Abstract

A perturbative expression up to fourth order in k0σ (k0 = 2π/λ; σ being the rms of the surface height and λ the wavelength) has been obtained for the mean diffuse intensity from a one-dimensional random rough surface that has normal statistics and a Gaussian correlation function for s polarization. For p polarization it is not possible to obtain this expression because of the existence of certain resonances; thus the calculations must be restricted to second order in k0σ. Perturbative calculations were derived from the Rayleigh hypothesis and also from the extinction theorem. The expression for the diffuse component of the mean scattered intensity was the same for p waves in both cases up to second order in k0σ. For s waves the equality was obtained up to fourth order in k0σ. Comparisons with exact numerical results and with those obtained by using the Kirchhoff approximation are made. This comparison allows us to establish assessments on the validity of the perturbative solution and to obtain some new interesting facts. In addition, the behavior of the diffuse halo at small σ/λ as a function of the correlation length T, the angle of incidence θ0, and the polarization is discussed. The validity of the Rayleigh hypothesis is also studied.

© 1990 Optical Society of America

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References

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    [CrossRef]
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  4. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).
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    [CrossRef]
  6. M. Nieto-Vesperinas, “Depolarization of EM waves scattered from slightly rough random surfaces: a study by means of the extinction theorem,” J. Opt. Soc. Am. 72, 539–547 (1982).
    [CrossRef]
  7. G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
    [CrossRef]
  8. G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
    [CrossRef]
  9. F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
    [CrossRef]
  10. R. Schiffer, “Reflectivity of a slightly rough surfaces,” Appl. Opt. 26, 704–712 (1987).
    [CrossRef] [PubMed]
  11. N. García, V. Celli, M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
    [CrossRef]
  12. M. Nieto-Vesperinas, N. García, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
    [CrossRef]
  13. M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Hafner, New York, 1963).
  14. J. Shen, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
    [CrossRef]
  15. D. P. Winebrener, A. Ishimaru, “Investigation of a surface field phase-perturbation technique for scattering from rough surfaces,” Radio. Sci. 20, 161–170 (1985).
    [CrossRef]
  16. D. P. Winebrener, A. Ishimaru, “Application of the phase-perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2293 (1985).
    [CrossRef]
  17. C. Eftimiu, G. V. Welland, “The use of Padé approximants in rough surface scattering,” IEEE Trans. Antennas Propag. AP-35, 721–727 (1987).
    [CrossRef]
  18. 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]
  19. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
    [CrossRef]
  20. J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
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  21. 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).
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  23. G. R. Valenzuela, J. W. Wright, J. C. Leader, “Comments on the relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,” IEEE Trans. Antennas Propag. AP-20, 536–539 (1972).
    [CrossRef]
  24. D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antennas Propag. AP-35, 120–122 (1987).
    [CrossRef]
  25. M. Nieto-Vesperinas, “Radiometry of rough surfaces,” Opt. Acta 29, 961–971 (1982).
    [CrossRef]
  26. V. M. Agranovich, D. L. Mills, eds., Surface Polaritons (North-Holland, Amsterdam, 1982).
  27. H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).
  28. V. Celli, A. A. Maradudin, A. M. Marvin, A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
    [CrossRef]
  29. P. Tran, V. Celli, “Monte Carlo calculations of backscattering enhancement for a randomly rough grating,” J. Opt. Soc. Am. A 5, 1635–1637 (1988).
    [CrossRef]
  30. N. García, E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
    [CrossRef]
  31. B. A. Lippmann, “Note on the theory of gratings,” J. Opt. Soc. Am. 43, 408–409 (1953).
    [CrossRef]
  32. J. L. Urestky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. (NY) 33, 400–427 (1965).
    [CrossRef]
  33. R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infinement conducteur. Note,” C. R. Acad. Sci. B 262, 468–471 (1966).
  34. R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
    [CrossRef]
  35. R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface. II,” Proc. Cambridge Philos. Soc. 69, 217–224 (1971).
    [CrossRef]
  36. R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
    [CrossRef]
  37. J. P. Rossi, D. Maystre, “Rigorous numerical study of speckle patterns for two dimensional, random microrough surfaces,” Opt. Eng. 25, 613–617 (1986).
    [CrossRef]
  38. A. Wirgin, “On Rayleigh’s theory of sinusoidal gratings,” Opt. Acta 27, 1671–1692 (1980).
    [CrossRef]
  39. A. Wirgin, “Scattering from sinusoidal gratings: an evaluation of the Kirchhoff approximation,” J. Opt.Soc. Am. 73, 1028–1041 (1983).
    [CrossRef]
  40. N. García, J. Ibañez, J. Solana, N. Cabrera, “On the ‘quantum rainbow’ in surface scattering,” Solid State Commun. 20, 1159–1163 (1976).
    [CrossRef]
  41. N. García, “The scattering of He atoms from a hard corrugated surface model using the GR method. I.” J. Chem. Phys. 67, 897–916 (1977).
    [CrossRef]
  42. N. García, N. Cabrera, “New methods for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576–589 (1978).
    [CrossRef]
  43. A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), p. 11.
  44. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, London, 1973), pp. 150–159.
  45. J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), p. 307.
  46. M. F. Chen, A. K. Fung, “A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models,” Radio Sci. 23, 163–170 (1988).
    [CrossRef]
  47. E. I. Thorsos, D. R. Jackson, “The validity of the pertubative approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86, 261–277 (1989).
    [CrossRef]
  48. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, p. 811.

1989 (3)

1988 (3)

M. F. Chen, A. K. Fung, “A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models,” Radio Sci. 23, 163–170 (1988).
[CrossRef]

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

P. Tran, V. Celli, “Monte Carlo calculations of backscattering enhancement for a randomly rough grating,” J. Opt. Soc. Am. A 5, 1635–1637 (1988).
[CrossRef]

1987 (4)

C. Eftimiu, G. V. Welland, “The use of Padé approximants in rough surface scattering,” IEEE Trans. Antennas Propag. AP-35, 721–727 (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]

D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antennas Propag. AP-35, 120–122 (1987).
[CrossRef]

R. Schiffer, “Reflectivity of a slightly rough surfaces,” Appl. Opt. 26, 704–712 (1987).
[CrossRef] [PubMed]

1986 (1)

J. P. Rossi, D. Maystre, “Rigorous numerical study of speckle patterns for two dimensional, random microrough surfaces,” Opt. Eng. 25, 613–617 (1986).
[CrossRef]

1985 (4)

V. Celli, A. A. Maradudin, A. M. Marvin, A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
[CrossRef]

G. S. Brown, “A comparison of approximate theories for scattering from rough surfaces,” Wave Motion 7, 195–205 (1985).
[CrossRef]

D. P. Winebrener, A. Ishimaru, “Investigation of a surface field phase-perturbation technique for scattering from rough surfaces,” Radio. Sci. 20, 161–170 (1985).
[CrossRef]

D. P. Winebrener, A. Ishimaru, “Application of the phase-perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2293 (1985).
[CrossRef]

1984 (1)

N. García, E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[CrossRef]

1983 (1)

A. Wirgin, “Scattering from sinusoidal gratings: an evaluation of the Kirchhoff approximation,” J. Opt.Soc. Am. 73, 1028–1041 (1983).
[CrossRef]

1982 (2)

1981 (2)

M. Nieto-Vesperinas, N. García, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

J. C. Leader, “The relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,” IEEE Trans. Antennas Propag. AP-19, 786–788 (1981).

1980 (2)

J. Shen, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[CrossRef]

A. Wirgin, “On Rayleigh’s theory of sinusoidal gratings,” Opt. Acta 27, 1671–1692 (1980).
[CrossRef]

1979 (1)

N. García, V. Celli, M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
[CrossRef]

1978 (1)

N. García, N. Cabrera, “New methods for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576–589 (1978).
[CrossRef]

1977 (3)

N. García, “The scattering of He atoms from a hard corrugated surface model using the GR method. I.” J. Chem. Phys. 67, 897–916 (1977).
[CrossRef]

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

1976 (1)

N. García, J. Ibañez, J. Solana, N. Cabrera, “On the ‘quantum rainbow’ in surface scattering,” Solid State Commun. 20, 1159–1163 (1976).
[CrossRef]

1973 (1)

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

1972 (1)

G. R. Valenzuela, J. W. Wright, J. C. Leader, “Comments on the relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,” IEEE Trans. Antennas Propag. AP-20, 536–539 (1972).
[CrossRef]

1971 (1)

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface. II,” Proc. Cambridge Philos. Soc. 69, 217–224 (1971).
[CrossRef]

1969 (1)

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
[CrossRef]

1967 (1)

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

1966 (1)

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infinement conducteur. Note,” C. R. Acad. Sci. B 262, 468–471 (1966).

1965 (1)

J. L. Urestky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. (NY) 33, 400–427 (1965).
[CrossRef]

1953 (1)

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

Bass, F. G.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).

Brown, G. S.

G. S. Brown, “A comparison of approximate theories for scattering from rough surfaces,” Wave Motion 7, 195–205 (1985).
[CrossRef]

J. A. de Santo, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics XXIII, E. Wolf, ed. (North-Holland, Amsterdam, 1986), pp. 3–62.

Cabrera, N.

N. García, N. Cabrera, “New methods for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576–589 (1978).
[CrossRef]

N. García, J. Ibañez, J. Solana, N. Cabrera, “On the ‘quantum rainbow’ in surface scattering,” Solid State Commun. 20, 1159–1163 (1976).
[CrossRef]

Cadilhac, M.

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infinement conducteur. Note,” C. R. Acad. Sci. B 262, 468–471 (1966).

Celli, V.

P. Tran, V. Celli, “Monte Carlo calculations of backscattering enhancement for a randomly rough grating,” J. Opt. Soc. Am. A 5, 1635–1637 (1988).
[CrossRef]

V. Celli, A. A. Maradudin, A. M. Marvin, A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
[CrossRef]

N. García, V. Celli, M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
[CrossRef]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Chen, M. F.

M. F. Chen, A. K. Fung, “A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models,” Radio Sci. 23, 163–170 (1988).
[CrossRef]

de Santo, J. A.

J. A. de Santo, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics XXIII, E. Wolf, ed. (North-Holland, Amsterdam, 1986), pp. 3–62.

Eftimiu, C.

C. Eftimiu, G. V. Welland, “The use of Padé approximants in rough surface scattering,” IEEE Trans. Antennas Propag. AP-35, 721–727 (1987).
[CrossRef]

Erdélyi, A.

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), p. 11.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, p. 811.

Fuks, I. M.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).

Fung, A. K.

M. F. Chen, A. K. Fung, “A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models,” Radio Sci. 23, 163–170 (1988).
[CrossRef]

García, N.

N. García, E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[CrossRef]

M. Nieto-Vesperinas, N. García, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

N. García, V. Celli, M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
[CrossRef]

N. García, N. Cabrera, “New methods for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576–589 (1978).
[CrossRef]

N. García, “The scattering of He atoms from a hard corrugated surface model using the GR method. I.” J. Chem. Phys. 67, 897–916 (1977).
[CrossRef]

N. García, J. Ibañez, J. Solana, N. Cabrera, “On the ‘quantum rainbow’ in surface scattering,” Solid State Commun. 20, 1159–1163 (1976).
[CrossRef]

Hill, N. R.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Holliday, D.

D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antennas Propag. AP-35, 120–122 (1987).
[CrossRef]

Ibañez, J.

N. García, J. Ibañez, J. Solana, N. Cabrera, “On the ‘quantum rainbow’ in surface scattering,” Solid State Commun. 20, 1159–1163 (1976).
[CrossRef]

Ishimaru, A.

D. P. Winebrener, A. Ishimaru, “Investigation of a surface field phase-perturbation technique for scattering from rough surfaces,” Radio. Sci. 20, 161–170 (1985).
[CrossRef]

D. P. Winebrener, A. Ishimaru, “Application of the phase-perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2293 (1985).
[CrossRef]

Jackson, D. R.

E. I. Thorsos, D. R. Jackson, “The validity of the pertubative approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86, 261–277 (1989).
[CrossRef]

Kendall, M. G.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Hafner, New York, 1963).

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), p. 307.

Leader, J. C.

J. C. Leader, “The relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,” IEEE Trans. Antennas Propag. AP-19, 786–788 (1981).

G. R. Valenzuela, J. W. Wright, J. C. Leader, “Comments on the relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,” IEEE Trans. Antennas Propag. AP-20, 536–539 (1972).
[CrossRef]

Lippmann, B. A.

Maradudin, A. A.

Marvin, A.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Marvin, A. M.

Maystre, D.

J. P. Rossi, D. Maystre, “Rigorous numerical study of speckle patterns for two dimensional, random microrough surfaces,” Opt. Eng. 25, 613–617 (1986).
[CrossRef]

McGurn, A. R.

Méndez, E. R.

Michel, T.

Millar, R. F.

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface. II,” Proc. Cambridge Philos. Soc. 69, 217–224 (1971).
[CrossRef]

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, p. 811.

Nieto-Vesperinas, M.

Petit, R.

R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infinement conducteur. Note,” C. R. Acad. Sci. B 262, 468–471 (1966).

Raether, H.

H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Rossi, J. P.

J. P. Rossi, D. Maystre, “Rigorous numerical study of speckle patterns for two dimensional, random microrough surfaces,” Opt. Eng. 25, 613–617 (1986).
[CrossRef]

Schiffer, R.

Shen, J.

J. Shen, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[CrossRef]

Solana, J.

N. García, J. Ibañez, J. Solana, N. Cabrera, “On the ‘quantum rainbow’ in surface scattering,” Solid State Commun. 20, 1159–1163 (1976).
[CrossRef]

Soto-Crespo, J. M.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).

Stoll, E.

N. García, E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[CrossRef]

Stuart, A.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Hafner, New York, 1963).

Thorsos, E. I.

E. I. Thorsos, D. R. Jackson, “The validity of the pertubative approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 86, 261–277 (1989).
[CrossRef]

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

Toigo, F.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Tran, P.

Urestky, J. L.

J. L. Urestky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. (NY) 33, 400–427 (1965).
[CrossRef]

Valenzuela, G. R.

G. R. Valenzuela, J. W. Wright, J. C. Leader, “Comments on the relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,” IEEE Trans. Antennas Propag. AP-20, 536–539 (1972).
[CrossRef]

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

Watson, G. N.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, London, 1973), pp. 150–159.

Welland, G. V.

C. Eftimiu, G. V. Welland, “The use of Padé approximants in rough surface scattering,” IEEE Trans. Antennas Propag. AP-35, 721–727 (1987).
[CrossRef]

Whittaker, E. T.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, London, 1973), pp. 150–159.

Winebrener, D. P.

D. P. Winebrener, A. Ishimaru, “Application of the phase-perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2293 (1985).
[CrossRef]

D. P. Winebrener, A. Ishimaru, “Investigation of a surface field phase-perturbation technique for scattering from rough surfaces,” Radio. Sci. 20, 161–170 (1985).
[CrossRef]

Wirgin, A.

A. Wirgin, “Scattering from sinusoidal gratings: an evaluation of the Kirchhoff approximation,” J. Opt.Soc. Am. 73, 1028–1041 (1983).
[CrossRef]

A. Wirgin, “On Rayleigh’s theory of sinusoidal gratings,” Opt. Acta 27, 1671–1692 (1980).
[CrossRef]

Wright, J. W.

G. R. Valenzuela, J. W. Wright, J. C. Leader, “Comments on the relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,” IEEE Trans. Antennas Propag. AP-20, 536–539 (1972).
[CrossRef]

Ann. Phys. (NY) (1)

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Figures (14)

Fig. 1
Fig. 1

Illustration of the scattering geometry.

Fig. 2
Fig. 2

s-Wave mean diffuse intensity from three random surfaces having three different T values at three different angles of incidence. Solid curves, exact numerical method; dotted curves, SO intensity; dashed curves, FO intensity. The upper tip marks the specular direction.

Fig. 3
Fig. 3

Mean diffuse intensity from three random surfaces with different σ values at three different angles of incidence. Solid curves, KA intensity; dotted curves, SO intensity for s waves; short-dashed curves, SO intensity for p waves; long-dashed curves, FO intensity for s waves.

Fig. 4
Fig. 4

s-Wave mean diffuse intensity for a surface with σ/λ = 0.2 and T/λ = 4.8 at θ0 = 70° and θ0 = 80°. Solid curves, exact numerical intensity; dashed curves, FO intensity; dotted curves, KA intensity.

Fig. 5
Fig. 5

Plots of σ/λ versus T/λ marking the zones of validity of the SO perturbative solutions at four angles of incidence: (a) 0.2 < T/λ < 1, (b) 1 < T/λ < 10.

Fig. 6
Fig. 6

Plots of σ/λ versus the angle of incidence θ0 at three values of T/λ marking the zones of validity of the SO perturbative solution.

Fig. 7
Fig. 7

TISE calculated in FO versus T/λ at different values of θ0. At each θ0 the value of σ/λ is linked to the corresponding T/λ by the curves of Figs. 5(a) and 5(b).

Fig. 8
Fig. 8

s-Wave mean diffuse intensity for a random surface with σ/λ = 0.07 and T = 0.15λ at θ0 = 0°, 40°, and 70°. Solid curves, exact numerical intensity; dotted curves, SO intensity; dashed curves, FO intensity. The TISE obtained through the Monte Carlo method is also given; it is represented by the area below the solid curve.

Fig. 9
Fig. 9

Same as in Fig. 8 for σ/λ = T/λ = 0.1 and θ0 = 0°, 30°, 50°.

Fig. 10
Fig. 10

Mean diffuse intensity for two surfaces with T/λ = 0.5 and σ/λ = 0.03 (the left column) and σ/λ = 0.04 (the right column) at three angles of incidence. Solid curves, exact numerical calculations for p waves; dotted curves, SO intensity for p waves; dashed curves, FO intensity for s waves.

Fig. 11
Fig. 11

Same as in Fig. 10 for T/λ = 0.2.

Fig. 12
Fig. 12

Energy contained in the coherent specular peak for p waves as a function of the angle of incidence for two random surfaces with σ/λ = 0.03 and T/λ = 0.5 (open circles) and 0.2 (filled circles).

Fig. 13
Fig. 13

s-Wave mean diffuse intensity from a random rough surface with T = 0.4λ and σ = 0.15λ at θ0 = 0° and 40°. Solid curves, exact numerical method; dotted curves, RT intensity.

Fig. 14
Fig. 14

Plot of σ/T versus T/λ marking the zone for which the Rayleigh method described in the text gives the same results as the exact numerical method. The dashed line corresponds to σ/T = 0.224.

Tables (1)

Tables Icon

Table 1 TISE Obtained from the Monte Carlo Method for those Parameters of σ, T, and θ0 Represented in Fig. 1a

Equations (100)

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I s ( θ ) d = I 2 ( θ ) d + I 4 ( θ ) d ,
I 2 ( θ ) d = F s ( θ , θ 0 ) 16 π 2 σ 2 T π / λ 3 × exp { [ T π ( sin θ sin θ 0 ) / λ ] 2 } ,
F s ( θ , θ 0 ) = cos 2 θ cos θ 0 .
I p ( θ ) d = F p ( θ , θ 0 ) 16 π 2 σ 2 T π / λ 3 × exp { [ T π ( sin θ sin θ 0 ) / λ ] 2 } ,
F p ( θ , θ 0 ) = ( 1 sin θ sin θ 0 ) 2 / cos θ 0 .
I KA ( θ ) d = k [ 1 + cos ( θ + θ 0 ) ] 2 π cos θ 0 ( cos θ 0 + cos θ ) 2 0 cos ( υ x T ) × ( exp { υ z 2 [ σ 2 C ( τ ) ] } exp [ ( σ υ z ) 2 ] ) d τ ,
υ x = k 0 ( sin θ 0 sin θ ) ,
υ z = k 0 ( cos θ 0 cos θ ) ,
E SO = π / 2 π / 2 I 2 ( θ ) d d θ ,
E FO = E SO + π / 2 π / 2 I 4 ( θ ) d d θ .
| E FO E SO | E FO 0.05 .
h / d 0.0713 .
H = σ ,
D = T π 2 .
σ / T 0.224 .
h / d < 0.39 0.0177 ( d / λ ) when d / λ 15 .
σ / T < 1.225 0.0786 T / λ when T / λ 3.376 .
exp [ i q 0 D ( y ) ] = exp { i [ Q n y + q n D ( y ) ] } A E ( Q n ) ,
[ ( d D / d y ) K + q 0 ] exp [ i q 0 D ( y ) ] = A H ( Q n ) × [ ( d D / d y ) ( Q n + K ) + q n ] exp [ i Q n y + i q n D ( y ) ] ,
Q n = k 0 n / d ,
q n = [ k 0 2 ( K + Q n ) 2 ] 1 / 2 .
I n = ( q n / q 0 ) | A ( Q n ) | 2 ,
I n in = I n to ( q n / q 0 ) | A ( Q n ) | 2 ,
I n to = ( q n / q 0 ) | A ( Q n ) | 2 .
I ( θ ) = I n / δ θ ,
δ θ = | ( θ n + θ n 1 ) / 2 ( θ n + 1 + θ n ) / 2 | .
k 0 = ( 0 , K , q 0 ) = ( 0 , k 0 sin θ 0 , k 0 sin θ 0 ) .
E 0 ( r ) = A 0 exp [ i ( K y q 0 z ) ] ,
E ( r ) = E 0 ( r ) π k 0 c L J ( y ) H 0 ( 1 ) ( k 0 | r r | ) × [ 1 + ( α d D / d y ) 2 ] 1 / 2 d y , z > α D ( y ) ,
E 0 ( r ) = π k 0 c L J ( y ) H 0 ( 1 ) ( k 0 | r r | ) × [ 1 + ( α d D / d y ) 2 ] 1 / 2 d y , z > α D ( y ) ,
H 0 ( 1 ) ( k 0 | r r | ) = 1 π d Q × exp [ i ( K + Q ) ( y y ) ] exp ( i q | z z | ) q ,
q = q ( Q ) = ( q 0 2 Q 2 2 K Q ) 1 / 2 .
E ( r ) = E 0 ( r ) d Q exp [ i ( Q + K ) y ] exp ( iqz ) 1 q L d y × exp ( i Q y ) exp [ i q α D ( y ) ] F ( y , α ) z > α D max ,
exp ( i q 0 z ) d Q 1 q exp ( iQy ) ] exp ( iqz ) L d y × exp ( i Q y ) exp [ i q α D ( y ) ] F ( y , α ) z > α D min ,
F ( y , α ) = k 0 c J ( y ) [ 1 + ( α d D / d y ) 2 ] 1 / 2 exp ( i K y ) .
δ ( Q ) = 1 q L d y exp ( i Q y ) exp [ i q α D ( y ) ] F ( y , α ) ,
E ( r ) = E 0 ( r ) + d Q exp [ i ( Q + K ) y ] exp ( iqz ) A ( Q ) , z > α D max ,
A ( Q ) = 1 q L d y exp ( i Q y ) exp [ i q α D ( y ) ] F ( y , α ) .
F ( y , α ) = 0 F ( n ) ( y ) α n / n ! ,
A ( Q , α ) = 0 A ( n ) ( Q ) α n / n !
δ ( Q ) = F ( 0 ) ( Q ) / q ,
A ( 0 ) ( Q ) = F ( 0 ) ( Q ) / q ,
F ( 1 ) ( Q ) = i q [ D ( Q ) * F ( 0 ) ( Q ) ] ,
A ( 1 ) ( Q ) = 2 i D ( Q ) * F ( 0 ) ( Q ) ,
F ( 2 ) ( Q ) = 2 i q [ F ( 1 ) ( Q ) * D ( Q ) ] + q 2 { [ D 2 ( y ) ] ( Q ) * F ( 0 ) ( Q ) } ,
A ( 2 ) ( Q ) = 4 i F ( 1 ) ( Q ) * D ( Q ) ,
A ( 3 ) ( Q ) = 6 i D ( Q ) * F ( 2 ) ( Q ) 2 i q 2 { [ D 3 ( y ) ] ( Q ) * F ( 0 ) ( Q ) ] .
A ( 0 ) ( Q ) = δ ( Q ) ,
A ( 1 ) ( Q ) = i q 0 π D ( Q ) ,
A ( 2 ) ( Q ) = 2 q 0 π { [ q D ( Q ) ] * D ( Q ) } ,
A ( 3 ) ( Q ) = 6 i q 0 π ( D ( Q ) * { q [ q D ( Q ) ] * D ( Q ) } ) + 3 i q 0 π ( D ( Q ) * { q 2 [ D 2 ( y ) ] ( Q ) } ) i q 0 q 2 π [ D 3 ( y ) ] ( Q ) .
f ( Q ) [ f ( y ) ] Q = L d y exp ( iQy ) f ( y ) ,
f ( y ) = 1 2 π d Q exp ( iQy ) f ( Q ) ,
f ( Q ) * g ( Q ) = 1 2 π d Q f ( Q Q ) g ( Q ) .
I ( θ ) = 2 π k 0 cos 2 θ L cos θ 0 | A ( Q ) | 2 ,
I ( θ ) d = 2 π k 0 cos 2 θ L cos θ 0 { A ( 1 ) ( Q ) A ( 1 ) * ( Q ) + 1 3 Re A ( 1 ) * ( Q ) A 3 ) Q + 1 4 [ A ( 2 ) ( Q ) A ( 2 ) * ( Q ) | A ( 2 ) ( Q ) | 2 } ,
D ( y ) D ( y ) D ( y ) D ( y ) = D ( y ) D ( y ) D ( y ) D ( y ) + D ( y ) D ( y ) D ( y ) D ( y ) + D ( y ) D ( y ) D ( y ) D ( y ) .
D ( Q ) D ( Q ) D ( Q ) D ( Q ) = ( 2 π ) 2 [ δ ( Q + Q ) δ ( Q + Q ) W ( Q ) W ( Q ) + δ ( Q + Q ) δ ( Q + Q ) W ( Q ) W ( Q ) + δ ( Q + Q ) δ ( Q + Q ) W ( Q ) W ( Q ) ] ,
W ( Q ) = d τ exp ( i Q τ ) C ( τ ) .
A ( 1 ) ( Q ) A ( 1 ) * ( Q ) = 4 L q 0 2 ( 2 π ) 2 W ( Q ) .
A ( 2 ) ( Q ) A ( 2 ) * ( Q ) = 16 q 0 2 ( 2 π ) 2 [ δ 2 ( Q ) d Q d Q q q * W ( Q ) W ( Q ) + L 2 π d Q | q | 2 W ( Q Q ) W ( Q ) + L 2 π d Q q ( Q Q ) q ( Q ) * W ( Q Q ) W ( Q ) ] .
A ( 1 ) * ( Q ) A ( 3 ) ( Q ) = 12 q 0 2 L ( 2 π ) 2 { σ 2 W ( Q ) ( q 0 2 q 2 ) W ( Q ) π q 2 W ( Q Q ) d Q + W ( Q ) π [ q 0 q W ( Q ) d Q + q ( Q ) q ( Q + Q ) W ( Q ) d Q + q q W ( Q Q ) d Q ] } ,
I ( θ ) d = 2 k 0 3 cos 2 θ cos θ 0 π × ( W ( Q ) + 1 2 π d Q | q | 2 W ( Q Q ) W ( Q ) + 1 2 π q ( Q Q ) q * ( Q ) W ( Q Q ) W ( Q ) d Q + W ( Q ) × { σ 2 ( q 0 2 + q 2 ) + 1 π q 2 W ( Q Q ) d Q 1 π Re [ q 0 q W ( Q ) d Q + q q W ( Q Q ) d q + q ( Q ) q ( Q + Q ) W ( Q ) d Q ] } ) ,
α D ( y ) α D ( y + τ ) = c ( τ ) = σ 2 exp ( τ 2 / T 2 )
W ( Q ) = σ 2 π T exp ( T 2 Q 2 / 4 ) .
I ( θ ) d = I 2 ( θ ) d + I 4 ( θ ) d ,
I 2 ( θ ) d = σ 2 16 π 2 cos 2 θ cos θ 0 λ 3 π T × exp [ T 2 π 2 λ 2 ( sin θ sin θ 0 ) 2 ]
I 4 ( θ ) d = 16 π 2 cos 2 θ cos θ 0 λ 3 [ 4 π 3 σ 4 T 2 λ 3 exp [ T 2 π 2 2 λ 2 ( sin θ sin θ 0 ) 2 ] × ( π λ 3 4 2 π 3 T 3 + ( X 0 2 1 ) λ T π 2 + 2 1 X 0 1 + X 0 [ 1 ( x + X 0 ) 2 ] × exp [ 2 ( T π x ) 2 λ 2 ] d x + + { [ 1 + ( x + X 0 ) 2 ] 1 / 2 } * × [ 1 ( x X 0 ) 2 ] 1 / 2 exp [ 2 ( T π x ) 2 λ 2 ] d x ) + σ 4 T 4 π 2 λ 2 exp { [ T π ( sin θ sin θ 0 ) / λ ] 2 } × ( π ( cos 2 θ + cos 2 θ ) π λ 2 ( T π ) 2 2 π T cos θ 0 / λ × 1 1 ( 1 x 2 ) 1 / 2 exp { [ T π ( sin θ 0 x ) / λ ] 2 } d x 2 π T cos θ / λ × 1 1 ( 1 x 2 ) 1 / 2 exp { [ T π ( sin θ x ) / λ ] 2 } d x 2 π T λ Re + ( 1 x 2 ) 1 / 2 [ 1 ( sin θ sin θ 0 x ) 2 ] 1 / 2 × exp { [ T π ( sin θ x ) / λ ] 2 } d x ) ] .
H 0 ( r ) = exp [ i ( K y q 0 z ) ] ,
H ( r ) = H 0 ( r ) + π i c L J ( y ) d y × iq + i α ( d D / d y ) ( K + Q ) q exp [ i ( K + Q ) ( y y ) ] × exp { i q [ z α D ( y ) ] } d Q , z > α D max .
H 0 ( r ) = π i c L J ( y ) d y d Q i q + i α ( d D / d y ) ( K + Q ) q × exp [ i ( K + Q ) ( y y ) ] exp { i q [ z α D ( y ) ] } z < α D min ,
H ( r ) = H 0 ( r ) + d Q exp [ i ( Q + K ) y ] exp ( iqz ) A ( Q ) , z > α D max ,
A ( Q , α ) = 1 q L d y [ q + α ( d D / d y ) ( K + Q ) ] × exp ( i Q y ) exp [ i q α D ( y ) ] G ( y , α ) .
G ( y , α ) = π c J ( y , α ) exp ( i K y ) .
δ ( Q ) = 1 q L d y [ q + α ( d D / d y ) ( K + Q ) ] × exp ( i Q y ) exp [ i q α D ( y ) ] G ( y , α ) .
A ( 0 ) ( Q ) = δ ( Q ) ,
A ( 1 ) ( Q ) = i π [ q + ( K + Q ) q q ] D ( Q ) ,
A ( 2 ) ( Q ) = 2 q π { D ( Q ) * [ q + ( Q + K ) Q q D ( Q ) ] } 2 ( K + Q ) π q { [ Q D ( Q ) ] * [ q + ( K + Q ) Q q D ( Q ) ] } ,
A ( 3 ) ( Q ) = 6 iqD ( Q ) * G ( 2 ) ( Q ) + i q 3 π [ D 3 ( y ) ] ( Q ) + 6 i ( K + Q ) q { [ Q D ( Q ) ] * G ( 2 ) ( Q ) } + 3 i π ( K + Q ) q { [ Q D ( Q ) ] * [ D 2 ( y ) ] ( Q ) } .
G ( 2 ) ( Q ) = q π ( D ( Q ) * { D ( Q ) * [ q + ( K + Q ) Q q D ( Q ) ] } ) q 2 2 π [ D 2 ( y ) ] ( Q ) + i ( K + Q ) π { [ Q D ( Q ) ] * D ( Q ) } + K + Q π q { [ Q D ( Q ) ] * [ q + ( K + Q ) Q q D ( Q ) ] } .
A ( 2 ) ( Q ) = 2 q π { D ( Q ) * [ ( K + Q ) Q q D ( Q ) ] } 2 K + Q π Q { [ Q D ( Q ) ] * [ ( K + Q ) Q q D ( Q ) ] } 2 q π { D ( Q ) * [ Q D ( Q ) ] } 2 K + Q π q { [ Q D ( Q ) ] * [ q D ( Q ) ] } .
F = q π 2 d Q D ( Q Q ) D ( Q ) ( K + Q ) Q q ,
F F * = q 2 π 4 d Q d Q ( K + Q ) Q ( K + Q ) Q ( q q * ) × D ( Q ) D ( Q Q ) D * ( Q ) D * ( Q Q ) .
F F * = F F * + 4 q 2 π 2 ( K + Q ) Q ( K + Q Q ) ( Q Q ) q ( Q ) q ( Q Q ) W ( Q Q ) W ( Q ) × d Q + 4 q 2 π 2 ( K + Q ) 2 Q 2 | q ( Q ) | 2 W ( Q ) W ( Q Q ) d Q .
I p ( θ ) d = 16 π 2 ( 1 sin θ sin θ 0 ) 2 cos θ 0 λ 3 W ( Q ) .
E [ y , α D ( y ) ] = 0 .
E 0 [ y , α D ( y ) ] = d Q exp [ i Q + K ) y ] × exp [ i q α D ( y ) ] A ( Q ) ,
E 0 [ y , α D ( y ) ] = exp { i [ K y q 0 α D ( y ) ] } .
exp [ i q 0 α D ( y ) ] = d Q exp ( iQy ) exp [ i q α D ( y ) ] A ( Q ) .
A ( 0 ) ( Q ) = δ ( Q ) ,
A ( 1 ) ( Q ) = i π q 0 D ( Q ) ,
A ( 2 ) ( Q ) = 2 q 0 π D ( Q ) * [ q D ( Q ) ] ,
A ( 3 ) ( Q ) = 3 i q 0 π [ D 2 ( y ) ] ( Q ) * [ q 2 D ( Q ) ] i q 0 3 π [ D 3 ( y ) ] ( Q ) 6 i q 0 π D ( Q ) * ( q { D ( Q ) * [ q D ( Q ) ] } ) .
Dif ( Q ) = A ET ( 3 ) ( Q ) A R ( 3 ) ( Q ) = i q 0 2 π 2 d Q ( 4 Q 2 + 8 K Q 12 K Q 6 Q Q ) × [ D 2 ( y ) ] ( Q ) D ( Q Q ) .
[ H ( r ) n ] z = α D ( y ) = 0 ,
n = [ 0 , α d D ( y ) / d y , 1 ] .
[ q 0 + K α d D ( y ) d y ] exp [ i q 0 α D ( y ) ] = d Q exp ( iQy ) × exp [ i q α D ( y ) ] [ q ( Q + K ) α d D ( y ) d y ] A ( Q ) ,
A ( 0 ) ( Q ) = δ ( Q ) ,
A ( 1 ) ( Q ) = i π ( q 0 2 Q K q ) D ( Q ) ,
A ( 2 ) ( Q ) = 21 q { D ( Q ) * [ q 2 A 1 ( Q ) ] } + 2 i q { [ Q D ( Q ) ] * [ ( Q + K ) A 1 ( Q ) ] } .

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