Abstract

It is well known that refraction of light at interfaces leads to a beam displacement for oblique incidence of light onto a slab of material. In ray optics and for homogeneous isotropic materials, the sign of this beam displacement is strictly identical to the sign of the refractive index. Our numerical calculations reveal negative beam displacements from state-of-the-art double-fishnet-type photonic metamaterials. This holds true for the “main” polarization corresponding to a negative phase velocity for normal incidence as well as for the “secondary” polarization with positive phase velocity. To understand and interpret these results, we also present exact analytical calculations for thin metal films showing that, in wave optics, the sign of the beam displacement (i.e., the sign of refraction) is generally not identical to the sign of the refractive index.

© 2007 Optical Society of America

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  1. V. G. Veselago, "The electrodynamics of substances with simultaneuously negative values of ? and ?," Sov. Phys. Uspekhi 10, 509-514 (1968).
    [CrossRef]
  2. V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1, 41-48 (2006).Q1
    [CrossRef]
  3. C. M. Soukoulis, S. Linden, and M. Wegener, "Negative Refractive Index at Optical Wavelengths," Science 315, 47-49 (2007).
    [CrossRef] [PubMed]
  4. K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
    [CrossRef]
  5. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verfication of a Negative Index of Refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  6. G. Dolling, M. Wegener, and S. Linden, "Realization of a three-functional-layer negative-index photonic metamaterial," Opt. Lett. 32, 551-553 (2007).
    [CrossRef] [PubMed]
  7. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial," Science 312, 892-894 (2006).
    [CrossRef] [PubMed]
  8. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
    [CrossRef] [PubMed]
  9. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007).
    [CrossRef]
  10. X. L. Chen, M. He, Y. Du, W. Y. Wang, and D. F. Zhang, "Negative refraction: An intrinsic property of uniaxial crystals," Phys. Rev. B 72, 113111 (2005).
    [CrossRef]
  11. E. Shamonina and L. Solymar, "Properties of magnetically coupled metamaterial elements," J. Magn. Magn. Mater. 300, 38-43 (2006).
    [CrossRef]
  12. G. Dolling, M. Wegener, A. Sch¨adle, S. Burger, and S. Linden, "Observation of magnetization waves in negativeindex photonic metamaterials," Appl. Phys. Lett. 89, 231118 (2006).
    [CrossRef]
  13. F. Goos and H. H¨anchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Physik 1, 333-346 (1947).
    [CrossRef]
  14. F. Goos and H. H¨anchen, "Neumessung des Strahlversetzungseffektes bei Totalreflexion," Ann. Physik 5, 251- 252 (1949).
    [CrossRef]
  15. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-H¨anchen effect," Phys. Rev. E 62, 7330-7339 (2000).
    [CrossRef]
  16. S. Zhang, W. Fan, K. J. Malloy, S. R. Brueck, N. C. Panoiu, and R. M. Osgood, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4930 (2005).
    [CrossRef] [PubMed]
  17. S. Burger, R. Klose, A. Sch¨adle, F. Schmidt, and L. Zschiedrich, "FEM modelling of 3D photonic crystals and photonic crystal waveguides," Proc. SPIE Vol. 5728, 164-173 (2005).
    [CrossRef]
  18. L. Zschiedrich, R. Klose, A. Sch¨adle, and F. Schmidt, "A new finite element realization of the Perfectly Matched Layer Method for Helmholtz scattering problems on polygonal domains in 2D," J. Comput Appl. Math. 188, 12-32 (2006).
    [CrossRef]
  19. L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, "Advanced Finite Element Method for Nano-Resonators," Proc. SPIE Vol. 6115, 164-174 (2006).
  20. P. B. Johnson and R. W. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  21. E. Hecht, Optics (Addison Wesley, 2001).
  22. J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  23. M. Wegener, G. Dolling, and S. Linden, "Backward waves moving forward," Nature Mater. 6, 475-476 (2007).
    [CrossRef]

2007 (5)

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative Refractive Index at Optical Wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

M. Wegener, G. Dolling, and S. Linden, "Backward waves moving forward," Nature Mater. 6, 475-476 (2007).
[CrossRef]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007).
[CrossRef]

G. Dolling, M. Wegener, and S. Linden, "Realization of a three-functional-layer negative-index photonic metamaterial," Opt. Lett. 32, 551-553 (2007).
[CrossRef] [PubMed]

2006 (7)

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial," Science 312, 892-894 (2006).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

L. Zschiedrich, R. Klose, A. Sch¨adle, and F. Schmidt, "A new finite element realization of the Perfectly Matched Layer Method for Helmholtz scattering problems on polygonal domains in 2D," J. Comput Appl. Math. 188, 12-32 (2006).
[CrossRef]

L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, "Advanced Finite Element Method for Nano-Resonators," Proc. SPIE Vol. 6115, 164-174 (2006).

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1, 41-48 (2006).Q1
[CrossRef]

E. Shamonina and L. Solymar, "Properties of magnetically coupled metamaterial elements," J. Magn. Magn. Mater. 300, 38-43 (2006).
[CrossRef]

G. Dolling, M. Wegener, A. Sch¨adle, S. Burger, and S. Linden, "Observation of magnetization waves in negativeindex photonic metamaterials," Appl. Phys. Lett. 89, 231118 (2006).
[CrossRef]

2005 (3)

X. L. Chen, M. He, Y. Du, W. Y. Wang, and D. F. Zhang, "Negative refraction: An intrinsic property of uniaxial crystals," Phys. Rev. B 72, 113111 (2005).
[CrossRef]

S. Burger, R. Klose, A. Sch¨adle, F. Schmidt, and L. Zschiedrich, "FEM modelling of 3D photonic crystals and photonic crystal waveguides," Proc. SPIE Vol. 5728, 164-173 (2005).
[CrossRef]

S. Zhang, W. Fan, K. J. Malloy, S. R. Brueck, N. C. Panoiu, and R. M. Osgood, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4930 (2005).
[CrossRef] [PubMed]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verfication of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

2000 (2)

J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-H¨anchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneuously negative values of ? and ?," Sov. Phys. Uspekhi 10, 509-514 (1968).
[CrossRef]

1949 (1)

F. Goos and H. H¨anchen, "Neumessung des Strahlversetzungseffektes bei Totalreflexion," Ann. Physik 5, 251- 252 (1949).
[CrossRef]

1947 (1)

F. Goos and H. H¨anchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Physik 1, 333-346 (1947).
[CrossRef]

Brueck, S. R.

Burger, S.

L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, "Advanced Finite Element Method for Nano-Resonators," Proc. SPIE Vol. 6115, 164-174 (2006).

S. Burger, R. Klose, A. Sch¨adle, F. Schmidt, and L. Zschiedrich, "FEM modelling of 3D photonic crystals and photonic crystal waveguides," Proc. SPIE Vol. 5728, 164-173 (2005).
[CrossRef]

Busch, K.

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Chen, X. L.

X. L. Chen, M. He, Y. Du, W. Y. Wang, and D. F. Zhang, "Negative refraction: An intrinsic property of uniaxial crystals," Phys. Rev. B 72, 113111 (2005).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Dolling, G.

M. Wegener, G. Dolling, and S. Linden, "Backward waves moving forward," Nature Mater. 6, 475-476 (2007).
[CrossRef]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007).
[CrossRef]

G. Dolling, M. Wegener, and S. Linden, "Realization of a three-functional-layer negative-index photonic metamaterial," Opt. Lett. 32, 551-553 (2007).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial," Science 312, 892-894 (2006).
[CrossRef] [PubMed]

G. Dolling, M. Wegener, A. Sch¨adle, S. Burger, and S. Linden, "Observation of magnetization waves in negativeindex photonic metamaterials," Appl. Phys. Lett. 89, 231118 (2006).
[CrossRef]

Du, Y.

X. L. Chen, M. He, Y. Du, W. Y. Wang, and D. F. Zhang, "Negative refraction: An intrinsic property of uniaxial crystals," Phys. Rev. B 72, 113111 (2005).
[CrossRef]

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial," Science 312, 892-894 (2006).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

Fan, W.

Goos, F.

F. Goos and H. H¨anchen, "Neumessung des Strahlversetzungseffektes bei Totalreflexion," Ann. Physik 5, 251- 252 (1949).
[CrossRef]

F. Goos and H. H¨anchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Physik 1, 333-346 (1947).
[CrossRef]

He, M.

X. L. Chen, M. He, Y. Du, W. Y. Wang, and D. F. Zhang, "Negative refraction: An intrinsic property of uniaxial crystals," Phys. Rev. B 72, 113111 (2005).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Kettner, B.

L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, "Advanced Finite Element Method for Nano-Resonators," Proc. SPIE Vol. 6115, 164-174 (2006).

Klose, R.

L. Zschiedrich, R. Klose, A. Sch¨adle, and F. Schmidt, "A new finite element realization of the Perfectly Matched Layer Method for Helmholtz scattering problems on polygonal domains in 2D," J. Comput Appl. Math. 188, 12-32 (2006).
[CrossRef]

S. Burger, R. Klose, A. Sch¨adle, F. Schmidt, and L. Zschiedrich, "FEM modelling of 3D photonic crystals and photonic crystal waveguides," Proc. SPIE Vol. 5728, 164-173 (2005).
[CrossRef]

Kwok, C. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-H¨anchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Lai, H. M.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-H¨anchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Linden, S.

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative Refractive Index at Optical Wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

M. Wegener, G. Dolling, and S. Linden, "Backward waves moving forward," Nature Mater. 6, 475-476 (2007).
[CrossRef]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007).
[CrossRef]

G. Dolling, M. Wegener, and S. Linden, "Realization of a three-functional-layer negative-index photonic metamaterial," Opt. Lett. 32, 551-553 (2007).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial," Science 312, 892-894 (2006).
[CrossRef] [PubMed]

Loo, Y. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-H¨anchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Malloy, K. J.

Mingaleev, S.

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Osgood, R. M.

Panoiu, N. C.

Pendry, J. B.

J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Schmidt, F.

L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, "Advanced Finite Element Method for Nano-Resonators," Proc. SPIE Vol. 6115, 164-174 (2006).

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verfication of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Shalaev, V. M.

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1, 41-48 (2006).Q1
[CrossRef]

Shamonina, E.

E. Shamonina and L. Solymar, "Properties of magnetically coupled metamaterial elements," J. Magn. Magn. Mater. 300, 38-43 (2006).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verfication of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verfication of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Solymar, L.

E. Shamonina and L. Solymar, "Properties of magnetically coupled metamaterial elements," J. Magn. Magn. Mater. 300, 38-43 (2006).
[CrossRef]

Soukoulis, C. M.

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative Refractive Index at Optical Wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial," Science 312, 892-894 (2006).
[CrossRef] [PubMed]

Tkeshelashvili, L.

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneuously negative values of ? and ?," Sov. Phys. Uspekhi 10, 509-514 (1968).
[CrossRef]

von Freymann, G.

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Wang, W. Y.

X. L. Chen, M. He, Y. Du, W. Y. Wang, and D. F. Zhang, "Negative refraction: An intrinsic property of uniaxial crystals," Phys. Rev. B 72, 113111 (2005).
[CrossRef]

Wegener, M.

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative Refractive Index at Optical Wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007).
[CrossRef]

G. Dolling, M. Wegener, and S. Linden, "Realization of a three-functional-layer negative-index photonic metamaterial," Opt. Lett. 32, 551-553 (2007).
[CrossRef] [PubMed]

M. Wegener, G. Dolling, and S. Linden, "Backward waves moving forward," Nature Mater. 6, 475-476 (2007).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

G. Dolling, M. Wegener, A. Sch¨adle, S. Burger, and S. Linden, "Observation of magnetization waves in negativeindex photonic metamaterials," Appl. Phys. Lett. 89, 231118 (2006).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial," Science 312, 892-894 (2006).
[CrossRef] [PubMed]

Xu, B. Y.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-H¨anchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Zhang, D. F.

X. L. Chen, M. He, Y. Du, W. Y. Wang, and D. F. Zhang, "Negative refraction: An intrinsic property of uniaxial crystals," Phys. Rev. B 72, 113111 (2005).
[CrossRef]

Zhang, S.

Zschiedrich, L.

L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, "Advanced Finite Element Method for Nano-Resonators," Proc. SPIE Vol. 6115, 164-174 (2006).

L. Zschiedrich, R. Klose, A. Sch¨adle, and F. Schmidt, "A new finite element realization of the Perfectly Matched Layer Method for Helmholtz scattering problems on polygonal domains in 2D," J. Comput Appl. Math. 188, 12-32 (2006).
[CrossRef]

Ann. Physik (2)

F. Goos and H. H¨anchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Physik 1, 333-346 (1947).
[CrossRef]

F. Goos and H. H¨anchen, "Neumessung des Strahlversetzungseffektes bei Totalreflexion," Ann. Physik 5, 251- 252 (1949).
[CrossRef]

Appl. Phys. Lett. (1)

G. Dolling, M. Wegener, A. Sch¨adle, S. Burger, and S. Linden, "Observation of magnetization waves in negativeindex photonic metamaterials," Appl. Phys. Lett. 89, 231118 (2006).
[CrossRef]

J. Comput Appl. Math. (1)

L. Zschiedrich, R. Klose, A. Sch¨adle, and F. Schmidt, "A new finite element realization of the Perfectly Matched Layer Method for Helmholtz scattering problems on polygonal domains in 2D," J. Comput Appl. Math. 188, 12-32 (2006).
[CrossRef]

J. Magn. Magn. Mater. (1)

E. Shamonina and L. Solymar, "Properties of magnetically coupled metamaterial elements," J. Magn. Magn. Mater. 300, 38-43 (2006).
[CrossRef]

Nature Mater. (1)

M. Wegener, G. Dolling, and S. Linden, "Backward waves moving forward," Nature Mater. 6, 475-476 (2007).
[CrossRef]

Nature Photon. (1)

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1, 41-48 (2006).Q1
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rep. (1)

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Phys. Rev. B (2)

X. L. Chen, M. He, Y. Du, W. Y. Wang, and D. F. Zhang, "Negative refraction: An intrinsic property of uniaxial crystals," Phys. Rev. B 72, 113111 (2005).
[CrossRef]

P. B. Johnson and R. W. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Phys. Rev. E (1)

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-H¨anchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Proc. SPIE Vol. (2)

S. Burger, R. Klose, A. Sch¨adle, F. Schmidt, and L. Zschiedrich, "FEM modelling of 3D photonic crystals and photonic crystal waveguides," Proc. SPIE Vol. 5728, 164-173 (2005).
[CrossRef]

L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, "Advanced Finite Element Method for Nano-Resonators," Proc. SPIE Vol. 6115, 164-174 (2006).

Science (3)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verfication of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative Refractive Index at Optical Wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial," Science 312, 892-894 (2006).
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Sov. Phys. Uspekhi (1)

V. G. Veselago, "The electrodynamics of substances with simultaneuously negative values of ? and ?," Sov. Phys. Uspekhi 10, 509-514 (1968).
[CrossRef]

Other (1)

E. Hecht, Optics (Addison Wesley, 2001).

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Figures (4)

Fig. 1.
Fig. 1.

Light impinges from vacuum onto a slab (gray area) of an isotropic and homogeneous material of thickness d under oblique incidence with an angle α as indicated. In ray optics, a positive real part of the refractive index Re(n) > 0 leads to a positive beam displacement Δx > 0 (left panel), whereas the displacement is negative (right panel) for a negative refractive index Re(n) < 0. The shift of the transmitted beam, s x , is also indicated.

Fig. 2.
Fig. 2.

Illustration of one unit cell of a single layer of the well-known so-called double-fishnet design of negative-index photonic metamaterials. The parameters are indicated. The yellow regions are the metal (Au), the blue region is the dielectric spacer (MgF2). The structure is located on a glass substrate (not shown).

Fig. 3.
Fig. 3.

Calculated response for the sample structure shown in Fig.2 and for the geometry illustrated in Fig. 1. Results are shown for the “main“ polarization (solid) and for the “secondary” polarization (dashed). The configurations are illustrated by the inset. The arrows indicate the polarization, i.e., the orientation of the electric-field vector of the incident light. (a) Intensity transmittance spectra for the angles of incidence as indicated, (b) retrieved values of the complex quantity n for normal incidence, (c) calculated beam displacements Δx for the complete microscopic theory, and (d) Δx obtained from the parameters obtained from the normal-incidence retrieval. The symbols in (a) and (c) are calculated, the curves are guides to the eye.

Fig. 4.
Fig. 4.

(a) Beam displacements versus angle of incidence a obtained from exact analytical calculations (curves) and complete numerical calculations (symbols) for p- (solid) and s-polarized (dashed) light, respectively, for oblique incidence onto a free-standing 25-nm thin silver film. The two different incident wavelengths are 532 nm (green curves) and 650 nm (red curves). (b) Corresponding (intensity) transmittance spectra. Note the excellent agreement between analytical and numerical results.

Equations (2)

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E t = Z t Z i 2 γt γ i cos ( kd ) γ i γ t γ m i sin ( kd ) γ m i sin ( kd ) + γ t cos ( kd ) E i
E t = 2 γ i γ i cos ( kd ) γ i γ t γ m i sin ( kd ) γ m i sin ( kd ) + γ t cos ( kd ) E i .

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