## A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics

Optics Express, Vol. 18, Issue 6, pp. 6327-6332 (2010)

http://dx.doi.org/10.1364/OE.18.006327

Acrobat PDF (213 KB)

### Abstract

A design method was proposed for transforming the scattering cross section’s shape of PEC objects to other arbitrary shapes based on transformation optics. The parameters of the transformer which is a kind of metamaterial being tightly covered to the original PEC object can be determined by the transformation expressions derived in the paper. With the method, the virtual PEC objects with their definite shape of the scattering cross section can be produced visually through the original PEC objects with a designed transformer. The validation was made by comparing the distribution of electromagnetic field of the PEC objects with and without transformer by means of finite-element method. Some examples are also given for demonstrating the effectiveness of the method.

© 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. 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]

7. L. Lin, W. Wang, J. Cui, C. Du, and X. Luo, “Design of electromagnetic refractor and phase transformer using coordinate transformation theory,” Opt. Express **16**(10), 6815–6821 (2008). [CrossRef] [PubMed]

3. 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]

8. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science **292**(5514), 77–79 (2001). [CrossRef] [PubMed]

9. 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]

10. 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]

11. H. Chen, Z. Liang, P. Yao, X. Jiang, H. Ma, and C. T. Chan, “Extending the bandwidth of electromagnetic cloaks,” Phys. Rev. B **76**(24), 241104 (2007). [CrossRef]

5. T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express **16**(22), 18545–18550 (2008). [CrossRef] [PubMed]

12. H. Chen, X. Zhang, X. Luo, H. Ma, and C. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. **10**(11), 113016 (2008). [CrossRef]

## 2. Principle

*D*and

*D’*, which are called virtual domain and physical domain respectively, can be defined for the virtual PEC object and original PEC object. A given coordinate transformation

*D*to

*D’*. If the permittivity and permeability tensors in virtual domain are

*ε*and

^{ij}*μ*, the relative permittivity and permeability tensors in the physical domain can be calculated by the prescription [13

^{ij}13. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**(21), 9794–9804 (2006). [CrossRef] [PubMed]

12. H. Chen, X. Zhang, X. Luo, H. Ma, and C. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. **10**(11), 113016 (2008). [CrossRef]

## 3. Simulation and result

*3GHz*and unit amplitude, and the inner and outer radii of the shell are

*R*and

_{1}= 0.2m*R*, respectively. The side length of the square in virtual space is

_{2}= 0.3m*2a = 0.2m*.

*r>R*, one can find that they are almost equivalent. If someone observes the device in the region

_{2}*r>R*, he will see a PEC square prism instead of a cylindrical one. Obviously, the simulations accord with our design very well and the shape of the scattering cross section has been changed from circular to square in the example.

_{2}## 4. Expanded model for the transformer

*ρ(θ)*is the scattering cross section’s boundary of the PEC in virtual space, and the virtual domain is the region between the curve

*ρ(θ)*and the circle with radius of

*R*. The general coordinate transformation equations can be expressed aswhere

_{2}*ρ(θ)*is continuous and piecewise differentiable. This transformation maps the virtual domain to the physical domain which is defined in the region between the two circles with radius

*R*and

_{1}*R*. The permittivity and permeability tensors in physical domain can be calculated based on the transformation optics as described in Section 2. If

_{2}*ρ(θ)*cannot be differentiable all over the function domain (

*0≤θ<2π*), it is necessary to divide the function domain into several subdomains, in which

*ρ(θ)*is differentiable so that the

*ρ’(θ)*is significative. Thus the relative tensors can be calculated in each subdomain, and tensors of the whole domain are available then. The transformer is just a medium with the calculated tensors of permittivity and permeability. If a PEC cylinder with radius

*R*wears the transformer, its scattering cross section’s boundary would become the curve

_{1}*ρ(θ)*instead of its real shape. As the function

*ρ(θ)*is arbitrary, we can change the scattering cross section of a PEC cylinder to various shapes utilizing designed transformers.

*R*,

_{1}*R*in Eq. (8) to

_{2}*R*as long as

_{1}(θ) R_{2}(θ)*R*is less than

_{1}(θ)*R*all over the function domain (

_{2}(θ)*0≤θ<2π*).

## 5. Conclusion

## Acknowledgment

## 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. 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 |

4. | 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 Maxwells equations,” Photon. Nanostruct. Fundam. Appl. |

5. | T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express |

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

7. | L. Lin, W. Wang, J. Cui, C. Du, and X. Luo, “Design of electromagnetic refractor and phase transformer using coordinate transformation theory,” Opt. Express |

8. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

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

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

11. | H. Chen, Z. Liang, P. Yao, X. Jiang, H. Ma, and C. T. Chan, “Extending the bandwidth of electromagnetic cloaks,” Phys. Rev. B |

12. | H. Chen, X. Zhang, X. Luo, H. Ma, and C. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. |

13. | D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express |

14. | M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(230.0230) Optical devices : Optical devices

(260.2110) Physical optics : Electromagnetic optics

(160.3918) Materials : Metamaterials

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: November 18, 2009

Revised Manuscript: February 27, 2010

Manuscript Accepted: February 28, 2010

Published: March 12, 2010

**Citation**

Guishan Yuan, Xiaochun Dong, Qiling Deng, Hongtao Gao, Chunheng Liu, Yueguang Lu, and Chunlei Du, "A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics," Opt. Express **18**, 6327-6332 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-6327

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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. 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]
- 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 Maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]
- T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008). [CrossRef] [PubMed]
- H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007). [CrossRef]
- L. Lin, W. Wang, J. Cui, C. Du, and X. Luo, “Design of electromagnetic refractor and phase transformer using coordinate transformation theory,” Opt. Express 16(10), 6815–6821 (2008). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
- 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]
- 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]
- H. Chen, Z. Liang, P. Yao, X. Jiang, H. Ma, and C. T. Chan, “Extending the bandwidth of electromagnetic cloaks,” Phys. Rev. B 76(24), 241104 (2007). [CrossRef]
- H. Chen, X. Zhang, X. Luo, H. Ma, and C. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 (2008). [CrossRef]
- D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006). [CrossRef] [PubMed]
- M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008). [CrossRef]

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