Abstract

Modeling meta-surfaces as thin metamaterial layers with continuously varying bulk parameters, we employed a rigorous mode-expansion theory to study the scattering properties of such systems. We found that a meta-surface with a linear reflection-phase profile could redirect an impinging light to a non-specular channel with nearly 100% efficiency, and a meta-surface with a parabolic reflection-phase profile could focus incident plane wave to a point image. Under certain approximations, our theory reduces to the local response model (LRM) established for such problems previously, but our full theory has overcome the energy non-conservation problems suffered by the LRM. Microwave experiments were performed on realistic samples to verify the key theoretical predictions, which match well with full-wave simulations.

© 2013 Optical Society of America

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  3. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308(5721), 534–537 (2005).
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  4. D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express13(6), 2127–2134 (2005).
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  5. S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt.50(9), 1419–1430 (2003).
  6. J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99(6), 063908 (2007).
    [CrossRef] [PubMed]
  7. U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
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  9. 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,” Science314(5801), 977–980 (2006).
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  12. Y. Lai, H. Y. 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).
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  13. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9(5), 387–396 (2010).
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  14. U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(24), 243901 (2007).
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  16. O. Paul, B. Reinhard, B. Krolla, R. Beigang, and M. Rahm, “Gradient index metamaterial based on slot elements,” Appl. Phys. Lett.96(24), 241110 (2010).
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  17. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010).
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  18. B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express18(19), 20321–20333 (2010).
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  23. P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
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  24. F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities,” Nano Lett.12(3), 1702–1706 (2012).
    [CrossRef] [PubMed]
  25. N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett.12(12), 6328–6333 (2012).
    [CrossRef] [PubMed]
  26. L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett.12(11), 5750–5755 (2012).
    [CrossRef] [PubMed]
  27. F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett.12(9), 4932–4936 (2012).
    [CrossRef] [PubMed]
  28. X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun.3, 1198 (2012).
    [CrossRef] [PubMed]
  29. X. Li, S. Y. Xiao, B. G. Cai, Q. He, T. J. Cui, and L. Zhou, “Flat metasurfaces to focus electromagnetic waves in reflection geometry,” Opt. Lett.37(23), 4940–4942 (2012).
    [CrossRef] [PubMed]
  30. D. Berry, R. Malech, and W. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propag.11(6), 645–651 (1963).
    [CrossRef]
  31. S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory,” Opt. Lett.37(12), 2391–2393 (2012).
    [CrossRef] [PubMed]
  32. F. L. Zhang, Q. Zhao, L. Kang, J. Zhou, and D. Lippens, “Experimental verification of isotropic and polarization properties of high permittivity-based metamaterial,” Phys. Rev. B80(19), 195119 (2009).
    [CrossRef]
  33. J. M. Hao, L. Zhou, and C. T. Chan, “An effective-medium model for high-impedance surfaces,” Appl. Phys. A Mater. Sci. Process.87(2), 281–284 (2007).
    [CrossRef]
  34. D. Sievenpiper, L. Zhang, R. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47(11), 2059–2074 (1999).
    [CrossRef]
  35. 35. These reflection channels could also be understood as the Floquet modes diffracted by our super-periodic system.
  36. In our computational approach, we have to set the number of sub-cells divided identical to the number plane waves adopted in region (both are 2N + 1), in order to ensure that the number of restraints equals to that of variables.
  37. For two boundary indexes, we have the following off-diagonal matrix elementsH1,2N+1=μM,1xγ, H2N+1,1=μM,2N+1xγ according to the periodic boundary condition.
  38. Since the super-cell length L is very large, the distribution of those discretized kxr,n is almost continuous. Thus, in what follows, we use ρ(kxr) to represent ρ(kxr,n) for simplicity.
  39. C. Qu, S. Y. Xiao, S. L. Sun, Q. He, and L. Zhou, “A theoretical study on the conversion efficiencies of gradient meta-surfaces,” Europhys. Lett.101(5), 54002 (2013).
    [CrossRef]
  40. A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett.13(2), 829–834 (2013).
    [CrossRef] [PubMed]
  41. M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials5(4), 178–205 (2011).
    [CrossRef]
  42. P. Sheng, “Wave scattering formalism,” in Introduction to Wave Scattering, Localization and Macroscopic Phenomena, R. Hull, R. M. Osgood, eds. (Springer, 2006).
  43. EastFDTD v2.0 Beta, DONGJUN Science and Technology Co., China.
  44. For the ξ = 0.4k0 sample, a super cell contains 10 pairs of “H” (altogether 20 ones) in one supercell, with L1 values of those 10 pairs set as 1.3 mm, 2.68 mm, 2.98 mm, 3.14 mm, 3.24 mm, 3.36 mm, 3.48 mm, 3.66 mm, 4.08 mm, and 5.7 mm. For the ξ = 0.8k0 sample, a super cell contains 10 “H” in one super cell with L1 parameters the same as the case of ξ = 0.4k0.
  45. The gain of the employed double-ridged horn antenna is about 14dB~15dB in this frequency region.

2013

C. Qu, S. Y. Xiao, S. L. Sun, Q. He, and L. Zhou, “A theoretical study on the conversion efficiencies of gradient meta-surfaces,” Europhys. Lett.101(5), 54002 (2013).
[CrossRef]

A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett.13(2), 829–834 (2013).
[CrossRef] [PubMed]

2012

S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory,” Opt. Lett.37(12), 2391–2393 (2012).
[CrossRef] [PubMed]

X. Li, S. Y. Xiao, B. G. Cai, Q. He, T. J. Cui, and L. Zhou, “Flat metasurfaces to focus electromagnetic waves in reflection geometry,” Opt. Lett.37(23), 4940–4942 (2012).
[CrossRef] [PubMed]

S. L. Sun, Q. He, S. Y. 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]

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science335(6067), 427 (2012).
[CrossRef] [PubMed]

S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
[CrossRef] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities,” Nano Lett.12(3), 1702–1706 (2012).
[CrossRef] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett.12(12), 6328–6333 (2012).
[CrossRef] [PubMed]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett.12(11), 5750–5755 (2012).
[CrossRef] [PubMed]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett.12(9), 4932–4936 (2012).
[CrossRef] [PubMed]

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun.3, 1198 (2012).
[CrossRef] [PubMed]

2011

M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials5(4), 178–205 (2011).
[CrossRef]

N. F. 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,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

2010

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9(5), 387–396 (2010).
[CrossRef] [PubMed]

O. Paul, B. Reinhard, B. Krolla, R. Beigang, and M. Rahm, “Gradient index metamaterial based on slot elements,” Appl. Phys. Lett.96(24), 241110 (2010).
[CrossRef]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010).
[CrossRef] [PubMed]

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express18(19), 20321–20333 (2010).
[CrossRef] [PubMed]

2009

F. L. Zhang, Q. Zhao, L. Kang, J. Zhou, and D. Lippens, “Experimental verification of isotropic and polarization properties of high permittivity-based metamaterial,” Phys. Rev. B80(19), 195119 (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]

Y. Lai, H. Y. 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]

2007

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99(6), 063908 (2007).
[CrossRef] [PubMed]

J. M. Hao, L. Zhou, and C. T. Chan, “An effective-medium model for high-impedance surfaces,” Appl. Phys. A Mater. Sci. Process.87(2), 281–284 (2007).
[CrossRef]

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(24), 243901 (2007).
[CrossRef] [PubMed]

A. O. Pinchuk and G. C. Schatz, “Metamaterials with gradient negative index of refraction,” J. Opt. Soc. Am. A24(10), A39–A44 (2007).
[CrossRef] [PubMed]

2006

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

2005

D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express13(6), 2127–2134 (2005).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308(5721), 534–537 (2005).
[CrossRef] [PubMed]

2003

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt.50(9), 1419–1430 (2003).

2001

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292(5514), 77–79 (2001).
[CrossRef] [PubMed]

2000

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

1999

D. Sievenpiper, L. Zhang, R. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47(11), 2059–2074 (1999).
[CrossRef]

1963

D. Berry, R. Malech, and W. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propag.11(6), 645–651 (1963).
[CrossRef]

Abashin, M.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(24), 243901 (2007).
[CrossRef] [PubMed]

Aieta, F.

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities,” Nano Lett.12(3), 1702–1706 (2012).
[CrossRef] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett.12(12), 6328–6333 (2012).
[CrossRef] [PubMed]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett.12(9), 4932–4936 (2012).
[CrossRef] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

N. F. 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,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Albooyeh, M.

M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials5(4), 178–205 (2011).
[CrossRef]

Alexopolous, N. G.

D. Sievenpiper, L. Zhang, R. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47(11), 2059–2074 (1999).
[CrossRef]

Bai, B.

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun.3, 1198 (2012).
[CrossRef] [PubMed]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett.12(11), 5750–5755 (2012).
[CrossRef] [PubMed]

Beigang, R.

O. Paul, B. Reinhard, B. Krolla, R. Beigang, and M. Rahm, “Gradient index metamaterial based on slot elements,” Appl. Phys. Lett.96(24), 241110 (2010).
[CrossRef]

Berry, D.

D. Berry, R. Malech, and W. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propag.11(6), 645–651 (1963).
[CrossRef]

Blaikie, R. J.

Blanchard, R.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett.12(9), 4932–4936 (2012).
[CrossRef] [PubMed]

Boltasseva, A.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science335(6067), 427 (2012).
[CrossRef] [PubMed]

Bozhevolnyi, S. I.

A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett.13(2), 829–834 (2013).
[CrossRef] [PubMed]

Broas, R.

D. Sievenpiper, L. Zhang, R. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47(11), 2059–2074 (1999).
[CrossRef]

Cai, B. G.

Cai, W. S.

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

Capasso, F.

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities,” Nano Lett.12(3), 1702–1706 (2012).
[CrossRef] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett.12(12), 6328–6333 (2012).
[CrossRef] [PubMed]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett.12(9), 4932–4936 (2012).
[CrossRef] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

N. F. 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,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9(5), 387–396 (2010).
[CrossRef] [PubMed]

Y. Lai, H. Y. 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]

J. M. Hao, L. Zhou, and C. T. Chan, “An effective-medium model for high-impedance surfaces,” Appl. Phys. A Mater. Sci. Process.87(2), 281–284 (2007).
[CrossRef]

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99(6), 063908 (2007).
[CrossRef] [PubMed]

Chen, H.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9(5), 387–396 (2010).
[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]

Y. Lai, H. Y. 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, W. T.

S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
[CrossRef] [PubMed]

Chen, X.

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett.12(11), 5750–5755 (2012).
[CrossRef] [PubMed]

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun.3, 1198 (2012).
[CrossRef] [PubMed]

Chettiar, U. K.

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

Cui, T. J.

Cummer, S. A.

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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Cunningham, J.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(24), 243901 (2007).
[CrossRef] [PubMed]

Emani, N. K.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science335(6067), 427 (2012).
[CrossRef] [PubMed]

Eriksen, R. L.

A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett.13(2), 829–834 (2013).
[CrossRef] [PubMed]

Fainman, Y.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(24), 243901 (2007).
[CrossRef] [PubMed]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Gaburro, Z.

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities,” Nano Lett.12(3), 1702–1706 (2012).
[CrossRef] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett.12(12), 6328–6333 (2012).
[CrossRef] [PubMed]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett.12(9), 4932–4936 (2012).
[CrossRef] [PubMed]

N. F. 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,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Gajic, R.

Genevet, P.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett.12(9), 4932–4936 (2012).
[CrossRef] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett.12(12), 6328–6333 (2012).
[CrossRef] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities,” Nano Lett.12(3), 1702–1706 (2012).
[CrossRef] [PubMed]

N. F. 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,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Guo, G.-Y.

S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
[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]

Hao, J. M.

J. M. Hao, L. Zhou, and C. T. Chan, “An effective-medium model for high-impedance surfaces,” Appl. Phys. A Mater. Sci. Process.87(2), 281–284 (2007).
[CrossRef]

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99(6), 063908 (2007).
[CrossRef] [PubMed]

He, Q.

C. Qu, S. Y. Xiao, S. L. Sun, Q. He, and L. Zhou, “A theoretical study on the conversion efficiencies of gradient meta-surfaces,” Europhys. Lett.101(5), 54002 (2013).
[CrossRef]

S. L. Sun, Q. He, S. Y. 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. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
[CrossRef] [PubMed]

X. Li, S. Y. Xiao, B. G. Cai, Q. He, T. J. Cui, and L. Zhou, “Flat metasurfaces to focus electromagnetic waves in reflection geometry,” Opt. Lett.37(23), 4940–4942 (2012).
[CrossRef] [PubMed]

Hingerl, K.

Huang, L.

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett.12(11), 5750–5755 (2012).
[CrossRef] [PubMed]

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun.3, 1198 (2012).
[CrossRef] [PubMed]

Ikeda, K.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(24), 243901 (2007).
[CrossRef] [PubMed]

Isic, G.

Jiang, T.

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99(6), 063908 (2007).
[CrossRef] [PubMed]

Jin, G.

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun.3, 1198 (2012).
[CrossRef] [PubMed]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett.12(11), 5750–5755 (2012).
[CrossRef] [PubMed]

Juan, T.-K.

S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
[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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Kang, L.

F. L. Zhang, Q. Zhao, L. Kang, J. Zhou, and D. Lippens, “Experimental verification of isotropic and polarization properties of high permittivity-based metamaterial,” Phys. Rev. B80(19), 195119 (2009).
[CrossRef]

Kats, M. A.

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett.12(12), 6328–6333 (2012).
[CrossRef] [PubMed]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett.12(9), 4932–4936 (2012).
[CrossRef] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities,” Nano Lett.12(3), 1702–1706 (2012).
[CrossRef] [PubMed]

N. F. 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,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Kennedy, W.

D. Berry, R. Malech, and W. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propag.11(6), 645–651 (1963).
[CrossRef]

Kildishev, A. V.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science335(6067), 427 (2012).
[CrossRef] [PubMed]

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

Kong, J. A.

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99(6), 063908 (2007).
[CrossRef] [PubMed]

Krishnamoorthy, A.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(24), 243901 (2007).
[CrossRef] [PubMed]

Krolla, B.

O. Paul, B. Reinhard, B. Krolla, R. Beigang, and M. Rahm, “Gradient index metamaterial based on slot elements,” Appl. Phys. Lett.96(24), 241110 (2010).
[CrossRef]

Kundtz, N.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010).
[CrossRef] [PubMed]

Kung, W.-T.

S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
[CrossRef] [PubMed]

Lai, Y.

Y. Lai, H. Y. 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]

Larouche, S.

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

Levy, U.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(24), 243901 (2007).
[CrossRef] [PubMed]

Li, G.

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun.3, 1198 (2012).
[CrossRef] [PubMed]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett.12(11), 5750–5755 (2012).
[CrossRef] [PubMed]

Li, X.

X. Li, S. Y. Xiao, B. G. Cai, Q. He, T. J. Cui, and L. Zhou, “Flat metasurfaces to focus electromagnetic waves in reflection geometry,” Opt. Lett.37(23), 4940–4942 (2012).
[CrossRef] [PubMed]

S. L. Sun, Q. He, S. Y. 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]

Liao, C. Y.

S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
[CrossRef] [PubMed]

Lin, J.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

Lippens, D.

F. L. Zhang, Q. Zhao, L. Kang, J. Zhou, and D. Lippens, “Experimental verification of isotropic and polarization properties of high permittivity-based metamaterial,” Phys. Rev. B80(19), 195119 (2009).
[CrossRef]

Malech, R.

D. Berry, R. Malech, and W. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propag.11(6), 645–651 (1963).
[CrossRef]

Melville, D. O. S.

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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Morits, D.

M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials5(4), 178–205 (2011).
[CrossRef]

Mühlenbernd, H.

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett.12(11), 5750–5755 (2012).
[CrossRef] [PubMed]

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun.3, 1198 (2012).
[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]

Ni, X.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science335(6067), 427 (2012).
[CrossRef] [PubMed]

Nielsen, M. G.

A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett.13(2), 829–834 (2013).
[CrossRef] [PubMed]

Paul, O.

O. Paul, B. Reinhard, B. Krolla, R. Beigang, and M. Rahm, “Gradient index metamaterial based on slot elements,” Appl. Phys. Lett.96(24), 241110 (2010).
[CrossRef]

Pendry, J. B.

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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt.50(9), 1419–1430 (2003).

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

Pinchuk, A. O.

Pors, A.

A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett.13(2), 829–834 (2013).
[CrossRef] [PubMed]

Qiu, C. W.

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun.3, 1198 (2012).
[CrossRef] [PubMed]

Qu, C.

C. Qu, S. Y. Xiao, S. L. Sun, Q. He, and L. Zhou, “A theoretical study on the conversion efficiencies of gradient meta-surfaces,” Europhys. Lett.101(5), 54002 (2013).
[CrossRef]

Rahm, M.

O. Paul, B. Reinhard, B. Krolla, R. Beigang, and M. Rahm, “Gradient index metamaterial based on slot elements,” Appl. Phys. Lett.96(24), 241110 (2010).
[CrossRef]

Ramakrishna, S. A.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt.50(9), 1419–1430 (2003).

Ran, L. X.

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99(6), 063908 (2007).
[CrossRef] [PubMed]

Reinhard, B.

O. Paul, B. Reinhard, B. Krolla, R. Beigang, and M. Rahm, “Gradient index metamaterial based on slot elements,” Appl. Phys. Lett.96(24), 241110 (2010).
[CrossRef]

Schatz, G. C.

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Scully, M. O.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

Shalaev, V. M.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science335(6067), 427 (2012).
[CrossRef] [PubMed]

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9(5), 387–396 (2010).
[CrossRef] [PubMed]

Sievenpiper, D.

D. Sievenpiper, L. Zhang, R. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47(11), 2059–2074 (1999).
[CrossRef]

Simovski, C. R.

M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials5(4), 178–205 (2011).
[CrossRef]

Smith, D. R.

S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory,” Opt. Lett.37(12), 2391–2393 (2012).
[CrossRef] [PubMed]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010).
[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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292(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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Stewart, W. J.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt.50(9), 1419–1430 (2003).

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Sun, S.

S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
[CrossRef] [PubMed]

Sun, S. L.

C. Qu, S. Y. Xiao, S. L. Sun, Q. He, and L. Zhou, “A theoretical study on the conversion efficiencies of gradient meta-surfaces,” Europhys. Lett.101(5), 54002 (2013).
[CrossRef]

S. L. Sun, Q. He, S. Y. 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]

Tan, Q.

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett.12(11), 5750–5755 (2012).
[CrossRef] [PubMed]

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun.3, 1198 (2012).
[CrossRef] [PubMed]

Tetienne, J.-P.

N. F. 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,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Tsai, D.-P.

S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
[CrossRef] [PubMed]

Vasic, B.

Wang, C.-M.

S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
[CrossRef] [PubMed]

Wiltshire, M. C. K.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt.50(9), 1419–1430 (2003).

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. Y.

C. Qu, S. Y. Xiao, S. L. Sun, Q. He, and L. Zhou, “A theoretical study on the conversion efficiencies of gradient meta-surfaces,” Europhys. Lett.101(5), 54002 (2013).
[CrossRef]

S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
[CrossRef] [PubMed]

S. L. Sun, Q. He, S. Y. 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]

X. Li, S. Y. Xiao, B. G. Cai, Q. He, T. J. Cui, and L. Zhou, “Flat metasurfaces to focus electromagnetic waves in reflection geometry,” Opt. Lett.37(23), 4940–4942 (2012).
[CrossRef] [PubMed]

Xu, Q.

S. L. Sun, Q. He, S. Y. 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]

Yablonovitch, E.

D. Sievenpiper, L. Zhang, R. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47(11), 2059–2074 (1999).
[CrossRef]

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S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
[CrossRef] [PubMed]

Yu, N.

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett.12(12), 6328–6333 (2012).
[CrossRef] [PubMed]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett.12(9), 4932–4936 (2012).
[CrossRef] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

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[CrossRef] [PubMed]

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N. F. 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,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Yuan, Y.

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99(6), 063908 (2007).
[CrossRef] [PubMed]

Zentgraf, T.

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun.3, 1198 (2012).
[CrossRef] [PubMed]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett.12(11), 5750–5755 (2012).
[CrossRef] [PubMed]

Zhang, F. L.

F. L. Zhang, Q. Zhao, L. Kang, J. Zhou, and D. Lippens, “Experimental verification of isotropic and polarization properties of high permittivity-based metamaterial,” Phys. Rev. B80(19), 195119 (2009).
[CrossRef]

Zhang, L.

D. Sievenpiper, L. Zhang, R. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47(11), 2059–2074 (1999).
[CrossRef]

Zhang, S.

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun.3, 1198 (2012).
[CrossRef] [PubMed]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett.12(11), 5750–5755 (2012).
[CrossRef] [PubMed]

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N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308(5721), 534–537 (2005).
[CrossRef] [PubMed]

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Y. Lai, H. Y. 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]

Zhao, Q.

F. L. Zhang, Q. Zhao, L. Kang, J. Zhou, and D. Lippens, “Experimental verification of isotropic and polarization properties of high permittivity-based metamaterial,” Phys. Rev. B80(19), 195119 (2009).
[CrossRef]

Zhou, J.

F. L. Zhang, Q. Zhao, L. Kang, J. Zhou, and D. Lippens, “Experimental verification of isotropic and polarization properties of high permittivity-based metamaterial,” Phys. Rev. B80(19), 195119 (2009).
[CrossRef]

Zhou, L.

C. Qu, S. Y. Xiao, S. L. Sun, Q. He, and L. Zhou, “A theoretical study on the conversion efficiencies of gradient meta-surfaces,” Europhys. Lett.101(5), 54002 (2013).
[CrossRef]

S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
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S. L. Sun, Q. He, S. Y. 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]

X. Li, S. Y. Xiao, B. G. Cai, Q. He, T. J. Cui, and L. Zhou, “Flat metasurfaces to focus electromagnetic waves in reflection geometry,” Opt. Lett.37(23), 4940–4942 (2012).
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Appl. Phys. A Mater. Sci. Process.

J. M. Hao, L. Zhou, and C. T. Chan, “An effective-medium model for high-impedance surfaces,” Appl. Phys. A Mater. Sci. Process.87(2), 281–284 (2007).
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Europhys. Lett.

C. Qu, S. Y. Xiao, S. L. Sun, Q. He, and L. Zhou, “A theoretical study on the conversion efficiencies of gradient meta-surfaces,” Europhys. Lett.101(5), 54002 (2013).
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J. Mod. Opt.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt.50(9), 1419–1430 (2003).

J. Opt. Soc. Am. A

Metamaterials

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Nano Lett.

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[CrossRef] [PubMed]

S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett.12(12), 6223–6229 (2012).
[CrossRef] [PubMed]

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities,” Nano Lett.12(3), 1702–1706 (2012).
[CrossRef] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett.12(12), 6328–6333 (2012).
[CrossRef] [PubMed]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett.12(11), 5750–5755 (2012).
[CrossRef] [PubMed]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett.12(9), 4932–4936 (2012).
[CrossRef] [PubMed]

Nat. Commun.

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun.3, 1198 (2012).
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Nat. Mater.

S. L. Sun, Q. He, S. Y. 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).
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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]

Y. Lai, H. Y. 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).
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[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

N. F. 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,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Other

35. These reflection channels could also be understood as the Floquet modes diffracted by our super-periodic system.

In our computational approach, we have to set the number of sub-cells divided identical to the number plane waves adopted in region (both are 2N + 1), in order to ensure that the number of restraints equals to that of variables.

For two boundary indexes, we have the following off-diagonal matrix elementsH1,2N+1=μM,1xγ, H2N+1,1=μM,2N+1xγ according to the periodic boundary condition.

Since the super-cell length L is very large, the distribution of those discretized kxr,n is almost continuous. Thus, in what follows, we use ρ(kxr) to represent ρ(kxr,n) for simplicity.

P. Sheng, “Wave scattering formalism,” in Introduction to Wave Scattering, Localization and Macroscopic Phenomena, R. Hull, R. M. Osgood, eds. (Springer, 2006).

EastFDTD v2.0 Beta, DONGJUN Science and Technology Co., China.

For the ξ = 0.4k0 sample, a super cell contains 10 pairs of “H” (altogether 20 ones) in one supercell, with L1 values of those 10 pairs set as 1.3 mm, 2.68 mm, 2.98 mm, 3.14 mm, 3.24 mm, 3.36 mm, 3.48 mm, 3.66 mm, 4.08 mm, and 5.7 mm. For the ξ = 0.8k0 sample, a super cell contains 10 “H” in one super cell with L1 parameters the same as the case of ξ = 0.4k0.

The gain of the employed double-ridged horn antenna is about 14dB~15dB in this frequency region.

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Figures (10)

Fig. 1
Fig. 1

(a) Geometry of the system under study. (b) Discretized model for the inhomogeneous structure.

Fig. 2
Fig. 2

Material properties of meta-surfaces with different ξ designed based the (a) [ ε M y = μ M x ] model and (b) the [ ε M y =1, μ M x ] model. (c) and (d): Calculated scattering coefficients |ρ( k x r ) | 2 versus k x r for different meta-surfaces.

Fig. 3
Fig. 3

(a) Calculated scattering coefficients |ρ( k x r ) | 2 of the ξ=0.8 k 0 meta-surface designed with the [ ε M y = μ M x ] model, under illuminations of TE-polarized input wave with different parallel wave-vectors. (b) Parallel wave-vector k x r of the reflected beam as functions that of the indent beam k x r , calculated by the mode-expansion theory for two meta-surfaces with different ξ and a PEC (with ξ = 0).

Fig. 4
Fig. 4

(a) Working scheme of the flat meta-surface lens. (b) Distributions of parameter values and reflection-phases Φ for the meta-surface. (c) Calculated |ρ( k x r ) | 2 spectrum for the designed lens under TE normal incident excitation. (d) Calculated E-field distribution for the waves scattered by the meta-surface.

Fig. 5
Fig. 5

Calculated G functions in discretized versions for different eigenvalues qz for the model with f(x) = 1 + ξx/2k0d. Dashed lines represent the x positions satisfying Eq. (41). Here, ξ = 0.4k0, d = λ /20, L = 200Ls, Ls = 2π / ξ with λ being the working wavelength.

Fig. 6
Fig. 6

The reflection efficiency R for (a) meta-surfaces with different ξ /k0 under normal-incidence excitations and for (b) a ξ = 0.4k0 meta-surface illuminated by plane waves with different k x in , calculated by the rigorous mode-expansion theory (black solid lines) and the LRM (red dotted lines). Here we adopted the impedance-matched model ε M y = μ M x =1+ξx/2 k 0 d for all meta-surfaces studied.

Fig. 7
Fig. 7

(a) Distributions of μ M x (x) for meta-surfaces with ξ = 0.4k0 (circles) and ξ = 0.8k0 (triangles), designed based on the stepwise [ ε M y =1, μ M x ] models. (b) Scattering coefficients |ρ( k x r ) | 2 versus k x r for meta-surfaces with properties depicted in (a), calculated by the mode-expansion theory.

Fig. 8
Fig. 8

FDTD-retrieved μeff parameter (line) for HIS’ consisting of periodic arrangements of unit cells depicted in the inset, with different values of central bar length L1. Scatters represent those units adopted in designing the ξ = 0.4k0 model. Other parameters Px, Py, d, w and L2 are fixed as 2.5 mm, 6 mm, 1 mm, 0.5 mm, and 2 mm. The working frequency is 15 GHz.

Fig. 9
Fig. 9

(a) Picture of part of the fabricated ξ = 0.4k0 sample. (b) Schematics of the FF characterization. Measured (scatters) and simulated (lines) scattering patterns, | S 21 | 2 , for the samples with (c) ξ = 0.4k0 and (d) ξ = 0.8k0. In our experiments, we cannot measure the reflection signals within the angle region of θ r θ i (grey area) where the two antennas touch each other.

Fig. 10
Fig. 10

Measured (blue line) and simulated (green circles) scattering patterns for (a) the ξ = 0.4k0 meta-surface illuminated by an TE wave with incident angle 30° and (b) the ξ = 0.8k0 meta-surface for incident angle 45°. Red lines denote the incident angle, and the grey area denotes the angle region where we cannot measure the reflected signals. (c) k x r k x in relations obtained by experiments (crosses), simulations (triangles) and model (lines) for meta-surfaces with ξ = 0.4k0 (green) ξ = 0.8k0 (blue) and a flat PEC surface(black). The shadowed area denotes the region where negative reflection happens.

Equations (44)

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

ε M (x)=( ε M x (x) 0 0 0 ε M y (x) 0 0 0 ε M z ), μ M (x)=( μ M x (x) 0 0 0 μ M y (x) 0 0 0 μ M z ),
E in ( r )= e i( k x in x k z in z ) ( 0 1 0 ), H in ( r )= 1 Z 0 e i( k x in x k z in z) ( k z in / k 0 0 k x in / k 0 ).
E ref,n ( r )= e i( k x r,n x + k z r,n z) ( 0 1 0 ), H ref,n ( r )= 1 Z 0 e i( k x r,n x + k z r,n z) ( k z r,n / k 0 0 k x r,n / k 0 ),
{ E I = E in + n E ref,n ρ( k x r,n ) H I = H in + n H ref,n ρ( k x r,n ) ,
ε M 1 (x)×[ μ M 1 (x)(× E (r))]= ω 2 E (r),
E y ± ( q z ,x,z)=G( q z ,x) e i q z z ,
μ M x (x) μ M z d 2 G( q z ,x) d x 2 +[ k 0 2 ε M y (x) μ M x (x) q z 2 ]G( q z ,x)=0.
H x ± ( q z ,x,z)= 1 iω μ 0 μ M x (x) E y z = q z k 0 Z 0 μ M x (x) G( q z ,x) e i q z z
H z ± ( q z ,x,z)= 1 iω μ 0 μ M z E y x = e i q z z i k 0 Z 0 μ M z G( q z ,x) x .
μ M,m x G( q z ,m1)γ+[ k 0 2 ε M,m y μ M,m x μ M,m x γ]G( q z ,m) + μ M,m+1 x G( q z ,m+1)γ= q z 2 G( q z ,m).
m' H mm' G m' = q z 2 G m ,
H mm' =( k 0 2 μ M,m x ε M,m y 2 μ M,m x γ) δ mm' + μ M,m x γ δ m,m'1 + μ M,m x γ δ m,m'+1 .
E II ( r )= j [ C + ( q z,j ) E + ( q z,j , r )+ C ( q z,j ) E ( q z,j , r ) ] , H II ( r )= j [ C + ( q z,j ) H + ( q z,j , r )+ C ( q z,j ) H ( q z,j , r ) ] ,
E y II (x,y,z=d)= j [ C + ( q z,j ) e i q z d + C ( q z,j ) e i q z d ]G( q z,j ,x) 0.
C ( q z,j )= C + ( q z,j ) e i2 q z d .
E y I = E y II , H x I = H x II , at z=0.
{ e i k x in x m + n ρ( k x r,n ) e i k x r,n x m = j G( q z,j ,m)[ C + ( q z,j )+ C ( q z,j ) ] k z in k 0 e i k x in x m n ρ( k x r,n ) k z r,n k 0 e i k x r,n x m = j q z,j μ M,m x k 0 G( q z,j ,m)[ C + ( q z,j ) C ( q z,j ) ] .
ρ( k x r,n )= j A 1 ( k x in , q z,j )B( q z,j , k x r,n ) ,
{ A( q z,j , k x in )= k z r,n k z r,n + k z in S( q z,j , k x r,n )+ k z in k z r,n + k z in S'( q z,j , k x r,n ) B( q z,j , k x r,n )= k z in k z r,n + k z in S( q z,j , k x n ) k z in k z r,n + k z in S'( q z,j , k x r,n ) ,
{ S( q z,j , k x r,n )= 1 L m h(1 e i2 q z,j d )G( q z,j ,m) e i k x r,n mh S'( q z,j , k x r,n )= 1 L m h q z,j μ M,m x k z in (1+ e i2 q z,j d )G( q z,j ,m) e i k x r,n mh
{ S( q z,j , k x r,n )= 1 L m h(1+ e i2 q z,j d )G( q z,j ,m) e i k x r,n mh S'( q z,j , k x r,n )= 1 L m h q z,j ε M,m x k z 0 (1 e i2 q z,j d )G( q z,j ,m) e i k x r,n mh ,
H mm' =( k 0 2 ε M,m x μ M,m y 2 ε M,m x γ) δ mm' + ε M,m x γ δ m,m'1 + ε M,m x γ δ m,m'+1 .
Φ(x)= Φ 0 +ξx,
ε M y (x)= μ M x (x)=1+κx
Φ(x)= cos 1 { [ ε M y + μ M x tan 2 ( ε M y μ M x k 0 d) ]/[ ε M y + μ M x tan 2 ( ε M y μ M x k 0 d) ] }.
k x r =ξ+ k x in
R c = | ρ( k x r ) | 2 cos θ r = | ρ( k x r ) | 2 1 ( ξ/ k 0 ) 2 .
Φ(x)= Φ 0 k 0 x 2 + l focus 2 + k 0 l focus
ε M y (x)= μ M x (x)=v+( l focus x 2 + l focus 2 )/2d,
δ( k x r k x in )+ρ( k x r )= 0 d q z C + ( q z ) 1 2π + (1 e i2 q z d )G( q z ,x) e i k x r x dx ,
δ( k x r k x in )ρ( k x r )= 0 d q z C + ( q z ) 1 2π + (1+ e i2 q z d )G( q z ,x) e i k x r x dx ,
δ( k x r k x in )= 1 2π + e i k x r x dx 0 d q z G( q z ,x) C + ( q z ) .
0 d q z G( q z ,x) C + ( q z ) = e i k x in x .
C + ( q z )= 1 2π + G * ( q z ,x) e i k x in x dx .
ρ( k x r )= 1 (2π) 2 + e i k x r x' dx' + e i k x in x dx 0 G( q z ,x') e i2 q z d G * ( q z ,x)d q z ,
ρ( k x r )= 1 (2π) 2 + dx' + dx 0 d q z 0 d q z × k x r | x' x' |G| q z q z |P| q z q z | G * |xx| k x in .
{ V q z , k x in = 1 2π + dx q z | G * |xx| k x in V k x r , q z * = 1 2π + dx k x r |xx|G| q z ,
ρ( k x r )= T k x r , k x in = q z , q z V k x r , q z * P q z , q z V q z , k x in .
q z = k 0 μ M x (x) ε M y (x) = k 0 f(x)
G( q z ,x)=δ( x f 1 ( q z / k 0 ) ) ,
ρ( k x r )= 1 2π + dx k x r |x r local (x)x| k x in ,
E sca (x)= r local (x)x| k x in .
ρ LRM ( k x r )= e 2i k 0 d δ( k x r k x in ξ).
R= |ρ( k x ) | 2 k 0 2 k x 2 d k x k 0 2 ( k x in ) 2 ,

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