, in our approach, we divide the virtual space of N segments into N 2 triangles and then apply two steps to obtain the desired goal. Figure 1 depicts the schematic diagram of the proposed open-coating concentrator device in which a square shaped virtual space of total four segments are further divided into eight different triangular regions labeled as ... 1, 2, 3 8. In the first step, the region 1 of ∆ABC and the region 2 of ∆CDB in the virtual space x y z ( , , ) is folded into region ∆ ′ ′ A B C labeled as ′ 1 and ∆ ′ CDB labeled as ′ 2 in the physical space ′ ′ ′ x y z ( , , ), respectively. Similarly, the other regions in virtual space labeled by ... 2, 3 8 are accordingly folded into colored regions 2′, 3′… 8′ to design the concentrator. It can be seen in Fig. 1, that the air-gap came to exist between the concentrator-region and concentration-coating. Thus, by taking advantage of this facility, in the second step, a compression and folding method is applied by compressing the bigger dashed lines into the compressed region I. For example, here, we applied compression technique to regions ′ 7 , ′ 8 and the whole segment is compressed into regions as ″ 7 and ″ 8 respectively, so that the coating materials turn into open that do not require to enclose the concentration-region.
In the second step of Fig. 1, consequently, we compressed the whole black dashed lined space into the region I (represented by x y z ( , , ) 1 1 1 ) and region II (denoted by x y z ( , , ) 2 ). Whereas, the material parameters of region II are obtained from the first step of Fig. 1. Thereafter, to overcome the discontinuity occur due to the compression, the compressed region is further coated with the folded regions such as region III (denoted by x y z ( , 3 ), region IV (denoted by x y z ( , 4 ), and region V (denoted by x y z ( , , ) 5 5 5 ). For detail, taking the second quadrant as an example of region III, the ∆ ′ PQA is folded into ∆PQR which can be seen in the purple color of zoom-in view. Similarly, region IV and region V in the first quadrant are obtained by folding the space of ′ QtA C into the light green colored region of QtRS and ∆PtC 1 into the golden colored region of ∆PtS 1 . Due to the symmetric structure, other parts of these complementary regions in third and fourth quadrants can be achieved by rotating all of the corresponding tensors by π. The materials used in folded regions are known as complementary materials that are also used in lens designs to compensate the space 44 . x n , y n , and z n indicate the coordinate system of each region, where = n 1, 2, 3, 4 and 5. In the following, full wave simulations of finite element method (COMSOL) are performed in 2D by adopting the transverse electric (TE) mode at the frequency of GHz 3 . The geometric structure parameter for the both steps in SI units are as follows: the total width of the squares = .
In Fig. 2, we consider to examine the working performance of proposed open-coating concentrator with the proposed close-coating concentrator under the illumination of plane waves with the two different incident angles of 0° and 90°, respectively. The electric field (E z ) distributions in all cases ( Fig. 2(a-c)) indicate that the electric field's amplitude in the concentrator region is equivalent to that in free space near the outer boundary of concentrator. In Fig. 2a, the working functionality of the designed closed-coating concentrator is similar to that in all the previous reported concentrators 21 . The beauty of proposed concentrator is that the − z direction space is unchanged and concentrator region compresses the space in − x y plane only. Therefore, the total energy flow is towards the concentration region and that region is fully covered with the properly designed coated materials, as shown in Fig. 2a. Another case is observed in Fig. 2b, to design a concentrator with open-coating materials and by comparing the field patterns of Fig. 2a and Fig. 2b, it is evident that the concentration region can be made open for the outside observers while the overall performance remains unchanged. Furthermore, to prove the proposed device have no limitations of incident waves' direction, the incident waves have applied at 90° angle, as seen in ( 0 4, 0) with the electric field intensity of V 1 . A tradeoff is that the material parameters of the above example involves negative values of anisotropic and homogeneous materials that have two windows in-front of the concentrator region and these windows can be increased as per demand for remote controlling phenomenon. Since the designed concentrator can only store EM energy that raise a question that whether we are able to rotate and store EM energy simultaneously, for example, make a device that can be used for solar cells and for antenna applications as well. As an answer, we further extend the concept of open-coating concentrator into the open-coating rotator first and then combine these two devices to obtain the combinational effect.
The schematic diagram of the open-coating rotator is given in Fig. 3 with two different steps. At the first step, the ∆ABC of region 1   In easy words, the ∆ABC is folded into ∆ ′ ′ A B C and so on. Meanwhile, the bigger square of radius c is compressed into a smaller square region of radius a. It should be clearly noticed that the air gap exists between the coated material and concentrator region. In the second step, a multi-folded transformation method is applied to create a window. As an example, after applying the second transformation function, the regions ′ 7 and ′ 8 turns into ″ 7 and ″ 8 , respectively, that is compressed and shifted towards the compressed region and that region is further coated with the folding regions. is compressed into region I of radius a and that compressed region also contain some mapped regions. As a reference, we compressed the segments of ′ 6 and ′ 7 into the compressed region II and they became as ″ 6 and ″ 7 , respectively. Therefore, the folding materials to cover the compressed region is obtained by folding the region ) . Hereafter, the proposed designs are analyzed by using COMSOL Multiphysics in transverse electric (TE) mode at the frequency of c m 0 0625 . The obtained material parameters for each region can be seen in method summary.
The simulation results are shown in Fig. 4(a-c). Figure 4a shows the rotator's field pattern in which the rotation region is fully encapsulated with the homogenous coating materials. Anyway, the functionality of our proposed rotator is equivalent to previously reported work 20 . Meanwhile, the Fig. 4b is the case of open-coating rotator in which the rotation region is open to the outer world. It can be clearly seen from Fig. 4(a,b) that the incident plane waves are impinging the devices with 0° angle and the devices rotate the EM waves with 90° angle in the destined region. In order to validate the bi-directional property of our proposed design, the incident wave is propagating along the 90° angle and as a result, the open-coating rotator's functionality is efficiently achieved, which can be seen in Fig. 4c.
Compared with the open-coating concentrator as discussed previously in Fig. 1, in this case, the EM waves are rotated but not concentrated. By taking advantage from these two proposed designs, here, we plan to merge the first step of Fig. 1 into the first step of Fig. 3, which can be clarified in Fig. 5. In simple words, the outer coating layers are composed by rotators as the rotation region is free space, so in that free space the concentrator device is merged. The red lines are used to differentiate the concentrator and rotator boundaries. This bi-functional device is proposed to have the rotation and concentration of EM waves at the same time and we assume that device as a virtual space for the next step to make it like open-coating device. The procedure to make it open, again the second step of Fig. 3 is recalled. The geometric structure, constitutive parameters and working frequency of this particular device are same as described earlier for Figs 2, 4. For more details see method summary.
After that, simulations are performed and the results are shown in Fig. 6(a-c). Figure 6a validates the working performance of proposed bi-functional device that is rotating and concentrating the EM field at the same time in the destined region. In this case, the destined region is fully covered with the combination of proposed rotator (Fig. 3) and proposed concentrator (Fig. 1). Figure 6b is a case of open-coating device in which the destined region is open to the outer world. It can be observed in Fig. 6(a,b) that the incident plane waves are impinging the devices with 0° angle and the device rotate and concentrate the EM waves with 90° angle in the destined region. In Fig. 6c, the incident wave propagates along the 90° angle and the efficiency of proposed device remains unchanged.
It can be observed that there exist some negligible distortions in EM waves due to the resonance of the incident wave between the positive index material and negative index metamaterials to compensate each other. In easy words, the ∆ABC is mapped into ∆ ′ ′ ′ A B C and so on. The material parameters of the proposed rotator designed from this method are homogeneous and easy to fabricate. In the second step, bigger dashed lines circle is compressed into region I and region II (contained mapped medium of ′ 6 and ′ 7 regions), and then isotropic folded material is used to coat the compressed region.
Theoretically, there should be no distortions according to the wave theory because of the existence of both folding space and resonance. However, for simulation case, it requires finest mesh to vanish the wave's distortions.

Conclusion
Based on multi-folded TO, we proposed devices of homogeneous and anisotropic materials to manipulate the EM waves and further created air-gap between the coated materials in order to make destined region open to outer  open-coating bi-functional device that first rotates and then concentrates the EM waves at the same time while the concentrator region is not fully enclosed with the coating materials. In the first step, the proposed bi-functional device is achieved by combining the first step of both Fig. 1 and Fig. 3, distinguished by the red lines boundaries. For second step, Fig. 3  world. Meanwhile, the structure of open-coating devices are composed by mapped EM medium merged inside the compressed region that is further coated with negative index materials. The simulation results validated each case of different proposed structures and verified that the dielectric region can store and rotate the EM waves without any physical connectivity of coating materials. These features of proposed concept will be very helpful for future remote controlling applications in microwave and optical engineering.

Methods
Before moving to the next steps, we consider the mapping shown in Fig. 7, which uniquely maps every point x y ( , ) of Fig. 7a in a new region ′ ′ x y ( , ) as shown in Fig. 7b. The transformation material of the new space ′ ′ x y ( , ) can be related to its original space x y ( , ) by: where J is the Jacobean tensor and for triangular shape only in 2D transformation, the Jacobean matrix can be found as: In accordance to achieve the ultimate goal, in this paper, we focused on to achieve homogenous materials by using triangular shape regions. Thus, Eqs (1,2) will play an important role in designing the proposed devices.
We start from the first step of open-coating concentrator as mentioned in Fig. 1. Here, the bigger square of size = . c m 0 25 is compressed into = .
a m 0 17 with the following transformation equations:  Thus, the constitutive parameters of that region will become as: ε µ ′ = ′ = .
. diag[1, 1, (0 25/0 17) ] a a 2 . In the same step, when ∆ABC of region 1 and ∆CDB of region 2 is folded into region ∆ ′ ′ A B C named as ′ 1 and ∆ ′ CDB named as ′ 2 respectively, the coordinates parameters of each point is known so this phenomenon is similar as mentioned in Fig. 7. So, by taking help from Eqs (1-2), the constitutive parameters will become as: The same procedure is applied to find the material parameters of all other regions such as: In the second step (Fig. 1), the bigger black dashed lines are compressed with the following transformation equations: where κ is the compression ratio of . 0 7. So, the material properties of region I and region II will become as: . After that folding transformation is applied and for each region as described in Fig. 1 (second step), the material parameters will become as:  III  III  IV  IV  V  V where as, the subscripts of ε′ and µ′ indicate the material parameters of that concerned region only. It can be clearly observed from the constitutive parameters that the inhomogeneity of designed concentrator is completely removed, and only anisotropy is required to obtain the desired functionality. For open-coating rotator (Fig. 3), in the first step, the conventional rotator is designed by simply mapping the region 1 as ∆ABC into region ∆ ′ ′ ′ A B C of label ′ 1 and similarly, other regions labeled as ... 2, 3 8 in virtual space are accordingly transformed into colored regions ′ ′... ′ 2 , 3 8 . Thus, by seeking help from Eqs (1,2), the constitutive parameters of the device will become as: In the second step (Fig. 3), the bigger dashed lines circle is compressed into smaller blue colored circle with the following transformation equations: So, the material properties of region I and region II will become as:  I  II  II  II  II  2  7  7  2  6  6  2 Similarly the, material parameters of region III are obtained from ref. 13  For open-coating bi-functional device (Fig. 5), the constitutive parameters for both first and second step will remain similar as that of the obtained material parameters of Figs 1 and 3 except the folding regions. It should be noticed that the second step for this proposed design is achieved by recalling the Fig. 3 (second step), with the same geometric and material parameters except the value of = − .