## Approximation approach of designing practical cloaks with arbitrary shapes

Optics Express, Vol. 16, Issue 20, pp. 15449-15454 (2008)

http://dx.doi.org/10.1364/OE.16.015449

Acrobat PDF (161 KB)

### Abstract

The approach of designing cloaks with arbitrary shapes was investigated. The coordinate transformation was considered as a homeomorphous topological mapping and the related geometrical description was proposed by introducing position surfaces and tracing lines. Then an approximation approach was presented, which considers a close shape as a polyhedron and performs the spatial compression in a general method. As an example of this approach, we deduced the material parameter equations for two-dimensional polygon cloaks and confirmed the results through numerical simulations. Our approach opens up possibilities of designing practical cloaks with arbitrary shapes through numerical or/and semi-analytical methods.

© 2008 Optical Society of America

## 1. Introduction

*et al*. [1

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

2. U. Leonhardt, “Optical conformal mapping,” Science **312**, 1777–1780 (2006). [CrossRef] [PubMed]

3. U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. **8**, 118 (2006). [CrossRef]

4. S. A. Cummer, B. I. Popa, D. Schurig, and D. R. Smith, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E **74**, 036621 (2006). [CrossRef]

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 **314**, 977–980 (2006). [CrossRef] [PubMed]

6. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. **6**, 87–95 (2008). [CrossRef]

8. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. **90**, 241105 (2007). [CrossRef]

9. G. W. Milton, M. Brian, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. **8**, 248 (2006). [CrossRef]

12. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, J. Pendry, M. Rahm, and A. Starr, “Scattering Theory Derivation of a 3D Acoustic Cloaking Shell,” Phys. Rev. Lett. **100**, 024301 (2008). [CrossRef] [PubMed]

16. B. Zhang, H. Chen, B.-I. Wu, J. A. Kong, and Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device,” Phys. Rev. Lett. **100**, 063904 (2008). [CrossRef] [PubMed]

17. P. Yao, Z. Liang, and X. Jiang, “Limitation of the electromagnetic cloak with dispersive material,” Appl. Phys. Lett. **92**, 031111 (2008). [CrossRef]

6. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. **6**, 87–95 (2008). [CrossRef]

18. H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, J. Chen, and Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A **77**, 013825 (2008). [CrossRef]

21. Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Invisibility cloaks for irregular particles using coordinate transformations,” Opt. Express **16**, 6134–6145 (2008). [CrossRef] [PubMed]

22. W. Yan, M. Yan, Z. Ruan, and M. Qiu , “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. **10**, 043040 (2008). [CrossRef]

23. W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped Objects,” Phys. Rev. E **77**, 066607 (2008). [CrossRef]

## 2. Position surfaces and Tracing Lines

22. W. Yan, M. Yan, Z. Ruan, and M. Qiu , “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. **10**, 043040 (2008). [CrossRef]

23. W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped Objects,” Phys. Rev. E **77**, 066607 (2008). [CrossRef]

*S*(

*x*,

*y*,

*z*)=0, and that the parameter function of TLs is

*L*:

*x*=

*ϕ*(

*t*),

*y*=

*φ*(

*t*),

*z*=ψ(

*t*). Thus the arc length of a TL from the original point (

*ϕ*(

*t*

_{0}),

*φ*(

*t*

_{0}),

*ψ*(

*t*

_{0}) to the considered point (

*ϕ*(

*t*),

*φ*(

*t*),

*ψ*(

*t*)) is:

*l*

*and*

_{a}*l*

*are the arc lengths of a TL from the original point to the point where the TL intercross with the inner and outer shells of the cloak, respectively. Equation (2) guarantees that all the points on a TL are squeezed according to their sequence on the arc between the inner and the outer shells of the cloak. With respect to the functions of PSs and TLs, we obtain the coordinate transformation by substituting equation (1) into (2):*

_{b}## 3. Approximation approach

## 4. Design of 2D cloaks

*x*

*,*

_{b}*y*

*) of the triangle hemline. Let spatial compression being performed as equation (2), we then obtain the coordinate transformation in a selected triangle as follows:*

_{b}*x*

*,*

_{a}*y*

*) is the end position of the inner shell cut by the selected triangle. By using the method prescribed by Rahm*

_{a}*et al*. [6

6. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. **6**, 87–95 (2008). [CrossRef]

*α*′(

*x*′,

*y*′)=

*Aα*(

*x*,

*y*)

*A*

*/det*

^{T}*A*, where

*α*(

*x*,

*y*) (before transformation) and

*α*′(

*x*′,

*y*′) (after transformation), respectively, denote the permittivity or permeability tensors, and A is the matrix whose elements are as follows:

*k*,

*x*

*and*

_{b}*y*

*the right values determined by their respective hemline positions. Obviously, these equations are useful to design an arbitrarily shaped cloak approximated by a polygon and are convenient for numerical simulations without any further coordinate transformation since they are already expressed in Cartesian systems.*

_{b}## 5. Conclusion

## Acknowledgments

## References and links

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

2. | U. Leonhardt, “Optical conformal mapping,” Science |

3. | U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. |

4. | S. A. Cummer, B. I. Popa, D. Schurig, and D. R. Smith, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E |

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. | M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. |

7. | 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. |

8. | H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. |

9. | G. W. Milton, M. Brian, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. |

10. | S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. |

11. | H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. |

12. | S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, J. Pendry, M. Rahm, and A. Starr, “Scattering Theory Derivation of a 3D Acoustic Cloaking Shell,” Phys. Rev. Lett. |

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

14. | Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal Cylindrical Cloak: Perfect but Sensitive to Tiny Perturbations,” Phys. Rev. Lett. |

15. | Baile Zhang, Hongsheng Chen, Bae-Ian Wu, Yu Luo, Lixin Ran, and Jin Au Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B |

16. | B. Zhang, H. Chen, B.-I. Wu, J. A. Kong, and Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device,” Phys. Rev. Lett. |

17. | P. Yao, Z. Liang, and X. Jiang, “Limitation of the electromagnetic cloak with dispersive material,” Appl. Phys. Lett. |

18. | H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, J. Chen, and Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A |

19. | D.-H. Kwon and D. H. Werner , “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. |

20. | W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Lin, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys. |

21. | Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Invisibility cloaks for irregular particles using coordinate transformations,” Opt. Express |

22. | W. Yan, M. Yan, Z. Ruan, and M. Qiu , “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. |

23. | W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped Objects,” Phys. Rev. E |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(230.0230) Optical devices : Optical devices

(260.2110) Physical optics : Electromagnetic optics

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 3, 2008

Revised Manuscript: July 14, 2008

Manuscript Accepted: August 26, 2008

Published: September 16, 2008

**Citation**

Hua Ma, Shaobo Qu, Zhuo Xu, and Jiafu Wang, "Approximation approach of designing practical cloaks with arbitrary shapes," Opt. Express **16**, 15449-15454 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15449

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### References

- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
- U. Leonhardt, "Notes on conformal invisibility devices," New J. Phys. 8, 118 (2006). [CrossRef]
- S. A. Cummer, B. I. Popa, D. Schurig, and D. R. Smith, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006). [CrossRef]
- 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, 977-980 (2006). [CrossRef] [PubMed]
- M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, "Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations," Photon. Nanostruct.: Fundam. Applic. 6, 87-95 (2008). [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, 063903 (2008). [CrossRef] [PubMed]
- H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett. 90, 241105 (2007). [CrossRef]
- G. W. Milton, M. Brian, and J. R. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006). [CrossRef]
- S. A. Cummer and D. Schurig, "One path to acoustic cloaking," New J. Phys. 9, 45 (2007). [CrossRef]
- H. Chen and C. T. Chan, "Acoustic cloaking in three dimensions using acoustic metamaterials," Appl. Phys. Lett. 91, 183518 (2007). [CrossRef]
- S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, J. Pendry, M. Rahm, and A. Starr, "Scattering Theory Derivation of a 3D Acoustic Cloaking Shell," Phys. Rev. Lett. 100, 024301 (2008). [CrossRef] [PubMed]
- H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007).
- Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal Cylindrical Cloak: Perfect but Sensitive to Tiny Perturbations," Phys. Rev. Lett. 99, 113903 (2007). [CrossRef] [PubMed]
- B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101(R)(2007). [CrossRef]
- B. Zhang, H. Chen, B.-I. Wu, and J. A. Kong, "Extraordinary surface voltage effect in the invisibility cloak with an active device," Phys. Rev. Lett. 100, 063904 (2008). [CrossRef] [PubMed]
- P. Yao, Z. Liang, and X. Jiang, "Limitation of the electromagnetic cloak with dispersive material," Appl. Phys. Lett. 92, 031111 (2008). [CrossRef]
- H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen and J. Wang, "Material parameter equation for elliptical cylindrical cloaks," Phys. Rev. A 77, 013825 (2008). [CrossRef]
- D.-H. Kwon and D. H. Werner, "Two-dimensional eccentric elliptic electromagnetic cloaks," Appl. Phys. Lett. 92, 013505 (2008). [CrossRef]
- W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, "Arbitrarily elliptical-cylindrical invisible cloaking," J. Phys. D: Appl. Phys. 41, 085504 (2008). [CrossRef]
- Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, "Invisibility cloaks for irregular particles using coordinate transformations," Opt. Express 16, 6134-6145 (2008). [CrossRef] [PubMed]
- W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformations make perfect invisibility cloaks with arbitrary shape," New J. Phys. 10, 043040 (2008). [CrossRef]
- W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, "Analytical design of conformally invisible cloaks for arbitrarily shaped Objects," Phys. Rev. E 77, 066607 (2008). [CrossRef]

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