Abstract

We employed both theoretical calculations and experiments to study the nonlinear responses in optical metamaterials. The spectra of second-harmonic generations measured on a fishnet metamaterial are in quantitative agreements with calculations based on full-wave numerical simulations combined with field integrations, both exhibiting ~80 times enhancements at the magnetic resonance frequency. Our calculations explained several interesting features observed experimentally, and suggested an optimal metamaterial structure to yield the strongest nonlinear signals.

© 2011 OSA

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  1. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
    [CrossRef] [PubMed]
  2. 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]
  3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [CrossRef] [PubMed]
  4. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
    [CrossRef] [PubMed]
  5. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
    [CrossRef] [PubMed]
  6. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
    [CrossRef] [PubMed]
  7. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
    [CrossRef] [PubMed]
  8. M. W. Klein, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Single-slit split-ring resonators at optical frequencies: limits of size scaling,” Opt. Lett. 31(9), 1259–1261 (2006).
    [CrossRef] [PubMed]
  9. W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett. 96(10), 107401 (2006).
    [CrossRef] [PubMed]
  10. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from re〉ection and transmission coef□cients,” Phys. Rev. B 65(19), 195104 (2002).
    [CrossRef]
  11. S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69(24), 241101 (2004).
    [CrossRef]
  12. J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005).
    [CrossRef] [PubMed]
  13. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [CrossRef] [PubMed]
  14. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
    [CrossRef] [PubMed]
  15. H. Q. Lei Zhou, “Li, Y. Q. Qin, Z. Y. Wei, and C. T. Chan, “Directive emissions from subwavelength metamaterial-based cavities,” Appl. Phys. Lett. 86, 101101 (2005).
  16. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
    [CrossRef]
  17. M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
    [CrossRef] [PubMed]
  18. M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15(8), 5238–5247 (2007).
    [CrossRef] [PubMed]
  19. M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials: erratum,” Opt. Express 16(11), 8055 (2008).
    [CrossRef]
  20. E. Kim, F. Wang, W. Wu, Z. Yu, and Y. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
    [CrossRef]
  21. Simulations were performed using the package CONCERTO 7.0, Vector Field Limited, England, (2008).
  22. Handbook of Optical Constants of Solids I, edited by E. D. Palik (Academic, San Diego, 1998).
  23. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
    [CrossRef]
  24. J. Rudnick and E. A. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B 4(12), 4274–4290 (1971).
    [CrossRef]
  25. D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626 (2004).
    [CrossRef]
  26. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1973).
  27. This is the simplest way to consider the local field corrections for the SH radiations. Those electric dipoles generated deep inside the structure (say, located at the inner surfaces sandwiched between two silver layers) do not radiate efficiently, and therefore, we do not include their contribution.
  28. BecauseχS,ζξξ(2)=χS,ζηη(2) = 0 from the free electron gas model for Ag, the local field component parallel to the surface cannot induce a non-vanishing pz(2) on the surface. However, a weak local field component perpendicular to the surface is actually present when the S-polarized input impinges on the nanostructure.

2008 (2)

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials: erratum,” Opt. Express 16(11), 8055 (2008).
[CrossRef]

2007 (1)

2006 (5)

M. W. Klein, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Single-slit split-ring resonators at optical frequencies: limits of size scaling,” Opt. Lett. 31(9), 1259–1261 (2006).
[CrossRef] [PubMed]

W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett. 96(10), 107401 (2006).
[CrossRef] [PubMed]

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

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[CrossRef] [PubMed]

2005 (4)

J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005).
[CrossRef] [PubMed]

H. Q. Lei Zhou, “Li, Y. Q. Qin, Z. Y. Wei, and C. T. Chan, “Directive emissions from subwavelength metamaterial-based cavities,” Appl. Phys. Lett. 86, 101101 (2005).

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[CrossRef] [PubMed]

2004 (4)

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69(24), 241101 (2004).
[CrossRef]

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626 (2004).
[CrossRef]

2002 (1)

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from re〉ection and transmission coef□cients,” Phys. Rev. B 65(19), 195104 (2002).
[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 (2)

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

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

1971 (1)

J. Rudnick and E. A. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B 4(12), 4274–4290 (1971).
[CrossRef]

Averitt, R. D.

W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett. 96(10), 107401 (2006).
[CrossRef] [PubMed]

Basov, D. N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

Burger, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

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,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Economou, E. N.

J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005).
[CrossRef] [PubMed]

Enkrich, C.

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[CrossRef] [PubMed]

M. W. Klein, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Single-slit split-ring resonators at optical frequencies: limits of size scaling,” Opt. Lett. 31(9), 1259–1261 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Fang, N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

Feth, N.

Firsov, A. A.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[CrossRef] [PubMed]

Geim, A. K.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[CrossRef] [PubMed]

Gleeson, H. F.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[CrossRef] [PubMed]

Grigorenko, A. N.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[CrossRef] [PubMed]

Highstrete, C.

W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett. 96(10), 107401 (2006).
[CrossRef] [PubMed]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Kafesaki, M.

J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005).
[CrossRef] [PubMed]

Khrushchev, I. Y.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[CrossRef] [PubMed]

Kim, E.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

Klein, M. W.

Koschny, T.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Koschny, Th.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005).
[CrossRef] [PubMed]

Krause, D.

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626 (2004).
[CrossRef]

Lee, M.

W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett. 96(10), 107401 (2006).
[CrossRef] [PubMed]

Lei Zhou, H. Q.

H. Q. Lei Zhou, “Li, Y. Q. Qin, Z. Y. Wei, and C. T. Chan, “Directive emissions from subwavelength metamaterial-based cavities,” Appl. Phys. Lett. 86, 101101 (2005).

Linden, S.

M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials: erratum,” Opt. Express 16(11), 8055 (2008).
[CrossRef]

M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15(8), 5238–5247 (2007).
[CrossRef] [PubMed]

M. W. Klein, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Single-slit split-ring resonators at optical frequencies: limits of size scaling,” Opt. Lett. 31(9), 1259–1261 (2006).
[CrossRef] [PubMed]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Markos, P.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from re〉ection and transmission coef□cients,” Phys. Rev. B 65(19), 195104 (2002).
[CrossRef]

McPeake, D.

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69(24), 241101 (2004).
[CrossRef]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

O’Brien, S.

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69(24), 241101 (2004).
[CrossRef]

Padilla, W. J.

W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett. 96(10), 107401 (2006).
[CrossRef] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

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,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

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

J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005).
[CrossRef] [PubMed]

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69(24), 241101 (2004).
[CrossRef]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

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

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[CrossRef]

Petrovic, J.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[CrossRef] [PubMed]

Ramakrishna, S. A.

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69(24), 241101 (2004).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[CrossRef]

Rogers, C. T.

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626 (2004).
[CrossRef]

Rudnick, J.

J. Rudnick and E. A. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B 4(12), 4274–4290 (1971).
[CrossRef]

Schmidt, F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

Schultz, S.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from re〉ection and transmission coef□cients,” Phys. Rev. B 65(19), 195104 (2002).
[CrossRef]

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, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

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

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Shen, Y.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

Smith, D. R.

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

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from re〉ection and transmission coef□cients,” Phys. Rev. B 65(19), 195104 (2002).
[CrossRef]

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, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Soukoulis, C. M.

M. W. Klein, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Single-slit split-ring resonators at optical frequencies: limits of size scaling,” Opt. Lett. 31(9), 1259–1261 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from re〉ection and transmission coef□cients,” Phys. Rev. B 65(19), 195104 (2002).
[CrossRef]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Stern, E. A.

J. Rudnick and E. A. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B 4(12), 4274–4290 (1971).
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[CrossRef]

Taylor, A. J.

W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett. 96(10), 107401 (2006).
[CrossRef] [PubMed]

Teplin, C. W.

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626 (2004).
[CrossRef]

Vier, D. C.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Wang, F.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

Wegener, M.

M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials: erratum,” Opt. Express 16(11), 8055 (2008).
[CrossRef]

M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15(8), 5238–5247 (2007).
[CrossRef] [PubMed]

M. W. Klein, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Single-slit split-ring resonators at optical frequencies: limits of size scaling,” Opt. Lett. 31(9), 1259–1261 (2006).
[CrossRef] [PubMed]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Wu, W.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

Yen, T. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

Yu, Z.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

Zhang, X.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

Zhang, Y.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[CrossRef] [PubMed]

Zhou, J.

J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Zhou, J. F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

Zschiedrich, L.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

H. Q. Lei Zhou, “Li, Y. Q. Qin, Z. Y. Wei, and C. T. Chan, “Directive emissions from subwavelength metamaterial-based cavities,” Appl. Phys. Lett. 86, 101101 (2005).

IEEE Trans. Microw. Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[CrossRef]

J. Appl. Phys. (1)

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626 (2004).
[CrossRef]

Nature (1)

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (5)

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

J. Rudnick and E. A. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B 4(12), 4274–4290 (1971).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from re〉ection and transmission coef□cients,” Phys. Rev. B 65(19), 195104 (2002).
[CrossRef]

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69(24), 241101 (2004).
[CrossRef]

Phys. Rev. Lett. (5)

J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

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

W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett. 96(10), 107401 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

Science (6)

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[CrossRef] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (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]

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

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Other (5)

Simulations were performed using the package CONCERTO 7.0, Vector Field Limited, England, (2008).

Handbook of Optical Constants of Solids I, edited by E. D. Palik (Academic, San Diego, 1998).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1973).

This is the simplest way to consider the local field corrections for the SH radiations. Those electric dipoles generated deep inside the structure (say, located at the inner surfaces sandwiched between two silver layers) do not radiate efficiently, and therefore, we do not include their contribution.

BecauseχS,ζξξ(2)=χS,ζηη(2) = 0 from the free electron gas model for Ag, the local field component parallel to the surface cannot induce a non-vanishing pz(2) on the surface. However, a weak local field component perpendicular to the surface is actually present when the S-polarized input impinges on the nanostructure.

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Figures (9)

Fig. 1
Fig. 1

(a) Cross section of the broad wire of the □shnet structure; (b) SEM image of the □shnet structure.

Fig. 2
Fig. 2

Linear transmission, reflection and absorption spectra of the double-fishnet metamaterial, obtained by experiments (red circles) and FDTD simulations (blue lines).

Fig. 3
Fig. 3

Normalized electric field distribution on the middle symmetrical x-z plane inside the realistic double-fishnet structure, calculated by FDTD simulations at the magnetic resonance frequency.

Fig. 4
Fig. 4

FDTD calculated (lines) and measured (symbols) SHG spectra of the fishnet structure for different polarization combinations: (a) P in, P out; (b) P in, S out; (c) S in, S out; (d) S in, P out. The incident angle is kept at α = 45 and the incident planes are specified in the legend for different cases. Inset shows the measured quadratic dependence of the SHG signal on the incident pump power.

Fig. 5
Fig. 5

Normalized local fields at the Ag/SiO2 edge in the inclined side plane as functions of the fundamental wavelength, calculated by FDTD simulations assuming (a) S-wave incident in the y-z plane and (b) P-wave incident in the x-z plane.

Fig. 6
Fig. 6

Distributions of normalized local fields (solid line: E x , l o c ; dashed line: E z , l o c ) along the symmetry lines on the top and the side wall Ag surface (see left panel), calculated by FDTD simulations for the fishnet structures with (a) + (b) θ = 71 , and (c) + (d) θ = 90 .

Fig. 7
Fig. 7

FDTD simulated SHG spectra of the perfect fishnet structure (with vertical side walls) for different polarization combinations: (a) P in P out, (b) P in S out, (c) S in S out, and (d) S in P out. Here, the incident angle is kept as α = 45 and the incident planes are specified in the legend for symbols and lines, respectively.

Fig. 8
Fig. 8

(a) Normalized local-field resonance enhancements at the upper Ag/SiO2 edge on the side wall as functions of inclination angle θ of the fishnet structure, calculated by FDTD simulations assuming a P-wave input. (b) FDTD calculated peak PP-SHG value as a function of θ, with the red star indicating the experimental situation.

Fig. 9
Fig. 9

PP-SHG spectra of the fishnet structure, obtained by FDTD simulations, experimental measurements and the analytical model.

Equations (7)

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

p ( 2 ) = L ( r , 2 ω ) : χ S ( 2 ) : [ E l o c ( r , ω ) ] 2 d r
S σ ' σ S r e f = | e ^ σ p ( 2 ) ( σ , σ ' ) | 2 | e ^ p [ p ( 2 ) ( p , p ) ] r e f | 2
[ p ( 2 ) ( p , p ) ] r e f = A L ( r , 2 ω ) : χ S ( 2 ) : [ L ( r , ω ) E p ( ω ) ] 2 ,
L ( r , ω ) A ( r ) + B ( r ) ω ω r + i Γ
p ( 2 ) ( 2 ω ) = Χ ( 2 ) : E ( ω ) E ( ω )
p z ( 2 ) ~ { χ ( 2 ) ( E z , l o c cos θ + E x , l o c sin θ ) 2 cos θ 2 χ | | ( 2 ) ( E x , l o c cos θ E z , l o c sin θ ) ( E z , l o c cos θ + E x , l o c sin θ ) sin θ } d S ,
S P P ( ω ) = [ B + C ω ω 1 + i Γ + D ω ω 2 + i Γ ] 4 ,

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