Abstract

For a conducting surface with one-dimensional roughness, we compare experimental and theoretical results for the four unique elements of the Stokes-scattering matrix that provide a complete description of the diffusely scattered light. The rough surface has been fabricated with new techniques and is strictly one dimensional, and scattered intensities at infrared wavelengths show clear backscattering enhancement that arises from multiple scattering within surface corrugations. To obtain theoretical results for the Stokes matrix elements, we numerically apply an impedance boundary-condition method, appropriate for the roughness and the high conductivity of the experimental surface, to a statistical ensemble of rough surfaces. The experimental surface has been found to have nearly Gaussian first-order height statistics, and experimental measurements of the matrix elements are compared with theoretical results for a surface consistent with a Gaussian process. These comparisons suggest that there is more multiple scattering in the experimental data than is accounted for by the theoretical calculations. We attribute this observation to the properties of the second derivative of the experimental surface, which are found to be inconsistent with those of a Gaussian process. In further calculations that take account of the unusual properties of the experimental surface, excellent agreement of theoretical and experimental results is obtained.

© 1993 Optical Society of America

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References

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  1. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  2. F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).
  3. G. S. Brown, “A comparison of approximate theories for scattering from random rough surfaces,” Wave Motion 7, 195–205 (1985).
    [CrossRef]
  4. J. A. DeSanto, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1986), Vol. 23.
    [CrossRef]
  5. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).
  6. M. Nieto-Vesperinas, J. C. Dainty, eds., Scattering in Volumes and Surfaces (North-Holland, Amsterdam, 1990).
  7. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).
  8. W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
    [CrossRef]
  9. J. M. Elson, J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,”J. Opt. Soc. Am. 69, 31–48 (1979).
    [CrossRef]
  10. Y. Wang, W. L. Wolfe, “Scattering from microrough surfaces: comparison of theory and experiment,”J. Opt. Soc. Am. 73, 1596–1602 (1983).
    [CrossRef]
  11. Y. Wang, W. L. Wolfe, “Further comparisons between surface scattering theory and measurements,” J. Opt. Soc. Am. A 1, 783–784 (1984).
    [CrossRef]
  12. J. M. Bennett, H. H. Hurt, J. P. Rahn, J. M. Elson, K. H. Guenther, M. Rasigni, F. Varnier, “Relation between optical scattering, microstructure and topography of thin silver films. 1: Optical scattering and topography,” Appl. Opt. 24, 2701–2711 (1985).
    [CrossRef] [PubMed]
  13. O. Hunderi, D. Beaglehole, “Study of the interaction of light with rough metal surfaces. II. Theory,” Phys. Rev. B 2, 321–329 (1970).
    [CrossRef]
  14. S. O. Sari, D. K. Cohen, K. D. Scherkoske, “Study of surface plasma-wave reflectance and roughness-induced scattering in silver foils,” Phys. Rev. B 21, 2162–2174 (1980).
    [CrossRef]
  15. 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]
  16. K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  17. E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
    [CrossRef]
  18. 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]
  19. 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]
  20. 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]
  21. T. Michel, A. A. Maradudin, E. R. Méndez, “Enhanced backscattering of light from a non-Gaussian random metal surface,” J. Opt. Soc. Am. B 6, 2438–2446 (1989).
    [CrossRef]
  22. 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).
    [CrossRef]
  23. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (NY.) 203, 255–307 (1990).
    [CrossRef]
  24. M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
    [CrossRef]
  25. E. I. Thorsos, D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 3, S165–S190 (1991).
    [CrossRef]
  26. J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
    [CrossRef]
  27. A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Media 3, S91–S107 (1991).
    [CrossRef]
  28. A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,”J. Acoust. Soc. Am. 88, 1877–1883 (1990).
    [CrossRef]
  29. Y. Q. Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
    [CrossRef]
  30. M. J. Kim, J. C. Dainty, A. T. Friberg, A. J. Sant, “Experimental study of enhanced backscattering from one- and two-dimensional random rough surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
    [CrossRef]
  31. J. C. Dainty, N. C. Bruce, A. J. Sant, “Measurements of light scattering by a characterized random rough surface,” Waves Random Media 3, S29–S39 (1991).
    [CrossRef]
  32. K. A. O’Donnell, M. E. Knotts, “Polarization-dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1126–1131 (1991).
    [CrossRef]
  33. T. R. Michel, M. E. Knotts, K. A. O’Donnell, “Stokes matrix of a one-dimensional perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 585–596 (1992).
    [CrossRef]
  34. Compare Figs. 6 and 8 of Ref. 32 with Figs. 2 and 3 of Ref. 33; also compare Figs. 4 and 5 with Figs. 9 and 10 of Ref. 33.
  35. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  36. R. Garcia-Molina, A. A. Maradudin, T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
    [CrossRef]
  37. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
  38. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).
  39. More precisely, for a zero-mean Gaussian process h(x), probability densities of arbitrary order are all invariant under the transformation h(x) → −h(x).
  40. R. J. Adler, The Geometry of Random Fields (Wiley, New York, 1981).

1992 (1)

1991 (5)

J. C. Dainty, N. C. Bruce, A. J. Sant, “Measurements of light scattering by a characterized random rough surface,” Waves Random Media 3, S29–S39 (1991).
[CrossRef]

K. A. O’Donnell, M. E. Knotts, “Polarization-dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1126–1131 (1991).
[CrossRef]

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

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
[CrossRef]

A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Media 3, S91–S107 (1991).
[CrossRef]

1990 (6)

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,”J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

Y. Q. Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
[CrossRef]

M. J. Kim, J. C. Dainty, A. T. Friberg, A. J. Sant, “Experimental study of enhanced backscattering from one- and two-dimensional random rough surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
[CrossRef]

R. Garcia-Molina, A. A. Maradudin, T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
[CrossRef]

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

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

1989 (3)

1988 (1)

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]

1987 (3)

1985 (4)

1984 (1)

1983 (1)

1980 (1)

S. O. Sari, D. K. Cohen, K. D. Scherkoske, “Study of surface plasma-wave reflectance and roughness-induced scattering in silver foils,” Phys. Rev. B 21, 2162–2174 (1980).
[CrossRef]

1979 (1)

1970 (1)

O. Hunderi, D. Beaglehole, “Study of the interaction of light with rough metal surfaces. II. Theory,” Phys. Rev. B 2, 321–329 (1970).
[CrossRef]

Adler, R. J.

R. J. Adler, The Geometry of Random Fields (Wiley, New York, 1981).

Bailey, W. M.

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Bass, F. G.

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

Beaglehole, D.

O. Hunderi, D. Beaglehole, “Study of the interaction of light with rough metal surfaces. II. Theory,” Phys. Rev. B 2, 321–329 (1970).
[CrossRef]

Beckmann, P.

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

Bennett, J. M.

Bickel, W. S.

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Brown, G. S.

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

J. A. DeSanto, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1986), Vol. 23.
[CrossRef]

Bruce, N. C.

J. C. Dainty, N. C. Bruce, A. J. Sant, “Measurements of light scattering by a characterized random rough surface,” Waves Random Media 3, S29–S39 (1991).
[CrossRef]

Celli, V.

Chen, J. S.

A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Media 3, S91–S107 (1991).
[CrossRef]

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,”J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

Cohen, D. K.

S. O. Sari, D. K. Cohen, K. D. Scherkoske, “Study of surface plasma-wave reflectance and roughness-induced scattering in silver foils,” Phys. Rev. B 21, 2162–2174 (1980).
[CrossRef]

Dainty, J. C.

J. C. Dainty, N. C. Bruce, A. J. Sant, “Measurements of light scattering by a characterized random rough surface,” Waves Random Media 3, S29–S39 (1991).
[CrossRef]

M. J. Kim, J. C. Dainty, A. T. Friberg, A. J. Sant, “Experimental study of enhanced backscattering from one- and two-dimensional random rough surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
[CrossRef]

DeSanto, J. A.

J. A. DeSanto, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1986), Vol. 23.
[CrossRef]

Elson, J. M.

Friberg, A. T.

Fuks, I. M.

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

Garcia-Molina, R.

R. Garcia-Molina, A. A. Maradudin, T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
[CrossRef]

Guenther, K. H.

Hunderi, O.

O. Hunderi, D. Beaglehole, “Study of the interaction of light with rough metal surfaces. II. Theory,” Phys. Rev. B 2, 321–329 (1970).
[CrossRef]

Hurt, H. H.

Ishimaru, A.

A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Media 3, S91–S107 (1991).
[CrossRef]

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,”J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

Jackson, D. R.

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

Jin, Y. Q.

Y. Q. Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
[CrossRef]

Kim, M. J.

Knotts, M. E.

Lax, M.

Y. Q. Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
[CrossRef]

Leskova, T. A.

R. Garcia-Molina, A. A. Maradudin, T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
[CrossRef]

Maradudin, A. A.

Marvin, A. M.

Mattsson, L.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

Maystre, D.

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

McGurn, A. R.

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

Méndez, E. R.

Michel, T.

Michel, T. R.

Nieto-Vesperinas, M.

O’Donnell, K. A.

Ogilvy, J. A.

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

Palik, E. D.

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

Phu, P.

A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Media 3, S91–S107 (1991).
[CrossRef]

Rahn, J. P.

Rasigni, M.

Saillard, M.

Sanchez-Gil, J. A.

Sant, A. J.

J. C. Dainty, N. C. Bruce, A. J. Sant, “Measurements of light scattering by a characterized random rough surface,” Waves Random Media 3, S29–S39 (1991).
[CrossRef]

M. J. Kim, J. C. Dainty, A. T. Friberg, A. J. Sant, “Experimental study of enhanced backscattering from one- and two-dimensional random rough surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
[CrossRef]

Sari, S. O.

S. O. Sari, D. K. Cohen, K. D. Scherkoske, “Study of surface plasma-wave reflectance and roughness-induced scattering in silver foils,” Phys. Rev. B 21, 2162–2174 (1980).
[CrossRef]

Scherkoske, K. D.

S. O. Sari, D. K. Cohen, K. D. Scherkoske, “Study of surface plasma-wave reflectance and roughness-induced scattering in silver foils,” Phys. Rev. B 21, 2162–2174 (1980).
[CrossRef]

Soto-Crespo, J. M.

Spizzichino, A.

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

Thorsos, E. I.

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

Varnier, F.

Wang, Y.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Wolfe, W. L.

Yoshitomi, K.

A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Media 3, S91–S107 (1991).
[CrossRef]

Am. J. Phys. (1)

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Ann. Phys. (NY.) (1)

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

Appl. Opt. (1)

J. Acoust. Soc. Am. (2)

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

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,”J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (9)

Y. Wang, W. L. Wolfe, “Further comparisons between surface scattering theory and measurements,” J. Opt. Soc. Am. A 1, 783–784 (1984).
[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]

K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[CrossRef]

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
[CrossRef]

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

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).
[CrossRef]

K. A. O’Donnell, M. E. Knotts, “Polarization-dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1126–1131 (1991).
[CrossRef]

T. R. Michel, M. E. Knotts, K. A. O’Donnell, “Stokes matrix of a one-dimensional perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 585–596 (1992).
[CrossRef]

M. J. Kim, J. C. Dainty, A. T. Friberg, A. J. Sant, “Experimental study of enhanced backscattering from one- and two-dimensional random rough surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

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

Opt. Lett. (2)

Phys. Rep. (1)

R. Garcia-Molina, A. A. Maradudin, T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
[CrossRef]

Phys. Rev. B (3)

Y. Q. Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
[CrossRef]

O. Hunderi, D. Beaglehole, “Study of the interaction of light with rough metal surfaces. II. Theory,” Phys. Rev. B 2, 321–329 (1970).
[CrossRef]

S. O. Sari, D. K. Cohen, K. D. Scherkoske, “Study of surface plasma-wave reflectance and roughness-induced scattering in silver foils,” Phys. Rev. B 21, 2162–2174 (1980).
[CrossRef]

Wave Motion (1)

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

Waves Random Media (3)

A. Ishimaru, J. S. Chen, P. Phu, K. Yoshitomi, “Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement,” Waves Random Media 3, S91–S107 (1991).
[CrossRef]

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

J. C. Dainty, N. C. Bruce, A. J. Sant, “Measurements of light scattering by a characterized random rough surface,” Waves Random Media 3, S29–S39 (1991).
[CrossRef]

Other (12)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

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

More precisely, for a zero-mean Gaussian process h(x), probability densities of arbitrary order are all invariant under the transformation h(x) → −h(x).

R. J. Adler, The Geometry of Random Fields (Wiley, New York, 1981).

Compare Figs. 6 and 8 of Ref. 32 with Figs. 2 and 3 of Ref. 33; also compare Figs. 4 and 5 with Figs. 9 and 10 of Ref. 33.

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

J. A. DeSanto, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1986), Vol. 23.
[CrossRef]

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

M. Nieto-Vesperinas, J. C. Dainty, eds., Scattering in Volumes and Surfaces (North-Holland, Amsterdam, 1990).

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

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

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

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Figures (24)

Fig. 1
Fig. 1

Scans of the scattered intensity as a function of angle (degrees) in a direction orthogonal to the plane of incidence; θi ≅ 0.5°, incident wavelength λ = 1.152 μm. The dashed curve is the result for the approximately one-dimensional surface of Refs. 32 and 33 (multiplied by a factor of 10), and the solid curve is the result for the surface fabricated with the new techniques. The dashed curve is nearly symmetric, and the notch is due to obstruction by a mirror in the optical system.

Fig. 2
Fig. 2

Height correlation function Cx) 〈h(x)h(x + Δx)〉 (solid curve) calculated from the measured rough-surface profile h(x). The dashed curve is a comparison with a Gaussian correlation function with e−1 width a = 3.43 μm and height σ2, where σ = 1.73 μm. The statistical errors (the standard deviations determined from the higher-order moments of the data) are 1.5% of σ2 at Δx = 0, 1.3% at Δx = 2.0 μm, and 1.1% at Δx = 20.0 μm. The slightly negative foot of Cx) is not statistically significant.

Fig. 3
Fig. 3

Probability density of surface height, p(h), calculated from the profilometry of the experimental rough surface (solid curve). The dashed curve is a comparison with a Gaussian probability density with standard deviation σ = 1.73 μm.

Fig. 4
Fig. 4

Scattering of an incident wave by a surface with one-dimensional roughness. The direction of the incident beam is orthogonal to the groove direction.

Fig. 5
Fig. 5

Diagram of the scattering instrument used in the measurements of rough-surface scattering. The grooves of the rough surface are oriented in the vertical direction. The incident beam is brought over the detector and, after orientation of its polarization direction, is incident on the rough surface. The detector arm may be moved out of the plane of the figure for measurement of the intensity at all scattering angles. M denotes a mirror, P denotes a linear polarizer, and W denotes a half-wave plate.

Fig. 6
Fig. 6

For θi = 0° and λ = 1.152 μm, the unique matrix elements s11, s12, s33, and s34 and the +45°-polarized intensity I+. Experimental results (solid curves) are compared with numerical results (dashed curves; surface impedance method, surface roughness consistent with a Gaussian process).

Fig. 7
Fig. 7

Same as Fig. 6 but for θi = 10°.

Fig. 8
Fig. 8

Same as Fig. 6 but for θi = 30°.

Fig. 9
Fig. 9

For θi = 0° and λ = 3.392 μm, the unique matrix elements s11, s12, s33, and s34 and the +45°-polarized intensity I+. Experimental results (solid curves) are compared with numerical results (dashed curves; surface impedance method, surface roughness consistent with a Gaussian process).

Fig. 10
Fig. 10

Same as Fig. 9 but for θi = 10°.

Fig. 11
Fig. 11

Same as Fig. 9 but for θi = 30°.

Fig. 12
Fig. 12

For θi = 0° and λ = 1.152 μm, the unique matrix elements s11, s12, s33, and s34 and the +45°-polarized intensity I+. Numerical results that employ the surface impedance method (solid curves) are compared with results for the perfect conductor (dashed curves); the surface roughness is consistent with a Gaussian process.

Fig. 13
Fig. 13

Same as Fig. 12 but for θi = 30°.

Fig. 14
Fig. 14

For θi = 0° and λ = 3.392 μm, the unique matrix elements s11, s12, s33, and s34 and the +45°-polarized intensity I+. Numerical results that employ the surface impedance method (solid curves) are compared with results for the perfect conductor (dashed curves); the surface roughness is consistent with a Gaussian process.

Fig. 15
Fig. 15

Same as Fig. 14 but for θi 30°.

Fig. 16
Fig. 16

Segment of one of the profilometer scans of the surface that illustrates the sharp peaks and the broad valleys of the experimental surface. The shape of the profilometer stylus employed is also indicated (60° wedge angle, 0.4-μm blunt tip).

Fig. 17
Fig. 17

Probability density of surface slope, p(h′), calculated from the profilometry of the experimental rough surface (solid curve). The dashed curve is a comparison with a Gaussian probability density (with root-mean-square slope 2 σ / a, as would be expected for a height correlation function of Gaussian form).

Fig. 18
Fig. 18

Probability density of the second derivative of the surface, p(h″), calculated from the profilometry of the experimental rough surface (solid curve). The dashed curve is a comparison with a Gaussian probability density (with root-mean-square second derivative 2 3 σ / a 2, as would be expected for a Gaussian height correlation function).

Fig. 19
Fig. 19

For θi = 0° and λ = 1.152 μm, the unique matrix elements s11, s12, s33, and s34 and the +45°-polarized intensity I+. Experimental results (solid curves) are compared with numerical results (dashed curves; surface impedance method applied to profiled surface).

Fig. 20
Fig. 20

Same as Fig. 19 but for θi = 10°.

Fig. 21
Fig. 21

Same as Fig. 19 but for θi = 30°.

Fig. 22
Fig. 22

For θi = 0° and λ = 3.392 μm, the unique matrix elements s11, s12, s33, and s34 and the +45°-polarized intensity I+. Experimental results (solid curves) are compared with numerical results (dashed curves; surface impedance method applied to profiled surface).

Fig. 23
Fig. 23

Same as Fig. 22 but for θi = 10°.

Fig. 24
Fig. 24

Same as Fig. 22 but for θi = 30°.

Tables (1)

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Table 1 Total Scattered Powers Contained in p, s, and +45° Polarization States

Equations (9)

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S = [ s 11 s 12 0 0 s 12 s 11 0 0 0 0 s 33 s 34 0 0 - s 34 s 33 ] ,
V + = [ 1 0 1 0 ] ,
V scat = [ s 11 s 12 s 33 - s 34 ] ,
s 11 = I p + I s = I + + I - = I R + I L ,             s 12 = I p - I s , s 33 = I + - I - ,             - s 34 = I R - I L ,
ψ α ( x ) = 2 ψ α ( x ) inc + P 2 π S d s [ n G 0 ( x x ) ] ψ α ( x ) - 1 2 π S d s G 0 ( x x ) n ψ α ( x ) ,
0 = ψ α ( x ) + P 2 π S d s [ n G ( x x ) ] ψ α ( x ) - β 2 π S d s G ( x x ) n ψ α ( x ) ,
ψ α ( x ) = 2 ψ α ( x ) inc + P 2 π S d s [ n G 0 ( x x ) ] ψ α ( x ) - 1 2 π S d s G 0 ( x x ) Z α ( x ) ψ α ( x ) ,
n ψ α ( x ) = Z α ( x ) ψ α ( x ) ,
Z α ( x ) = 1 d β ( 1 + d 2 ζ ( x 1 ) { 1 + [ ζ ( x 1 ) ] 2 } 3 / 2 - d 2 8 [ ζ ( x 1 ) ] 2 { 1 + [ ζ ( x 1 ) ] 2 } 3 ) ,

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