## A near-perfect invisibility cloak constructed with homogeneous materials

Optics Express, Vol. 17, Issue 26, pp. 23410-23416 (2009)

http://dx.doi.org/10.1364/OE.17.023410

Acrobat PDF (314 KB)

### Abstract

A near-perfect, non-singular cylindrical invisibility cloak with diamond cross section is achieved by a two-step coordinate transformation. A small line segment is stretched and then blown up into a diamond space, and finally the cloak consisting of four kinds and eight blocks of homogeneous transformation media is obtained. Numerical simulations confirm the well performance of the cloak. The operation bandwidth of the cloak is also investigated. Our scheme is promising to create a simple and well-performed cloak in practice.

© 2009 OSA

## 1. Introduction

15. G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett. **94**(4), 041904 (2009). [CrossRef]

16. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. **100**(6), 063903 (2008). [CrossRef] [PubMed]

17. J. S. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. **101**(20), 203901 (2008). [CrossRef] [PubMed]

18. E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A **79**(6), 063825 (2009). [CrossRef]

14. W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, “Invisibility Cloak without Singularity,” Appl. Phys. Lett. **93**(19), 194102 (2008). [CrossRef]

## 2. Derivation of the material parameters of the diamond shaped cloak

*x, y, z*), the transitional Cartesian space (

*x*',

*y*',

*z*'), and the final transformed Cartesian space (

*x*”,

*y*”,

*z*”), respectively. In the original space, a line segment with a length of 2

*a*and midpoint at origin is parallel to the

*x*axis. The big diamond area depicted in Fig. 1(a) outlines the space to be transformed. The first step is aim to stretch the line segment from 2

*a*to 2

*b*. For this purpose, a small diamond space of which the line segment is one of the diagonal lines is stretched in

*x*direction by

*b*/

*a*times, while the space between the two diamonds is compressed homogeneously in

*x*direction. As shown in Fig. 1(b), the transformed big diamond in space (

*x*',

*y*',

*z*') is divided into eight parts I-VIII by the stretched small diamond and the coordinate axes. For each part, we can give the transformation equations. Take the two parts (region I and II) in the first quadrant as examples, in region I:in region II:

*y*direction. Figure 1(c) shows the final transformed space. Finally, the transformation from the transitional Cartesian space (

*x*',

*y*',

*z*') to the final transformed Cartesian space (

*x*”,

*y*”,

*z*”) in region II is expressed as:

*x*” =

*x*' and

*y*” =

*y*'. At the end of the transformation, a cloak containing eight blocks of material is obtained whose outer boundary and inner boundary are both diamond shaped. Combining the transformations of the two steps, the total transformation in each block is thus obtained. We still take the two regions in the first quadrant as examples,in region I:in region II:

*M*=

*b*/

*a*,

*T*=

*e*/

*b*and

*K*= 1-

*e*/

*d*. The material parameters of the rest parts of the cloak can also be obtained similarly. Interestingly, the parameters in all regions are constants once the shape of the cloak and the equivalent line segment has been fixed, because no other variables are involved in the material parameters tensors in this case. Clearly, there is also no singularity in the material parameters. From Fig. 1(c), one can learn that the cloak is constructed with eight blocks of materials. Moreover, only four kinds of materials are needed indeed because the cloak is symmetrical with respect to the origin, i.e., the constitutive parameters in regions I, II, III and IV are identical to that of regions V, VI, VII and VIII, respectively. The above characteristics of the cloak highly simplify the construction of the device.

## 3. Numerical simulations and discussion

*ε*,

_{xx}*ε*,

_{xy}*ε*,

_{yx}*ε*and

_{yy}*μ*components of the material parameters tensor are relevant. Due to the symmetry of the material parameters tensor, we have

_{zz}*ε*=

_{xy}*ε*. So we only need

_{yx}*ε*,

_{xx}*ε*,

_{xy}*ε*and

_{yy}*μ*components for the simulations. The constants defining the shape of the diamond cloak are set as

_{zz}*b*= 1 m,

*c*= 2 m,

*d*= 4 m and

*e*= 1 m. The half length of the equivalent line segment

*a*is chosen to be a small value of 0.15 m. A perfect electric conducting (PEC) cylinder is fitted into the cloaked region. Figure 2(a) shows the magnetic field distribution when a TM polarized plane wave is impinging on the cloak from left to right. The wave fronts are bent around the cloaked area in the left part, and then restored their original directions by the right part, which is similar to most of the cloak deduced by the coordinate transformation method. The fields outside the cloak are almost unperturbed except tiny scattering due to the imperfection of the cloak. If the cloaked object is not a PEC cylinder which was just fitted into the cloaked region, the EM field will penetrate into the core of the device. In this case, the cloaking performance should vary with the properties such as the size, position and material parameters of the cloaked object. To avoid this, a PEC lining is usually added to the inner wall of the cloak shell.

2. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **74**(3), 036621 (2006). [CrossRef] [PubMed]

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] [PubMed]

*a*= 0.5 m. To further qualify these results, the total scattering cross sections versus

*a*are plotted in Fig. 4(a) when the incident wave is perpendicular, parallel and at a 45 degree angle to the +

*x*axis respectively. From Fig. 4(a) we learn that the longer the equivalent line segment, the stronger scattering the cloak will induce. With the increasing of the wave incident angle, the total scattering also increases quickly. However, the scattering can be kept at a very low level so long as the equivalent line segment is short.

*η*represents

_{j}*j*component of permittivity or permeability tensor, and

*f*(

_{j}*ω*) =

*ω*

_{p}^{2}/(

*ω*

_{aj}^{2}-

*ω*

^{2}-

*iωγ*) are the Lorentzian dispersive functions, where

*γ*is the damping factor,

*ω*is the plasma frequency,

_{p}*ω*is the atom resonated frequency. We set

_{aj}*ω*= 2π × 0.15 GHz,

_{0}*ω*= 4

_{p}*ω*,

_{0}*γ*= 0.01

*ω*,

_{0}*ω*= 0.6

_{aj}*ω*when

_{0}*η*< 1 and

_{j}*ω*= 1.4

_{aj}*ω*when

_{0}*η*≥ 1. Figure 4(b) shows the normalized total scattering cross section varying with the frequencies. At frequency of 0.15 GHz, the total scattering is quite small. However, as the EM wave frequency deviates from that point, the total scattering increases drastically because of the dispersion and the losses brought by the dispersion. In spite of that, the cloak is found to perform well in a certain range of frequencies. In the range of 0.13-0.17 GHz, the total scattering cross section can be reduced 50% at least. The result suggests that the diamond-shaped cloak do work at multiple frequencies other than a single frequency like the singular ones, though the bandwidth is limited due to the dispersion and losses of the cloak. It is imaginable that different dispersion parameters or different materials as well as the parameters of the cloak such as the length of the equivalent line segment may lead to different bandwidth, which we will discuss elsewhere.

_{j}## 4. Summary

## Acknowledgements

## References and links

1. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

2. | S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

3. | F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. |

4. | H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. |

5. | D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science |

6. | W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, “Nonmagnetic Cloak with Minimized Scattering,” Appl. Phys. Lett. |

7. | W. Yan, M. Yan, and M. Qiu, “Non-Magnetic Simplified Cylindrical Cloak with Suppressed Zeroth Order Scattering,” Appl. Phys. Lett. |

8. | B. I. Popa and S. A. Cummer, “Cloaking with Optimized Homogeneous Anisotropic Layers,” Phys. Rev. A |

9. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical Cloaking with Metamaterials,” Nat. Photonics |

10. | B. L. Zhang, H. S. Chen, and B. I. Wu, “Limitations of high-order transformation and incident angle on simplified invisibility cloaks,” Opt. Express |

11. | M. Yan, Z. Ruan, and M. Qiu, “Scattering characteristics of simplified cylindrical invisibility cloaks,” Opt. Express |

12. | G. Isic, R. Gajic, B. Novakovic, Z. V. Popovic, and K. Hingerl, “Radiation and scattering from imperfect cylindrical electromagnetic cloaks,” Opt. Express |

13. | M. Yan, Z. Ruan, and M. Qiu, “Cylindrical invisibility cloak with simplified material parameters is inherently visible,” Phys. Rev. Lett. |

14. | W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, “Invisibility Cloak without Singularity,” Appl. Phys. Lett. |

15. | G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett. |

16. | M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. |

17. | J. S. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. |

18. | E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A |

19. | K. Guven, E. Saenz, R. Gonzalo, E. Ozbay, and S. Tretyakov, “Electromagnetic cloaking with canonical spiral inclusions,” N. J. Phys. |

20. | H. Y. Chen and C. T. Chan, “Electromagnetic wave manipulation by layered systems using the transformation media concept,” Phys. Rev. B |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(260.2110) Physical optics : Electromagnetic optics

(160.3918) Materials : Metamaterials

(230.3205) Optical devices : Invisibility cloaks

**ToC Category:**

Physical Optics

**History**

Original Manuscript: September 21, 2009

Revised Manuscript: October 30, 2009

Manuscript Accepted: November 9, 2009

Published: December 7, 2009

**Citation**

Wei Li, Jianguo Guan, Zhigang Sun, Wei Wang, and Qingjie Zhang, "A near-perfect invisibility cloak constructed with homogeneous materials," Opt. Express **17**, 23410-23416 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-23410

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### References

- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
- S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621 (2006). [CrossRef] [PubMed]
- F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32(9), 1069–1071 (2007). [CrossRef] [PubMed]
- 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] [PubMed]
- D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]
- W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, “Nonmagnetic Cloak with Minimized Scattering,” Appl. Phys. Lett. 91(11), 111105 (2007). [CrossRef]
- W. Yan, M. Yan, and M. Qiu, “Non-Magnetic Simplified Cylindrical Cloak with Suppressed Zeroth Order Scattering,” Appl. Phys. Lett. 93(2), 021909 (2008). [CrossRef]
- B. I. Popa and S. A. Cummer, “Cloaking with Optimized Homogeneous Anisotropic Layers,” Phys. Rev. A 79(2), 023806 (2009). [CrossRef]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical Cloaking with Metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]
- B. L. Zhang, H. S. Chen, and B. I. Wu, “Limitations of high-order transformation and incident angle on simplified invisibility cloaks,” Opt. Express 16(19), 14655–14660 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14655 . [CrossRef] [PubMed]
- M. Yan, Z. Ruan, and M. Qiu, “Scattering characteristics of simplified cylindrical invisibility cloaks,” Opt. Express 15(26), 17772–17782 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-26-17772 . [CrossRef] [PubMed]
- G. Isic, R. Gajic, B. Novakovic, Z. V. Popovic, and K. Hingerl, “Radiation and scattering from imperfect cylindrical electromagnetic cloaks,” Opt. Express 16(3), 1413–1422 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-1413 . [CrossRef] [PubMed]
- M. Yan, Z. Ruan, and M. Qiu, “Cylindrical invisibility cloak with simplified material parameters is inherently visible,” Phys. Rev. Lett. 99(23), 233901 (2007). [CrossRef] [PubMed]
- W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, “Invisibility Cloak without Singularity,” Appl. Phys. Lett. 93(19), 194102 (2008). [CrossRef]
- G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett. 94(4), 041904 (2009). [CrossRef]
- M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008). [CrossRef] [PubMed]
- J. S. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]
- E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A 79(6), 063825 (2009). [CrossRef]
- K. Guven, E. Saenz, R. Gonzalo, E. Ozbay, and S. Tretyakov, “Electromagnetic cloaking with canonical spiral inclusions,” N. J. Phys. 10(11), 115037 (2008). [CrossRef]
- H. Y. Chen and C. T. Chan, “Electromagnetic wave manipulation by layered systems using the transformation media concept,” Phys. Rev. B 78(5), 054204 (2008). [CrossRef]

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