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Abstract
By virtue of eliminating the characteristic signals of physical objects, invisibility cloaks can also prevent the objects from being detected by instruments using electrical current. However, most existing designs of direct current cloaks cannot be directly applied in the natural environment. In this paper, we present a three-dimensional direct current cloak by solving Laplace equation in electric conduction. The cloak is composed of a bilayer structure with homogeneous and isotropic bulk materials. The designed structure, having withstood both conventional electrical potential measurements and tests by a commercially-available detector, is highly effective in suppressing current characteristic signals in close simulation of natural environmental conditions. The cloak is expected to facilitate the study of characteristic signal control in medical and geological research in the field of current detection, such as tumor diagnosis and mineral exploration.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Over time, invisibility cloak [1] has long gone from a mysterious magic in fantasy novels to a technological reality. Due to the great development of metamaterials [2,3], the controlling methods for vector fields, particularly the electromagnetic fields [4–10] and acoustic fields [11,12], have gained full momentum. Transformation optics (TO) theory [5] provides a powerful tool for the design of invisibility cloaks. However, original TO design for electromagnetic waves usually requires the support of double-negative and bianisotropic materials, which increases difficulties of production. Later, researchers improved TO method with quasi-conformal mapping [13], which allows the composition of cloaks to be inhomogeneous but isotropic materials [14–16]. Cloaking devices proposed above generally have a large size. By two-dimensionalizing bulk metamaterials, metasurface [17] is obtained with ultrathin feature. Metasurface-based devices [18–21] are making great progress, but the structures are still somewhat complicated on some simple occasions. Scattering cancellation method [22] enables homogeneous and isotropic materials to be applied in cloaking design [23–25]. However, scattering cancellation cloak usually requires the object size to satisfy electrically small in electromagnetic waves.
The technical bottleneck gave rise to the emergence of metamaterials for scalar fields, also known as Laplacian metamaterials [26]. In addition to achieving invisibility properties of objects in thermal conduction fields [27–32], magnetostatic fields [33–35] and electric current fields [36–43], cloaking devices in scalar fields can also demonstrate the feasibility based on some complicated theoretical models, since only one physical parameter affects the flow of the field. However, the studies on cloaking have long focused on magnetostatic fields rather than cloaks in electric current fields. Most previous electric current cloaks draw an analogy between resistor network on a printed circuit board and a certain conductor material [37–40,44]. This resistor network structure has such advantage that it can provide accurate electrical parameters which match well with the theoretical model. However, this structure is deemed as impractical because of little resemblance to nature scenario. A few direct current invisibility cloaks [41,42] had been developed with bulk materials, but these developments were confined only in two-dimensional (2D) or in cylindrical coordinate system structures. Three-dimensional (3D) direct current invisibility cloaks with bulk materials still remains an unresolved puzzle.
Extending 2D direct current cloak to 3D case involves several challenges and possible drawbacks. First, the designed cloak needs to match the background material. In most of the previous works, their structures were fabricated with large dimension metallic materials, with poorly adjustable conductivity and relatively limited range of selections. Moreover, using metal as a background material still lacks resemblance to nature scenario. To this end, a model of 3D direct current invisibility cloak with bulk materials is proposed in this paper and experimentally demonstrated on the basis of solving above problems. It is unnecessary for this model to acquire the electrical parameters of a hidden object in advance, making it theoretically possible to suppress the characteristic signal of any objects. This model also owns a flexible fabrication process and can be considered as an ideal 3D direct current cloak solution.
2. Theory
Figure 1(a) shows the application scenario for a typical 3D electric current cloak. Consider a background environment consisting of some non-insulating material, such as non-pure water with electrical conductivity $\sigma > 0 \; {\mathrm{\mu} \mathrm{S/cm}}$, where the section of the background material has high and low potentials at both ends. Obviously, electric currents will be generated in this background material due to the electrical potential difference. At this point, an object is placed in this background material. If the object is a good conductor with a large difference in conductivity from the background material such as a metal sphere, then the magnitude and direction of the electric currents at each point in the background material will change. Naturally, the electric potential at each point will also change, which will be more easily measured by the instrument. If these changes can be measured, the presence of the metal sphere can be detected by external detectors. Therefore, we design a practical 3D invisibility cloak that can cancel the distortion and disturbance of the electric currents by the good conductor and thus hide the presence of the object.
Fig. 1. (a) An application schematic for a typical 3D electric current cloak. (b) The designed cloak structure.
The scattering cancellation theory of electromagnetic fields suggests that the scattering of one object can be cancelled by another one, and the whole assembly can be seen as transparent. Based on similar mathematical derivation, if there are two objects in the direct current field with one’s distortion cancelling the other’s, the whole assembly can also be seen as invisible. Such cloaks can be easily prepared with the bilayer design in which all component materials are homogeneous and isotropic. Since the spatial relationships involved in the model are complicated, we make an analogy between the designed spherical cloak and the Earth for the convenience of better understanding. As shown in Fig. 1(b), the bilayer 3D cloak is composed of a “core”, a “mantle” and a “crust”. The “core” region ($r < {R_1}$, $\sigma = {\sigma _1}$) represents the cloaked object, while the “mantle” region (${R_1} < r < {R_2}$, $\sigma = {\sigma _2}$) and “crust” region (${R_2} < r < {R_3}$, $\sigma = {\sigma _3}$) represent the isolation layer and the matched layer, respectively. One of the great circle cross-sections of our spherical cloak is parallel to the horizon, and the located plane is called the “ecliptic plane”.
The electric potential satisfies Laplace equation:
According to the separation of variables method, the electric potential at each point in space can be expressed as:
If we set the inner layer as a perfect insulation material, i.e., ${\sigma _2} = 0$, we obtain the mathematical relations between geometric parameters of the cloak ${R_2}$, ${R_3}$ and electrical parameters ${\sigma _3}$, ${\sigma _4}$ (${\sigma _{bg}}$):
where m is the cubic ratio of the radii of the bilayer cloaking device, i.e., $m = {\left( {\frac{{{R_3}}}{{{R_2}}}} \right)^3}$. The detailed derivation can be found in Supplement 1. The above calculation shows that the electrical parameters of this 3D bilayer cloak are determined by the cubic ratio of outer and inner layer dimensions when the background conductivity is known.To demonstrate the availability of this 3D direct current cloak, the following simulation setup is proceeded to numerically verify the performance of the designed structure. The cloak is a 3D structure, and at the location where the cross-section is larger, the distortion level on the surrounding currents is higher. Therefore, as long as the designed structure has a good cloaking effect for the largest cross-section, it can be predicted that the cloaking effect at other locations will be even better. Thus, we choose to observe a cross-section on the ecliptic plane. We attach the left side to $20\textrm{ V}$ high potential, and the right side is set to a low potential $0\textrm{ V}$. The electrical parameters are given as:${R_1} = 38.1\textrm{ mm}$, ${R_2} = 42\textrm{ mm}$, ${R_3} = 43\textrm{ mm}$, ${\sigma _1} = 1.5 \times {10^5}\textrm{ S/m}$, ${\sigma _2} = 1 \times {10^{ - 5}}\textrm{ S/m}$, ${\sigma _3} = 6452\,{ \mathrm{\mu} \mathrm{S/cm}}$ and ${\sigma _{bg}} = 300\,{ \mathrm{\mu} \mathrm{S/cm}}$.The simulation is performed by the Finite Element Method analysis software COMSOL Multiphysics. Figure 2(a) shows the simulation results of currents flowing through homogeneous and isotropic background resulting in a uniform equipotential surface. Figure 2(b) represents a bare object placed in the background, affecting the electric current flow with equipotential surface distorted. Figure 2(c) demonstrates that the designed 3D cloak is able to restore the current around the object, ‘restore’ the distorted equipotential surface to uniform. Since any object inside the “core” region $r < {R_1}$ will not be affected by the external environment, electric potential in that region maintained 0V as Fig. 2(d), which is the initial value of the simulation setup.
Fig. 2. Simulated potential distribution of with currents flowing from left side to right side on the ecliptic plane of the system when existing (a) a homogeneous and isotropic background only, (b) a good conductor only, and (c) a hidden object with a 3D cloak. (a-c) show the potential data on the x-z plane of their insets. (d) shows the zoom-in figure of the cloaked region. (e) The electric potential on the line $y = 72\textrm{ mm}$. The blue squares and the orange triangles represent the cloaking and background cases, respectively, while the red circles are for the object only case.
To quantitatively verify the effectiveness of the invisibility cloak, we extract the electric potential data from the position $y = 72\textrm{ mm}$ for the three cases to obtain a clearer view. As shown in Fig. 2(e), this invisibility cloak is capable of concealing a good conductor (${\sigma _1} = 1.5 \times {10^5}\textrm{ S/m}$) very well. Moreover, this cloak is capable of making an invisible region that conceals different objects from electric current detector. More results can be found in Supplement 1.
3. Experiment
To demonstrate the feasibility of the above cloak design concept in the experiment, we further fabricated the sample of our 3D direct current cloak with bulk materials. In the experiment, the electrical parameters of each object are consistent with above simulations. The background material uses plain water, whose conductivity can be fine-tuned with salt. The cloaked object is chosen as a copper alloy, which has a strong distortion effect on the electric current flow, and the cloaking effect of the device can be easily observed. The bilayer cloak consists of two layers of materials, the inner isolation layer is made of insulating resin, which can be fabricated with 3D printing technology. The outer layer uses conductive silicone, whose electrical conductivity can be tuned by doping with different conductive particles, such as graphite. Such doping technique gives us a powerful tool to manufacture the “crust” matching layer of the cloak.
In order to obtain the data for the same scenario as the simulation cases above, measurements need to be made on the ecliptic plane of the whole structure. The experimental setup is shown in Fig. 3(a), where the left and right sides of the background medium are supplied with $20\textrm{ V}$ and $0\textrm{ V}$ respectively by a direct current power supply, and the potential probe is maintained in the ecliptic plane for step measurements. The measured results for three cases, namely homogeneous and isotropic background, a bare object without cloak and with the 3D cloak, are shown in Fig. 3(b), (c) and (d), respectively. The quantitative results at position $y = 72\textrm{ mm}$ are given in Fig. 3(e), which prove that the fabricated 3D invisibility cloak does provide a better concealment of the direct current field.
Fig. 3. (a) Experiment of potential measurement setup. (b-d) Measured experimental results of potential distribution on the ecliptic plane of the system when existing (b) a homogeneous and isotropic background only, (c) a good conductor only, and (d) the hidden object with a 3D cloak. (e) The measured electric potential on the line $y = 72\textrm{ mm}$. The blue squares and the orange triangles represent the cloaking and background cases, respectively, while the red circles are for the object only case.
In the above experiments, we have made precise and detailed measurements of the electric potential distribution. To better validate the practicality of this design, we added measurements by commercially available instruments, which enable a rough evaluation of the cloaking effectiveness intuitively. The experimental setup is shown in Fig. 4(a). This loose-measuring instrument has two probes, and it applies the same method as ohmmeter to estimate the average electrical conductivity in the surrounding area of the probes. These two probes provide a voltage difference, generating a weak current in the material. If a different object is present in the background material, the current flow through the area will vary. By deploying our cloak, the current is restored to the condition without the existence of the object. The instrument will display conductivity readings that approximate to the background, and the object can be regarded as hidden.
Fig. 4. (a) The schematic of the conductivity measurement experiment setup. (b-c) conductivity distribution in the surrounding area when (b) only a bare object (c) when the 3D current cloak covers the object in the background material.
Figure 4(b) and (c) show the measured average electrical conductivity results of the area by this instrument. The original background material has a conductivity of ${\sigma _{bg}} = 300\,{ \mathrm{\mu} \mathrm{S/cm}}$. As is shown in Fig. 4(b), the conductivity of the surrounding area rises due to the presence of the good conductor object. In Fig. 4(c), the fabricated 3D invisibility cloak can suppress the effect on the average conductivity of the surrounding area to a certain level, which proves the practical value of our design.
We may observe that in both Fig. 3(b-d) and Fig. 4(b-c) experimental results, there exist errors compared to the ideal values. Errors may come from the following aspects: the preparation of conductive silicone is not precise enough to fully meet the required conductivity, leading to incorrect matching of electrical parameters; The doping of the conductive particles in the conductive silicone layer is not homogeneous enough, so the matching effect in different directions is not exactly the same; During the measurement, there is displacement error when moving the probe, and the probe deformation also causes the shift in the measurement position.
4. Conclusion
In this paper, a 3D direct current cloak is designed and experimentally demonstrated. It fills the gap of applying 3D bulk material to the steady current field invisibility cloak. Through the ingenious bilayer design, this invisibility cloak can eliminate the influence of an object on the surrounding electric current, thus hiding the existence of the object. The materials used in the designed cloak, such as conductive silicone with customizable electrical conductivity and insulating resin of certain shape, are mature bulk materials for mass production in modern industry. In addition, we have also verified the cloaking effect of the produced structure using commercially available conductivity measurement tools. Therefore, this design provides a practical 3D direct current cloak solution and takes one step forward in the technology maturity of producing an actual direct current invisibility cloak. We expect that the proposed cloaking device can be applied in potential applications in medical and geological studies of characteristic signal control.
Funding
National Natural Science Foundation of China (11961141010, 61625502, 61975176, 62071423); Natural Science Foundation of Zhejiang Province (LQ21F050002, LY19F010015); Fundamental Research Funds for the Central Universities; Youth Talent Lifting Project of the China Association for Science and Technology (2020-JCJQ-QT-016); China Postdoctoral Science Foundation (2020M671720).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. T. H. Chen, B. Zheng, C. Qian, and H. S. Chen, “Progress of novel electromagnetics cloaking research,” Acta Phys. Sin. 69(15), 154104 (2020). [CrossRef]
2. V. G. Veselago, “Reviews of Topical Problems: the Electrodynamics of Substances with Simultaneously Negative Values of epsilon and mu,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]
3. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef]
4. H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99(6), 063903 (2007). [CrossRef]
5. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef]
6. H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N. I. Zheludev, and B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nat. Commun. 4(1), 2652 (2013). [CrossRef]
7. F. Sun, Y. Zhang, J. Evans, and S. He, “A Camouflage Device Without Metamaterials,” Prog. Electromagn. Res. 165, 107–117 (2019). [CrossRef]
8. P. Nochian and Z. Atlasbaf, “A Novel Single Layer Ultra-Wideband Metamaterial Absorber,” Prog. Electromagn. Res. Lett. 93, 107–114 (2020). [CrossRef]
9. T. Cai, G. Wang, S. Tang, H. Xu, J. Duan, H. Guo, F. Guan, S. Sun, Q. He, and L. Zhou, “High-Efficiency and Full-Space Manipulation of Electromagnetic Wave Fronts with Metasurfaces,” Phys. Rev. Appl. 8(3), 034033 (2017). [CrossRef]
10. T. Cai, S. Tang, B. Zheng, G. Wang, W. Ji, C. Qian, Z. Wang, E. Li, and H. Chen, “Ultrawideband chromatic aberration-free meta-mirrors,” Adv. Photonics 3(1), 016001 (2020). [CrossRef]
11. S. Zhang, C. Xia, and N. Fang, “Broadband Acoustic Cloak for Ultrasound Waves,” Phys. Rev. Lett. 106(2), 024301 (2011). [CrossRef]
12. L. Zigoneanu, B. I. Popa, and S. A. Cummer, “Three-dimensional broadband omnidirectional acoustic ground cloak,” Nat. Mater. 13(4), 352–355 (2014). [CrossRef]
13. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef]
14. H. F. Ma, T. J. Cui, W. X. Jiang, X. Y. Zhou, and X. M. Yang, “Compact-sized and broadband carpet cloak and free-space cloak,” Opt. Express 17(22), 19947–19959 (2009). [CrossRef]
15. I. Koh, I. Seo, Y. Kim, and Y. Lee, “Design method for broadband free-space electromagnetic cloak based on isotropic material for size reduction and enhanced invisibility,” Opt. Express 24(20), 22708–22717 (2016). [CrossRef]
16. Y. Kim, T. Deng, W. X. Jiang, T. J. Cui, Y. Lee, and C. W. Qiu, “Robust Control of a Multifrequency Metamaterial Cloak Featuring Intrinsic Harmonic Selection,” Phys. Rev. Appl. 10(4), 044027 (2018). [CrossRef]
17. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011). [CrossRef]
18. X. Ni, Z. J. Wong, M. Mrejen, Y. Wang, and X. Zhang, “An ultrathin invisibility skin cloak for visible light,” Science 349(6254), 1310–1314 (2015). [CrossRef]
19. J. Zhang, X. Hu, H. Chen, and F. Gao, “Designer surface plasmons enable terahertz cherenkov radiation,” Prog. Electromagn. Res. 169, 25–32 (2020). [CrossRef]
20. H. Chu, Q. Li, B. Liu, J. Luo, S. Sun, Z. H. Hang, L. Zhou, and Y. Lai, “A hybrid invisibility cloak based on integration of transparent metasurfaces and zero-index materials,” Light: Sci. Appl. 7(1), 50 (2018). [CrossRef]
21. P. Xie, G. M. Wang, H. P. Li, Y. W. Wang, and B. Zong, “Wideband RCS reduction of high gain fabry-perot antenna employing a receiver-transmitter metasurface,” Prog. Electromagn. Res. 169, 103–115 (2020). [CrossRef]
22. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72(1), 016623 (2005). [CrossRef]
23. Y. Lai, H. Chen, Z.-Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102(9), 093901 (2009). [CrossRef]
24. B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103(15), 153901 (2009). [CrossRef]
25. D. Rainwater, A. Kerkhoff, K. Melin, J. C. Soric, G. Moreno, and A. Alù, “Experimental verification of three-dimensional plasmonic cloaking in free-space,” New J. Phys. 14(1), 013054 (2012). [CrossRef]
26. T. Han and C. W. Qiu, “Transformation Laplacian metamaterials: recent advances in manipulating thermal and dc fields,” J. Opt. 18(4), 044003 (2016). [CrossRef]
27. C. Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett. 92(25), 251907 (2008). [CrossRef]
28. T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction,” Appl. Phys. Lett. 93(11), 114103 (2008). [CrossRef]
29. R. Schittny, M. Kadic, S. Guenneau, and M. Wegener, “Experiments on Transformation Thermodynamics: Molding the Flow of Heat,” Phys. Rev. Lett. 110(19), 195901 (2013). [CrossRef]
30. T. H. Chen, F. Yang, and Z. L. Mei, “A simple and flexible thermal illusion device and its experimental verification,” Phys. Status Solidi 212(8), 1746–1750 (2015). [CrossRef]
31. Y. Li, X. Shen, Z. Wu, J. Huang, Y. Chen, Y. Ni, and J. Huang, “Temperature-Dependent Transformation Thermotics: From Switchable Thermal Cloaks to Macroscopic Thermal Diodes,” Phys. Rev. Lett. 115(19), 195503 (2015). [CrossRef]
32. Y. Li, K. J. Zhu, Y. G. Peng, W. Li, T. Yang, H. X. Xu, H. Chen, X. F. Zhu, S. Fan, and C. W. Qiu, “Thermal meta-device in analogue of zero-index photonics,” Nat. Mater. 18(1), 48–54 (2019). [CrossRef]
33. S. Narayana and Y. Sato, “DC Magnetic Cloak,” Adv. Mater. 24(1), 71–74 (2012). [CrossRef]
34. F. Gömöry, M. Solovyov, J. Souc, C. Navau, J. Prat-Camps, and A. Sanchez, “Experimental realization of a magnetic cloak,” Science 335(6075), 1466–1468 (2012). [CrossRef]
35. C. Navau, J. Prat-Camps, and A. Sanchez, “Magnetic Energy Harvesting and Concentration at a Distance by Transformation Optics,” Phys. Rev. Lett. 109(26), 263903 (2012). [CrossRef]
36. A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas. 24(2), 413–419 (2003). [CrossRef]
37. F. Yang, Z. L. Mei, T. Y. Jin, and T. J. Cui, “dc Electric Invisibility Cloak,” Phys. Rev. Lett. 109(5), 053902 (2012). [CrossRef]
38. Q. Ma, Z. L. Mei, S. K. Zhu, T. Y. Jin, and T. J. Cui, “Experiments on Active Cloaking and Illusion for Laplace Equation,” Phys. Rev. Lett. 111(17), 173901 (2013). [CrossRef]
39. W. Xiang Jiang, C. Yang Luo, Z. Lei Mei, and T. Jun Cui, “An ultrathin but nearly perfect direct current electric cloak,” Appl. Phys. Lett. 102(1), 014102 (2013). [CrossRef]
40. F. Yang, Z. L. Mei, X. Y. Yang, T. Y. Jin, and T. J. Cui, “A negative conductivity material makes a dc invisibility cloak hide an object at a distance,” Adv. Funct. Mater. 23(35), 4306–4310 (2013). [CrossRef]
41. T. Han, H. Ye, Y. Luo, S. P. Yeo, J. Teng, S. Zhang, and C.-W. Qiu, “Manipulating DC Currents with Bilayer Bulk Natural Materials,” Adv. Mater. 26(21), 3478–3483 (2014). [CrossRef]
42. L. Zeng, “Bi-layer steady state current cloak,” Phys. Lett. A 378(13), 923–926 (2014). [CrossRef]
43. G. Fujii and Y. Akimoto, “dc electric cloak concentrator via topology optimization,” Phys. Rev. E 102(3), 033308 (2020). [CrossRef]
44. T. Chen, B. Zheng, Y. Yang, L. Shen, Z. Wang, F. Gao, E. Li, Y. Luo, T. J. Cui, and H. Chen, “Direct current remote cloak for arbitrary objects,” Light: Sci. Appl. 8(1), 30 (2019). [CrossRef]
