Abstract

We derive an analytical expression for the scattering of an s-polarized plane wave from a perfectly conducting self-affine one-dimensional surface in the framework of the Kirchhoff approximation. We show that most of the results can be recovered by means of a scaling analysis. We identify the typical slope taken over one wavelength as the relevant parameter controlling the scattering process. We compare our predictions with direct numerical simulations performed on surfaces of varying roughness parameters and confirm the broad range of applicability of our description up to very large roughness. Finally we verify that a nonzero electrical resistivity, provided that it is small, does not invalidate our results.

© 2001 Optical Society of America

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  1. J. W. S. Rayleigh, The Theory of Sound (Dover, New York, 1945).
  2. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett. 12, 979–981 (1987).
    [CrossRef] [PubMed]
  3. E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random rough surfaces,” Opt. Commun. 61, 91–95 (1987).
    [CrossRef]
  4. A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
    [CrossRef] [PubMed]
  5. J. A. Sanchez-Gil, “Coupling, resonance transmission, and tunneling of surface-plasmon polaritons through metallic gratings of finite length,” Phys. Rev. B 53, 10317–10327 (1996).
    [CrossRef]
  6. S. C. Kitson, W. L. Barnes, J. L. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2670–2673 (1996).
    [CrossRef] [PubMed]
  7. F. Pincemin, J. J. Greffet, “Propagation and localization of a surface plasmon polariton on a finite grating,” J. Opt. Soc. Am. B 13, 1499–1509 (1996).
    [CrossRef]
  8. J. A. Sanchez-Gil, A. A. Maradudin, “Competition between Anderson localization and leakage of surface-plasmon polaritons on randomly rough periodic metal surfaces,” Phys. Rev. B 56, 1103–1106 (1997).
    [CrossRef]
  9. C. S. West, K. A. O’Donnell, “Observation of backscattering enhancement from polaritons on a rough metal surface,” J. Opt. Soc. Am. A 12, 390–397 (1995).
    [CrossRef]
  10. A. Sentenac, J. J. Greffet, “Mean-field theory of light scattering by one-dimensional rough surfaces,” J. Opt. Soc. Am. A 15, 528–532 (1998).
    [CrossRef]
  11. S. L. Broschat, E. I. Thorsos, “An investigation of the small slope approximation for scattering from rough surfaces. i: theory,” J. Acoust. Soc. Am. 97, 2082–2093 (1995).
    [CrossRef]
  12. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (IOP, Bristol, UK, 1991).
  13. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1963).
  14. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1975).
  15. E. Bouchaud, “Scaling properties of cracks,” J. Phys. Condens. Matter 9, 4319–4344 (1997).
    [CrossRef]
  16. P. Meakin, Fractals, Scaling and Growth Far from Equilibrium (Cambridge U. Press, Cambridge, UK, 1998).
  17. F. Plouraboué, M. Boehm, “Multi-scale roughness transfer in cold metal rolling,” Tribol. Int. 32, 45–57 (1999).
    [CrossRef]
  18. M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
    [CrossRef]
  19. D. L. Jaggard, X. Sun, “Fractal surface scattering: a generalized Rayleigh solution,” J. Appl. Phys. 68(11), 5456–5462 (1990).
    [CrossRef]
  20. M. K. Shepard, R. A. Brackett, R. E. Arvidson, “Self-affine (fractal) topography: surface parametrization and radar scattering,” J. Geophys. Res. 100, E6, 11709–11718 (1995).
    [CrossRef]
  21. P. E. McSharry, P. J. Cullen, D. Moroney, “Wave scattering by a two-dimensional band limited fractal surface based on a perturbation of the Green’s function,” J. Appl. Phys. 78(12), 6940–6948 (1995).
    [CrossRef]
  22. N. Lin, H. P. Lee, S. P. Lim, K. S. Lee, “Wave scattering from fractal surfaces,” J. Mod. Opt. 42, 225–235 (1995).
    [CrossRef]
  23. J. Chen, T. K. Y. Lo, H. Leung, J. Litva, “The use of fractals for modeling EM waves scattering from rough sea surface,” IEEE Trans. Geosci. 34(4), 966–972 (1996).
    [CrossRef]
  24. C. J. R. Sheppard, “Scattering by fractal surfaces with an outer-scale,” Opt. Commun. 122, 178–188 (1996).
    [CrossRef]
  25. J. A. Sánchez-Gil, J. V. Garcı́a-Ramos, “Far field intensity of electromagnetic waves scattered from random, self-affine fractal metal surfaces,” Waves Random Media 7, 285–293 (1997).
    [CrossRef]
  26. J. A. Sánchez-Gil, J. V. Garcı́a-Ramos, “Calculations of the direct electromagnetic enhancement in surface enhanced raman scattering on random self-affine fractal metal surfaces,” J. Chem. Phys. 108, 1317–1325 (1998).
  27. Y-P Zhao, C. F. Cheng, G. C. Wang, T. M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. 409, L703–L708 (1998).
    [CrossRef]
  28. E. Jakeman, “Scattering by fractals” in Fractals in Physics, L. Pietronero, E. Tossati, eds. (Elsevier, Amsterdam, 1986), pp. 55–60.
  29. D. L. Jordan, R. C. Hollins, E. Jakeman, A. Prewett, “Visible and infra-red scattering from well-characterized surfaces,” Surf. Topog. 1, 27–36 (1988).
  30. H. N. Yang, T. M. Lu, G. C. Wang, “Diffraction from surface growth fronts,” Phys. Rev. B 47, 3911–3922 (1993).
    [CrossRef]
  31. Y. P. Zhao, G. C. Wang, T. M. Lu, “Diffraction from non-Gaussian rough surfaces,” Phys. Rev. B 55, 13938–13952 (1997).
    [CrossRef]
  32. Y. P. Zhao, I. Wu, C. F. Cheng, U. Block, G. C. Wang, T. M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
    [CrossRef]
  33. I. Simonsen, D. Vandembroucq, S. Roux, “Wave scattering from self-affine surfaces,” Phys. Rev. E 61, 5914–5917 (2000).
    [CrossRef]
  34. J. Schmittbuhl, J. P. Vilotte, S. Roux, “Reliability of self-affine measurements,” Phys. Rev. E 51, 131–147 (1995).
    [CrossRef]
  35. I. Simonsen, A. Hansen, O. M. Nes, “Determination of Hurst exponents by use of the wavelet transform,” Phys. Rev. E 58, 2779–2787 (1998).
    [CrossRef]
  36. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–307 (1990).
    [CrossRef]
  37. P. Lévy, Théorie de l’Addition des Variables Aléatoires (Gauthier-Villars, Paris, 1937).
  38. B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sum of Independent Random Variables (Addison Wesley, Reading, Mass., 1954).
  39. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–82 (1988).
    [CrossRef]
  40. E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 3, S165–S190 (1991).
    [CrossRef]
  41. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).
  42. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).
  43. J. Feder, Fractals (Plenum, New York, 1988).
  44. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

2000 (1)

I. Simonsen, D. Vandembroucq, S. Roux, “Wave scattering from self-affine surfaces,” Phys. Rev. E 61, 5914–5917 (2000).
[CrossRef]

1999 (1)

F. Plouraboué, M. Boehm, “Multi-scale roughness transfer in cold metal rolling,” Tribol. Int. 32, 45–57 (1999).
[CrossRef]

1998 (5)

J. A. Sánchez-Gil, J. V. Garcı́a-Ramos, “Calculations of the direct electromagnetic enhancement in surface enhanced raman scattering on random self-affine fractal metal surfaces,” J. Chem. Phys. 108, 1317–1325 (1998).

Y-P Zhao, C. F. Cheng, G. C. Wang, T. M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. 409, L703–L708 (1998).
[CrossRef]

I. Simonsen, A. Hansen, O. M. Nes, “Determination of Hurst exponents by use of the wavelet transform,” Phys. Rev. E 58, 2779–2787 (1998).
[CrossRef]

Y. P. Zhao, I. Wu, C. F. Cheng, U. Block, G. C. Wang, T. M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[CrossRef]

A. Sentenac, J. J. Greffet, “Mean-field theory of light scattering by one-dimensional rough surfaces,” J. Opt. Soc. Am. A 15, 528–532 (1998).
[CrossRef]

1997 (4)

J. A. Sanchez-Gil, A. A. Maradudin, “Competition between Anderson localization and leakage of surface-plasmon polaritons on randomly rough periodic metal surfaces,” Phys. Rev. B 56, 1103–1106 (1997).
[CrossRef]

J. A. Sánchez-Gil, J. V. Garcı́a-Ramos, “Far field intensity of electromagnetic waves scattered from random, self-affine fractal metal surfaces,” Waves Random Media 7, 285–293 (1997).
[CrossRef]

Y. P. Zhao, G. C. Wang, T. M. Lu, “Diffraction from non-Gaussian rough surfaces,” Phys. Rev. B 55, 13938–13952 (1997).
[CrossRef]

E. Bouchaud, “Scaling properties of cracks,” J. Phys. Condens. Matter 9, 4319–4344 (1997).
[CrossRef]

1996 (5)

J. Chen, T. K. Y. Lo, H. Leung, J. Litva, “The use of fractals for modeling EM waves scattering from rough sea surface,” IEEE Trans. Geosci. 34(4), 966–972 (1996).
[CrossRef]

C. J. R. Sheppard, “Scattering by fractal surfaces with an outer-scale,” Opt. Commun. 122, 178–188 (1996).
[CrossRef]

J. A. Sanchez-Gil, “Coupling, resonance transmission, and tunneling of surface-plasmon polaritons through metallic gratings of finite length,” Phys. Rev. B 53, 10317–10327 (1996).
[CrossRef]

S. C. Kitson, W. L. Barnes, J. L. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2670–2673 (1996).
[CrossRef] [PubMed]

F. Pincemin, J. J. Greffet, “Propagation and localization of a surface plasmon polariton on a finite grating,” J. Opt. Soc. Am. B 13, 1499–1509 (1996).
[CrossRef]

1995 (6)

C. S. West, K. A. O’Donnell, “Observation of backscattering enhancement from polaritons on a rough metal surface,” J. Opt. Soc. Am. A 12, 390–397 (1995).
[CrossRef]

M. K. Shepard, R. A. Brackett, R. E. Arvidson, “Self-affine (fractal) topography: surface parametrization and radar scattering,” J. Geophys. Res. 100, E6, 11709–11718 (1995).
[CrossRef]

P. E. McSharry, P. J. Cullen, D. Moroney, “Wave scattering by a two-dimensional band limited fractal surface based on a perturbation of the Green’s function,” J. Appl. Phys. 78(12), 6940–6948 (1995).
[CrossRef]

N. Lin, H. P. Lee, S. P. Lim, K. S. Lee, “Wave scattering from fractal surfaces,” J. Mod. Opt. 42, 225–235 (1995).
[CrossRef]

S. L. Broschat, E. I. Thorsos, “An investigation of the small slope approximation for scattering from rough surfaces. i: theory,” J. Acoust. Soc. Am. 97, 2082–2093 (1995).
[CrossRef]

J. Schmittbuhl, J. P. Vilotte, S. Roux, “Reliability of self-affine measurements,” Phys. Rev. E 51, 131–147 (1995).
[CrossRef]

1993 (1)

H. N. Yang, T. M. Lu, G. C. Wang, “Diffraction from surface growth fronts,” Phys. Rev. B 47, 3911–3922 (1993).
[CrossRef]

1991 (1)

E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 3, S165–S190 (1991).
[CrossRef]

1990 (2)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–307 (1990).
[CrossRef]

D. L. Jaggard, X. Sun, “Fractal surface scattering: a generalized Rayleigh solution,” J. Appl. Phys. 68(11), 5456–5462 (1990).
[CrossRef]

1989 (1)

1988 (2)

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–82 (1988).
[CrossRef]

D. L. Jordan, R. C. Hollins, E. Jakeman, A. Prewett, “Visible and infra-red scattering from well-characterized surfaces,” Surf. Topog. 1, 27–36 (1988).

1987 (2)

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random rough surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett. 12, 979–981 (1987).
[CrossRef] [PubMed]

1979 (1)

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

Arvidson, R. E.

M. K. Shepard, R. A. Brackett, R. E. Arvidson, “Self-affine (fractal) topography: surface parametrization and radar scattering,” J. Geophys. Res. 100, E6, 11709–11718 (1995).
[CrossRef]

Barnes, W. L.

S. C. Kitson, W. L. Barnes, J. L. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2670–2673 (1996).
[CrossRef] [PubMed]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1963).

Berry, M. V.

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[CrossRef]

Block, U.

Y. P. Zhao, I. Wu, C. F. Cheng, U. Block, G. C. Wang, T. M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[CrossRef]

Boehm, M.

F. Plouraboué, M. Boehm, “Multi-scale roughness transfer in cold metal rolling,” Tribol. Int. 32, 45–57 (1999).
[CrossRef]

Bouchaud, E.

E. Bouchaud, “Scaling properties of cracks,” J. Phys. Condens. Matter 9, 4319–4344 (1997).
[CrossRef]

Brackett, R. A.

M. K. Shepard, R. A. Brackett, R. E. Arvidson, “Self-affine (fractal) topography: surface parametrization and radar scattering,” J. Geophys. Res. 100, E6, 11709–11718 (1995).
[CrossRef]

Broschat, S. L.

S. L. Broschat, E. I. Thorsos, “An investigation of the small slope approximation for scattering from rough surfaces. i: theory,” J. Acoust. Soc. Am. 97, 2082–2093 (1995).
[CrossRef]

Chen, J.

J. Chen, T. K. Y. Lo, H. Leung, J. Litva, “The use of fractals for modeling EM waves scattering from rough sea surface,” IEEE Trans. Geosci. 34(4), 966–972 (1996).
[CrossRef]

Cheng, C. F.

Y. P. Zhao, I. Wu, C. F. Cheng, U. Block, G. C. Wang, T. M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[CrossRef]

Y-P Zhao, C. F. Cheng, G. C. Wang, T. M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. 409, L703–L708 (1998).
[CrossRef]

Cullen, P. J.

P. E. McSharry, P. J. Cullen, D. Moroney, “Wave scattering by a two-dimensional band limited fractal surface based on a perturbation of the Green’s function,” J. Appl. Phys. 78(12), 6940–6948 (1995).
[CrossRef]

Feder, J.

J. Feder, Fractals (Plenum, New York, 1988).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Garci´a-Ramos, J. V.

J. A. Sánchez-Gil, J. V. Garcı́a-Ramos, “Calculations of the direct electromagnetic enhancement in surface enhanced raman scattering on random self-affine fractal metal surfaces,” J. Chem. Phys. 108, 1317–1325 (1998).

J. A. Sánchez-Gil, J. V. Garcı́a-Ramos, “Far field intensity of electromagnetic waves scattered from random, self-affine fractal metal surfaces,” Waves Random Media 7, 285–293 (1997).
[CrossRef]

Gnedenko, B. V.

B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sum of Independent Random Variables (Addison Wesley, Reading, Mass., 1954).

Greffet, J. J.

Hansen, A.

I. Simonsen, A. Hansen, O. M. Nes, “Determination of Hurst exponents by use of the wavelet transform,” Phys. Rev. E 58, 2779–2787 (1998).
[CrossRef]

Hollins, R. C.

D. L. Jordan, R. C. Hollins, E. Jakeman, A. Prewett, “Visible and infra-red scattering from well-characterized surfaces,” Surf. Topog. 1, 27–36 (1988).

Jackson, D. R.

E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 3, S165–S190 (1991).
[CrossRef]

Jaggard, D. L.

D. L. Jaggard, X. Sun, “Fractal surface scattering: a generalized Rayleigh solution,” J. Appl. Phys. 68(11), 5456–5462 (1990).
[CrossRef]

Jakeman, E.

D. L. Jordan, R. C. Hollins, E. Jakeman, A. Prewett, “Visible and infra-red scattering from well-characterized surfaces,” Surf. Topog. 1, 27–36 (1988).

E. Jakeman, “Scattering by fractals” in Fractals in Physics, L. Pietronero, E. Tossati, eds. (Elsevier, Amsterdam, 1986), pp. 55–60.

Jordan, D. L.

D. L. Jordan, R. C. Hollins, E. Jakeman, A. Prewett, “Visible and infra-red scattering from well-characterized surfaces,” Surf. Topog. 1, 27–36 (1988).

Kitson, S. C.

S. C. Kitson, W. L. Barnes, J. L. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2670–2673 (1996).
[CrossRef] [PubMed]

Kolmogorov, A. N.

B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sum of Independent Random Variables (Addison Wesley, Reading, Mass., 1954).

Lee, H. P.

N. Lin, H. P. Lee, S. P. Lim, K. S. Lee, “Wave scattering from fractal surfaces,” J. Mod. Opt. 42, 225–235 (1995).
[CrossRef]

Lee, K. S.

N. Lin, H. P. Lee, S. P. Lim, K. S. Lee, “Wave scattering from fractal surfaces,” J. Mod. Opt. 42, 225–235 (1995).
[CrossRef]

Leung, H.

J. Chen, T. K. Y. Lo, H. Leung, J. Litva, “The use of fractals for modeling EM waves scattering from rough sea surface,” IEEE Trans. Geosci. 34(4), 966–972 (1996).
[CrossRef]

Lévy, P.

P. Lévy, Théorie de l’Addition des Variables Aléatoires (Gauthier-Villars, Paris, 1937).

Lim, S. P.

N. Lin, H. P. Lee, S. P. Lim, K. S. Lee, “Wave scattering from fractal surfaces,” J. Mod. Opt. 42, 225–235 (1995).
[CrossRef]

Lin, N.

N. Lin, H. P. Lee, S. P. Lim, K. S. Lee, “Wave scattering from fractal surfaces,” J. Mod. Opt. 42, 225–235 (1995).
[CrossRef]

Litva, J.

J. Chen, T. K. Y. Lo, H. Leung, J. Litva, “The use of fractals for modeling EM waves scattering from rough sea surface,” IEEE Trans. Geosci. 34(4), 966–972 (1996).
[CrossRef]

Lo, T. K. Y.

J. Chen, T. K. Y. Lo, H. Leung, J. Litva, “The use of fractals for modeling EM waves scattering from rough sea surface,” IEEE Trans. Geosci. 34(4), 966–972 (1996).
[CrossRef]

Lu, T. M.

Y. P. Zhao, I. Wu, C. F. Cheng, U. Block, G. C. Wang, T. M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[CrossRef]

Y-P Zhao, C. F. Cheng, G. C. Wang, T. M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. 409, L703–L708 (1998).
[CrossRef]

Y. P. Zhao, G. C. Wang, T. M. Lu, “Diffraction from non-Gaussian rough surfaces,” Phys. Rev. B 55, 13938–13952 (1997).
[CrossRef]

H. N. Yang, T. M. Lu, G. C. Wang, “Diffraction from surface growth fronts,” Phys. Rev. B 47, 3911–3922 (1993).
[CrossRef]

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1975).

Maradudin, A. A.

J. A. Sanchez-Gil, A. A. Maradudin, “Competition between Anderson localization and leakage of surface-plasmon polaritons on randomly rough periodic metal surfaces,” Phys. Rev. B 56, 1103–1106 (1997).
[CrossRef]

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–307 (1990).
[CrossRef]

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef] [PubMed]

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–307 (1990).
[CrossRef]

McSharry, P. E.

P. E. McSharry, P. J. Cullen, D. Moroney, “Wave scattering by a two-dimensional band limited fractal surface based on a perturbation of the Green’s function,” J. Appl. Phys. 78(12), 6940–6948 (1995).
[CrossRef]

Meakin, P.

P. Meakin, Fractals, Scaling and Growth Far from Equilibrium (Cambridge U. Press, Cambridge, UK, 1998).

Méndez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–307 (1990).
[CrossRef]

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef] [PubMed]

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random rough surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–307 (1990).
[CrossRef]

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef] [PubMed]

Moroney, D.

P. E. McSharry, P. J. Cullen, D. Moroney, “Wave scattering by a two-dimensional band limited fractal surface based on a perturbation of the Green’s function,” J. Appl. Phys. 78(12), 6940–6948 (1995).
[CrossRef]

Nes, O. M.

I. Simonsen, A. Hansen, O. M. Nes, “Determination of Hurst exponents by use of the wavelet transform,” Phys. Rev. E 58, 2779–2787 (1998).
[CrossRef]

Nieto-Vesperinas, M.

O’Donnell, K. A.

C. S. West, K. A. O’Donnell, “Observation of backscattering enhancement from polaritons on a rough metal surface,” J. Opt. Soc. Am. A 12, 390–397 (1995).
[CrossRef]

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random rough surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (IOP, Bristol, UK, 1991).

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

Pincemin, F.

Plouraboué, F.

F. Plouraboué, M. Boehm, “Multi-scale roughness transfer in cold metal rolling,” Tribol. Int. 32, 45–57 (1999).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Prewett, A.

D. L. Jordan, R. C. Hollins, E. Jakeman, A. Prewett, “Visible and infra-red scattering from well-characterized surfaces,” Surf. Topog. 1, 27–36 (1988).

Rayleigh, J. W. S.

J. W. S. Rayleigh, The Theory of Sound (Dover, New York, 1945).

Roux, S.

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[CrossRef]

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[CrossRef]

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J. A. Sánchez-Gil, J. V. Garcı́a-Ramos, “Calculations of the direct electromagnetic enhancement in surface enhanced raman scattering on random self-affine fractal metal surfaces,” J. Chem. Phys. 108, 1317–1325 (1998).

J. A. Sánchez-Gil, J. V. Garcı́a-Ramos, “Far field intensity of electromagnetic waves scattered from random, self-affine fractal metal surfaces,” Waves Random Media 7, 285–293 (1997).
[CrossRef]

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[CrossRef]

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[CrossRef]

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J. A. Sanchez-Gil, A. A. Maradudin, “Competition between Anderson localization and leakage of surface-plasmon polaritons on randomly rough periodic metal surfaces,” Phys. Rev. B 56, 1103–1106 (1997).
[CrossRef]

J. A. Sanchez-Gil, “Coupling, resonance transmission, and tunneling of surface-plasmon polaritons through metallic gratings of finite length,” Phys. Rev. B 53, 10317–10327 (1996).
[CrossRef]

H. N. Yang, T. M. Lu, G. C. Wang, “Diffraction from surface growth fronts,” Phys. Rev. B 47, 3911–3922 (1993).
[CrossRef]

Y. P. Zhao, G. C. Wang, T. M. Lu, “Diffraction from non-Gaussian rough surfaces,” Phys. Rev. B 55, 13938–13952 (1997).
[CrossRef]

Phys. Rev. E (3)

I. Simonsen, D. Vandembroucq, S. Roux, “Wave scattering from self-affine surfaces,” Phys. Rev. E 61, 5914–5917 (2000).
[CrossRef]

J. Schmittbuhl, J. P. Vilotte, S. Roux, “Reliability of self-affine measurements,” Phys. Rev. E 51, 131–147 (1995).
[CrossRef]

I. Simonsen, A. Hansen, O. M. Nes, “Determination of Hurst exponents by use of the wavelet transform,” Phys. Rev. E 58, 2779–2787 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

S. C. Kitson, W. L. Barnes, J. L. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2670–2673 (1996).
[CrossRef] [PubMed]

Surf. Sci. (1)

Y-P Zhao, C. F. Cheng, G. C. Wang, T. M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. 409, L703–L708 (1998).
[CrossRef]

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J. A. Sánchez-Gil, J. V. Garcı́a-Ramos, “Far field intensity of electromagnetic waves scattered from random, self-affine fractal metal surfaces,” Waves Random Media 7, 285–293 (1997).
[CrossRef]

E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 3, S165–S190 (1991).
[CrossRef]

Other (12)

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

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P. Meakin, Fractals, Scaling and Growth Far from Equilibrium (Cambridge U. Press, Cambridge, UK, 1998).

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (IOP, Bristol, UK, 1991).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1963).

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1975).

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Figures (8)

Fig. 1
Fig. 1

Scattering geometry considered in this paper.

Fig. 2
Fig. 2

Mean DRC Rs/θ versus scattering angle θ for a perfectly conducting self-affine surface. The plotted curves are the prediction of Eqs. (17). The Hurst exponent in all cases is H=0.7, and the topothesies l range from l=10-2λ [s(λ)=0.016] down to 10-6λ [s(λ)=0.25], as indicated in the figures. The incident angles were (a) θ0=0° and (b) θ0=50°.

Fig. 3
Fig. 3

Full single-scattering solution (solid curve), Eqs. (17), for the mean differential reflection coefficient versus scattering angle θ for a perfectly conducting self-affine surface compared with its specular (short dashed curve) and diffuse (long dashed curve) expansions as given by relations (19) and (24), respectively. The surface parameters used were H=0.7 and l=10-4λ [s(λ)=0.063], and the light was incident normally onto the rough surface.

Fig. 4
Fig. 4

Comparison plotted in (a) linear and (b) linear-log scale of the mean DRC Rs/θ versus scattering angle θ for a perfectly conducting self-affine surface obtained by a rigorous numerical-simulation approach (solid curves) and therefore including all possible multiple-scattering processes, and the single-scattering results obtained from Eqs. (17) (dashed curves). The surface parameters were H=0.7 and l=10-4λ [s(λ)=0.063] with λ=612.7 nm. The angles of the incident light were 0° and 50° as indicated in the figure.

Fig. 5
Fig. 5

Same as Figs. 2 (single-scattering results), but now with a rigorous numerical-simulation approach (see text for details) that incorporates all higher-order scattering processes.

Fig. 6
Fig. 6

Specular peak amplitude, Rs/θ|θ=θ0, and its half-width at half-maximum, w(H, l/λ, θ0) as a function of topothesy l. The angle of incidence was in both cases θ0=0°. The solid lines are analytical results obtained from relations (20) and (21), and the circles (amplitudes) and the squares (widths) were obtained from the numerical simulations results shown in Fig. 5(a).

Fig. 7
Fig. 7

Rescaled version of the rigorous numerical-simulation results shown in Fig. 5(b). Only the data corresponding to θ<50° are included. In the rescaled coordinates all data (symbols) should within the single-scattering approximation collapse onto a Lévy distribution of order 2H (solid curve).

Fig. 8
Fig. 8

Same as Fig. 4(b), but now with a real metal (silver) instead of a perfect conductor. The value of the dielectric constant of silver at the wavelength of the incident light (λ=612.7 nm) was ε(ω)=-17.2+0.50i.

Equations (48)

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ΔxμΔx,
ΔζμHΔζ,
p(Δζ; Δx)d(Δζ)=p(μHΔζ; μΔx)d(μHΔζ).
σ(Δx)=l1-HΔxH
s(Δx)=lΔx1-H.
σ(Δx)=σ(λ)ΔxλH=λs(λ)ΔxλH,
P(k)=-ζ(x)exp(ikx)dx2k-1-2H.
p(Δζ; Δx)=λH-12πs(λ)ΔxH exp-12 λH-1Δζs(λ)ΔxH2.
x2+z2+ω2c2Φ(x, z|ω)=0,
Φ(x, z|ω)=Φ0(x, z|ω)+- dq2πR(q|k)×exp[iqx+iα0(q, ω)z]
Φ0(x, z|ω)=exp[ikx-iα0(k, ω)z]
α0(q, ω)=(ω/c)2-q2,
Rα0(q, ω)>0,Iα0(q, ω)>0.
k=ωc sin θ0,
q=ωc sin θ,
α0(k, ω)=ωc cos θ0,
α0(q, ω)=ωc cos θ.
Rsθ=1L ω2πc cos2 θcos θ0|R(q|k)|2.
R(q|k)=-i2α0(q, ω) -L/2L/2dx exp[-iqx-iα0(q, ω)ζ(x)]×N0(x|ω),
N0(x|ω)=2nΦ0(x, z|ω)|z=ζ(x).
Rsθ=ω2πc 1cos θ0 cos[(θ+θ0)/2]cos[(θ-θ0)/2]2×-L/2L/2dv expi ωc(sin θ-sin θ0)vΩ(v),
Ω(v)=exp-i ωc(cos θ+cos θ0)Δζ(v),
Ω(v)=-dz exp-i ωc(cos θ+cos θ0)zp(z; v)=exp-ωc cos θ+cos θ02s(λ)λ1-HvH2.
u=vωc cos θ+cos θ02s(λ)λ1-H1/H,
Rsθ=s(λ)-1/Ha-(1/H-1)2 cos θ0 cos θ+θ02cos3 θ-θ02×L2H2 tan θ-θ02a1/H-1s(λ)1/H,
a=2π2 cos θ+θ02 cos θ-θ02,
Lα(x)=12π -dk exp(ikx)exp(-|k|α).
Lα(x)=1παΓ1α1-Γ(3/α)2Γ(1/α)x2+O(x4).
Rsθθ=θ0+δθ=Γ(1/2H)22πH(22π cos θ0)1/H-1s(λ)1/H×1+δθ 1-2H2H tan θ0+(δθ)24×1H-(2H-1)(1-H)2H2 tan2 θ0-Γ(3/2H)Γ(1/2H)(22π cos θ0)2/H-2s(λ)2/H.
Rsθθ=θ0Γ(1/2H)22πH(22π cos θ0)1/H-1s(λ)1/H
w(H, s(λ), θ0)2Γ(1/2H)Γ(3/2H)1/2(22π cos θ0)1/H-1×s(λ)1/H.
Δθ0-1-2HH Γ(1/2H)Γ(3/2H)×tan θ0(22π cos θ0)2/H-2s(λ)2/H1-4H4H tan θ0w2[H, s(λ), θ0].
Lα(x)=Γ(1+α)π|x|1+α sinαπ2+O1|x|1+2α,
RsθΓ(1+2H)sin(πH)(4π)2H-1 s(λ)2cos θ0 cos θ+θ023-2Hsin θ-θ021+2H.
Δcang=(sin θ-sin θ0)Δx,
Δcrough=(cos θ+cos θ0)Δz.
δang=λsin θ-sin θ0,
δrough=λ(cos θ+cos θ0)1/Hs(λ)-1/H.
χ=δroughδang=sin θ-sin θ0(cos θ+cos θ0)1/Hs(λ)-1/H.
w[2s(λ)]1/H(cos θ0)1/H-1,
Rsθθ=θ01w[2s(λ)]-1/H(cos θ0)1-1/H.
Rsθ=(cos θ0)1-1/Hs(λ)1/HΨ(χ).
Ψ(χ)1,(χ1).
RsθP2πλ(sin θ-sin θ0),
Ψ(χ)χ-1-2H,(χ1).
N(x|ω)=2N0(x|ω)-2P  dx×nG0(x, z|x, z)|z=ζ(x)N(x|ω).
N(x|ω)=nΦ(x, z|ω)|z=ζ(x),
G0(x, z|x, z)=iπH0(1)ωc|r-r|,

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