Abstract

Under the restrictions that the mapping functions of transformation are defined in extended two-dimensional (2D) forms and the incident waves are 2D propagating fields, the conditions for non-reflecting boundaries in a finite-embedded coordinate transformation metamaterial slab are derived. By exploring several examples, including some reported in the literatures and some novel ones developed in this study, we show that our approach can be efficiently used to determine the condition for a finite-embedded coordinate transformed metamaterial slab to be non-reflecting.

© 2010 OSA

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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. A. Ward and J. Pendry, “Refraction and geometry in Maxwell's equations,” J. Mod. Opt. 43, 773–793 (1996).
    [CrossRef]
  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. J. S. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
    [CrossRef] [PubMed]
  5. 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(2), 024301 (2008).
    [CrossRef] [PubMed]
  6. S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100(12), 123002 (2008).
    [CrossRef] [PubMed]
  7. H. Y. Chen, X. D. Luo, H. R. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 (2008).
    [CrossRef] [PubMed]
  8. M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008).
    [CrossRef] [PubMed]
  9. M. Rahm, D. Schurig, D. Roberts, S. Cummer, D. Smith, and J. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
    [CrossRef]
  10. N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express 16(26), 21215–21222 (2008).
    [CrossRef] [PubMed]
  11. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. 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]
  12. 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(6), 063903 (2008).
    [CrossRef] [PubMed]
  13. W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background,” arXiv:0806.3231v1 (2008).
  14. L. Bergamin, “Electromagnetic fields and boundary conditions at the interface of generalized transformation media,” Phys. Rev. A 80(6), 063835 (2009).
    [CrossRef]
  15. I.-M. Lee, “Study on the transmission characteristics of the optical waves in photonic metamaterials,” PhD Dissertation (School of Electrical Engineering, Seoul National University, Seoul, Korea, 2009).
  16. P. Zhang, Y. Jin, and S. He, “Inverse transformation optics and reflection analysis for two-dimensional finite embedded coordinate transformation,” arXiv:0906.2038v2 (2009).
  17. I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B 81(12), 125124 (2010).
    [CrossRef]
  18. “Comsol multiphysics” (Comsol AB), < http://www.comsol.com >.

2010

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B 81(12), 125124 (2010).
[CrossRef]

2009

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. 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]

L. Bergamin, “Electromagnetic fields and boundary conditions at the interface of generalized transformation media,” Phys. Rev. A 80(6), 063835 (2009).
[CrossRef]

2008

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(6), 063903 (2008).
[CrossRef] [PubMed]

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

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(2), 024301 (2008).
[CrossRef] [PubMed]

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100(12), 123002 (2008).
[CrossRef] [PubMed]

M. Rahm, D. Schurig, D. Roberts, S. Cummer, D. Smith, and J. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[CrossRef]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008).
[CrossRef] [PubMed]

H. Y. Chen, X. D. Luo, H. R. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 (2008).
[CrossRef] [PubMed]

N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express 16(26), 21215–21222 (2008).
[CrossRef] [PubMed]

2006

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]

1996

A. Ward and J. Pendry, “Refraction and geometry in Maxwell's equations,” J. Mod. Opt. 43, 773–793 (1996).
[CrossRef]

Allen, J.

Alù, A.

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B 81(12), 125124 (2010).
[CrossRef]

Bergamin, L.

L. Bergamin, “Electromagnetic fields and boundary conditions at the interface of generalized transformation media,” Phys. Rev. A 80(6), 063835 (2009).
[CrossRef]

Castaldi, G.

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B 81(12), 125124 (2010).
[CrossRef]

Chan, C. T.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. 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]

H. Y. Chen, X. D. Luo, H. R. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 (2008).
[CrossRef] [PubMed]

Chen, H. Y.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. 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]

H. Y. Chen, X. D. Luo, H. R. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 (2008).
[CrossRef] [PubMed]

Cummer, S.

M. Rahm, D. Schurig, D. Roberts, S. Cummer, D. Smith, and J. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[CrossRef]

N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express 16(26), 21215–21222 (2008).
[CrossRef] [PubMed]

Cummer, S. A.

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(2), 024301 (2008).
[CrossRef] [PubMed]

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(6), 063903 (2008).
[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]

Engheta, N.

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B 81(12), 125124 (2010).
[CrossRef]

Galdi, V.

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B 81(12), 125124 (2010).
[CrossRef]

Gallina, I.

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B 81(12), 125124 (2010).
[CrossRef]

Genov, D. A.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100(12), 123002 (2008).
[CrossRef] [PubMed]

Han, D. Z.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. 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]

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]

Kundtz, N.

Lai, Y.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. 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]

Li, J. S.

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

Luo, X. D.

Ma, H. R.

Mock, J. 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]

Ng, J.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. 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]

Pendry, J.

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(2), 024301 (2008).
[CrossRef] [PubMed]

M. Rahm, D. Schurig, D. Roberts, S. Cummer, D. Smith, and J. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[CrossRef]

A. Ward and J. Pendry, “Refraction and geometry in Maxwell's equations,” J. Mod. Opt. 43, 773–793 (1996).
[CrossRef]

Pendry, J. B.

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

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(6), 063903 (2008).
[CrossRef] [PubMed]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008).
[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]

Popa, B. I.

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(2), 024301 (2008).
[CrossRef] [PubMed]

Rahm, M.

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(2), 024301 (2008).
[CrossRef] [PubMed]

M. Rahm, D. Schurig, D. Roberts, S. Cummer, D. Smith, and J. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[CrossRef]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008).
[CrossRef] [PubMed]

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(6), 063903 (2008).
[CrossRef] [PubMed]

Roberts, D.

M. Rahm, D. Schurig, D. Roberts, S. Cummer, D. Smith, and J. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[CrossRef]

Roberts, D. A.

Schurig, D.

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(6), 063903 (2008).
[CrossRef] [PubMed]

M. Rahm, D. Schurig, D. Roberts, S. Cummer, D. Smith, and J. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[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(2), 024301 (2008).
[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]

Smith, D.

M. Rahm, D. Schurig, D. Roberts, S. Cummer, D. Smith, and J. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[CrossRef]

Smith, D. R.

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(2), 024301 (2008).
[CrossRef] [PubMed]

N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express 16(26), 21215–21222 (2008).
[CrossRef] [PubMed]

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(6), 063903 (2008).
[CrossRef] [PubMed]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008).
[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]

Starr, A.

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(2), 024301 (2008).
[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]

Sun, C.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100(12), 123002 (2008).
[CrossRef] [PubMed]

Ward, A.

A. Ward and J. Pendry, “Refraction and geometry in Maxwell's equations,” J. Mod. Opt. 43, 773–793 (1996).
[CrossRef]

Xiao, J. J.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. 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]

Zhang, S.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100(12), 123002 (2008).
[CrossRef] [PubMed]

Zhang, X.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100(12), 123002 (2008).
[CrossRef] [PubMed]

Zhang, Z. Q.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. 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]

J. Mod. Opt.

A. Ward and J. Pendry, “Refraction and geometry in Maxwell's equations,” J. Mod. Opt. 43, 773–793 (1996).
[CrossRef]

Opt. Express

Photonics Nanostruct. Fundam. Appl.

M. Rahm, D. Schurig, D. Roberts, S. Cummer, D. Smith, and J. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[CrossRef]

Phys. Rev. A

L. Bergamin, “Electromagnetic fields and boundary conditions at the interface of generalized transformation media,” Phys. Rev. A 80(6), 063835 (2009).
[CrossRef]

Phys. Rev. B

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B 81(12), 125124 (2010).
[CrossRef]

Phys. Rev. Lett.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. 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]

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(6), 063903 (2008).
[CrossRef] [PubMed]

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

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(2), 024301 (2008).
[CrossRef] [PubMed]

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100(12), 123002 (2008).
[CrossRef] [PubMed]

Science

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]

Other

“Comsol multiphysics” (Comsol AB), < http://www.comsol.com >.

W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background,” arXiv:0806.3231v1 (2008).

I.-M. Lee, “Study on the transmission characteristics of the optical waves in photonic metamaterials,” PhD Dissertation (School of Electrical Engineering, Seoul National University, Seoul, Korea, 2009).

P. Zhang, Y. Jin, and S. He, “Inverse transformation optics and reflection analysis for two-dimensional finite embedded coordinate transformation,” arXiv:0906.2038v2 (2009).

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Figures (5)

Fig. 4
Fig. 4

Field distributions for the skewed-and-expanded coordinate transformation.

Fig. 5
Fig. 5

Numerical results for the example of the 2D beam expander without any reflection. The normalized field distribution at z = 0 plane is depicted in (a). The results in (b) show the material parameters ( m ¯ ¯ ) along the line y = 5 λ .

Fig. 1
Fig. 1

Geometrical representation of a finite-embedded coordinate transformation.

Fig. 2
Fig. 2

Geometrical representation of (a) beam shifter and (b) beam expander.

Fig. 3
Fig. 3

Transformed geometrical grid made by the skewed and expanded transformation given by Eq. (42).

Equations (72)

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

( x ' , y ' , z ' ) = ( F ( x , y , z ) , G ( x , y , z ) , H ( x , y , z ) ) ,
F ( x , y , z ) = F ( x , y ) ,     G ( x , y , z ) = G ( x , y ) ,     H ( x , y , z ) = h ( x , y ) z .
m ¯ ¯ i ' j ' = A i i ' A j j ' m ¯ ¯ i j / det ( A i i ' ) ,
A i i ' = x i ' / x i .
m ¯ ¯ ' = [ a b c b d e c e f ]
( m ¯ ¯ ' ) 1 = 1 Δ [ ( d f e 2 ) ( b f c e ) ( b e c d ) ( b f c e ) ( a f c 2 ) ( a e b c ) ( b e c d ) ( a e b c ) ( a d b 2 ) ] = 1 Δ [ Δ j i ] ,
Δ = a d f a e 2 b 2 f c 2 d + 2 b c e
k 0 2 Δ 2 S E + k 0 2 n 2 E = 0 ,
S = [ Δ 12 κ z Δ 13 κ y Δ 11 κ z + Δ 13 κ x , I I Δ 11 κ y Δ 12 κ x , I I Δ 22 κ z Δ 23 κ y Δ 12 κ z + Δ 23 κ x , I I Δ 12 κ y Δ 22 κ x , I I Δ 23 κ z Δ 33 κ y Δ 13 κ z + Δ 33 κ x , I I Δ 13 κ y Δ 23 κ x , I I ] 2 ,
n I I 2 ( Δ n I I 2 + a κ x , I I 2 + d κ y 2 + f κ z 2 + 2 b κ x , I I κ y + 2 c κ z κ x , I I + 2 e κ y κ z ) 2 / Δ 2 = 0.
Δ 33 ( Δ 22 κ x , I I Δ 12 κ y ) 2 = Δ 22 n I I 2 Δ 2 + Δ 22 ( Δ 23 κ x , I I Δ 13 κ y ) 2 Δ 33 ( Δ 11 Δ 22 Δ 12 2 ) κ y 2 .
H z , I = exp [ j ( k x , I x + k y y ) ] + r exp [ j ( k x , I x + k y y ) ] ,
H z , I I = t exp [ j ( k x , I I x + k y y ) ] ,
E y , I = k x , I k 0 ε I { exp [ j ( k x , I x + k y y ) ] r exp [ j ( k x , I x + k y y ) ] } .
E y , I I = Δ 22 k x , I I Δ 12 k y k 0 ε I I Δ t exp [ j ( k x , I I x + k y y ) ] .
r = Δ ε I I κ x , I ε I ( Δ 22 κ x , I I Δ 12 κ y ) Δ ε I I κ x , I + ε I ( Δ 22 κ x , I I Δ 12 κ y ) ,
t = 2 Δ ε I I κ x , I Δ ε I I κ x , I + ε I ( Δ 22 κ x , I I Δ 12 κ y ) .
Δ ε I I κ x , I = ε I ( Δ 22 κ x , I I Δ 12 κ y ) .
( A x x ' A y z ' A y x ' A x z ' ) ( A x x ' A x z ' + A y x ' A y z ' ) = 0.
Δ 22 2 μ I I 2 = Δ 33 μ I 2 ,
( A z z ' ) 2 [ ( A x x ' ) 2 + ( A y x ' ) 2 ] 2 μ I I 2 = ( A x x ' A y y ' A y x ' A x y ' ) 4 μ I 2 ,
( A z z ' ) [ ( A x x ' ) 2 + ( A y x ' ) 2 ] = ± ρ ( A x x ' A y y ' A y x ' A x y ' ) 2 ,
Δ 22 2 μ I I 2 = Δ 33 μ I I I 2 .
( A z z ' ) [ ( A x x ' ) 2 + ( A y x ' ) 2 ] | i = ± ρ i ( A x x ' A y y ' A y x ' A x y ' ) 2 | i .
A x z ' = A y z ' = 0.
x ' ( x , y ) = A ( x ) + B ( y ) + F ( x ) G ( y ) + c 1 ,
y ' ( x , y ) = C ( x ) + D ( y ) + H ( x ) K ( y ) + c 2 ,
z ' ( x , y ) = c 3 z ,
c 3 [ ( A x + F x G ) 2 + ( B y + F G y ) 2 ] | i = ± ρ i [ ( A x + F x G ) ( D y + H K y ) ( B y + F G y ) ( C x + H x K ) ] 2 | i ,
x ' = x ,     y ' = y + tan θ x ,     z ' = c 3 z ,
A ( x ) = x ,     F ( x ) = 1 ,     C ( x ) = x tan θ ,     D ( y ) = y ,     K ( y ) = 1 ,
c 3 = ± ρ i .
x ' = x ,     y ' = m ( x ) y ,     z ' = c 3 z ,
m ( x ) = ( M 1 ) ( x d ) / d + M ,
A ( x ) = x ,     F ( x ) = 1 ,     H ( x ) = m ( x ) ,     K ( y ) = y ,
c 3 = ± ρ i [ m ( x i ) ] 2 .
x ' = α x + β y ,     y ' = χ x + δ y ,     z ' = z ,
A ( x ) = α x ,     B ( y ) = β y ,     C ( x ) = χ x ,     D ( y ) = δ y ,
( α 2 + β 2 ) = ( α δ β χ ) 2 ,
( α 2 + β 2 ) = ( α 2 + β 2 ) 2 .
x ' = cos θ x + sin θ y ,     y ' = sin θ x + cos θ y ,     z ' = z ,
x ' = 3 x + 4 y ,     y ' = x + 3 y ,     z ' = z ,
m ¯ ¯ ' = [ 5 3 0 3 2 0 0 0 0.2 ] ,
x ' = a u ( x ) ,     y ' = y / u x ( x ) + v ( x ) ,     z ' = z ,
A ( x ) = a u ( x ) ,     C ( x ) = v ( x ) ,     H ( x ) = 1 / u x ( x ) ,     K ( y ) = y ,
( a u x ) 2 = ( a u x / u x ) 2 ,
u x = ± 1 ,
x ' = a u x ( x ) ,     y ' = y / u ( x ) + v ( x ) ,     z ' = z ,
A ( x ) = a u x ( x ) ,     C ( x ) = v ( x ) ,     H ( x ) = 1 / u ( x ) ,     K ( y ) = y
( a u x x ) 2 = ( a u x x / u ) 2 ,
u = ± 1
x ' = x ,     y ' = m ( x ) y ,     z ' = c ( x ) z .
m ¯ ¯ ' = 1 c m [ 1 m x y c x z m x y ( m x y ) 2 + m 2 m x y c x z c x z m x y c x z ( c x z ) 2 + c 2 ] .
c ( x ) = ± ρ i [ m ( x ) ] 2 .
Δ ε I I κ x , I = ε I ( Δ 22 κ x , I I Δ 12 κ y ) .
κ x , I 2 = n I 2 κ y 2 ,
[ ( Δ 22 Δ 33 Δ 23 2 ) κ x , I I + ( Δ 23 Δ 13 Δ 12 Δ 33 ) κ y ] 2 = Δ 2 [ ( Δ 22 Δ 33 Δ 23 2 ) n I I 2 Δ 33 κ y 2 ] .
[ ( Δ 23 Δ 13 Δ 12 Δ 33 ) 2 + Δ 2 Δ 33 ] κ y 2 = 0.
κ x , I I = 1 Δ 22 Δ 33 Δ 23 2 { ( Δ 23 Δ 13 Δ 12 Δ 33 ) κ y ± Δ [ ( Δ 22 Δ 33 Δ 23 2 ) n I I 2 Δ 33 κ y 2 ] 1 / 2 } .
( Δ 22 Δ 33 Δ 23 2 ) 2 Δ 2 ε I I 2 ( n I 2 κ y 2 ) = ε I 2 { Δ 23 ( Δ 12 Δ 23 Δ 22 Δ 13 ) κ y ± Δ Δ 22 [ ( Δ 22 Δ 33 Δ 23 2 ) n I I 2 Δ 33 κ y 2 ] 1 / 2 } 2 .
A B κ y 2 = [ C κ y ± ( D + E κ y 2 ) 1 / 2 ] 2 ,
A D ( B + C 2 E ) κ y 2 = ± 2 C κ y ( D + E κ y 2 ) 1 / 2 .
( A D ) 2 2 [ ( A D ) ( B + C 2 + E ) + 2 C 2 D ] κ y 2 + [ ( B + C 2 + E ) 2 4 C 2 E ] κ y 4 = 0.
A D = 0 ,
C D = 0 ,
( B + C 2 + E ) 2 4 C 2 E = 0.
Δ 23 ( Δ 12 Δ 23 Δ 22 Δ 13 ) = 0     or     Δ 22 2 ( Δ 22 Δ 33 Δ 23 2 ) = 0 ,
( Δ 22 Δ 33 Δ 23 2 ) ε I I 2 n I 2 = Δ 22 2 ε I 2 n I I 2 ,
( Δ 22 Δ 33 Δ 23 2 ) 2 ε I I 2 = Δ 22 2 Δ 33 ε I 2 .
Δ 33 n I 2 = ( Δ 22 Δ 33 Δ 23 2 ) n I I 2 .
Δ 33 μ I 2 = Δ 22 2 μ I I 2 .
( A x x ' A y z ' A y x ' A x z ' ) ( A x x ' A x z ' + A y x ' A y z ' ) = 0.

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