Electrostatic Field Invisibility Cloak

The invisibility cloak has been drawing much attention due to its new concept for manipulating many physical fields, from oscillating wave fields (electromagnetic, acoustic and elastic) to static magnetic fields, dc electric fields, and diffusive fields. Here, an electrostatic field invisibility cloak has been theoretically investigated and experimentally demonstrated to perfectly hide two dimensional objects without disturbing their external electrostatic fields. The desired cloaking effect has been achieved via both cancelling technology and transformation optics (TO). This study demonstrates a novel way for manipulating electrostatic fields, which shows promise for a wide range of potential applications.

plots the dependence of the relative dielectric constant ε ε / b 2 for the outer layer on the radii ratio / c b. In our study, castor oil with the dielectric constant of 4.3 is used as the background medium. Figure 1b also gives the required geometry parameters when the outer layer is air ε ( = . ) 1 0 or Teflon ε ( = . ) 2 1 , respectively. In our study, air is chosen as the outer layer, thus the required radii ratio / c b is 1.3. We choose a steel layer (SL) with = .
a 1 3 cm, and = . b m 1 5 c as the inner layer. Note that other good conductors can also be used. The geometry parameters for the air layer can be determined as: = . b 1 5 cm, and = .
c 1 95 cm. To verify the theoretical prediction, numerical simulations were carried out based on Multiphysics Comsol. Here three cases are discussed: a) background medium (castor oil); b) castor oil + steel layer (SL); c) castor oil + air layer (AL). In the simulations, the size of the modelling area is 15 × 15 cm 2 , and − 1000 V potential is applied to two edges to generate uniform electrostatic field. Figure 2a-d gives the simulation results, where the electric field distribution and isopotential lines are plotted. Figure 2a-c, describe case a), case b) and case c), respectively, and Fig. 2d provides the results for the designed bilayer cloak. As seen in Fig. 2a, a uniform electric field and gradient potential can be generated. Figure 2b,c illustrate that the steel layer repels the isopotential lines and protects the interior from the external field, while air layer attracts the isopotential lines. For both cases, the isopotential lines and electric field are seriously distorted. In the case of the bilayer cloak, however, the electric field travels around the inner domain without any disturbance, as depicted in Fig. 2d. In contrast, the distortion for electric field and isopotential only occurs in air layer. Therefore, the inner domain is protected from the external field and thereby a perfect cloak is obtained. Experimental demonstration of bilayer cloak. This fabricated bilayer cloak shown in Fig. 3 is composed of commercially available steel tubing with following dimensions: inner radius a = 1.3 cm, outer radius b = 1.5 cm and height h = 5 cm, and it is placed in a photosensitive resin container. To construct the bilayer cloak described above, the steel shell tube is further wrapped by a photosensitive resin (ε = 2.0) shell container with the following dimensions: inner radius b = 1.5 cm, outer radius c = 1.95 cm and height h = 5 cm. The photosensitive resin shell tube can be fabricated by SL process 33 . The thickness of the container tube wall is 0.45 mm. The shell container tube is filled with air, and a bilayer coaxial tube cloak is obtained. Simulations show the presence of the container tube produces very little influence on the performance.
In the experiment, the bigger container is filled with castor oil and two copper plates are used as electrodes. The electrodes are applied with − 1000 V by electrostatic generator to create an electrostatic field in x-direction. The performance of cloak can be evaluated by measuring the electric field distribution along the line 2.1 cm from the center of bilayer cloak. Clearly, the simulated distribution of electric field for the homogeneous dielectric medium (castor oil) is uniform, with the value of 6666 V/m. The presence of the steel layer tube and air layer tube causes the distortion of electric field, which can be confirmed by the position-dependent electric field. As seen in Fig. 4a, the electric field near the steel tube is strong and the direction of the electric field has a significant change. The maximum electric field can be determined to be 10,426 V/m. For the air shell tube, the electric field decreases near the shell to a minimum value of 4,661 V/m. For the bilayer coaxial cloak, the external field is almost undisturbed and one can obtain uniform electric field with the value about 6660 V/m. Thus, good cloak performance has been achieved.
In the measurement, an electrostatic instrument is used to quantify the corresponding electric field. The detailed information for electrostatic measuring instrument can be found in Methods, where the current readings in ampere meter are positive to the electrostatic field detected by the probe. Therefore, one can characterize the electrostatic field distribution by obtaining the corresponding current at a position. The measured results are presented in Fig. 4b, where the measured current distributions are in good agreement with the simulated electric field distributions, thus validating the feasibility of our scheme. It is noteworthy that the deviation can be attributed to the fabrication and measurement.
Carpet electrostatic cloak. In addition to scattering cancelling method, the TO theory can also be used to obtain cloaking. As shown in Fig. 5a, the x-z PEC plane is connected to ground. In the transformation, the AOB is stretched to AC'B, while ACB remains unchanged. Thus, by placing the appropriate materials in the region AC'BCA, one can make the space of AC'BA invisible, then a carpet cloak is achieved. According to the theoretical analysis in Methods, the required components of dielectric constant tensor in the ′ ′ x y system are: The rotation between the new and original coordinate system is Here, α β α = ( − )/ k tan tan tan , and τ β = tan . 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 the other one is smaller. Thus we employ air ε ( = . ) 1 0 r and ultrapure water ε ( = . ) 80 0 r to fabricate such a metamaterial. In our study, the geometrical parameters for the carpet cloak are: AB = 2a = 20 cm, OC = a = 10 cm, OC' = 0.5a = 5cm. As a result, one can obtain that: α = tan 1, β = . tan 0 5. The designed metamaterial is given in Fig. 5b, where the filling ratio of the air is about 88%. Simulations are carried out to characterize the performance of the designed carpet cloak. In the simulations, −1000 V potential is biased between the two electrodes to generate nearly uniform electric field. The simulation results for the electric field and potential are shown in the Fig. 6a-c. Figure 6a shows the process how the uniform electric field is generated between the two electrodes, while Fig. 6b shows that the presence of isosceles triangular shaped PEC ridge causes serious distortion of the electric field and potential lines. The simulation results for the carpet cloak are provided in the Fig. 6c, where the distortion of the cloak disappears and only occurs in the carpet cloak, indicating good cloaking performance. As schematically shown in    7, the fabricated carpet cloak is a multilayer-groove structure, where the grooves are alternately filled with ultrapure water and air. The performance of the carpet cloak can also be evaluated by the electric field intensity along the dash lines as shown in Fig. 5b. The simulated electric field is presented in Fig. 8a. The electric field is uniform and has the value 6666 V/m. When the isosceles triangular shaped PEC ridge is placed in contact with the electrode, the electric field is strongly distorted. However, when the PEC is wrapped by carpet cloak, the distortion is cancelled and the electric field becomes uniform again. The measured results are shown in Fig. 8b, where the measured current shows good agreement with the simulated electric field, indicating the feasibility of our proposed scheme.

Discussion
For EM wave propagation, the electric and magnetic fields couple to each other, causing great difficulties for the practical realization of invisibility cloaking in free space. The previous experimental works are usually classified into two categories. The first one is based on TO method, which however requires anisotropic, inhomogeneous, and even singular parameters for the magnetic and electric permittivity. Although a reduced scheme has been successfully proposed to obtain a cloak in free space, it is difficult to be extended to applications with high frequencies and three-dimensional configuration 3 . Another one is based on the scattering cancelling technology, which can avoid the problems of the TO-based cloak 34 . However, it is still imperfect, since only some scattering terms are cancelled. Thus, there is still a long way to go before a perfect cloak is obtained. However, as for electrostatic fields, realizing a perfect cloak (the cylindrical case for 2D or spherical one for 3D) in free space is easy. As demonstrated above, the perfect cloak can be achieved by directly solving the electrostatic equation. Using a bilayer structure, the cloak can be made with two kinds of naturally occurring materials. Although this bilayer cloak can only work for the two-dimensional case, it can be easily extended to three-dimensional ones. In addition, due to its simple configuration, it can be easily scalable. It's worth mentioning that the bilayer cloak also has shortcoming: it only works for the inhomogeneous electric field, and the cloaking effect will be poor if a point source is used. This can be explained easily according to the previous work on static magnetic fields 12 and thermal fields 21 , both of which show that the cloak performance under a pointlike diffuse source can be improved when the thickness of the bilayer cloak is reduced. The carpet invisibility cloak directly confirms the feasibility of the TO method for the electrostatic field. This powerful mathematical tool, with the combination of metamaterial, would provide a broad platform for the design of new devices. It is worth mentioning that the wavelength is infinite for the static case, which means that there is no subwavelength limits, thus the practical realization would be greatly simplified.
In summary, using cancelling technology and the transformation optics method, we demonstrate electrostatic field cloaks that can shield a specified region from the external field without any disturbance. These cloaks with homogeneous dielectric constants, can be readily obtained with naturally occurring materials. In addition, the simple structure can be easily extended to micro-nanoscale and three-dimensional configurations, thereby greatly enhancing practical realization and to enable applications like non-destructive detection. More importantly, our concept for manipulation of the electrostatic field can also be extended to other devices, such as, concentrators, rotators and illusion, which may find applications in various fields.

Methods
The theoretical analysis for bilayer cloak. Figure 9 schematically illustrates the corresponding two-dimensional (2D) physical model of coaxial tubes where a uniform electric field E is produced from high potential to low potential. In the considered space, the electric potential is governed by the 2D Laplace's equation φ ∇ = 0 2 , which can be expressed as    The theoretical analysis for carpet cloak. In the transformation (see Fig. 5a), the AOB is stretched to AC'B, while ACB keeps unchanged. Thus, by placing the appropriate materials into the region AC'BCA, 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 Here ε is the dielectric constant of the background medium. α β α = ( − )/ k tan tan tan , and τ β = tan . For 2D case, only in-plane parameters are considered and they form a symmetric 2 × 2 matrix. This matrix can be further diagonalized in the ′ ′ Electric field measurement instrument. The circuit of electrostatic field intensity measurement instrument is shown in Fig. 10. Electric field induction signal is detected using the 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 electrostatic field around the probe of measurement instrument. There is no current getting through resistance R3, which cuts off BG2. Therefore, collector current of BG2 is zero, ampere meter is zero and the circuit is in the stationary state. When there is electrostatic field around the probe of measurement instrument, the charge begin to accumulate in probe because of electrostatic 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 BG2 and the current amplified by BG2 is measured though ampere meter. The probe of measurement instrument can induct different quantity of electric charge in different position of electrostatic field because of different electrostatic field strength. Therefore, there is different collector current of BG2. The relative electrostatic field can be measured with this method.