Electrostatic Field Invisibility Cloak Chuwen Lan 1,, Yuping Yang 3, Ji Zhou 1 *, Bo Li * 1 State Key Laboratory of New Ceramics and Fine Processing, School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Advanced Materials Institute, Shenzhen Graduate School, Tsinghua University, Shenzhen, China 3 School of science, Minzu University of China, Beijing, 100081, China * Corresponding author: zhouji@mail.tsinghua.edu.cn, libo@mail.tsinghua.edu.cn, Abstract: Invisibility cloak is currently attracting much attention due to its special avenue for manipulation of physical fields,which has been demonstrated in various waves like electromagnetic field, acoustic field and elastic wave, as well as scalar fields, such as thermal field, static magnetic field and dc electric field. Here, we report on the first theoretical and experimental realization of electrostatic field invisibility cloak to shield a certain region without disturbing the external electrostatic field. Both scattering cancelling technology and transformation optics (TO) method were employed to achieve this goal. This work provides a novel method for manipulation of electrostatic field, which may find potential applications in broad areas. Keywords:Electrostatic field; invisibility cloak; scattering cancelling technology; transformation optics.
Introduction The term of invisibility cloak refers to a device which has the capability of shielding the object without disturbing the external physical field. Usually, it is considered impossible and can only occur in fiction and legend. However, this perspective has been changed due to the theoretical prediction and experimental demonstration of an invisibility cloak in 006 [1-]. Since that, it has emerged as a hot research topic due to its novel concept for manipulation of electromagnetic wave [1-8]. In the past years, considerable effort was focused on this field and much progress has been made. Motived by the great achievement in electromagnetic wave, it is rapidly extended to other waves like mechanical wave [9], elastic wave [10], and matter waves [11]. Recently, this concept has been applied to scalar fields like magnetic field [1, 13], dc electric field [14, 15], thermal field [16-0] and mass diffusion [1-4]. The first magnetic field cloak was theoretically investigated by A. Sanchez and then demonstrated experimentally with superconductors and ferromagnetic material by two groups [1, 13]. Based on the form invariance of the heat conduction equation, the transformation thermodynamics was successfully developed and demonstrated to cloak diffusive heat flow [16, 18]. The transient thermal cloak was also experimentally realized through a rescaled diffusion equation [0]. Using artificially structured material (multilayered composite), homogeneous thermal cloak was theoretically and experimentally demonstrated [16]. In addition, three-dimensional thermal cloak was also realized by using bilayer structure [19, 1]. Using the network of resistors in analogy to conductive materials, dc electric cloak based on a transformation-optics has been implemented by Mei [14]. In addition, bilayer dc electric cloak using naturally occurring conductive materials was successfully realized by Han [15]. However, the possibility for cloaking the electrostatic field, which is different from the one for dc electric field where carriers are required, is still unexplored. Here, we report the first electrostatic field invisibility cloak to shield a certain region without disturbing the external electrostatic field, and we further experimentally realize this device to confirm the proposed methodology. Our
contribution is twofold. First, we demonstrated the electrostatic field invisibility cloak for what we believe to be the first time. It should be remarked that the electrostatic field invisibility cloak is different from the dc electric cloak where carriers are required. In addition, the proposed methods, namely, scattering cancelling technology and transform optics (TO) method, can also be extended to other devices like concentrator, illusion, and rotator, which would bring much potential applications in various fields. Results We start with electrostatic field invisibility cloak based on cancelling technology method. Figure1(a) schematically illustrates the corresponding physical model where a uniform electric field E is applied in x-direction with displacement current D E from high potential to low potential. In the considered space, the electric potential is governed by the Laplace s equation 0, which can be expressed as i m i m i m m m1 [ A r B r ]cos m (1) where A and B (i=1,, 3) are constants to be determined, and i represents the i m i m potential in different regions: i=1 for region I, i= for region II ( b r c ) and i=3 for region III ( r c). Figure1. (a) The physical model for electrostatic field invisibility cloak. The uniform electric field is generated from the high potential (H) to low potential (L). (b) The required relative
dielectric constant for electrostatic field invisibility cloak with radii ratio c/ bof the outer layer of cloak. The material candidates for achieving cloak are marked by dash lines. Taking into account that 1 should be finite when r 0, one can obtain that B. In addition, 3 should tend to E0r cos when r, one only needs to 1 1 0 consider m 1. Furthermore, the electric potential and the normal component of electric field vector are continuous across the interfaces, one can obtain that i rb, c i 1 rb, c r r i i1 i rb, c i1 rb, c () Here, 3 b, where b is the electric dielectric constant of the background. Combine the Eq. (1) and Eq. (), one can obtain B E c ( M M ) ( M M ) ( ) ( ) 3 1 b 1 1 0 M1 M b M1 M (3) 1 where M1 c (1 ), 1 M b (1 ). By making B 3 1 0, one can obtain c b ( 1)( b) ( )( ) 1 b (4) Consequently, one can get the electrostatic field invisibility cloak according to the Eq. (4). Here, considering the inner layer of cloak is made of PEC or metal conductor, which has dielectric constant of 1 at static case. It should be noted that the PEC or metal conductor is chosen due to its excellent electrostatic shielding performance. Since 1, Eq. (4) can be expressed as c b 1( b) ( ) 1 b (5) which can be further expressed as: c b ( b) ( ) b (6)
As a result, the required dielectric constant for the outer layer of the cloak can be obtained: c b ( ) b c b Figure 1(b) plots the dependence of relative dielectric constant (7) / b for the outer layer on the radii ratio c/ b. In our study, the castor oil with dielectric constant of 4.3 was used as background medium. Figure 1(b) marks the required geometry parameters when the outer layer is made of air ( 1.0 ) and Teflon (.1), respectively. According to Eq. (7), the required radii ratio c/ b for air layer is 1.3. We choose steel layer with a 1.3cm b 1.5cm as inner layer. The geometry parameters for air layer can be determined as: b 1.5cm c 1.95cm. In order to confirm the theoretical prediction, simulations based on Multiphysics Comsol were carried out. We also discuss three comparison cases: a) homogeneous background medium (castor oil); b) steel layer; c) air layer. In the simulations, computational domain was set as 15cm 15cm and -1000V potential was applied to two edges to generate uniform electrostatic field. Figure. The simulation results of electric field intensity and isopotential lines for different cases:
(a) homogeneous dielectric medium (castor oil). (b) high dielectric constant material (PEC shell). (c) low dielectric constant material (air shell). The dotted white lines denote the measuring lines in the experiments. Figure shows the simulation results, where the electric field distribution and isopotential lines are plotted. Figure (a) corresponds to the homogeneous dielectric medium (castor oil). Figure (a)(b) correspond to PEC shell and air layer, respectively, while the Figure (d) provides the result for the designed bilayer cloak. In the simulations, symmetry boundary condition was applied to the y-direction and -1000V potential was applied to the x-direction. As shown in Figure (a), a uniform electric field and gradient potential can be generated between the high potential and low potential. Figure (b) and Figure (c) illustrate that the layer made of high dielectric constant material (PEC) repels the isopotential lines while the one made of low dielectric constant material attracts the isopotential lines. For both cases, the isopotential lines and electric field are seriously distorted. However, as for the bilayer cloak, the electric field travels around the inner domain as nothing happens. Instead, the distortion for electric field and isopotential only occurs in the air layer. Therefore, the inner domain is protected from the external field and a perfect cloak is obtained. We have fabricated this bilayer cloak, which is shown in Figure 3(a). The steel shell with inner radius a=1.3cm, outer radius b=1.5cm and height h=5cm is placed in a PMMA container, whose geometrical parameters for the container are provided in this picture. The steel shell is further wrapped of an air layer, which is actually a PMMA shell container with inner radius a=1.95cm and height h=5cm. All the other parameters are the same to the ones in the simulations. As a result, a bilayer cloak is achieved. In the experiment, the castor oil is filled in the PMMA container and two copper plates are used as electrodes. The electrodes were applied with -1000V by electrostatic generator to generate electrostatic field along the x-direction.
Figure3 Experimental demonstration of electrostatic field invisibility cloak. (a) The paragraph of fabricated sample. (b) Schematic illustration for realization of electrostatic field invisibility cloak. (c) The schematic diagram for electrostatic measuring instrument. Figure4 (a) Simulated results of electric field intensity along the white dash line for the cases: air
shell, metal shell (PEC shell) and bilayer cloak (air shell and metal shell). (b) The measured results of current in the electrostatic field measurement instrument for corresponding cases, respectively. The performance of cloak can be evaluated by obtaining the electric field intensity distribution along the line.1cm from the center of bilayer cloak as depicted in Figure 4. Clearly, the simulated electric field intensity distribution for the homogeneous dielectric medium (castor oil) is uniform, and can be determined as 6666V/m. The presence of materials made of high dielectric constant (PEC) and low dielectric constant (air layer) causes the distortion of electric field, which can be confirmed by the angle dependent electric field intensity. As one can see in Figure 4(a), the electric field intensity near the PEC shell is seriously disturbed and greatly enhanced. The maximum electric field intensity can be determined as 1046 V/m. As for air shell, the electric field intensity decreases near the shell and has a minimum value of 4661 V/m. As for the bilayer cloak, the external field is undisturbed, thus one can achieve uniform electric field intensity with value about 6660 V/m. Clearly, good cloak performance has obtained. In the measurement, we use the electrostatic measuring instrument to character the corresponding electric field. Note that the potential distribution is not measured since it is a challenge to obtain the potential distribution. This is very different from the dc current electric field where potential can be readily measured. The corresponding schematic diagram is shown in Figure 3b, where the current readings in ampere meter are positive to the electrostatic field intensity detected by the probe. The detailed information for electrostatic measuring instrument can be found in Method. Therefore, one can character the electrostatic field intensity distribution by obtaining the corresponding current in a certain position. The measured results are presented in Figure 4b, where the measured current distributions agree well with the simulated electric field intensity distributions, thus validating the feasibility of our scheme.
Figure 5. (a) The transformation model for the carpet cloak, where the line AOB is stretched to AC B while the line ACB keeps unchanged. (b) The designed carpet cloak. The metamaterial multilayer structure is composed of air and distilled water alternately stacked sheets. The rotation angle between new and old coordinate systems is. (c) The paragraph of fabricated sample. In addition to scattering cancelling method, the TO theory can also be employed to obtain cloak, which is schematically illustrated in Figure 4. As shown in this picture, the x-axis is a PEC plane connected to ground. In the transformation, the AOB is stretched to AC B, while ACB keeps unchanged. Thus, by placing the appropriate materials into the region AC BC, one can make the space of AC BA invisible, thus a carpet cloak is achieved. According to the TO theory, one can obtain the required dielectric constant T A A ' (8) det( A) ( x ', y ', z ') where A ( x, y, z) is the Jacobian matrix. Here, the transformation equation is Then the required dielectric constant can be determined as x' x y ' ky ( a x) z' z (9)
1 0 k k ( k ) ' 0 k k 1 0 0 k Here is the dielectric constant of the background medium. k (tan tan ) / tan, and tan. For D case, only in-plane parameters are considered and they form a symmetric matrix. This matrix can be further diagonalized in the constant tensor are (10) xysystem, ' ' where the corresponding components of dielectric ( k 1) ( k 1) 4 xx ' ' k yy ' ' ( k 1) ( k 1) 4 k The rotation between the new and original coordinate system is 1 arctan k 1 (11) (1) Clearly, the required material for the carpet cloak is homogenous but anisotropic. To achieve this anisotropy, one can use the metamaterial multilayer structure. Note that one component of the required dielectric constant is larger than background medium and another one is smaller than the background medium. Thus we employ air ( r 1.0 ) and distilled water ( 80.0) to fabricate such metamaterial. In our study, r the geometrical parameters for carpet cloak are: AB=a=0cm, OC=0.5a=5cm, OC =0.5a=5cm. As a result, one can obtain that: tan =1, tan =0.5. The designed metamaterial is presented in Figure 5b, where the filling ratio of the air is about 88%. Simulations were carried out to character the performance of the designed carpet cloak. In the simulations, -1000V potential was applied between the two electrodes to generate nearly uniform electric field. The simulation results for the electric field and potential is shown in the Figure 6. Figure 6a presents the uniform electric field is generated between the two electrodes.
Figure 8. The simulation results for electric field intensity and isopotential lines: (a) homogeneous dielectric medium (background material) (b) cone-shape PEC without cloak. (c) cone-shape PEC with carpet cloak. Figure 6b shows that the presence of cone-shaped PEC causes serious distortion of electric field and potential lines. The simulation result for the carpet cloak is provided in the Figure 6c, where the distortion of the cloak is cancelled and only occurs in the carpet cloak, revealing a good performance. In fact, the performance of the carpet cloak can also be evaluated by the electric field intensity along the dash lines. The simulated results for electric field intensity are provided in Figure 8a. As shown in this picture, the electric field intensity is uniform and can be determined by 6666V/m. When the cone-shape PEC is placed on the ground, the electric field intensity is
serious distorted. However, when the PEC is wrapped of carpet cloak, the distortion is cancelled and the electric field intensity becomes uniform again, revealing a good cloak performance. The measured results are shown in Figure 8b, where the measured current shows good agreement with the simulated electric field intensity, indicating the feasibility of our proposed scheme. Figure8 (a) Simulated results of electric field intensity along the white dash line for the cases. (b) The measured results of current in the electrostatic field measurement instrument for corresponding cases, respectively. Discussion In conclusion, using scattering cancelling technology and transformation optics (TO) method, we have realized electrostatic field cloaks that can protect a certain region from the external field as if nothing happens. The proposed cloaks require homogeneous dielectric constant which can be readily obtained with naturally occurring materials, meanwhile the simple structure can be easily extended to micro-nanoscale and three-dimensional configuration, thus greatly enhances the practical realization and would enable applications like non-destructive detection. More importantly, our concept for manipulation of electrostatic field can also be extended to other devices like concentrator, rotator and illusion, which may find considerable applications in various fields.
This work was supported by the National Natural Science Foundation of China under Grant Nos. 5103003, 1174198, 51191 and 6175176, National High Technology Research and Development Program of China under Grant No. 01AA030403, Beijing Municipal Natural Science Program under Grant No. Z14110000414001, and the Science and technology plan of Shenzhen city under grant JCYJ01061915711509,JC0110518080A and CXZZ01303164541915. Reference 1 Leonhardt, U. Optical conformal mapping. Science 31, 1777-1780, (006). Pendry, J. B., Schurig, D. & Smith, D. R. Controlling electromagnetic fields. Science 31, 1780-178, (006). 3 Schurig, D. et al. Metamaterial electromagnetic cloak at microwave frequencies. Science 314, 977-980, (006). 4 Liu, R., Ji, C., Mock, J. J., Chin, J. Y. & Cui, T. J. Broadband ground-plane cloak. Science 33, 366-369, (009). 5 Valentine, J., Li, J. S., Zentgraf, T., Bartal, G. & Zhang, X. An optical cloak made of dielectrics. Nature Mater. 8, 568-571, (009). 6 Ma, H. F. & Cui, T. J. Three-dimensional broadband ground-plane cloak made of metamaterials. Nat. Commun. 1, 1, (010). 7 Zhang, B., Luo, Y., Liu, X. & Barbastathis, G. Macroscopic Invisibility Cloak for Visible Light. Phys. Rev. Lett. 106, 033901, (011). 8 Chen, X. et al. Macroscopic invisibility cloaking of visible light. Nat. Commun., 1176, (011). 9 Zhang, S., Xia, C. & Fang, N. Broadband Acoustic Cloak for Ultrasound Waves. Phys. Rev. Lett. 106, 04301, (011). 10 Farhat, M., Guenneau, S. & Enoch, S. Ultrabroadband Elastic Cloaking in Thin Plates. Phys. Rev. Lett. 103, 04301, (009). 11 Zhang, S., Genov, D. A., Sun, C. & Zhang, X. Cloaking of Matter Waves. Phys. Rev. Lett. 100, 1300, (008). 1 Gomory, F., Solovyov, M., Souc, J., Navau, C. & Prat-Camps, J. Experimental realization of a magnetic cloak. Science 335, 1466-1468, (01). 13 Narayana, S. & Sato, Y. DC magnetic cloak. Adv. Mater. 4, 71-74, (01). 14 Yang, F., Mei, Z. L., Jin, T. Y. & Cui, T. J. dc Electric Invisibility Cloak. Phys. Rev. Lett. 109, 05390, (01). 15 Han, T. C., Ye, H. P., Luo, Y., Yeo, S. P., Teng, J. H.; Zhang, S.; Qiu, C. W. Manipulating DC Currents with Bilayer Bulk Natural Materials, Adv. Mater. 6, 3478-3483, (014). 16 Guenneau, S., Amra, C. & Veynante, D. Transformation thermodynamics: cloaking and concentrating heat flux. Opt. Express 0, 807-818, (01). 17 Narayana, S. & Sato, Y. Heat Flux Manipulation with Engineered Thermal
Materials. Phys. Rev. Lett. 108, 14303, (01). 18 Schittny, R., Kadic, M., Guenneau, S. & Wegener, M. Experiments on Transformation Thermodynamics: Molding the Flow of Heat. Phys. Rev. Lett. 110, 195901, (013). 19 Han, T. C., Bai X., Gao D. L., Thong J.T. L., Li B. W. & Qiu C. W. Experimental Demonstration of a Bilayer Thermal Cloak. Phys. Rev. Lett. 11.05430, (014). 0 Ma Y.G., Lan L., Jiang W., Sun F. and He S. L., A transient thermal cloak experimentally realized through a rescaled diffusion equation with anisotropic thermal diffusivity, NPG Asia Materials 5, 73, (013). 1 Xu H. Y., Shi X. H., Gao F., Sun H. D. & Zhang B. L. Ultrathin Three-Dimensional Thermal Cloak. Phys. Rev. Lett. 11.054301, (014). Guenneau S. & Puvirajesinghe T. M. Fick s second law transformed: One path to cloaking in mass diffusion, J. R. Soc. Interface 10, 0130106 (013). 3 R. Schittny et al. Invisibility Cloaking in a Diffusive Light Scattering Medium, Science 345, 47 (014). 4 Zeng L. and Song R. Controlling chloride ions diffusion in concrete, Sci. Rep. 3, 3359 (013). Method The circuit of electrostatic field intensity measurement instrument is shown in Figure M1. Electric field induction signal is detected using the characteristic of field-effect tube which has very high input resistance and is very sensitive to electric field induction around it. After switch K is thrown, the source of field-effect tube BG1 and the voltage between drains is lower when there is no electro static field around the probe of measurement instrument. There is no current getting through resistance R3, which cuts off BG. Therefore, collector current of BG is zero, ampere meter is zero and the circuit is in the stationary state. When there is electro static field around the probe of measurement instrument, the charge begin to accumulate in probe because of electro static induction. The bias voltage produced between both ends of resistance R changes the internal resistance of BG1 source and the drain, which results in changes of the whole circuit state. There is current through resistance R3 after breaking over BG and the current amplified by BG is measured though ampere meter. The probe of measurement instrument can induct different quantity of electric charge in different position of electro static field because of different electro static field strength. Therefore, there is different collector current of BG. The electrostatic field intensity can be measured relatively with this method.
FigureM1 The schematic diagram for electrostatic measuring instrument