Abstract

Optic-null medium (ONM), an electromagnetic (EM) space representing optically nothing, has many interesting applications but is difficult to realize practically due to its extreme EM parameters. Here we demonstrate that a holey metallic plate with periodic array of subwavelength apertures can well mimic an ONM. We develop an effective-medium theory to extract the EM parameters of the designed ONM, and employ full-wave simulations to demonstrate its optical functionalities. Microwave experiments, in excellent agreement with full-wave simulations, are performed to illustrate several applications of the ONM, including the radiation cancellation effect and the hyperlensing effect.

© 2013 Optical Society of America

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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. 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]
  4. Y. Lai, J. Ng, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Illusion optics,” Front. Phys. China 5(3), 308–318 (2010).
    [CrossRef]
  5. H. Y. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
    [CrossRef]
  6. 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]
  7. W. X. Jiang, T. J. Cui, H. F. Ma, X. Y. Zhou, and Q. Cheng, “Cylindrical-to-plane-wave conversion via embedded optical transformation,” Appl. Phys. Lett. 92(26), 261903 (2008).
    [CrossRef]
  8. Y. Shen, K. Ding, W. Sun, and L. Zhou, “A chirality switching device designed with transformation optics,” Opt. Express 18(20), 21419–21426 (2010).
    [CrossRef] [PubMed]
  9. S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8(12), 4243–4247 (2008).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  13. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [CrossRef] [PubMed]
  14. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
    [CrossRef] [PubMed]
  15. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
    [CrossRef] [PubMed]
  16. S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
    [CrossRef] [PubMed]
  17. COMSOL Multi-physics 3.5, developed by COMSOL ©, network license (2008).
  18. In our simulations, we tookΔ/b=10000.
  19. Here, we only present the field distributions for TE-polarized excitation since the TM case is quite similar to the TE case.
  20. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
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  21. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-Near-Zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
    [CrossRef] [PubMed]
  22. J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
    [CrossRef]
  23. I. C. Khoo, D. H. Werner, X. Liang, A. Diaz, and B. Weiner, “Nanosphere dispersed liquid crystals for tunable negative-zero-positive index of refraction in the optical and terahertz regimes,” Opt. Lett. 31(17), 2592–2594 (2006).
    [CrossRef] [PubMed]
  24. N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, “Metamaterials: electromagnetic enhancement at zero-index transition,” Opt. Lett. 33(20), 2350–2352 (2008).
    [CrossRef] [PubMed]
  25. J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. Condens. Matter 15(37), 6345–6364 (2003).
    [CrossRef]
  26. Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102(9), 093901 (2009).
    [CrossRef] [PubMed]
  27. W. Yan, M. Yan, and M. Qiu, “Generalized compensated bilayer structure from the transformation optics perspective,” J. Opt. Soc. Am. B 26(12), 32–35 (2009).
    [CrossRef]
  28. W. Yan, M. Yan, and M. Qiu, “Generalized nihility media from transformation optics,” J. Opt. 13(2), 024005 (2011).
    [CrossRef]
  29. J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
    [CrossRef] [PubMed]
  30. F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surface with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
    [CrossRef]
  31. S. Xiao, Q. He, X. Huang, and L. Zhou, “Super imaging with a plasmonic metamaterial: role of aperture shape,” Metamaterials (Amst.) 5(2–3), 112–118 (2011).
    [CrossRef]
  32. J. Bravo-Abad, F. J. García-Vidal, and L. Martín-Moreno, “Resonant transmission of light through finite chains of subwavelength holes in a metallic film,” Phys. Rev. Lett. 93(22), 227401 (2004).
    [CrossRef] [PubMed]
  33. Here one can also impose the condition that the two systems exhibit the same transmission properties, although the mathematics are a bit complicated.
  34. CONCERTO 7.0, developed by Vector Fields Ltd, England (2008).
  35. To estimate the working bandwidth, we define the frequencies at which the transmittance of our sample decreases to 0.5 as two boundaries of the working range.

2012 (2)

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[CrossRef] [PubMed]

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

2011 (3)

W. Yan, M. Yan, and M. Qiu, “Generalized nihility media from transformation optics,” J. Opt. 13(2), 024005 (2011).
[CrossRef]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[CrossRef] [PubMed]

S. Xiao, Q. He, X. Huang, and L. Zhou, “Super imaging with a plasmonic metamaterial: role of aperture shape,” Metamaterials (Amst.) 5(2–3), 112–118 (2011).
[CrossRef]

2010 (2)

Y. Shen, K. Ding, W. Sun, and L. Zhou, “A chirality switching device designed with transformation optics,” Opt. Express 18(20), 21419–21426 (2010).
[CrossRef] [PubMed]

Y. Lai, J. Ng, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Illusion optics,” Front. Phys. China 5(3), 308–318 (2010).
[CrossRef]

2009 (3)

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102(9), 093901 (2009).
[CrossRef] [PubMed]

W. Yan, M. Yan, and M. Qiu, “Generalized compensated bilayer structure from the transformation optics perspective,” J. Opt. Soc. Am. B 26(12), 32–35 (2009).
[CrossRef]

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]

2008 (4)

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]

W. X. Jiang, T. J. Cui, H. F. Ma, X. Y. Zhou, and Q. Cheng, “Cylindrical-to-plane-wave conversion via embedded optical transformation,” Appl. Phys. Lett. 92(26), 261903 (2008).
[CrossRef]

N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, “Metamaterials: electromagnetic enhancement at zero-index transition,” Opt. Lett. 33(20), 2350–2352 (2008).
[CrossRef] [PubMed]

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8(12), 4243–4247 (2008).
[CrossRef] [PubMed]

2007 (3)

A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. 32(23), 3432–3434 (2007).
[CrossRef] [PubMed]

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

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[CrossRef]

2006 (4)

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-Near-Zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[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]

I. C. Khoo, D. H. Werner, X. Liang, A. Diaz, and B. Weiner, “Nanosphere dispersed liquid crystals for tunable negative-zero-positive index of refraction in the optical and terahertz regimes,” Opt. Lett. 31(17), 2592–2594 (2006).
[CrossRef] [PubMed]

2005 (2)

F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surface with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
[CrossRef]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

2004 (3)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

J. Bravo-Abad, F. J. García-Vidal, and L. Martín-Moreno, “Resonant transmission of light through finite chains of subwavelength holes in a metallic film,” Phys. Rev. Lett. 93(22), 227401 (2004).
[CrossRef] [PubMed]

2003 (1)

J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. Condens. Matter 15(37), 6345–6364 (2003).
[CrossRef]

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]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Aieta, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Alù, A.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[CrossRef]

Bartal, G.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8(12), 4243–4247 (2008).
[CrossRef] [PubMed]

Bravo-Abad, J.

J. Bravo-Abad, F. J. García-Vidal, and L. Martín-Moreno, “Resonant transmission of light through finite chains of subwavelength holes in a metallic film,” Phys. Rev. Lett. 93(22), 227401 (2004).
[CrossRef] [PubMed]

Capasso, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Chan, C. T.

Y. Lai, J. Ng, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Illusion optics,” Front. Phys. China 5(3), 308–318 (2010).
[CrossRef]

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102(9), 093901 (2009).
[CrossRef] [PubMed]

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

Chen, H.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102(9), 093901 (2009).
[CrossRef] [PubMed]

Chen, H. Y.

Y. Lai, J. Ng, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Illusion optics,” Front. Phys. China 5(3), 308–318 (2010).
[CrossRef]

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

Cheng, Q.

W. X. Jiang, T. J. Cui, H. F. Ma, X. Y. Zhou, and Q. Cheng, “Cylindrical-to-plane-wave conversion via embedded optical transformation,” Appl. Phys. Lett. 92(26), 261903 (2008).
[CrossRef]

Cui, T. J.

W. X. Jiang, T. J. Cui, H. F. Ma, X. Y. Zhou, and Q. Cheng, “Cylindrical-to-plane-wave conversion via embedded optical transformation,” Appl. Phys. Lett. 92(26), 261903 (2008).
[CrossRef]

Cummer, S. A.

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]

Diaz, A.

Ding, K.

Engheta, N.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[CrossRef]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-Near-Zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[CrossRef] [PubMed]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Gabitov, I. R.

Gaburro, Z.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Gao, L.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

García-Vidal, F. J.

F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surface with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
[CrossRef]

J. Bravo-Abad, F. J. García-Vidal, and L. Martín-Moreno, “Resonant transmission of light through finite chains of subwavelength holes in a metallic film,” Phys. Rev. Lett. 93(22), 227401 (2004).
[CrossRef] [PubMed]

J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

Genevet, P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Genov, D.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8(12), 4243–4247 (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]

Han, S.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8(12), 4243–4247 (2008).
[CrossRef] [PubMed]

He, Q.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[CrossRef] [PubMed]

S. Xiao, Q. He, X. Huang, and L. Zhou, “Super imaging with a plasmonic metamaterial: role of aperture shape,” Metamaterials (Amst.) 5(2–3), 112–118 (2011).
[CrossRef]

Hou, B.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

Huang, X.

S. Xiao, Q. He, X. Huang, and L. Zhou, “Super imaging with a plasmonic metamaterial: role of aperture shape,” Metamaterials (Amst.) 5(2–3), 112–118 (2011).
[CrossRef]

Jiang, W. X.

W. X. Jiang, T. J. Cui, H. F. Ma, X. Y. Zhou, and Q. Cheng, “Cylindrical-to-plane-wave conversion via embedded optical transformation,” Appl. Phys. Lett. 92(26), 261903 (2008).
[CrossRef]

Kats, M. A.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Khoo, I. C.

Kildishev, A. V.

Lai, Y.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

Y. Lai, J. Ng, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Illusion optics,” Front. Phys. China 5(3), 308–318 (2010).
[CrossRef]

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102(9), 093901 (2009).
[CrossRef] [PubMed]

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]

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Leonhardt, U.

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

Li, X.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[CrossRef] [PubMed]

Liang, X.

Litchinitser, N. M.

Liu, Z.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8(12), 4243–4247 (2008).
[CrossRef] [PubMed]

Luo, J.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

Ma, H. F.

W. X. Jiang, T. J. Cui, H. F. Ma, X. Y. Zhou, and Q. Cheng, “Cylindrical-to-plane-wave conversion via embedded optical transformation,” Appl. Phys. Lett. 92(26), 261903 (2008).
[CrossRef]

Maimistov, A. I.

Martín-Moreno, L.

F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surface with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
[CrossRef]

J. Bravo-Abad, F. J. García-Vidal, and L. Martín-Moreno, “Resonant transmission of light through finite chains of subwavelength holes in a metallic film,” Phys. Rev. Lett. 93(22), 227401 (2004).
[CrossRef] [PubMed]

J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

Narimanov, E. E.

Ng, J.

Y. Lai, J. Ng, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Illusion optics,” Front. Phys. China 5(3), 308–318 (2010).
[CrossRef]

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. B.

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

F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surface with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
[CrossRef]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. Condens. Matter 15(37), 6345–6364 (2003).
[CrossRef]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Qiu, M.

W. Yan, M. Yan, and M. Qiu, “Generalized nihility media from transformation optics,” J. Opt. 13(2), 024005 (2011).
[CrossRef]

W. Yan, M. Yan, and M. Qiu, “Generalized compensated bilayer structure from the transformation optics perspective,” J. Opt. Soc. Am. B 26(12), 32–35 (2009).
[CrossRef]

Rahm, M.

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]

Ramakrishna, S. A.

J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. Condens. Matter 15(37), 6345–6364 (2003).
[CrossRef]

Sagdeev, R. Z.

Salandrino, A.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[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, 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. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

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]

Shen, Y.

Silveirinha, M.

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-Near-Zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[CrossRef] [PubMed]

Silveirinha, M. G.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[CrossRef]

Smith, D. R.

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

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[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]

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Sun, S.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[CrossRef] [PubMed]

Sun, W.

Tetienne, J.-P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Weiner, B.

Werner, D. H.

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

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]

Xiao, S.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[CrossRef] [PubMed]

S. Xiao, Q. He, X. Huang, and L. Zhou, “Super imaging with a plasmonic metamaterial: role of aperture shape,” Metamaterials (Amst.) 5(2–3), 112–118 (2011).
[CrossRef]

Xiong, Y.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8(12), 4243–4247 (2008).
[CrossRef] [PubMed]

Xu, P.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

Xu, Q.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[CrossRef] [PubMed]

Yan, M.

W. Yan, M. Yan, and M. Qiu, “Generalized nihility media from transformation optics,” J. Opt. 13(2), 024005 (2011).
[CrossRef]

W. Yan, M. Yan, and M. Qiu, “Generalized compensated bilayer structure from the transformation optics perspective,” J. Opt. Soc. Am. B 26(12), 32–35 (2009).
[CrossRef]

Yan, W.

W. Yan, M. Yan, and M. Qiu, “Generalized nihility media from transformation optics,” J. Opt. 13(2), 024005 (2011).
[CrossRef]

W. Yan, M. Yan, and M. Qiu, “Generalized compensated bilayer structure from the transformation optics perspective,” J. Opt. Soc. Am. B 26(12), 32–35 (2009).
[CrossRef]

Yu, N.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Zhang, X.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8(12), 4243–4247 (2008).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Zhang, Z. Q.

Y. Lai, J. Ng, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Illusion optics,” Front. Phys. China 5(3), 308–318 (2010).
[CrossRef]

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102(9), 093901 (2009).
[CrossRef] [PubMed]

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]

Zhou, L.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[CrossRef] [PubMed]

S. Xiao, Q. He, X. Huang, and L. Zhou, “Super imaging with a plasmonic metamaterial: role of aperture shape,” Metamaterials (Amst.) 5(2–3), 112–118 (2011).
[CrossRef]

Y. Shen, K. Ding, W. Sun, and L. Zhou, “A chirality switching device designed with transformation optics,” Opt. Express 18(20), 21419–21426 (2010).
[CrossRef] [PubMed]

Zhou, X. Y.

W. X. Jiang, T. J. Cui, H. F. Ma, X. Y. Zhou, and Q. Cheng, “Cylindrical-to-plane-wave conversion via embedded optical transformation,” Appl. Phys. Lett. 92(26), 261903 (2008).
[CrossRef]

Appl. Phys. Lett. (3)

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

W. X. Jiang, T. J. Cui, H. F. Ma, X. Y. Zhou, and Q. Cheng, “Cylindrical-to-plane-wave conversion via embedded optical transformation,” Appl. Phys. Lett. 92(26), 261903 (2008).
[CrossRef]

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[CrossRef]

Front. Phys. China (1)

Y. Lai, J. Ng, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Illusion optics,” Front. Phys. China 5(3), 308–318 (2010).
[CrossRef]

J. Opt. (1)

W. Yan, M. Yan, and M. Qiu, “Generalized nihility media from transformation optics,” J. Opt. 13(2), 024005 (2011).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surface with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
[CrossRef]

J. Opt. Soc. Am. B (1)

W. Yan, M. Yan, and M. Qiu, “Generalized compensated bilayer structure from the transformation optics perspective,” J. Opt. Soc. Am. B 26(12), 32–35 (2009).
[CrossRef]

J. Phys. Condens. Matter (1)

J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. Condens. Matter 15(37), 6345–6364 (2003).
[CrossRef]

Metamaterials (Amst.) (1)

S. Xiao, Q. He, X. Huang, and L. Zhou, “Super imaging with a plasmonic metamaterial: role of aperture shape,” Metamaterials (Amst.) 5(2–3), 112–118 (2011).
[CrossRef]

Nano Lett. (1)

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. 8(12), 4243–4247 (2008).
[CrossRef] [PubMed]

Nat. Mater. (1)

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. B (1)

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[CrossRef]

Phys. Rev. Lett. (6)

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-Near-Zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

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. Bravo-Abad, F. J. García-Vidal, and L. Martín-Moreno, “Resonant transmission of light through finite chains of subwavelength holes in a metallic film,” Phys. Rev. Lett. 93(22), 227401 (2004).
[CrossRef] [PubMed]

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102(9), 093901 (2009).
[CrossRef] [PubMed]

Science (7)

J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[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]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[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]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Other (6)

COMSOL Multi-physics 3.5, developed by COMSOL ©, network license (2008).

In our simulations, we tookΔ/b=10000.

Here, we only present the field distributions for TE-polarized excitation since the TM case is quite similar to the TE case.

Here one can also impose the condition that the two systems exhibit the same transmission properties, although the mathematics are a bit complicated.

CONCERTO 7.0, developed by Vector Fields Ltd, England (2008).

To estimate the working bandwidth, we define the frequencies at which the transmittance of our sample decreases to 0.5 as two boundaries of the working range.

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Figures (8)

Fig. 1
Fig. 1

Coordinate transformation that stretches an original flat space with thickness Δ ((a) and (d)) to a final space with thickness of b ((b) and (e)), which can in turn be replaced by a flat space filled with the optics-null medium ((c) and (f)). The upper row corresponds to Cartesian coordinate and the lower row to cylindrical coordinate.

Fig. 2
Fig. 2

(a) FEM-computed transmission amplitudes |T| for EM waves with different incident angle and polarization passing through a 2λ -thick slab of ONM (solid circles), ENZ (green dash line, with ε=0.1 ) or ZIM (red line), correspondingly. (b)-(d) FEM simulated electric field ( E y ) distributions for TE-polarized EM waves passing through a 2λ -thick ONM slab with parallel wave-vectors: (b) k x =0 (c) k x =0.5 k 0 and (d) k x =1.2 k 0 . Here, k 0 is the wave-vector in vacuum and the shadow areas represent the ONM.

Fig. 3
Fig. 3

(a) Geometry of a HMP. (b) Scheme of mapping a HMP to a homogeneous anisotropic effective medium.

Fig. 4
Fig. 4

(a) Designed HMP slab with parameters d = 20 mm, l1 = 10 mm, l2 = 5 mm, l3 = l4 = 4 mm, w = 1 mm and h = 50 mm. (b) Transmission amplitudes (red line and circles, left axis) and phases (blue line and circles, right axis) for EM waves with different parallel wave-vector and polarization passing through the designed HMP (circles, calculated by FDTD simulations) and the corresponding effective-medium slab (lines). (c) Distributions of |Ey| along z-axis for the system shined with a TE-polarized incident evanescent wave with k|| = π/3d (left panel) and a TM-polarized incident evanescent wave with k|| = π/2d (right panel), calculated by FDTD simulations (|Ey| is averaged over a unit cell) for realistic HMP (triangles) and the effective-medium slab (lines). Here, the shadow areas represent the HMP, and the working frequency is 2.0 GHz in all calculations.

Fig. 5
Fig. 5

(a) Picture of the fabricated sample with parameters L1 = 12 mm, L2 = 13 mm, L3 = 6 mm, Px = 18 mm, Py = 31 mm, w = 1 mm (b) Return loss (|S11|) spectra of a dipole antenna put on top of standing-alone ONM slabs (black), ONM slabs backed by a PEC substrate (red) and air gaps backed by a PEC substrate (green), obtained by experiment (symbols) and FDTD simulations (lines). Here the working frequency is 2.63 GHz.

Fig. 6
Fig. 6

(a) Top-view and (b) side-view pictures of the experiment sample. | E/ E ref | distributions for the hyperlensing effect realized by (c) a cylindrical ONM with parameter given by Eq. (2) and (d) our designed sample obtained by FDTD simulations at frequency 2.63 GHz.

Fig. 7
Fig. 7

(a,c) Measured and (b,d) simulated | E/ E ref | distributions at the outer surface of the hyperlens when two dipole antennas are placed on the inner surface of the hyperlens, fed by in-phase ((a) and (b)) and out-of-phase ((c) and (d)) signals. Black lines/symbols correspond to the case of replacing the sample by air. In both measurements and simulations, the | E | value at the position l imag =0 mm in the in-phase case is taken as the reference | E ref | .

Fig. 8
Fig. 8

FDTD simulated | E/ E ref | distributions on the (a) x-z plane, (b) source plane, and (c) image plane for the hyperlensing effect realized by a cylindrical ONM with R out =475mm and R in =235mm at the working frequency 2.63 GHz.

Equations (11)

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ε = μ =( Δ/b 0 0 0 Δ/b 0 0 0 b/Δ ) Δ0 ( 0 0 0 0 0 0 0 0 )
ε = μ =( 0 0 0 0 0 0 0 0 ),
ε eff = ε 0 ( ε eff || 0 0 0 ε eff || 0 0 0 ε eff ), μ eff = μ 0 ( μ eff || 0 0 0 μ eff || 0 0 0 μ eff ).
r 0 = | S 0 | 2 k z air / k z WG 1 | S 0 | 2 k z air / k z WG +1 ,
| S 0 | 2 = | u.c. d r || ( E || in ) E 0,|| WG | 2 u.c. d r || | E || in | 2 u.c. d r || | E 0,|| WG | 2
k z WG = ε h ω 2 ω c 2 /c,
r= μ eff || k z air / k z MTM 1 μ eff || k z air / k z MTM +1 ,
k z MTM = ( ω/c ) 2 ε eff || μ eff || k || 2 μ eff || / μ eff
ε eff || =( 1 ω c 2 / ω 2 ) ε h / | S 0 | 2 μ eff || = | S 0 | 2 . μ eff =
ε eff =.
ε eff = ε 0 ( 0 0 0 0 0 0 0 0 ), μ eff = μ 0 ( | S 0 | 2 0 0 0 | S 0 | 2 0 0 0 ).

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