Abstract

We report rigorous numerical calculations of the near field scattered from rough, one-dimensional self-affine fractal silver surfaces. We show that fractal lower-scale cutoff (decreased to the order of tens of nanometers) has a strong effect on excitation and strength of localized optical modes, leading to very large enhancements of the intensity (larger than 104) and fluctuations of the electric field.

© 2001 Optical Society of America

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  1. D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, J. S. Suh, and R. Botet, Phys. Rev. Lett. 72, 4149 (1994).
    [CrossRef] [PubMed]
  2. S. Bozhevolnyi, B. Vohnsen, I. I. Smolyaninov, and A. V. Zyats, Opt. Commun. 117, 417 (1995).
    [CrossRef]
  3. S. Bozhevolnyi, B. Vohnsen, A. V. Zyats, and I. I. Smolyaninov, Surf. Sci. 356, 268 (1996).
    [CrossRef]
  4. P. Zhang, T. L. Haslett, C. Douketis, and M. Moskovits, Phys. Rev. B 57, 15,513 (1998).
    [CrossRef]
  5. S. Bozhevolnyi, V. A. Markel, V. Coello, W. Kim, and V. M. Shalaev, Phys. Rev. B 58, 11,441 (1998).
    [CrossRef]
  6. V. A. Markel, V. M. Shalaev, P. Zhang, W. Huynh, L. Tay, T. L. Haslett, and M. Moskovits, Phys. Rev. B 59, 10,903 (1999).
    [CrossRef]
  7. K. Kneipp, Y. Wang, H. Kneipp, L. T. Perlman, I. Itzkan, R. R. Dasari, and M. S. Feld, Phys. Rev. Lett. 78, 1667 (1997).
    [CrossRef]
  8. H. Xu, E. J. Bjerneld, M. Käll, and L. Börjesson, Phys. Rev. Lett. 83, 4357 (1999).
    [CrossRef]
  9. M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, Phys. Rev. Lett. 72, 2486 (1994).
    [CrossRef] [PubMed]
  10. V. M. Shalaev, Phys. Rep. 272, 61 (1996).
    [CrossRef]
  11. E. Y. Poliakov, V. M. Shalaev, V. A. Markel, and R. Botet, Opt. Lett. 21, 1628 (1996).
    [CrossRef] [PubMed]
  12. J. A. Sánchez-Gil and J. V. García-Ramos, Opt. Commun. 134, 11 (1997).
    [CrossRef]
  13. J. A. Sánchez-Gil and J. V. García-Ramos, J. Chem. Phys. 108, 317 (1998).
  14. J. A. Sánchez-Gil, J. V. García-Ramos, and E. R. Méndez, Phys. Rev. B 62, 10515 (2000).
    [CrossRef]
  15. A. Mendoza-Suárez and E. R. Méndez, Appl. Opt. 36, 3521 (1997).
    [CrossRef]
  16. J. Feder, Fractals (Plenum, New York, 1988).
    [CrossRef]
  17. ξL here is the spatial distance between contiguous points as obtained from Voss’s algorithm. The actual lower-scale cutoff is slightly larger.12,16
  18. I. Simonsen, D. Vandembroucq, and S. Roux, Phys. Rev. E 61, 5914 (2000).
    [CrossRef]
  19. D. W. Lynch and W. R. Hunter, in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), p. 356.
  20. K. A. O’Donnell, R. Torre, and C. S. West, Opt. Lett. 21, 1738 (1996).
    [CrossRef]
  21. M. Leyva-Lucero, E. R. Méndez, T. A. Leskova, A. A. Maradudin, and J. Q. Lu, Opt. Lett. 21, 1809 (1996).
    [CrossRef] [PubMed]

2000 (2)

J. A. Sánchez-Gil, J. V. García-Ramos, and E. R. Méndez, Phys. Rev. B 62, 10515 (2000).
[CrossRef]

I. Simonsen, D. Vandembroucq, and S. Roux, Phys. Rev. E 61, 5914 (2000).
[CrossRef]

1999 (2)

V. A. Markel, V. M. Shalaev, P. Zhang, W. Huynh, L. Tay, T. L. Haslett, and M. Moskovits, Phys. Rev. B 59, 10,903 (1999).
[CrossRef]

H. Xu, E. J. Bjerneld, M. Käll, and L. Börjesson, Phys. Rev. Lett. 83, 4357 (1999).
[CrossRef]

1998 (3)

P. Zhang, T. L. Haslett, C. Douketis, and M. Moskovits, Phys. Rev. B 57, 15,513 (1998).
[CrossRef]

S. Bozhevolnyi, V. A. Markel, V. Coello, W. Kim, and V. M. Shalaev, Phys. Rev. B 58, 11,441 (1998).
[CrossRef]

J. A. Sánchez-Gil and J. V. García-Ramos, J. Chem. Phys. 108, 317 (1998).

1997 (3)

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perlman, I. Itzkan, R. R. Dasari, and M. S. Feld, Phys. Rev. Lett. 78, 1667 (1997).
[CrossRef]

A. Mendoza-Suárez and E. R. Méndez, Appl. Opt. 36, 3521 (1997).
[CrossRef]

J. A. Sánchez-Gil and J. V. García-Ramos, Opt. Commun. 134, 11 (1997).
[CrossRef]

1996 (5)

1995 (1)

S. Bozhevolnyi, B. Vohnsen, I. I. Smolyaninov, and A. V. Zyats, Opt. Commun. 117, 417 (1995).
[CrossRef]

1994 (2)

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, J. S. Suh, and R. Botet, Phys. Rev. Lett. 72, 4149 (1994).
[CrossRef] [PubMed]

M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, Phys. Rev. Lett. 72, 2486 (1994).
[CrossRef] [PubMed]

Bjerneld, E. J.

H. Xu, E. J. Bjerneld, M. Käll, and L. Börjesson, Phys. Rev. Lett. 83, 4357 (1999).
[CrossRef]

Börjesson, L.

H. Xu, E. J. Bjerneld, M. Käll, and L. Börjesson, Phys. Rev. Lett. 83, 4357 (1999).
[CrossRef]

Botet, R.

E. Y. Poliakov, V. M. Shalaev, V. A. Markel, and R. Botet, Opt. Lett. 21, 1628 (1996).
[CrossRef] [PubMed]

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, J. S. Suh, and R. Botet, Phys. Rev. Lett. 72, 4149 (1994).
[CrossRef] [PubMed]

Bozhevolnyi, S.

S. Bozhevolnyi, V. A. Markel, V. Coello, W. Kim, and V. M. Shalaev, Phys. Rev. B 58, 11,441 (1998).
[CrossRef]

S. Bozhevolnyi, B. Vohnsen, A. V. Zyats, and I. I. Smolyaninov, Surf. Sci. 356, 268 (1996).
[CrossRef]

S. Bozhevolnyi, B. Vohnsen, I. I. Smolyaninov, and A. V. Zyats, Opt. Commun. 117, 417 (1995).
[CrossRef]

Coello, V.

S. Bozhevolnyi, V. A. Markel, V. Coello, W. Kim, and V. M. Shalaev, Phys. Rev. B 58, 11,441 (1998).
[CrossRef]

Dasari, R. R.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perlman, I. Itzkan, R. R. Dasari, and M. S. Feld, Phys. Rev. Lett. 78, 1667 (1997).
[CrossRef]

Douketis, C.

P. Zhang, T. L. Haslett, C. Douketis, and M. Moskovits, Phys. Rev. B 57, 15,513 (1998).
[CrossRef]

Feder, J.

J. Feder, Fractals (Plenum, New York, 1988).
[CrossRef]

Feld, M. S.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perlman, I. Itzkan, R. R. Dasari, and M. S. Feld, Phys. Rev. Lett. 78, 1667 (1997).
[CrossRef]

García-Ramos, J. V.

J. A. Sánchez-Gil, J. V. García-Ramos, and E. R. Méndez, Phys. Rev. B 62, 10515 (2000).
[CrossRef]

J. A. Sánchez-Gil and J. V. García-Ramos, J. Chem. Phys. 108, 317 (1998).

J. A. Sánchez-Gil and J. V. García-Ramos, Opt. Commun. 134, 11 (1997).
[CrossRef]

George, T. F.

M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, Phys. Rev. Lett. 72, 2486 (1994).
[CrossRef] [PubMed]

Haslett, T. L.

V. A. Markel, V. M. Shalaev, P. Zhang, W. Huynh, L. Tay, T. L. Haslett, and M. Moskovits, Phys. Rev. B 59, 10,903 (1999).
[CrossRef]

P. Zhang, T. L. Haslett, C. Douketis, and M. Moskovits, Phys. Rev. B 57, 15,513 (1998).
[CrossRef]

Hunter, W. R.

D. W. Lynch and W. R. Hunter, in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), p. 356.

Huynh, W.

V. A. Markel, V. M. Shalaev, P. Zhang, W. Huynh, L. Tay, T. L. Haslett, and M. Moskovits, Phys. Rev. B 59, 10,903 (1999).
[CrossRef]

Itzkan, I.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perlman, I. Itzkan, R. R. Dasari, and M. S. Feld, Phys. Rev. Lett. 78, 1667 (1997).
[CrossRef]

Käll, M.

H. Xu, E. J. Bjerneld, M. Käll, and L. Börjesson, Phys. Rev. Lett. 83, 4357 (1999).
[CrossRef]

Kim, W.

S. Bozhevolnyi, V. A. Markel, V. Coello, W. Kim, and V. M. Shalaev, Phys. Rev. B 58, 11,441 (1998).
[CrossRef]

Kneipp, H.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perlman, I. Itzkan, R. R. Dasari, and M. S. Feld, Phys. Rev. Lett. 78, 1667 (1997).
[CrossRef]

Kneipp, K.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perlman, I. Itzkan, R. R. Dasari, and M. S. Feld, Phys. Rev. Lett. 78, 1667 (1997).
[CrossRef]

Kovacs, J.

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, J. S. Suh, and R. Botet, Phys. Rev. Lett. 72, 4149 (1994).
[CrossRef] [PubMed]

Leskova, T. A.

Leyva-Lucero, M.

Lu, J. Q.

Lynch, D. W.

D. W. Lynch and W. R. Hunter, in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), p. 356.

Maradudin, A. A.

Markel, V. A.

V. A. Markel, V. M. Shalaev, P. Zhang, W. Huynh, L. Tay, T. L. Haslett, and M. Moskovits, Phys. Rev. B 59, 10,903 (1999).
[CrossRef]

S. Bozhevolnyi, V. A. Markel, V. Coello, W. Kim, and V. M. Shalaev, Phys. Rev. B 58, 11,441 (1998).
[CrossRef]

E. Y. Poliakov, V. M. Shalaev, V. A. Markel, and R. Botet, Opt. Lett. 21, 1628 (1996).
[CrossRef] [PubMed]

Méndez, E. R.

Mendoza-Suárez, A.

Moskovits, M.

V. A. Markel, V. M. Shalaev, P. Zhang, W. Huynh, L. Tay, T. L. Haslett, and M. Moskovits, Phys. Rev. B 59, 10,903 (1999).
[CrossRef]

P. Zhang, T. L. Haslett, C. Douketis, and M. Moskovits, Phys. Rev. B 57, 15,513 (1998).
[CrossRef]

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, J. S. Suh, and R. Botet, Phys. Rev. Lett. 72, 4149 (1994).
[CrossRef] [PubMed]

Muratov, L. S.

M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, Phys. Rev. Lett. 72, 2486 (1994).
[CrossRef] [PubMed]

O’Donnell, K. A.

Pandey, L. N.

M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, Phys. Rev. Lett. 72, 2486 (1994).
[CrossRef] [PubMed]

Perlman, L. T.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perlman, I. Itzkan, R. R. Dasari, and M. S. Feld, Phys. Rev. Lett. 78, 1667 (1997).
[CrossRef]

Poliakov, E. Y.

Roux, S.

I. Simonsen, D. Vandembroucq, and S. Roux, Phys. Rev. E 61, 5914 (2000).
[CrossRef]

Sánchez-Gil, J. A.

J. A. Sánchez-Gil, J. V. García-Ramos, and E. R. Méndez, Phys. Rev. B 62, 10515 (2000).
[CrossRef]

J. A. Sánchez-Gil and J. V. García-Ramos, J. Chem. Phys. 108, 317 (1998).

J. A. Sánchez-Gil and J. V. García-Ramos, Opt. Commun. 134, 11 (1997).
[CrossRef]

Shalaev, V. M.

V. A. Markel, V. M. Shalaev, P. Zhang, W. Huynh, L. Tay, T. L. Haslett, and M. Moskovits, Phys. Rev. B 59, 10,903 (1999).
[CrossRef]

S. Bozhevolnyi, V. A. Markel, V. Coello, W. Kim, and V. M. Shalaev, Phys. Rev. B 58, 11,441 (1998).
[CrossRef]

E. Y. Poliakov, V. M. Shalaev, V. A. Markel, and R. Botet, Opt. Lett. 21, 1628 (1996).
[CrossRef] [PubMed]

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

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, J. S. Suh, and R. Botet, Phys. Rev. Lett. 72, 4149 (1994).
[CrossRef] [PubMed]

Simonsen, I.

I. Simonsen, D. Vandembroucq, and S. Roux, Phys. Rev. E 61, 5914 (2000).
[CrossRef]

Smolyaninov, I. I.

S. Bozhevolnyi, B. Vohnsen, A. V. Zyats, and I. I. Smolyaninov, Surf. Sci. 356, 268 (1996).
[CrossRef]

S. Bozhevolnyi, B. Vohnsen, I. I. Smolyaninov, and A. V. Zyats, Opt. Commun. 117, 417 (1995).
[CrossRef]

Stockman, M. I.

M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, Phys. Rev. Lett. 72, 2486 (1994).
[CrossRef] [PubMed]

Suh, J. S.

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, J. S. Suh, and R. Botet, Phys. Rev. Lett. 72, 4149 (1994).
[CrossRef] [PubMed]

Tay, L.

V. A. Markel, V. M. Shalaev, P. Zhang, W. Huynh, L. Tay, T. L. Haslett, and M. Moskovits, Phys. Rev. B 59, 10,903 (1999).
[CrossRef]

Torre, R.

Tsai, D. P.

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, J. S. Suh, and R. Botet, Phys. Rev. Lett. 72, 4149 (1994).
[CrossRef] [PubMed]

Vandembroucq, D.

I. Simonsen, D. Vandembroucq, and S. Roux, Phys. Rev. E 61, 5914 (2000).
[CrossRef]

Vohnsen, B.

S. Bozhevolnyi, B. Vohnsen, A. V. Zyats, and I. I. Smolyaninov, Surf. Sci. 356, 268 (1996).
[CrossRef]

S. Bozhevolnyi, B. Vohnsen, I. I. Smolyaninov, and A. V. Zyats, Opt. Commun. 117, 417 (1995).
[CrossRef]

Wang, Y.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perlman, I. Itzkan, R. R. Dasari, and M. S. Feld, Phys. Rev. Lett. 78, 1667 (1997).
[CrossRef]

Wang, Z.

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, J. S. Suh, and R. Botet, Phys. Rev. Lett. 72, 4149 (1994).
[CrossRef] [PubMed]

West, C. S.

Xu, H.

H. Xu, E. J. Bjerneld, M. Käll, and L. Börjesson, Phys. Rev. Lett. 83, 4357 (1999).
[CrossRef]

Zhang, P.

V. A. Markel, V. M. Shalaev, P. Zhang, W. Huynh, L. Tay, T. L. Haslett, and M. Moskovits, Phys. Rev. B 59, 10,903 (1999).
[CrossRef]

P. Zhang, T. L. Haslett, C. Douketis, and M. Moskovits, Phys. Rev. B 57, 15,513 (1998).
[CrossRef]

Zyats, A. V.

S. Bozhevolnyi, B. Vohnsen, A. V. Zyats, and I. I. Smolyaninov, Surf. Sci. 356, 268 (1996).
[CrossRef]

S. Bozhevolnyi, B. Vohnsen, I. I. Smolyaninov, and A. V. Zyats, Opt. Commun. 117, 417 (1995).
[CrossRef]

Appl. Opt. (1)

J. Chem. Phys. (1)

J. A. Sánchez-Gil and J. V. García-Ramos, J. Chem. Phys. 108, 317 (1998).

Opt. Commun. (2)

J. A. Sánchez-Gil and J. V. García-Ramos, Opt. Commun. 134, 11 (1997).
[CrossRef]

S. Bozhevolnyi, B. Vohnsen, I. I. Smolyaninov, and A. V. Zyats, Opt. Commun. 117, 417 (1995).
[CrossRef]

Opt. Lett. (3)

Phys. Rep. (1)

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

Phys. Rev. B (4)

P. Zhang, T. L. Haslett, C. Douketis, and M. Moskovits, Phys. Rev. B 57, 15,513 (1998).
[CrossRef]

S. Bozhevolnyi, V. A. Markel, V. Coello, W. Kim, and V. M. Shalaev, Phys. Rev. B 58, 11,441 (1998).
[CrossRef]

V. A. Markel, V. M. Shalaev, P. Zhang, W. Huynh, L. Tay, T. L. Haslett, and M. Moskovits, Phys. Rev. B 59, 10,903 (1999).
[CrossRef]

J. A. Sánchez-Gil, J. V. García-Ramos, and E. R. Méndez, Phys. Rev. B 62, 10515 (2000).
[CrossRef]

Phys. Rev. E (1)

I. Simonsen, D. Vandembroucq, and S. Roux, Phys. Rev. E 61, 5914 (2000).
[CrossRef]

Phys. Rev. Lett. (4)

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perlman, I. Itzkan, R. R. Dasari, and M. S. Feld, Phys. Rev. Lett. 78, 1667 (1997).
[CrossRef]

H. Xu, E. J. Bjerneld, M. Käll, and L. Börjesson, Phys. Rev. Lett. 83, 4357 (1999).
[CrossRef]

M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, Phys. Rev. Lett. 72, 2486 (1994).
[CrossRef] [PubMed]

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, J. S. Suh, and R. Botet, Phys. Rev. Lett. 72, 4149 (1994).
[CrossRef] [PubMed]

Surf. Sci. (1)

S. Bozhevolnyi, B. Vohnsen, A. V. Zyats, and I. I. Smolyaninov, Surf. Sci. 356, 268 (1996).
[CrossRef]

Other (3)

D. W. Lynch and W. R. Hunter, in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), p. 356.

J. Feder, Fractals (Plenum, New York, 1988).
[CrossRef]

ξL here is the spatial distance between contiguous points as obtained from Voss’s algorithm. The actual lower-scale cutoff is slightly larger.12,16

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Figures (3)

Fig. 1
Fig. 1

Surface (a) magnetic- and (b) electric field intensities (on a log10 scale) along the center half of the illuminated area that results from p-polarized scattering, with θ0=0°, λ=629.9 nm, and W=L/4cosθ0, from Ag fractal surfaces with D=1.9, δ=51.4 nm, and four values of ξL. The KA field intensity is also included (see text). (c) The corresponding surface profiles.

Fig. 2
Fig. 2

(a), (b) Same as in Fig.  1 (only the electric field intensity in a small part of the illuminated area is shown) but for rougher Ag fractal surfaces with δ=257 nm. (c) Near-field intensity image (on a log10 scale) in an area of 570 nm×570 nm, where a few localized optical modes are observed; the intensity of the mode at the center is >103 times that of the incident field.

Fig. 3
Fig. 3

Dependence of the enhancement factor of electric field intensity σ on ξL obtained from the results for Nrea=100 Ag fractal profiles with the same parameters as in Figs.  1 and 2: (a) mean, (b) local maximum, (c) fluctuations.

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