Abstract

We present experiments on second-harmonic generation from arrays of magnetic split-ring resonators and arrays of complementary split-ring resonators. In both cases, the fundamental resonance is excited by the incident femtosecond laser pulses under normal incidence, leading to comparably strong second-harmonic signals. These findings are discussed in terms of Babinet’s principle and in terms of a recently developed microscopic classical theory that leads to good agreement regarding the relative and the absolute nonlinear signal strengths. The hydrodynamic convective contribution is found to be the dominant source of second-harmonic generation—in contrast to a previous assignment [Science 313, 502 (2006) ].

© 2008 Optical Society of America

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  1. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
    [CrossRef]
  2. V. M. Shalaev, Nat. Photonics 1, 41 (2007).
    [CrossRef]
  3. C. M. Soukoulis, S. Linden, and M. Wegener, Science 315, 47 (2007).
    [CrossRef] [PubMed]
  4. J. E. Sipe and R. W. Boyd, Phys. Rev. A 46, 1614 (1992).
    [CrossRef] [PubMed]
  5. V. M. Shalaev, Phys. Rep. 272, 61 (1996).
    [CrossRef]
  6. M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, Science 313, 502 (2006).
    [CrossRef] [PubMed]
  7. M. W. Klein, M. Wegener, N. Feth, and S. Linden, Opt. Express 15, 5238 (2007).
    [CrossRef] [PubMed]
  8. M. W. Klein, M. Wegener, N. Feth, and S. Linden, Opt. Express 16, 8055 (2008).
    [CrossRef]
  9. S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Phys. Rev. Lett. 98, 167403 (2007).
    [CrossRef] [PubMed]
  10. T. Xu, X. Jiao, G. P. Zhang, and S. Blair, Opt. Express 15, 13894 (2007).
    [CrossRef] [PubMed]
  11. F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).
    [CrossRef] [PubMed]
  12. T. Zentgraf, T. P. Meyrath, A. Seidel, S. Kaiser, H. Giessen, C. Rockstuhl, and F. Lederer, Phys. Rev. B 76, 033407 (2007).
    [CrossRef]
  13. Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory of second-harmonic generation from magnetic metamaterials,”arXiv:0807.3531v1http://arxiv.org/abs/0807.3531v1 (2008).
  14. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, Th. Koschny, and C. M. Soukoulis, Phys. Rev. Lett. 95, 203901 (2005).
    [CrossRef] [PubMed]
  15. J. Freidberg, Plasma Physics and Fusion Energy (Cambridge, 2007).
    [CrossRef]

2008

2007

M. W. Klein, M. Wegener, N. Feth, and S. Linden, Opt. Express 15, 5238 (2007).
[CrossRef] [PubMed]

T. Xu, X. Jiao, G. P. Zhang, and S. Blair, Opt. Express 15, 13894 (2007).
[CrossRef] [PubMed]

V. M. Shalaev, Nat. Photonics 1, 41 (2007).
[CrossRef]

C. M. Soukoulis, S. Linden, and M. Wegener, Science 315, 47 (2007).
[CrossRef] [PubMed]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

T. Zentgraf, T. P. Meyrath, A. Seidel, S. Kaiser, H. Giessen, C. Rockstuhl, and F. Lederer, Phys. Rev. B 76, 033407 (2007).
[CrossRef]

2006

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, Science 313, 502 (2006).
[CrossRef] [PubMed]

2005

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, Th. Koschny, and C. M. Soukoulis, Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

2004

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

1999

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

1996

V. M. Shalaev, Phys. Rep. 272, 61 (1996).
[CrossRef]

1992

J. E. Sipe and R. W. Boyd, Phys. Rev. A 46, 1614 (1992).
[CrossRef] [PubMed]

Baena, J. D.

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Beruete, M.

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Blair, S.

Bonache, J.

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Boyd, R. W.

J. E. Sipe and R. W. Boyd, Phys. Rev. A 46, 1614 (1992).
[CrossRef] [PubMed]

Burger, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, Th. Koschny, and C. M. Soukoulis, Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Canfield, B. K.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

Enkrich, C.

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, Science 313, 502 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, Th. Koschny, and C. M. Soukoulis, Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Falcone, F.

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Feth, N.

Freidberg, J.

J. Freidberg, Plasma Physics and Fusion Energy (Cambridge, 2007).
[CrossRef]

Giessen, H.

T. Zentgraf, T. P. Meyrath, A. Seidel, S. Kaiser, H. Giessen, C. Rockstuhl, and F. Lederer, Phys. Rev. B 76, 033407 (2007).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Hoyer, W.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory of second-harmonic generation from magnetic metamaterials,”arXiv:0807.3531v1http://arxiv.org/abs/0807.3531v1 (2008).

Jiao, X.

Kaiser, S.

T. Zentgraf, T. P. Meyrath, A. Seidel, S. Kaiser, H. Giessen, C. Rockstuhl, and F. Lederer, Phys. Rev. B 76, 033407 (2007).
[CrossRef]

Kauranen, M.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

Klein, M. W.

Koch, S. W.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory of second-harmonic generation from magnetic metamaterials,”arXiv:0807.3531v1http://arxiv.org/abs/0807.3531v1 (2008).

Koschny, Th.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, Th. Koschny, and C. M. Soukoulis, Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Kujala, S.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

Laso, M. A.

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Lederer, F.

T. Zentgraf, T. P. Meyrath, A. Seidel, S. Kaiser, H. Giessen, C. Rockstuhl, and F. Lederer, Phys. Rev. B 76, 033407 (2007).
[CrossRef]

Linden, S.

M. W. Klein, M. Wegener, N. Feth, and S. Linden, Opt. Express 16, 8055 (2008).
[CrossRef]

M. W. Klein, M. Wegener, N. Feth, and S. Linden, Opt. Express 15, 5238 (2007).
[CrossRef] [PubMed]

C. M. Soukoulis, S. Linden, and M. Wegener, Science 315, 47 (2007).
[CrossRef] [PubMed]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, Science 313, 502 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, Th. Koschny, and C. M. Soukoulis, Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Liu, J.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory of second-harmonic generation from magnetic metamaterials,”arXiv:0807.3531v1http://arxiv.org/abs/0807.3531v1 (2008).

Lopetegi, T.

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Marques, R.

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Martin, F.

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Meyrath, T. P.

T. Zentgraf, T. P. Meyrath, A. Seidel, S. Kaiser, H. Giessen, C. Rockstuhl, and F. Lederer, Phys. Rev. B 76, 033407 (2007).
[CrossRef]

Moloney, J. V.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory of second-harmonic generation from magnetic metamaterials,”arXiv:0807.3531v1http://arxiv.org/abs/0807.3531v1 (2008).

Pendry, J. B.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Rockstuhl, C.

T. Zentgraf, T. P. Meyrath, A. Seidel, S. Kaiser, H. Giessen, C. Rockstuhl, and F. Lederer, Phys. Rev. B 76, 033407 (2007).
[CrossRef]

Schmidt, F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, Th. Koschny, and C. M. Soukoulis, Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Seidel, A.

T. Zentgraf, T. P. Meyrath, A. Seidel, S. Kaiser, H. Giessen, C. Rockstuhl, and F. Lederer, Phys. Rev. B 76, 033407 (2007).
[CrossRef]

Shalaev, V. M.

V. M. Shalaev, Nat. Photonics 1, 41 (2007).
[CrossRef]

V. M. Shalaev, Phys. Rep. 272, 61 (1996).
[CrossRef]

Sipe, J. E.

J. E. Sipe and R. W. Boyd, Phys. Rev. A 46, 1614 (1992).
[CrossRef] [PubMed]

Sorolla, M.

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

Soukoulis, C. M.

C. M. Soukoulis, S. Linden, and M. Wegener, Science 315, 47 (2007).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, Th. Koschny, and C. M. Soukoulis, Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Svirko, Y.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

Turunen, J.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

Wegener, M.

M. W. Klein, M. Wegener, N. Feth, and S. Linden, Opt. Express 16, 8055 (2008).
[CrossRef]

C. M. Soukoulis, S. Linden, and M. Wegener, Science 315, 47 (2007).
[CrossRef] [PubMed]

M. W. Klein, M. Wegener, N. Feth, and S. Linden, Opt. Express 15, 5238 (2007).
[CrossRef] [PubMed]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, Science 313, 502 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, Th. Koschny, and C. M. Soukoulis, Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Xu, T.

Zeng, Y.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory of second-harmonic generation from magnetic metamaterials,”arXiv:0807.3531v1http://arxiv.org/abs/0807.3531v1 (2008).

Zentgraf, T.

T. Zentgraf, T. P. Meyrath, A. Seidel, S. Kaiser, H. Giessen, C. Rockstuhl, and F. Lederer, Phys. Rev. B 76, 033407 (2007).
[CrossRef]

Zhang, G. P.

Zhou, J.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, Th. Koschny, and C. M. Soukoulis, Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Zschiedrich, L.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, Th. Koschny, and C. M. Soukoulis, Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

IEEE Trans. Microwave Theory Tech.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Nat. Photonics

V. M. Shalaev, Nat. Photonics 1, 41 (2007).
[CrossRef]

Opt. Express

Phys. Rep.

V. M. Shalaev, Phys. Rep. 272, 61 (1996).
[CrossRef]

Phys. Rev. A

J. E. Sipe and R. W. Boyd, Phys. Rev. A 46, 1614 (1992).
[CrossRef] [PubMed]

Phys. Rev. B

T. Zentgraf, T. P. Meyrath, A. Seidel, S. Kaiser, H. Giessen, C. Rockstuhl, and F. Lederer, Phys. Rev. B 76, 033407 (2007).
[CrossRef]

Phys. Rev. Lett.

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, Th. Koschny, and C. M. Soukoulis, Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Phys. Rev. Lett. 98, 167403 (2007).
[CrossRef] [PubMed]

Science

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, Science 313, 502 (2006).
[CrossRef] [PubMed]

C. M. Soukoulis, S. Linden, and M. Wegener, Science 315, 47 (2007).
[CrossRef] [PubMed]

Other

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory of second-harmonic generation from magnetic metamaterials,”arXiv:0807.3531v1http://arxiv.org/abs/0807.3531v1 (2008).

J. Freidberg, Plasma Physics and Fusion Energy (Cambridge, 2007).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Experiments on (a) split-ring resonators and (b) complementary split-ring resonators. The different rows (from top to bottom) show scanning electron micrographs, linear transmittance T (solid) and reflectance R (dashed) spectra (black), and the relative measured SHG signals [bars labeled with percents (red online)]. The corresponding rates Σ are the measured integral number of counts per second. The arrows indicate the incident linear polarization (dark), the measured linear polarization of the SHG [gray (red online)]—if sufficiently large. The shaded areas (blue online) in the linear spectra illustrate the exciting laser spectrum centered at the 1.5 μ m wavelength.

Fig. 2
Fig. 2

Theoretical results for (a) SRR and (b) CSRR that can directly be compared to the experimental results shown in Fig. 1. The absolute power conversion efficiencies η are quoted at the bottom.

Equations (1)

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j 2 ω ( 2 ) t = ε 0 ω pl 2 E 2 ω ( 2 ) ω coll j 2 ω ( 2 ) ε 0 e m ( E ( 1 ) ) E ( 1 ) e m j ( 1 ) B ( 1 ) + k = 1 3 r k j ( 1 ) j k ( 1 ) e n i .

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