The material properties of toroidal invisibility cloaks are derived based on the coordinate transformation method. The permittivity and permeability tensors for toroidal cloaks are substantially different from those for spherical cloaks, but quite similar to those for 2D cylindrical cloaks because a singularity is involved at the inner boundary in both the cases. The cloaking effect is confirmed by the electric field distribution in the vicinity of toroidal cloaks simulated from the generalized discrete-dipole approximation (DDA) method. This study extends the concept of electromagnetic cloaking of arbitrarily-shaped objects to a complex geometry.

© 2009 Optical Society of America

1. A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell’s equations," J. Mod. Opt. 43, 773-793 (1996). [CrossRef]
2. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
3. U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006). [CrossRef]
4. D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006). [CrossRef] [PubMed]
5. U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
6. U. Leonhardt, "Notes on conformal invisibility devices," New J. Phys. 8, 118 (2006). [CrossRef]
7. D. Schurig, J. B. Pendry, and D. R. Smith, "Transformation-designed optical elements," Opt. Express 15, 14772-14782 (2007). [CrossRef] [PubMed]
8. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006). [CrossRef]
9. H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interaction with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007). [CrossRef] [PubMed]
10. 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 (2007). [CrossRef]
11. 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]
12. D.-H. Kwon and D. H. Werner, "Two-dimensional eccentric elliptic electromagnetic cloaks," Appl. Phys. Lett. 92, 013505 (2008). [CrossRef]
13. 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]
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15. 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]
16. 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]
17. A. Nicolet, F. Zolla, and S. Guenneau, "Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section," Opt. Lett. 33, 1584-1586 (2008). [CrossRef] [PubMed]
18. C. Li and F. Li, "Two-dimensional electromagnetic cloaks with arbitrary geometries," Opt. Express 16, 13414-13420 (2008). [CrossRef] [PubMed]
19. C. Li, K. Yao, and F. Li, "Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries," Opt. Express 16, 19366-19374 (2008). [CrossRef]
20. 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]
21. H. Ma, S. Qu, Z. Xu, and J. Wang, "Approximation approach of designing practical cloaks with arbitrary shapes," Opt. Express 16, 15449-15454 (2008). [CrossRef] [PubMed]
22. J. Hu, X. Zhou and G. Hu, "Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation," Opt. Express 17, 1308-1320 (2009). [CrossRef] [PubMed]
23. 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]
24. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, "Two-dimensional metamaterial structure exhibiting reduced visibility at 500 nm," Opt. Lett. 33, 1342-1344 (2008). [CrossRef] [PubMed]
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26. P. Collins and J. McGuirk, "A novel methodology for deriving improved material parameter sets for simplified cylindrical cloaks," J. Opt. A 11, 015104 (2009). [CrossRef]
27. S. Xi, H. Chen, B. Zhang, B.-I. Wu, and J. A. Kong, "A route to quasi-perfect invisibility cylindrical cloaks without extreme values in the parameters," http://arxiv.org/abs/0812.1995 (2008).
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30. B. T. Draine and P. J. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994). [CrossRef]
31. Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, "Zero-backscatter cloak for aspherical particles using a generalized DDA formalism," Opt. Express 16, 2068-2079 (2008). [CrossRef] [PubMed]

Appl. Phys. Lett.

• D.-H. Kwon and D. H. Werner, "Two-dimensional eccentric elliptic electromagnetic cloaks," Appl. Phys. Lett. 92, 013505 (2008). [CrossRef]

Astrophys. J.

• E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973). [CrossRef]
• B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988). [CrossRef]

J. Mod. Opt.

• A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell’s equations," J. Mod. Opt. 43, 773-793 (1996). [CrossRef]

J. Opt. A

• P. Collins and J. McGuirk, "A novel methodology for deriving improved material parameter sets for simplified cylindrical cloaks," J. Opt. A 11, 015104 (2009). [CrossRef]

J. Opt. Soc. Am. A

• B. T. Draine and P. J. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994). [CrossRef]

J. Phys. D: Appl. Phys.

• 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]

Nat. Photon.

• W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shavlaev, "Optical cloaking with metamaterials," Nat. Photon. 1, 224-227 (2007). [CrossRef]

New J. Phys.

• U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006). [CrossRef]
• U. Leonhardt, "Notes on conformal invisibility devices," New J. Phys. 8, 118 (2006). [CrossRef]

New. J. Phys.

• 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]

Opt. Express

• H. Ma, S. Qu, Z. Xu, and J. Wang, "Approximation approach of designing practical cloaks with arbitrary shapes," Opt. Express 16, 15449-15454 (2008). [CrossRef] [PubMed]
• J. Hu, X. Zhou and G. Hu, "Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation," Opt. Express 17, 1308-1320 (2009). [CrossRef] [PubMed]
• Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, "Zero-backscatter cloak for aspherical particles using a generalized DDA formalism," Opt. Express 16, 2068-2079 (2008). [CrossRef] [PubMed]
• D. Schurig, J. B. Pendry, and D. R. Smith, "Transformation-designed optical elements," Opt. Express 15, 14772-14782 (2007). [CrossRef] [PubMed]
• D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006). [CrossRef] [PubMed]
• 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]
• C. Li and F. Li, "Two-dimensional electromagnetic cloaks with arbitrary geometries," Opt. Express 16, 13414-13420 (2008). [CrossRef] [PubMed]
• C. Li, K. Yao, and F. Li, "Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries," Opt. Express 16, 19366-19374 (2008). [CrossRef]

Opt. Lett.

• A. Nicolet, F. Zolla, and S. Guenneau, "Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section," Opt. Lett. 33, 1584-1586 (2008). [CrossRef] [PubMed]
• I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, "Two-dimensional metamaterial structure exhibiting reduced visibility at 500 nm," Opt. Lett. 33, 1342-1344 (2008). [CrossRef] [PubMed]

Photon. Nanostruct.: Fundam. Applic.

• 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]

Phys. Rev. A

• 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]

Phys. Rev. B

• 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 (2007). [CrossRef]

Phys. Rev. E

• 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. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006). [CrossRef]

Phys. Rev. Lett.

• H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interaction with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007). [CrossRef] [PubMed]

Science

• U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
• J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [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, 977-980 (2006). [CrossRef] [PubMed]

Other

• S. Xi, H. Chen, B. Zhang, B.-I. Wu, and J. A. Kong, "A route to quasi-perfect invisibility cylindrical cloaks without extreme values in the parameters," http://arxiv.org/abs/0812.1995 (2008).

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