## Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries

Optics Express, Vol. 18, Issue 6, pp. 6089-6096 (2010)

http://dx.doi.org/10.1364/OE.18.006089

Acrobat PDF (273 KB)

### Abstract

Recently, there are emerging demands for isotropic material parameters, arising from the broadband requirement of the functional devices. Since inverse Laplace’s equation with sliding boundary condition will determine a quasi-conformal mapping, and a quasi-conformal mapping will minimize the transformation material anisotropy, so in this work, the inverse Laplace’s equation with sliding boundary condition is proposed for quasi-isotropic transformation material design. Examples of quasi-isotropic arbitrary carpet cloak and waveguide with arbitrary cross sections are provided to validate the proposed method. The proposed method is very simple compared with other quasi-conformal methods based on grid generation tools.

© 2010 OSA

## 1. Introduction

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**(5781), 1780–1782 (2006). [CrossRef] [PubMed]

2. U. Leonhardt, “Optical conformal mapping,” Science **312**(5781), 1777–1780 (2006). [CrossRef] [PubMed]

3. D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. **5**(9), 687–692 (2009). [CrossRef]

4. Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. **102**(25), 253902 (2009). [CrossRef] [PubMed]

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**(5801), 977–980 (2006). [CrossRef] [PubMed]

6. C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express **16**(17), 13414–13420 (2008). [CrossRef] [PubMed]

11. 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**(8), 085504 (2008). [CrossRef]

12. A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. **33**(14), 1584–1586 (2008). [CrossRef] [PubMed]

13. J. Hu, X. M. Zhou, and G. K. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express **17**(3), 1308–1320 (2009). [CrossRef] [PubMed]

14. P. Zhang, Y. Jin, and S. He, “Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak,” Appl. Phys. Lett. **93**(24), 243502 (2008). [CrossRef]

15. J. Hu, X. M. Zhou, and G. K. Hu, “Nonsingular two dimensional cloak of arbitrary shape,” Appl. Phys. Lett. **95**(1), 011107 (2009). [CrossRef]

16. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science **323**(5910), 110–112 (2009). [CrossRef]

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

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

*et al.*[18

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

19. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science **323**(5912), 366–369 (2009). [CrossRef] [PubMed]

21. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. **8**(7), 568–571 (2009). [CrossRef] [PubMed]

*et al.*[13

13. J. Hu, X. M. Zhou, and G. K. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express **17**(3), 1308–1320 (2009). [CrossRef] [PubMed]

15. J. Hu, X. M. Zhou, and G. K. Hu, “Nonsingular two dimensional cloak of arbitrary shape,” Appl. Phys. Lett. **95**(1), 011107 (2009). [CrossRef]

## 2. Quasi-conformal transformation based on inverse Laplace’s equation

### 2.1 Preliminary

### 2.2 Variational form of inverse Laplace’s equation

13. J. Hu, X. M. Zhou, and G. K. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express **17**(3), 1308–1320 (2009). [CrossRef] [PubMed]

*g*is the determinant of the metric tensor

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

*et al*[22].

### 2.3 Sliding boundary conditions x i

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**(5781), 1780–1782 (2006). [CrossRef] [PubMed]

**101**(20), 203901 (2008). [CrossRef] [PubMed]

## 3 Applications and discussions

### 3.1 Carpet cloaks

**101**(20), 203901 (2008). [CrossRef] [PubMed]

**101**(20), 203901 (2008). [CrossRef] [PubMed]

12. A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. **33**(14), 1584–1586 (2008). [CrossRef] [PubMed]

**101**(20), 203901 (2008). [CrossRef] [PubMed]

*et al*. [19

19. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science **323**(5912), 366–369 (2009). [CrossRef] [PubMed]

*et al.*[18

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

### 3.2 Arbitrary waveguide

## 4. Conclusions

**101**(20), 203901 (2008). [CrossRef] [PubMed]

24. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express **17**(17), 14872–14879 (2009). [CrossRef] [PubMed]

25. J. Hu, X. M. Zhou, and G. K. Hu, “A numerical method for designing acoustic cloak with arbitrary shapes,” Comput. Mater. Sci. **46**(3), 708–712 (2009). [CrossRef]

## 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. | D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. |

4. | Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” 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. | C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express |

7. | M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. Pendry, “Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell’s Equations,” Photon. Nanostruct. Fundam. Appl. |

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

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

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

11. | 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. |

12. | A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. |

13. | J. Hu, X. M. Zhou, and G. K. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express |

14. | P. Zhang, Y. Jin, and S. He, “Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak,” Appl. Phys. Lett. |

15. | J. Hu, X. M. Zhou, and G. K. Hu, “Nonsingular two dimensional cloak of arbitrary shape,” Appl. Phys. Lett. |

16. | U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science |

17. | J. 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. | R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science |

20. | L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics |

21. | J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. |

22. | J. F. Thompson, B. K. Sdoni, and N. P. Weatherill, Handbook of grid generation, (CRC Press, New York, 1999). |

23. | P. Knupp, and S. Steinberg, Fundamentals of Grid Generation, (CRC Press, Boca Raton, 1994). |

24. | N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express |

25. | J. Hu, X. M. Zhou, and G. K. Hu, “A numerical method for designing acoustic cloak with arbitrary shapes,” Comput. Mater. Sci. |

**OCIS Codes**

(230.0230) Optical devices : Optical devices

(230.3205) Optical devices : Invisibility cloaks

**ToC Category:**

Physical Optics

**History**

Original Manuscript: December 22, 2009

Revised Manuscript: February 22, 2010

Manuscript Accepted: February 22, 2010

Published: March 11, 2010

**Citation**

Zheng Chang, Xiaoming Zhou, Jin Hu, and Gengkai Hu, "Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries," Opt. Express **18**, 6089-6096 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-6089

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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]
- U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
- D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009). [CrossRef]
- Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009). [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]
- C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express 16(17), 13414–13420 (2008). [CrossRef] [PubMed]
- M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. Pendry, “Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell’s Equations,” Photon. Nanostruct. Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]
- H. Ma, S. B. Qu, Z. Xu, J. Q. Zhang, B. W. Chen, and J. F. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77(1), 013825 (2008). [CrossRef]
- Y. You, G. W. Kattawar, P. W. Zhai, and P. Yang, “Invisibility cloaks for irregular particles using coordinate transformations,” Opt. Express 16(9), 6134–6145 (2008). [CrossRef] [PubMed]
- D. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92(1), 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(8), 085504 (2008). [CrossRef]
- A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. 33(14), 1584–1586 (2008). [CrossRef] [PubMed]
- J. Hu, X. M. Zhou, and G. K. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17(3), 1308–1320 (2009). [CrossRef] [PubMed]
- P. Zhang, Y. Jin, and S. He, “Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak,” Appl. Phys. Lett. 93(24), 243502 (2008). [CrossRef]
- J. Hu, X. M. Zhou, and G. K. Hu, “Nonsingular two dimensional cloak of arbitrary shape,” Appl. Phys. Lett. 95(1), 011107 (2009). [CrossRef]
- U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323(5910), 110–112 (2009). [CrossRef]
- J. 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]
- R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009). [CrossRef] [PubMed]
- L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009). [CrossRef]
- J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]
- J. F. Thompson, B. K. Sdoni, and N. P. Weatherill, Handbook of grid generation, (CRC Press, New York, 1999).
- P. Knupp, and S. Steinberg, Fundamentals of Grid Generation, (CRC Press, Boca Raton, 1994).
- N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17), 14872–14879 (2009). [CrossRef] [PubMed]
- J. Hu, X. M. Zhou, and G. K. Hu, “A numerical method for designing acoustic cloak with arbitrary shapes,” Comput. Mater. Sci. 46(3), 708–712 (2009). [CrossRef]

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