Abstract

We analytically study second-harmonic (SH) generation from thin metallic films with subwavelength noncentrosymmetric patterns. Because the thickness of the film is much smaller than the SH wavelength, retardation effects are negligible. The far-field SH intensities are thus dominated by an effective electric dipole. It is further demonstrated that internal bulk SH polarization density is comparable with its surface counterpart. The electric dipole, consequently, originates from the entire volume of the metallic membrane.

© 2009 Optical Society of America

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  1. F. Brown, R. E. Parks, and A. M. Sleeper, Phys. Rev. Lett. 14, 1029 (1965).
    [CrossRef]
  2. A. Liebsch, Electronic Excitations at Metal Surfaces (Plenum, 1997).
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    [CrossRef]
  4. K. Busch, G. von Freymann, S. Linden, S. F. Mingalaeev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
    [CrossRef]
  5. M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, Science 313, 502 (2006).
    [CrossRef] [PubMed]
  6. M. W. Klein, M. Wegener, N. Feth, and S. Linden, Opt. Express 15, 5238 (2007).
    [CrossRef] [PubMed]
  7. N. Feth, S. Linden, M. W. Klein, M. Decker, F. Niesler, Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, J. V. Moloney, and M. Wegener, Opt. Lett. 33, 1975 (2008).
    [CrossRef] [PubMed]
  8. B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, Nano Lett. 7, 1251 (2007).
    [CrossRef] [PubMed]
  9. S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Phys. Rev. Lett. 98, 167403 (2007).
    [CrossRef] [PubMed]
  10. S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Opt. Express 16, 17196 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  17. C. Anceau, S. Brasselet, J. Zyss, and P. Gadenne, Opt. Lett. 28, 713 (2003).
    [CrossRef] [PubMed]
  18. M. I. Stockman, D. J. Bergman, C. Anceau, S. Brasselet, and J. Zyss, Phys. Rev. Lett. 92, 057402 (2004).
    [CrossRef] [PubMed]
  19. E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, Phys. Rev. B 78, 113102 (2008).
    [CrossRef]

2009 (2)

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, Phys. Rev. B 79, 235109 (2009).
[CrossRef]

Y. Zeng and J. V. Moloney, Opt. Lett. 34, 1600 (2009).
[CrossRef] [PubMed]

2008 (4)

2007 (5)

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, Nano Lett. 7, 1251 (2007).
[CrossRef] [PubMed]

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

K. Busch, G. von Freymann, S. Linden, S. F. Mingalaeev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

M. Finazzi, P. Biagiona, M. Celebrano, and L. Duò, Phys. Rev. B 76, 125414 (2007).
[CrossRef]

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

2006 (1)

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

2004 (2)

J. I. Dadap, J. Shan, and T. F. Heinz, J. Opt. Soc. Am. B 21, 1328 (2004).
[CrossRef]

M. I. Stockman, D. J. Bergman, C. Anceau, S. Brasselet, and J. Zyss, Phys. Rev. Lett. 92, 057402 (2004).
[CrossRef] [PubMed]

2003 (2)

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef] [PubMed]

C. Anceau, S. Brasselet, J. Zyss, and P. Gadenne, Opt. Lett. 28, 713 (2003).
[CrossRef] [PubMed]

1997 (1)

A. Liebsch, Electronic Excitations at Metal Surfaces (Plenum, 1997).

1975 (1)

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

1965 (1)

F. Brown, R. E. Parks, and A. M. Sleeper, Phys. Rev. Lett. 14, 1029 (1965).
[CrossRef]

Anceau, C.

M. I. Stockman, D. J. Bergman, C. Anceau, S. Brasselet, and J. Zyss, Phys. Rev. Lett. 92, 057402 (2004).
[CrossRef] [PubMed]

C. Anceau, S. Brasselet, J. Zyss, and P. Gadenne, Opt. Lett. 28, 713 (2003).
[CrossRef] [PubMed]

Bachelier, G.

Bai, B.

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, Nano Lett. 7, 1251 (2007).
[CrossRef] [PubMed]

Benichou, E.

Bergman, D. J.

M. I. Stockman, D. J. Bergman, C. Anceau, S. Brasselet, and J. Zyss, Phys. Rev. Lett. 92, 057402 (2004).
[CrossRef] [PubMed]

Beversluis, M.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef] [PubMed]

Biagiona, P.

M. Finazzi, P. Biagiona, M. Celebrano, and L. Duò, Phys. Rev. B 76, 125414 (2007).
[CrossRef]

Bouhelier, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef] [PubMed]

Brasselet, S.

M. I. Stockman, D. J. Bergman, C. Anceau, S. Brasselet, and J. Zyss, Phys. Rev. Lett. 92, 057402 (2004).
[CrossRef] [PubMed]

C. Anceau, S. Brasselet, J. Zyss, and P. Gadenne, Opt. Lett. 28, 713 (2003).
[CrossRef] [PubMed]

Brevet, P. F.

Brown, F.

F. Brown, R. E. Parks, and A. M. Sleeper, Phys. Rev. Lett. 14, 1029 (1965).
[CrossRef]

Busch, K.

K. Busch, G. von Freymann, S. Linden, S. F. Mingalaeev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Canfield, B. K.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Opt. Express 16, 17196 (2008).
[CrossRef] [PubMed]

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

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, Nano Lett. 7, 1251 (2007).
[CrossRef] [PubMed]

Celebrano, M.

M. Finazzi, P. Biagiona, M. Celebrano, and L. Duò, Phys. Rev. B 76, 125414 (2007).
[CrossRef]

Dadap, J. I.

Decker, M.

Duò, L.

M. Finazzi, P. Biagiona, M. Celebrano, and L. Duò, Phys. Rev. B 76, 125414 (2007).
[CrossRef]

Enkrich, C.

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

Feth, N.

Finazzi, M.

M. Finazzi, P. Biagiona, M. Celebrano, and L. Duò, Phys. Rev. B 76, 125414 (2007).
[CrossRef]

Gadenne, P.

Hartschuh, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef] [PubMed]

Heinz, T. F.

Hoyer, W.

Husu, H.

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, Nano Lett. 7, 1251 (2007).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

Jonin, C.

Kauranen, M.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Opt. Express 16, 17196 (2008).
[CrossRef] [PubMed]

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

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, Nano Lett. 7, 1251 (2007).
[CrossRef] [PubMed]

Kim, E.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, Phys. Rev. B 78, 113102 (2008).
[CrossRef]

Klein, M. W.

Koch, S. W.

Kuittinen, M.

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, Nano Lett. 7, 1251 (2007).
[CrossRef] [PubMed]

Kujala, S.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Opt. Express 16, 17196 (2008).
[CrossRef] [PubMed]

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

Laukkanen, J.

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, Nano Lett. 7, 1251 (2007).
[CrossRef] [PubMed]

Liebsch, A.

A. Liebsch, Electronic Excitations at Metal Surfaces (Plenum, 1997).

Linden, S.

Liu, J.

Mingalaeev, S. F.

K. Busch, G. von Freymann, S. Linden, S. F. Mingalaeev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Moloney, J. V.

Niesler, F.

Novotny, L.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef] [PubMed]

Parks, R. E.

F. Brown, R. E. Parks, and A. M. Sleeper, Phys. Rev. Lett. 14, 1029 (1965).
[CrossRef]

Russier-Antoine, I.

Shan, J.

Shen, Y. R.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, Phys. Rev. B 78, 113102 (2008).
[CrossRef]

Sleeper, A. M.

F. Brown, R. E. Parks, and A. M. Sleeper, Phys. Rev. Lett. 14, 1029 (1965).
[CrossRef]

Stockman, M. I.

M. I. Stockman, D. J. Bergman, C. Anceau, S. Brasselet, and J. Zyss, Phys. Rev. Lett. 92, 057402 (2004).
[CrossRef] [PubMed]

Svirko, Y.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Opt. Express 16, 17196 (2008).
[CrossRef] [PubMed]

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

Tkeshelashvili, L.

K. Busch, G. von Freymann, S. Linden, S. F. Mingalaeev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Turunen, J.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, Opt. Express 16, 17196 (2008).
[CrossRef] [PubMed]

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

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, Nano Lett. 7, 1251 (2007).
[CrossRef] [PubMed]

von Freymann, G.

K. Busch, G. von Freymann, S. Linden, S. F. Mingalaeev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Wang, F.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, Phys. Rev. B 78, 113102 (2008).
[CrossRef]

Wegener, M.

Wu, W.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, Phys. Rev. B 78, 113102 (2008).
[CrossRef]

Yu, Z.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, Phys. Rev. B 78, 113102 (2008).
[CrossRef]

Zeng, Y.

Zyss, J.

M. I. Stockman, D. J. Bergman, C. Anceau, S. Brasselet, and J. Zyss, Phys. Rev. Lett. 92, 057402 (2004).
[CrossRef] [PubMed]

C. Anceau, S. Brasselet, J. Zyss, and P. Gadenne, Opt. Lett. 28, 713 (2003).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (2)

Nano Lett. (1)

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, Nano Lett. 7, 1251 (2007).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rep. (1)

K. Busch, G. von Freymann, S. Linden, S. F. Mingalaeev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Phys. Rev. B (3)

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, Phys. Rev. B 79, 235109 (2009).
[CrossRef]

M. Finazzi, P. Biagiona, M. Celebrano, and L. Duò, Phys. Rev. B 76, 125414 (2007).
[CrossRef]

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, Phys. Rev. B 78, 113102 (2008).
[CrossRef]

Phys. Rev. Lett. (4)

M. I. Stockman, D. J. Bergman, C. Anceau, S. Brasselet, and J. Zyss, Phys. Rev. Lett. 92, 057402 (2004).
[CrossRef] [PubMed]

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef] [PubMed]

F. Brown, R. E. Parks, and A. M. Sleeper, Phys. Rev. Lett. 14, 1029 (1965).
[CrossRef]

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

Science (1)

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

Other (2)

A. Liebsch, Electronic Excitations at Metal Surfaces (Plenum, 1997).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic of a square lattice of gold split-ring resonators. The thickness of the gold film is 25 nm. (b) Its linear spectra under an x-polarized normal incidence.

Fig. 2
Fig. 2

Effect of incident polarization angle θ on the far-zone SH electric field along the ± z direction, with θ = 0 corresponding to polarization along the x direction. The fitted functions employed are (a) cos 2 θ and (b) | sin   2 θ | . The averaged current density (proportional to the electric dipole) j is also shown. The amplitude of the incident FF electric field E 0 is 2 × 10 7   V / m , same as that used in the experiments [5]. The bulk plasma frequency of gold is taken as ω p = 1.367 × 10 16 s 1 ; the phenomenological collision frequency is γ = 6.478 × 10 13 s 1 [3].

Fig. 3
Fig. 3

(a), (b) Spatial distribution of SH current (with arbitrary units) with an x-polarized normal incidence. The resolution is limited to 2.5 nm owing to computational resource limitations. z ( x ) = 0 corresponds to the starting plane of the gold film along the z ( x ) direction. (c) Current distribution on the plane of z 0 = 12.5   nm .

Equations (12)

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× × E ( ω ) ω 2 c 2 E = i ω μ 0 f ( r ) j ( r , ω ) ,
E = i k η 4 π r e i k r ( n × p ) × n ,
p ( n ) = j ( x , y , z ) e i k ( n x x + n y y + n z z ) d x d y d z ,
E = i k η 4 π r e i k r ( p x x ̂ + p y y ̂ ) .
| 0 d z g ( z ) e i k z d z | 2     at   + z   direction ,
| 0 d z g ( z ) e i k z d z | 2     at   z   direction ,
g ( z ) = j x ( x , y , z ) d x d y ,
| 0 d z g ( z ) d z | 2 = | 0 d z j x ( x , y , z ) d x d y d z | 2 .
2 i k 0 d z 0 d z g ( z 1 ) g ( z 2 ) ( z 1 z 2 ) d z 1 d z 2 ,
Im { 4 k 0 d z g ( z ) z d z 0 d z g ( z ) d z } .
g ( z ) = x , y j y ( x , y , z ) ,     g s ( z ) = z z g ( z ) ,
l ( x ) = y j y ( x , y , z 0 ) ,     l s ( x ) = x x l ( x ) ,

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