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

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References

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  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]
  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. 6(1), 87–95 (2008).
    [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]
  6. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
    [CrossRef]
  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]
  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]
  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]
  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]
  14. M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
    [CrossRef]

2009 (2)

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]

2008 (5)

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]

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]

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
[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]

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]

2007 (2)

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
[CrossRef]

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]

2006 (4)

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]

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]

2001 (1)

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]

Bartal, G.

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]

Chan, C.

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]

Chan, C. T.

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
[CrossRef]

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]

Chen, H.

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

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 and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
[CrossRef]

Chin, J. Y.

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]

Cui, J.

Cui, T. J.

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]

Cummer, S. A.

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]

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]

Du, C.

Ji, C.

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]

Jiang, X.

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]

Justice, B. J.

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]

Leonhardt, U.

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

Li, J.

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]

Liang, Z.

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]

Lin, L.

Liu, R.

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]

Luo, X.

Ma, H.

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]

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, 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]

Mock, J. J.

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]

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]

Pendry, J. B.

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]

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]

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]

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

Psaltis, D.

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
[CrossRef]

Rahm, M.

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]

Roberts, D. A.

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]

Schultz, S.

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]

Schurig, D.

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]

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]

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]

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

Shelby, R. A.

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]

Smith, D. R.

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]

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]

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]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 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(5801), 977–980 (2006).
[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]

Starr, A. F.

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]

Tsang, M.

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
[CrossRef]

Valentine, J.

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]

Wang, W.

Yang, T.

Yao, P.

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]

Zentgraf, T.

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]

Zhang, X.

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

Appl. Phys. Lett. (1)

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
[CrossRef]

N. J. Phys. (1)

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]

Nat. Mater. (1)

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]

Opt. Express (3)

Photon. Nanostruct. Fundam. Appl. (1)

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]

Phys. Rev. B (2)

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]

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
[CrossRef]

Science (5)

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. 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]

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the space transformation for the transformer. The radii of outer and inner circles are R2 and R1 , respectively, and the sidelength of the virtual square is 2a. The transformation maps point A to point A’.

Fig. 2
Fig. 2

Snapshot of the total electric field. (a) (b) are the total electric field induced by PEC cylinder with radius R2 = 0.2m and by PEC square prism with side length 2a = 0.2m, respectively. (c) is the total electric field induced by our designed device transformer.

Fig. 3
Fig. 3

Snapshot of the total electric field. (a) (b) are the total electric field induced by PEC prisms whose cross section are triangular and cardioids-like (the boundary curve ρ(θ) = h(1 + cos(θ)), where h = 0.1m), respectively. (c) and (d) are the total electric field induced by the relative transformers.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

ε i ' j ' = [ det ( A i i ' ) ] 1 A i i ' A j j ' ε i j
μ i ' j ' = [ det ( A i i ' ) ] 1 A i i ' A j j ' μ i j
{ r ' = ( R 2 R 1 ) cos θ R 2 cos θ a ( r R 2 ) + R 2 θ ' = θ z ' = z
J = ( r ' r r ' r θ 0 r ' θ ' r r ' θ ' r θ 0 0 0 z ' z ) = ( a 11 a 12 0 0 a 22 0 0 0 1 )
a 11 = ( R 2 R 1 ) cos θ R 2 cos θ a a 12 = a ( R 2 R 1 ) sin θ ( R 2 cos θ a ) 2 r R 2 r a 22 = r ' r
ε ' ¯ = μ ' ¯ = ( a 11 2 + a 12 2 a 11 a 22 a 12 a 11 0 a 12 a 11 a 22 a 11 0 0 0 1 a 11 a 22 )
μ r r = a 11 2 + a 12 2 a 11 a 22     μ r θ = μ θ r = a 12 a 11 μ θ θ = a 22 a 11 ε z z = 1 a 11 a 22
{ r ' = R 2 R 1 R 2 ρ ( θ ) ( r R 2 ) + R 2 θ ' = θ z ' = z

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