Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Equivalence between positive and negative refractive index materials in electrostatic cloaks

Abstract

We investigate, both theoretically and numerically, the equivalence relationship between the positive and negative refraction index dielectric materials in electrostatic invisibility cloak. We have derived an analytical formula that enables fast calculate the corresponding positive dielectric constant from the negative refraction index material. The numerical results show that the negative refraction index material can be replaced by the positive refractive index materials in the static field cloak. This offers some new viewpoints for designing new sensing systems and devices in physics, colloid science, and engineering applications.

Introduction

In military and some scientific experiments, hiding one object from the environmental is a fundamental requirement1. Several studies have confirmed that using metamaterials or designing special structures can make the target cloaking or hiding in the environment2,3. The transformation optics and scattering cancellation-based cloaking should be two powerful tools4,5,6. The basic idea of transformation optics is to manipulate electromagnetic waves by precisely designing the refractive index and permeability of every point and every direction in space. Scattering cancellation-based cloaking mainly using metamaterial, meta-surfaces, graphene and/or plasmonic materials to eliminate the scattering field of target object, and it can also be used in some physical fields7,8,9,10,11,12,13,14,15,16,17,18. As some of the literatures suggests that each cloaking technique has its own advantages and disadvantages7,19, exploring the new methods, for example, the illusion optics, or the common materials that can also make targets invisible from different ways, will significantly promote the practical application of relevant research results20,21,22.

At presents some researchers have reported that the invisibility cloak can be widely used in electromagnetic waves23,24,25,26, mechanical waves27, elastic waves28,29, matter waves30, water waves31, magnetic fields32, DC magnetic or electric fields33,34,35, current36, and thermal fields37,38,39. Based on the metamaterial, people can also design an electrostatic field concentrator3, magnetic field concentrator40, asymmetric universal and invisible gateway41, the perfect lens42,43, perfect transmission channel44, general illusion device21, transparency coating45, dc electric concentrator46, tunable invisibility cloaking47, special imaging probe48 and so on48,49,50,51. However, the materials used in these devices are anisotropic, negative refractive index medium, chiral materials, even double-negative materials or field gain materials50,52,53. Some papers also discuss the tunable electromagnetically induced transparency metamaterial based on solid-state plasma49 and the broadband perfect absorption based on plasmonic nanoparticles54. It is well known that the negative refractive index materials or metamaterials are hard to be produced, and usually, its sizes are much larger than the targets55. Can we find alternatives to negative index materials?

Besides, the electric field commonly exists in the nature, and it may change the object’s physical characteristic56,57,58. Some conditions we need effectively manipulate the electric field, for example, the measurement of electrostatic phenomena in sandstorms and other aerosol weather59,60, neuro-medicine 61, the shielding of an static electromagnetic field for some special devices or sensors62, the enhancement of localized electric field63, polymer self-assembly properties induced by strong electric field64, etc. In these applications, the placed sensor will undoubtedly cause electric field perturbations, which may affect the experimental results. In contrast, the precise control of the electric field in the target test area helps improve the experimental accuracy. Therefore, it is an interesting and meaningful research topic to design a device that does not notably perturb the applied electric field, but it still keeps the original information of the incident field, or it can adjust the localized field exactly according to our will. Generally speaking, existing designs are based either on anisotropic negative refraction index materials or on the geometric dimensions of the cloak structure65,66,67. At present, material doping has been used to fabricate some new materials with specific properties68,69. If the isotropic positive refractive index materials can also be used to design the metamaterials, we think the cloaking will become more useful. Some references have introduced the positive refractive index and isotropic material to induce “invisibility” in the Rayleigh limit for two-dimensional objects16,67,70,71. Then, is there an equivalent transformation relationship between positive and negative refractive index materials?

To answer the above question, we are inspired by the electric potential distribution function of the coated sphere in a uniform static field58,72,73,74. Supposed that the inner and outer radius of a coated sphere are \(R_{1}\) and \(R_{2}\), respectively, the permittivity in the core and shell are \(\varepsilon_{1} ,\varepsilon_{2}\), respectively, and the permittivity of environment medium is \(\varepsilon_{m}\), as illustrated in Fig. 1. The applied electric field is along the z-axis, and its intensity is \(E_{0}\). For the selected physical model, the core can be thought as the target to be invisible, and the shell zone can be viewed as the functional device that needs to be designed.

The potential inside and outside the core–shell particles can be calculated through the equation \(\nabla^{2} \phi { = }0\) with separating variables method, and the equivalent dielectric constant \(\varepsilon_{equ}\) of the core–shell particle also can be obtained73. We need to make \(\varepsilon_{equ} { = }\varepsilon_{m}\), and then a new equation derived,

$$\xi (2\varepsilon_{2} + \varepsilon_{m} )\left( {\varepsilon_{1} - \varepsilon_{2} } \right){ = }(\varepsilon_{m} - \varepsilon_{2} )\left( {\varepsilon_{1} + 2\varepsilon_{2} } \right)$$
(1)
Figure 1
figure1

Schematic of a core–shell particle in a uniform electrostatic field.

By solving the above equation and set \(\beta = \left( {2\xi + 1} \right)\varepsilon_{1} - \left( {2 + \xi } \right)\varepsilon_{m}\),\(\xi = {{R_{1}^{3} } \mathord{\left/ {\vphantom {{R_{1}^{3} } {R_{2}^{3} }}} \right. \kern-\nulldelimiterspace} {R_{2}^{3} }}\), we can get two roots of \(\varepsilon_{2}\) for the Eq. (1) we defined them as \(\varepsilon_{21}^{a}\) and \(\varepsilon_{21}^{b}\), which is expressed as following.

$$\varepsilon_{21}^{a} = \frac{{ - \beta + \sqrt {\beta^{2} { + }8\left( {1 - \xi } \right)^{2} \varepsilon_{1} \varepsilon_{m} } }}{{4\left( {1 - \xi } \right)}}$$
(2)
$$\varepsilon_{21}^{b} = \frac{{ - \beta - \sqrt {\beta^{2} { + }8\left( {1 - \xi } \right)^{2} \varepsilon_{1} \varepsilon_{m} } }}{{4\left( {1 - \xi } \right)}}$$
(3)

In addition, we made the two roots \(\varepsilon_{21}^{a}\) and \(\varepsilon_{21}^{b}\) divided by \(\varepsilon_{m}\), then we can obtain the relative permittivity of the shell, and Eqs. (2) and (3) became to a new expression,

$$\upvarepsilon _{2}^{{\text{a}}} = \frac{{ - \beta_{1} + \sqrt {\beta_{1}^{2} + 8\left( {1 - \xi } \right)^{2} \varepsilon_{1r} } }}{{4\left( {1 - \xi } \right)}}$$
(4)
$$\upvarepsilon _{{2}}^{{\text{b}}} = \frac{{ - \beta_{1} - \sqrt {\beta_{1}^{2} { + }8\left( {1 - \xi } \right)^{2} \varepsilon_{1r} } }}{{4\left( {1 - \xi } \right)}}$$
(5)

here \(\beta_{1} = \left( {2\xi + 1} \right)\varepsilon_{1r} - \left( {2 + \xi } \right)\),\(\varepsilon_{1r} { = }{{\varepsilon_{1} } \mathord{\left/ {\vphantom {{\varepsilon_{1} } {\varepsilon_{m} }}} \right. \kern-\nulldelimiterspace} {\varepsilon_{m} }}\) is the relative permittivity of core. The numerical simulation results shown that \({\upvarepsilon }_{{2}}^{{\text{a}}}\) corresponds to an isotropic positive refractive index dielectric material, but \(\upvarepsilon _{{2}}^{{\text{b}}}\) corresponds to the negative refractive index material, which means there is a reciprocity relationship between the positive and negative refraction index dielectric materials. Considering the similarity of the two expressions in Eqs. (4) and (5), and add these two equations together. Through some simplify calculation, we can obtain the following formula,

$$\upvarepsilon _{{2}}^{{\text{a}}} = \frac{{\left( {2 + \xi } \right) - \left( {2\xi + 1} \right)\varepsilon_{1r} }}{{2\left( {1 - \xi } \right)}} -\upvarepsilon _{{2}}^{{\text{b}}}$$
(6)

Therefore, we can calculate the matching positive refractive index according to the negative refractive index for the electrostatic field invisibility cloak. That means if we have obtained the material permittivity of a cloak through the transformation optics or other method, we can deduce its equivalent positive refractive index parameters, which make it much simpler to design a required invisibility structure.

Results

To clearly verify the reliability of the above formulas, we calculate the electric potential of the particle surrounded by the designed cloak, whose permittivity is given by the solution of Eq. (4), and the equipotential lines are shown to investigate the perturbation of the particle to the applied electric field. The electric potentials are calculated by Eqs. (7)–(9). Here we set \(\varepsilon_{1r} = 2.0 + 0.1i\),\(R_{1} = 0.5\;{\text{m}}\), \(R_{2} = 0.7\;{\text{m}}\), the thickness of the shell \(dr = R_{2} - R_{1}\), \(E_{0} = 100\;{\text{V/m}}\). Substitute them into the Eqs. (4) and (5), we can obtain that \({\upvarepsilon }_{{2}}^{{\text{a}}} = 0.6575 - 0.0181i\) and \({\upvarepsilon }_{{2}}^{{\text{b}}} = - 1.5176 - 0.1179i\). The results are shown in Fig. 2. It can be seen from the Fig. 2a that the electric field around the object without the cloak covering are severely distorted, but from Fig. 2b, c we can find there are no any perturbation around the object. It means the spherical object will be perfectly "invisible" in the electric field. Therefore, besides the negative refractive index materials reported by some researchers, the non-negative refractive index materials also can make the object invisible in the electric field. Compared with Fig. 2a, the number of equipotential lines inside the sphere decreases in Fig. 2b (the color becomes lighter), but increases in Fig. 2c (the color becomes darker). So, we conclude that the cloak with permittivity \({\upvarepsilon }_{{2}}^{{\text{a}}}\) can shield the applied electric field to a certain extent, while the cloak with permittivity \({\upvarepsilon }_{{2}}^{{\text{b}}}\) can increase its internal electric potential. Based on these properties, we can design some special devices following the requirement of the experiment. For example, we can make the electrostatic field sensor coated with the material \({\upvarepsilon }_{{2}}^{{\text{a}}}\) to avoid the distortion of the external electric field caused by the sensor, and we also can design a more sensitive sensor with material \({\upvarepsilon }_{{2}}^{{\text{b}}}\) to measure a weak signal. Furthermore, we also find that when the imaginary part of the cloak’s permittivity is positive, the field distribution is the same as that when the imaginary part of cloak’s permittivity is negative, as shown in Fig. 3. So we can conclude that the imaginary part of the medium’s dielectric constant does not affect the response of the medium to the external electrostatic field.

Figure 2
figure2

The equipotential line of different spherical object in a uniform electric field.

Figure 3
figure3

The equipotential line of a layered object with different shell.

In order to analyze the effect of the core zone on the permittivity of the cloak, we set the permittivity of the core particle can be expressed by \(\varepsilon_{1r} = \varepsilon^{r} + i\varepsilon^{i}\). Figure 4 shows the effect of the real part of permittivity of the core zone on the permittivity of cloak. We have set \(\varepsilon^{r} = 2 \times n\) and \(\varepsilon^{i} = 0.1\), \(n\) is the magnification of the real part of permittivity, other parameters are the same as above. From the Fig. 4, we can find that with the increase of the parameter \(\varepsilon^{r}\), both the real part and the imaginary part of the dielectric constant \({\upvarepsilon }_{{2}}^{{\text{a}}}\) decreases exponentially, and finally tend to be a stable value. However, the dielectric constant \({\upvarepsilon }_{{2}}^{{\text{b}}}\) shows different changing rules, and its real and imaginary parts both increase exponentially, but its real part does not tend to be stable. In addition, we have observed that with the increase of the cloak thickness, the real part of the permittivity \({\upvarepsilon }_{{2}}^{{\text{a}}}\) increases continuously, but its imaginary part and the permittivity \({\upvarepsilon }_{{2}}^{{\text{b}}}\) both decrease continuously. So, the requirement of cloak’s permittivity can be improved by adjusting its thickness, which makes it possible to design a practical cloak. In addition, we also can use material doping to obtain the proper dielectric constant75,76.

Figure 4
figure4

Influence of \(\varepsilon^{r}\) on the permittivity of the cloak with different permittivity.

In Fig. 5 we make a similar discussion on the absorbent particle, with a permittivity \(\varepsilon^{r} = 2,\varepsilon^{i} = n \times 0.1\), \(n\) is the magnification of the imaginary part of core’s permittivity. Other parameters are the same as above. We find that with the increase of \(\varepsilon^{i}\), the real part of \({\upvarepsilon }_{{2}}^{{\text{a}}}\) decreases linearly, while its imaginary part is increasing linearly. However, the dielectric constant of the cloak designed by the negative refractive material \({\upvarepsilon }_{{2}}^{{\text{b}}}\) still increases linearly. However, no matter which one material is used, the change degree of the real part of the dielectric constant of the cloak is much less than that of its imaginary part. Besides, we also found that when the ratio between the inner radius and outer radius of the core–shell particle remains the same, the change of particle geometry size does not affect the dielectric constant of the required shell (cloak) medium.

Figure 5
figure5

Influence of \(\varepsilon^{i}\) on the permittivity of the cloak with different permittivity.

Discussion

In summary, based on the theoretical derivation and numerical simulation, an equivalence relationship between positive-permittivity and negative-permittivity materials in electric invisibility cloak is proposed. We also present a formula to realize the conversion between the positive and negative permittivity of the corresponding materials. The numerical results show that both positive-permittivity and negative-permittivity materials all can be used to achieve an electric invisibility cloak, and the positive-permittivity cloak can reduce the electric field inside it, while the negative-permittivity cloak can enhance the electric field inside it. In addition, we find that the permittivity of cloak is influenced by the physical parameters of core and the thickness of cloak. In terms of the equivalence of real physical field, this idea is feasible, and it also can be applied to other types of physical fields. Moreover, especially and importantly, we have demonstrated the conversion relationship between the positive-permittivity and negative-permittivity dielectric materials, further research is needed to determine whether similar relationships exist in other physical fields.

Methods

The distribution of electric potential inside and outside the particle can be represented as the following equations in the spherical coordinates73.

$$\phi_{1} = - AE_{0} r\cos \theta$$
(7)
$$\phi_{2} = - E_{0} (Br - Cr^{ - 2} )\cos \theta$$
(8)
$$\phi_{m} = - E_{0} (r - Dr^{ - 2} )\cos \theta$$
(9)

Supposed that \(x = {{\left( {\varepsilon_{1} - \varepsilon_{2} } \right)} \mathord{\left/ {\vphantom {{\left( {\varepsilon_{1} - \varepsilon_{2} } \right)} {\left( {\varepsilon_{1} + 2\varepsilon_{2} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\varepsilon_{1} + 2\varepsilon_{2} } \right)}},\xi = {{R_{1}^{3} } \mathord{\left/ {\vphantom {{R_{1}^{3} } {R_{2}^{3} }}} \right. \kern-\nulldelimiterspace} {R_{2}^{3} }}\), the parameters \(A,B,C,D\) can be calculated through the following expressions73,

$$\begin{gathered} A = \frac{{9\varepsilon_{m} \varepsilon_{2} }}{{(\varepsilon_{2} + 2\varepsilon_{m} )(\varepsilon_{1} + 2\varepsilon_{2} ) + 2\xi (\varepsilon_{2} - \varepsilon_{m} )(\varepsilon_{1} - \varepsilon_{2} )}}\quad B = \frac{{3\varepsilon_{m} (\varepsilon_{1} + 2\varepsilon_{2} )}}{{(\varepsilon_{2} + 2\varepsilon_{m} )(\varepsilon_{1} + 2\varepsilon_{2} ) + 2\xi (\varepsilon_{2} - \varepsilon_{m} )(\varepsilon_{1} - \varepsilon_{2} )}} \hfill \\ C = \frac{{3\varepsilon_{m} (\varepsilon_{1} + \varepsilon_{2} )r_{2}^{3} }}{{(\varepsilon_{2} + 2\varepsilon_{m} )(\varepsilon_{1} + 2\varepsilon_{2} ) + 2\xi (\varepsilon_{2} - \varepsilon_{m} )(\varepsilon_{1} - \varepsilon_{2} )}}\quad D = \frac{{\xi (2\varepsilon_{2} + \varepsilon_{m} )x + (\varepsilon_{2} - \varepsilon_{m} )}}{{2\xi (\varepsilon_{2} - \varepsilon_{m} )x + (\varepsilon_{2} + 2\varepsilon_{m} )}}r_{2}^{3} \hfill \\ \end{gathered}$$

the equivalent dielectric constant \(\varepsilon_{equ}\) of the core–shell particle also can be obtained,

$$\varepsilon_{equ} = \frac{1 + 2\xi x}{{1 - \xi x}}\varepsilon_{2}$$
(10)

In order to cancel the perturbation of the external field by the object, we need to make \(\varepsilon_{equ} { = }\varepsilon_{m}\) and then a new equation derived,

$$\xi (2\varepsilon_{2} + \varepsilon_{m} )x + (\varepsilon_{2} - \varepsilon_{m} ) = 0$$
(11)

Then through some simplified operation we can obtain the Eq. (1).

In addition, if we add the Eqs. (4) and (5) together, then we can obtain,

$${\upvarepsilon }_{{2}}^{{\text{a}}} + {\upvarepsilon }_{{2}}^{{\text{b}}} = \frac{{ - 2\beta_{1} }}{{4\left( {1 - \xi } \right)}} = \frac{{ - \beta_{1} }}{{2\left( {1 - \xi } \right)}}$$
(12)

Then substitute the expression of \(\beta_{1}\) in Eq. (12) we can obtain the Eq. (6).

The above derivation can also be applied to the cloak design in static magnetic fields, or the condition that the object size is much smaller than electromagnetic wavelength.

References

  1. 1.

    Alu, A. & Engheta, N. Cloaking a sensor. Phys. Rev. Lett. 102, 233901. https://doi.org/10.1103/PhysRevLett.102.233901 (2009).

    ADS  CAS  Article  PubMed  Google Scholar 

  2. 2.

    Cai, W., Chettiar, U. K., Kildishev, A. V. & Shalaev, V. M. Optical cloaking with metamaterials. Nat. Photonics 1, 224–227 (2007).

    ADS  CAS  Article  Google Scholar 

  3. 3.

    Liu, W. et al. Enhancement of electrostatic field by a metamaterial electrostatic concentrator. J. Alloys Compd. 724, 1064–1069 (2017).

    CAS  Article  Google Scholar 

  4. 4.

    Chen, H., Chan, C. T. & Sheng, P. Transformation optics and metamaterials. Nat. Mater. 9, 387–396 (2010).

    ADS  CAS  PubMed  Article  Google Scholar 

  5. 5.

    Pendry, J. B. Controlling electromagnetic fields. Science 312, 1780–1782 (2006).

    ADS  MathSciNet  CAS  PubMed  MATH  Article  Google Scholar 

  6. 6.

    Schurig, D., Pendry, J. B. & Smith, D. R. Calculation of material properties and ray tracing in transformation media. Opt. Express 14, 9794–9804 (2006).

    ADS  CAS  PubMed  Article  Google Scholar 

  7. 7.

    Chen, P. Y., Soric, J. & Alu, A. Invisibility and cloaking based on scattering cancellation. Adv. Mater. 24, 281–304 (2012).

    Google Scholar 

  8. 8.

    Schofield, R. S., Soric, J. C., Rainwater, D., Kerkhoff, A. & Alù, A. Scattering suppression and wideband tunability of a flexible mantle cloak for finite-length conducting rods. New J. Phys. 16, 063063 (2014).

    ADS  Article  Google Scholar 

  9. 9.

    Fleury, R. & Alù, A. Quantum cloaking based on scattering cancellation. Phys. Rev. B 87, 045423 (2013).

    ADS  Article  CAS  Google Scholar 

  10. 10.

    Farhat, M., Guenneau, S., Chen, P. Y., Alù, A. & Salama, K. N. Scattering cancellation-based cloaking for the Maxwell-Cattaneo heat waves. Phys. Rev. Appl. 11, 044089 (2019).

    ADS  CAS  Article  Google Scholar 

  11. 11.

    Alù, A. Mantle cloak: Invisibility induced by a surface. Phys. Rev. B 80, 245115 (2009).

    ADS  Article  CAS  Google Scholar 

  12. 12.

    Guild, M. D., Alù, A. & Haberman, M. R. Cloaking of an acoustic sensor using scattering cancellation. Appl. Phys. Lett. 105, 023510 (2014).

    ADS  Article  CAS  Google Scholar 

  13. 13.

    Chen, A. & Monticone, F. Active scattering-cancellation cloaking: broadband invisibility and stability constraints. IEEE Trans. Antennas Propag. 68, 1655–1664 (2020).

    ADS  Article  Google Scholar 

  14. 14.

    Xu, S., Wang, Y., Zhang, B. & Chen, H. Invisibility cloaks from forward design to inverse design. Sci. China Inf. Sci. 56, 1–11 (2013).

    ADS  CAS  Google Scholar 

  15. 15.

    Lee, J. Y. & Lee, R.-K. Hiding the interior region of core-shell nanoparticles with quantum invisible cloaks. Phys. Rev. B 89, 155425 (2014).

    ADS  Article  CAS  Google Scholar 

  16. 16.

    Lan, C., Yang, Y., Geng, Z., Li, B. & Zhou, J. Electrostatic field invisibility cloak. Sci. Rep. 5, 16416 (2015).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  17. 17.

    Cummer, S. A., Popa, B.-I., Schurig, D., Smith, D. R. & Pendry, J. Full-wave simulations of electromagnetic cloaking structures. Phys. Rev. E 74, 036621 (2006).

    ADS  Article  CAS  Google Scholar 

  18. 18.

    Luo, Y., Zhang, J., Chen, H. & Wu, B.-I. Full-wave analysis of prolate spheroidal and hyperboloidal cloaks. J. Phys. D Appl. Phys. 41, 235101 (2008).

    ADS  Article  CAS  Google Scholar 

  19. 19.

    Farhat, M. et al. Understanding the functionality of an array of invisibility cloaks. Phys. Rev. B 84, 235105 (2011).

    ADS  Article  CAS  Google Scholar 

  20. 20.

    Zhou, L., Huang, S., Wang, M., Hu, R. & Luo, X. While rotating while cloaking. Phys. Lett. A 383, 759–763. https://doi.org/10.1016/j.physleta.2018.11.041 (2019).

    ADS  CAS  Article  MATH  Google Scholar 

  21. 21.

    Lai, Y. et al. Illusion optics: the optical transformation of an object into another object. Phys. Rev. Lett. 102, 253902 (2009).

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  22. 22.

    Lin, J.-H., Yen, T.-J. & Huang, T.-Y. Design of annulus-based dielectric metamaterial cloak with properties of illusion optics. J. Opt. 22, 085101. https://doi.org/10.1088/2040-8986/ab9cdb (2020).

    ADS  CAS  Article  Google Scholar 

  23. 23.

    Choi, M. et al. A terahertz metamaterial with unnaturally high refractive index. Nature 470, 369–373 (2011).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  24. 24.

    Pendry, J. B., Fernández-Domínguez, A. I., Luo, Y. & Zhao, R. Capturing photons with transformation optics. Nat. Phys. 9, 518–522 (2013).

    CAS  Article  Google Scholar 

  25. 25.

    Ang, P. & Eleftheriades, G. V. Active cloaking of a non-uniform scatterer. Sci. Rep. 10, 2021 (2020).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  26. 26.

    Vitiello, A. et al. Waveguide characterization of S-band microwave mantle cloaks for dielectric and conducting objects. Sci. Rep. 6, 19716. https://doi.org/10.1038/srep19716 (2016).

    ADS  CAS  Article  PubMed  PubMed Central  Google Scholar 

  27. 27.

    Zhang, S., Xia, C. & Fang, N. Broadband acoustic cloak for ultrasound waves. Phys. Rev. Lett. 106, 024301 (2011).

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  28. 28.

    Farhat, M., Guenneau, S. & Enoch, S. Ultrabroadband elastic cloaking in thin plates. Phys. Rev. Lett. 103, 24301 (2009).

    ADS  Article  CAS  Google Scholar 

  29. 29.

    Quadrelli, D. E., Craster, R., Kadic, M. & Braghin, F. Elastic wave near-cloaking. Extreme Mech. Lett. 44, 101262. https://doi.org/10.1016/j.eml.2021.101262 (2021).

    Article  Google Scholar 

  30. 30.

    Zhang, S., Genov, D. A., Sun, C. & Zhang, X. Cloaking of matter waves. Phys. Rev. Lett. 100, 123002 (2008).

    ADS  PubMed  Article  CAS  Google Scholar 

  31. 31.

    Zou, S. et al. Broadband waveguide cloak for water waves. Phys. Rev. Lett. 123, 074501 (2019).

    ADS  CAS  PubMed  Article  Google Scholar 

  32. 32.

    Gomory, F. et al. Experimental realization of a magnetic cloak. Science 335, 1466–1468 (2012).

    ADS  PubMed  Article  CAS  Google Scholar 

  33. 33.

    Narayana, S. & Sato, Y. DC magnetic cloak. Adv. Mater. 24, 71–74 (2012).

    CAS  PubMed  Article  Google Scholar 

  34. 34.

    Yang, F., Mei, Z. L., Jin, T. Y. & Cui, T. J. DC electric invisibility cloak. Phys. Rev. Lett. 109, 053902 (2012).

    ADS  PubMed  Article  CAS  Google Scholar 

  35. 35.

    Alekseev, G. V., Tereshko, D. A. & Shestopalov, Y. V. Optimization approach for axisymmetric electric field cloaking and shielding. Inverse Probl. Sci. Eng. 29, 40–55. https://doi.org/10.1080/17415977.2020.1772780 (2020).

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Chen, T. et al. Direct current remote cloak for arbitrary objects. Light Sci. Appl. 8, 30 (2019).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  37. 37.

    Schittny, R., Kadic, M., Guenneau, S. & Wegener, M. Experiments on transformation thermodynamics: molding the flow of heat. Phys. Rev. Lett. 110, 195901 (2013).

    ADS  PubMed  Article  CAS  Google Scholar 

  38. 38.

    Guenneau, S., Amra, C. & Veynante, D. Transformation thermodynamics: cloaking and concentrating heat flux. Opt. Express 20, 8207–8218 (2012).

    ADS  PubMed  Article  Google Scholar 

  39. 39.

    Narayana, S. & Sato, Y. Heat flux manipulation with engineered thermal materials. Phys. Rev. Lett. 108, 214303 (2012).

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  40. 40.

    Madni, H. A. et al. A novel EM concentrator with open-concentrator region based on multi-folded transformation optics. Sci. Rep. 8, 9641 (2018).

    ADS  PubMed  PubMed Central  Article  CAS  Google Scholar 

  41. 41.

    Xu, Y. et al. Asymmetric universal invisible gateway. Opt. Express 28, 35363 (2020).

    ADS  PubMed  Article  Google Scholar 

  42. 42.

    Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966–3969 (2000).

    ADS  CAS  PubMed  Article  Google Scholar 

  43. 43.

    Pendry, J. B. Perfect cylindrical lenses. Opt. Express 11, 755–760 (2003).

    ADS  CAS  PubMed  Article  Google Scholar 

  44. 44.

    Sebbah, P. A channel of perfect transmission. Nat. Photonics 11, 337–339 (2017).

    ADS  CAS  Article  Google Scholar 

  45. 45.

    Alu, A. & Engheta, N. Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 72, 016623. https://doi.org/10.1103/PhysRevE.72.016623 (2005).

    ADS  CAS  Article  Google Scholar 

  46. 46.

    Jiang, W. X., Luo, C. Y., Ma, H. F., Mei, Z. L. & Cui, T. J. Enhancement of current density by dc electric concentrator. Sci. Rep. 2, 956 (2012).

    ADS  PubMed  PubMed Central  Article  CAS  Google Scholar 

  47. 47.

    Naserpour, M., Zapata-Rodriguez, C. J., Vukovic, S. M., Pashaeiadl, H. & Belic, M. R. Tunable invisibility cloaking by using isolated graphene-coated nanowires and dimers. Sci. Rep. 7, 12186 (2017).

    ADS  PubMed  PubMed Central  Article  CAS  Google Scholar 

  48. 48.

    Isakov, D., Stevens, C. J., Castles, F. & Grant, P. S. A split ring resonator dielectric probe for near-field dielectric imaging. Sci. Rep. 7, 2038. https://doi.org/10.1038/s41598-017-02176-3 (2017).

    ADS  CAS  Article  PubMed  PubMed Central  Google Scholar 

  49. 49.

    Chen, Q., Li, F., Zhang, D. & Zhang, H. Tunable electromagnetically induced transparency metamaterial based on solid-state plasma: from a narrow band to a broad one. J. Opt. Soc. Am. B 38, 1571–1578 (2021).

    ADS  Article  Google Scholar 

  50. 50.

    Zeng, L. & Song, R. Quantized chiral anomaly materials cloak. Sci. Rep. 7, 3253 (2017).

    ADS  PubMed  PubMed Central  Article  CAS  Google Scholar 

  51. 51.

    Sidhwa, H. H., Aiyar, R. P. R. C. & Kavehvash, Z. Cloaking of irregularly shaped bodies using coordinate transformation. Optik 197, 163201. https://doi.org/10.1016/j.ijleo.2019.163201 (2019).

    ADS  Article  Google Scholar 

  52. 52.

    Ramakrishna, S. A. Physics of negative refractive index materials. Rep. Prog. Phys. 68, 449 (2005).

    ADS  Article  Google Scholar 

  53. 53.

    Ramakrishna, S. A. & Tomasz, M. G. Physics and Applications of Negative Refractive Index Materials (SPIE Press, CRC Press, 2009).

    Google Scholar 

  54. 54.

    Bordo, V. G. Theory of light reflection and transmission by a plasmonic nanocomposite slab: emergence of broadband perfect absorption. J. Opt. Soc. Am. B 38, 1442–1451 (2021).

    ADS  Article  Google Scholar 

  55. 55.

    Ma, H. F., Jiang, W. X., Yang, X. M., Zhou, X. Y. & Cui, T. J. Compact-sized and broadband carpet cloak and free-space cloak. Opt. Express 17, 19947–19959 (2009).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  56. 56.

    Xingcai, L., Xing, M. & Dandan, L. Rayleigh approximation for the scattering of small partially charged sand particles. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 31, 1495–1501 (2014).

    Article  Google Scholar 

  57. 57.

    Li, X., Liu, D. & Min, X. The electric field in sandstorm can strongly affect the sand’s scattering properties. J. Quant. Spectrosc. Radiat. Transfer 149, 103–107 (2014).

    CAS  Article  Google Scholar 

  58. 58.

    Li, X., Wang, M., Zhang, B. & Ming, H. Scattering of small particles and the remote sensing of sandstorm by microwave (Electronic Industry Press, 2017).

    Google Scholar 

  59. 59.

    Zheng, X. Mechanics of Wind-Blown Sand Movements (Springer, 2009).

    Book  Google Scholar 

  60. 60.

    Zheng, X. Electrification of wind-blown sand: recent advances and key issues. Eur. Phys. J. E Soft Matter 36, 138 (2013).

    PubMed  Article  CAS  PubMed Central  Google Scholar 

  61. 61.

    Tofts, P. S. & Branston, N. M. The measurement of electric field, and the influence of surface charge, in magnetic stimulation. Electroencephalogr. Clin. Neurophysiol. Evoked Potentials Sect. 81, 238–239 (1991).

    CAS  Article  Google Scholar 

  62. 62.

    Jaroszewski, M., Thomas, S. & Rane, A. V. Advanced Materials for Electromagnetic Shielding: Fundamentals, Properties, and Applications (Wiely, 2019).

    Google Scholar 

  63. 63.

    Ibraheem, A.-N. et al. Effect of local field enhancement on the nonlinear terahertz response of a silicon-based metamaterial. Phys. Rev. B 88, 195203 (2013).

    ADS  Article  CAS  Google Scholar 

  64. 64.

    Yan, L.-T. & Xie, X.-M. Computational modeling and simulation of nanoparticle self-assembly in polymeric systems: structures, properties and external field effects. Prog. Polym. Sci. 38, 369–405 (2013).

    CAS  Article  Google Scholar 

  65. 65.

    Kadic, M., Milton, G. W., van Hecke, M. & Wegener, M. 3D metamaterials. Nat. Rev. Phys. 1, 198–210 (2019).

    Article  Google Scholar 

  66. 66.

    Askari, M. et al. Additive manufacturing of metamaterials: a review. Addit. Manuf. 36, 101562 (2020).

    Google Scholar 

  67. 67.

    Zhang, J., Liu, L., Luo, Y., Zhang, S. & Mortensen, N. A. Homogeneous optical cloak constructed with uniform layered structures. Opt. Express 19, 8625–8631 (2011).

    ADS  PubMed  Article  PubMed Central  Google Scholar 

  68. 68.

    Su, P., Ma, C. G., Brik, M. G. & Srivastava, A. M. A short review of theoretical and empirical models for characterization of optical materials doped with the transition metal and rare earth ions. Opt. Mater. 79, 129–136. https://doi.org/10.1016/j.optmat.2018.03.032 (2018).

    ADS  CAS  Article  Google Scholar 

  69. 69.

    Gupta, S. K., Sudarshan, K. & Kadam, R. M. Optical nanomaterials with focus on rare earth doped oxide. Mater. Today Commun. https://doi.org/10.1016/j.mtcomm.2021.102277 (2021).

    Article  Google Scholar 

  70. 70.

    Huang, Y., Feng, Y. & Jiang, T. Electromagnetic cloaking by layered structure of homogeneous isotropic materials. Opt. Express 15, 11133–11141 (2007).

    ADS  PubMed  Article  Google Scholar 

  71. 71.

    Qiu, C.-W., Hu, L., Xu, X. & Feng, Y. Spherical cloaking with homogeneous isotropic multilayered structures. Phys. Rev. E 79, 047602. https://doi.org/10.1103/PhysRevE.79.047602 (2009).

    ADS  CAS  Article  Google Scholar 

  72. 72.

    Bohren, C. F. & Huffman, D. R. Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983).

    Google Scholar 

  73. 73.

    Li, X. & Zhang, B. An equivalent solution for the electromagnetic scattering of multilayer particle. J. Quant. Spectrosc. Radiat. Transf. 129, 236–240 (2013).

    CAS  Article  Google Scholar 

  74. 74.

    Li, X., Wang, M. & Zhang, B. Equivalent medium theory of layered sphere particle with anisotropic shells. J. Quant. Spectrosc. Radiat. Transfer 179, 165–169. https://doi.org/10.1016/j.jqsrt.2016.03.008 (2016).

    ADS  CAS  Article  Google Scholar 

  75. 75.

    Markel, V. A. Maxwell Garnett approximation (advanced topics): tutorial. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 33, 2237–2255 (2016).

    ADS  PubMed  Article  Google Scholar 

  76. 76.

    Mallet, P., Guerin, C. A. & Sentenac, A. Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy. Phys. Rev. B 72, 014205 (2005).

    ADS  Article  CAS  Google Scholar 

Download references

Acknowledgements

This research was supported by a grant from National Natural Science Foundation of China (12064034, 11562017, 11302111), the Leading Talents Program of Science and Technology innovation in Ningxia Province, in China (2020GKLRLX08), the CAS Light of West China Program (XAB2017AW03), the Key Research and Development Program of Ningxia Province in China (2018BFH03004), The Natural Science Foundation of the Jiangsu Higher Education Institutions of China (19KJB560005). The authors express their sincere appreciation to the supports.

Author information

Affiliations

Authors

Contributions

X.L. introduced the research idea and participated in the data analysis, J.W. finished the calculation and prepared the manuscript, J.Z. participated in writing the manuscript. All authors reviewed the manuscript.

Corresponding author

Correspondence to Xingcai Li.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, X., Wang, J. & Zhang, J. Equivalence between positive and negative refractive index materials in electrostatic cloaks. Sci Rep 11, 20467 (2021). https://doi.org/10.1038/s41598-021-00124-w

Download citation

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links