Abstract

Measurements of the polarization dependence of the coherent and diffuse scatter from a set of seven conducting surfaces with strictly one-dimensional roughness are presented. The surfaces have been fabricated in gold with photoresist techniques and have been accurately characterized with stylus profilometry. The standard deviation of surface height σ varies between 0.25 and 1.73 μm throughout the series of surfaces, but all have height statistics that are close to Gaussian and a correlation length that is nearly fixed at 3.3 μm. The polarization dependence of the scattered intensity is fully specified by the four unique elements of the Stokes matrix, which are determined from six intensities measured with different polarization conditions. In studies of the coherent scatter for wavelength 3.392 μm, large differences are found between the p- and s-polarized intensities, and it is shown that the relative phase of the p and s coherent amplitudes is strongly dependent on σ. In the case of diffuse scatter, the wide range of behavior exhibited by the scattered intensities and matrix elements is demonstrated for wavelengths 1.152 and 3.392 μm. The rise of backscattering enhancement and associated polarization effects are seen as σ increases, and it is shown that surfaces with surprisingly modest slopes may produce backscattering enhancement. At high angles of incidence, large differences between the p- and s-polarized diffuse intensities are observed.

© 1994 Optical Society of America

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References

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  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]
  2. 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]
  3. J. M. Soto-Crespo, M. Nieto-Vesperinas, A. T. Friberg, “Scattering from slightly rough random surfaces: a detailed study on the validity of the small perturbation method,” J. Opt. Soc. Am. A 7, 1185–1201 (1990).
    [CrossRef]
  4. A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  5. J. Renau, J. A. Collinson, “Measurements of electromagnetic backscattering from known, rough surfaces,” Bell Syst. Tech. J. 44, 2203–2226 (1965).
  6. D. H. Hensler, “Light scattering from fused polycrystalline aluminum oxide surfaces,” Appl. Opt. 11, 2522–2528 (1972).
    [CrossRef] [PubMed]
  7. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 3, pp. 17–28.
  8. 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]
  9. 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]
  10. 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]
  11. J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep ref lection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
    [CrossRef]
  12. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
    [CrossRef]
  13. Y. Q. Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
    [CrossRef]
  14. 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]
  15. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989), Chap. 3, pp. 13–37.
  16. W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
    [CrossRef]
  17. H. E. Bennett, J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,”J. Opt. Soc. Am. 51, 123–129 (1961).
    [CrossRef]
  18. K. E. Torrance, E. M. Sparrow, R. C. Birkebak, “Polarization, directional distribution, and off-specular peak phenomena in light reflected from roughened surfaces,”J. Opt. Soc. Am. 56, 916–925 (1966).
    [CrossRef]
  19. L. H. Tanner, M. Fahoum, “A study of the surface parameters of ground and lapped metal surfaces, using specular and diffuse reflection of laser light,” Wear 36, 299–316 (1976).
    [CrossRef]
  20. J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
    [CrossRef]
  21. M. E. Knotts, T. R. Michel, K. A. O’Donnell, “Comparisons of theory and experiment in light scattering from a randomly rough surface,” J. Opt. Soc. Am. A 10, 928–941 (1993).
    [CrossRef]
  22. 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]
  23. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 46–54.
  24. M. E. Knotts, T. R. Michel, K. A. O’Donnell, “Angular correlation functions of polarized intensities scattered from a one-dimensionally rough surface,” J. Opt. Soc. Am. A 9, 1822–1831 (1992).
    [CrossRef]
  25. See, for example, M. Zelen, N. C. Severo, “Probability functions,” in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds. (National Bureau of Standards, Washington, D.C., 1964), pp. 927–930.
  26. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 10, pp. 554–555.
  27. G. S. Brown, “A stochastic Fourier transform approach to scattering from perfectly conducting randomly rough surfaces,”IEEE Trans. Antennas Propag. AP-30, 1135–1144 (1982).
    [CrossRef]
  28. E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985), p. 295.
  29. S. Solimeno, B. Crosignani, P. Diporto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986), Chap. 21, pp. 208–210.
  30. Ref. 26, Chap. 13, pp. 615–624.
  31. Ref. 7, Chap. 5, p. 81.
  32. N. Garcia, E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough sufaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
    [CrossRef]
  33. See Ref. 12, Sec. 6.
  34. G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
    [CrossRef]
  35. Backscattering enhancement has been theoretically predicted for surfaces with much weaker slopes, but the physical mechanisms appear to be different from cases discussed here. See Ref. 4.

1993 (1)

1992 (2)

1991 (2)

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]

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]

1990 (4)

1989 (1)

1987 (1)

1985 (4)

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

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]

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

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
[CrossRef]

1984 (2)

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

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

1982 (1)

G. S. Brown, “A stochastic Fourier transform approach to scattering from perfectly conducting randomly rough surfaces,”IEEE Trans. Antennas Propag. AP-30, 1135–1144 (1982).
[CrossRef]

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]

1976 (1)

L. H. Tanner, M. Fahoum, “A study of the surface parameters of ground and lapped metal surfaces, using specular and diffuse reflection of laser light,” Wear 36, 299–316 (1976).
[CrossRef]

1972 (1)

1966 (1)

1965 (1)

J. Renau, J. A. Collinson, “Measurements of electromagnetic backscattering from known, rough surfaces,” Bell Syst. Tech. J. 44, 2203–2226 (1965).

1961 (1)

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]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 3, pp. 17–28.

Bennett, H. E.

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]

Birkebak, R. C.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 10, pp. 554–555.

Brown, G.

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
[CrossRef]

Brown, G. S.

G. S. Brown, “A stochastic Fourier transform approach to scattering from perfectly conducting randomly rough surfaces,”IEEE Trans. Antennas Propag. AP-30, 1135–1144 (1982).
[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.

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
[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]

Collinson, J. A.

J. Renau, J. A. Collinson, “Measurements of electromagnetic backscattering from known, rough surfaces,” Bell Syst. Tech. J. 44, 2203–2226 (1965).

Crosignani, B.

S. Solimeno, B. Crosignani, P. Diporto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986), Chap. 21, pp. 208–210.

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]

Diporto, P.

S. Solimeno, B. Crosignani, P. Diporto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986), Chap. 21, pp. 208–210.

Elson, J. M.

Fahoum, M.

L. H. Tanner, M. Fahoum, “A study of the surface parameters of ground and lapped metal surfaces, using specular and diffuse reflection of laser light,” Wear 36, 299–316 (1976).
[CrossRef]

Friberg, A. T.

Garcia, N.

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

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 46–54.

Guenther, K. H.

Haller, M.

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
[CrossRef]

Hensler, D. H.

Hurt, H. H.

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]

Maradudin, A. A.

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

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
[CrossRef]

Marvin, A.

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
[CrossRef]

Mattsson, L.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989), Chap. 3, pp. 13–37.

McGurn, A. R.

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

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Méndez, E. R.

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

Michel, T.

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

Michel, T. R.

Nieto-Vesperinas, M.

O’Donnell, K. A.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985), p. 295.

Porteus, J. O.

Rahn, J. P.

Rasigni, M.

Renau, J.

J. Renau, J. A. Collinson, “Measurements of electromagnetic backscattering from known, rough surfaces,” Bell Syst. Tech. J. 44, 2203–2226 (1965).

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]

Severo, N. C.

See, for example, M. Zelen, N. C. Severo, “Probability functions,” in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds. (National Bureau of Standards, Washington, D.C., 1964), pp. 927–930.

Solimeno, S.

S. Solimeno, B. Crosignani, P. Diporto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986), Chap. 21, pp. 208–210.

Soto-Crespo, J. M.

Sparrow, E. M.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 3, pp. 17–28.

Stoll, E.

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

Tanner, L. H.

L. H. Tanner, M. Fahoum, “A study of the surface parameters of ground and lapped metal surfaces, using specular and diffuse reflection of laser light,” Wear 36, 299–316 (1976).
[CrossRef]

Torrance, K. E.

Varnier, F.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 10, pp. 554–555.

Zelen, M.

See, for example, M. Zelen, N. C. Severo, “Probability functions,” in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds. (National Bureau of Standards, Washington, D.C., 1964), pp. 927–930.

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. (1)

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

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

J. Renau, J. A. Collinson, “Measurements of electromagnetic backscattering from known, rough surfaces,” Bell Syst. Tech. J. 44, 2203–2226 (1965).

IEEE Trans. Antennas Propag. (1)

G. S. Brown, “A stochastic Fourier transform approach to scattering from perfectly conducting randomly rough surfaces,”IEEE Trans. Antennas Propag. AP-30, 1135–1144 (1982).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (8)

J. M. Soto-Crespo, M. Nieto-Vesperinas, A. T. Friberg, “Scattering from slightly rough random surfaces: a detailed study on the validity of the small perturbation method,” J. Opt. Soc. Am. A 7, 1185–1201 (1990).
[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]

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]

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]

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep ref lection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
[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. E. Knotts, T. R. Michel, K. A. O’Donnell, “Comparisons of theory and experiment in light scattering from a randomly rough surface,” J. Opt. Soc. Am. A 10, 928–941 (1993).
[CrossRef]

M. E. Knotts, T. R. Michel, K. A. O’Donnell, “Angular correlation functions of polarized intensities scattered from a one-dimensionally rough surface,” J. Opt. Soc. Am. A 9, 1822–1831 (1992).
[CrossRef]

Phys. Rev. B (5)

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
[CrossRef]

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[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]

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

Phys. Rev. Lett. (1)

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

Waves Random Media (1)

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]

Wear (1)

L. H. Tanner, M. Fahoum, “A study of the surface parameters of ground and lapped metal surfaces, using specular and diffuse reflection of laser light,” Wear 36, 299–316 (1976).
[CrossRef]

Other (11)

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 3, pp. 17–28.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 46–54.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989), Chap. 3, pp. 13–37.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985), p. 295.

S. Solimeno, B. Crosignani, P. Diporto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986), Chap. 21, pp. 208–210.

Ref. 26, Chap. 13, pp. 615–624.

Ref. 7, Chap. 5, p. 81.

See Ref. 12, Sec. 6.

See, for example, M. Zelen, N. C. Severo, “Probability functions,” in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds. (National Bureau of Standards, Washington, D.C., 1964), pp. 927–930.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 10, pp. 554–555.

Backscattering enhancement has been theoretically predicted for surfaces with much weaker slopes, but the physical mechanisms appear to be different from cases discussed here. See Ref. 4.

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Figures (23)

Fig. 1
Fig. 1

Histograms of surface height compared with Gaussian histograms with standard deviations from Table 1 (smooth curves). The histogram bin widths are A, 0.0116 μm; B, 0.0229 μm; C, 0.0400 μm; D, 0.0414 μm; E, 0.0570 μm; F, 0.0684 μm; and G, 0.0741 μm; tick marks on the y axis are spaced by 0.005.

Fig. 2
Fig. 2

Normalized correlation functions of surface height 〈h(x) h(x + Δx)〉/σ2 (solid curves) calculated from the measured rough surface profiles h(x) of surfaces A–G. Dashed curves show a comparison with Gaussian functions with e−1 width a from Table 1; tick marks on the y axis are spaced by 0.20.

Fig. 3
Fig. 3

Scattering by a surface with one-dimensional roughness. The directions of p and s polarization are shown for the incident and scattered waves as parallel and perpendicular to the plane of incidence, respectively. The incident and scattering angles θi and θs are positive as shown.

Fig. 4
Fig. 4

Coherent parts of the p- and s-polarized scattered intensities for rough surfaces A–G. The illumination wavelength is λ = 3.392 μm, and the curves labeled P are the intensities calculated from the Fresnel reflection coefficients for a planar gold interface. Perfect reflection of incident power is denoted by 0.5.

Fig. 5
Fig. 5

Coherent parts of the matrix elements s33 and s34 for rough surfaces A–G. The illumination wavelength is λ = 3.392 μm, and the curves labeled P are the matrix elements calculated from the Fresnel reflection coefficients for a planar gold interface.

Fig. 6
Fig. 6

Angular dependence of the matrix element s11 for surfaces A–G for θi = 0° and wavelength 1.152 μm. Tick marks on the y axis are spaced by 0.10, and backscattering enhancement is seen for surfaces E–G. For surface A, s11 has been divided by 3.0.

Fig. 7
Fig. 7

Same as in Fig. 6 but for wavelength 3.392 μm. Coherent components are apparent for surfaces A and B; backscattering enhancement is seen for surfaces E–G.

Fig. 8
Fig. 8

Angular dependence of the matrix element s12 for surfaces A–G for θi = 0° and wavelengths 1.152 μm (top) and 3.392 μm (bottom). Tick marks on they axis are spaced by 0.05.

Fig. 9
Fig. 9

For surfaces A–G, the +45°-polarized intensity I+ for θi = 0° and λ = 1.152 μm. For an incident +45° state of unit power, results correspond to total scattered powers of A, 0.00027; B, 0.0022; C, 0.0112; D, 0.0392; E, 0.0997; F, 0.178; and G, 0.300. Backscattering enhancement occurs for all surfaces with the exception of surface A (see Fig. 13).

Fig. 10
Fig. 10

For surfaces A–G, the +45°-polarized intensity I+ for θi = 0° and λ = 3.392 μm. For an incident +45° state of unit power, results correspond to total scattered powers of A, 0.00012; B, 0.015; C, 0.034; D, 0.070; E, 0.144; F, 0.237; and G, 0.350. The maximum at θs = 0° for surface A is a coherent reflection, and the maximum for B is due to a coherent reflection as well as backscattering enhancement.

Fig. 11
Fig. 11

Angular dependence of the matrix element s34 for surfaces A–G for θi = 0° and wavelengths 1.152 μm (top) and 3.392 μm (bottom). Tick marks on the y axis are spaced by 0.05.

Fig. 12
Fig. 12

Matrix element s11 as in Fig. 6 but for θi = 10°. Backscattering enhancement is seen for surfaces E–G.

Fig. 13
Fig. 13

For surfaces A–G, the +45°-polarized intensity I+ for θi = 10° and λ = 1.152 μm. For an incident +45° state of unit power, results correspond to total scattered powers of A, 0.00037; B, 0.0038; C, 0.0153; D, 0.0400; E, 0.0963; F, 0.166; and G, 0.279.

Fig. 14
Fig. 14

Angular dependence of the matrix element s34 for surfaces A–G for θi = 10° and wavelength 1.152 μm. Tick marks on the y axis are spaced by 0.05.

Fig. 15
Fig. 15

For a normally incident wave with +45° polarization, the polarized (top) and randomly polarized (bottom) scattered intensities for surfaces A–G. The wavelength is 1.152 μm.

Fig. 16
Fig. 16

Angular dependence of the matrix element s11 for surfaces A–G for θi = 30° and wavelength 1.152 μm. Tick marks on the y axis are spaced by 0.10. For surface A, s11 has been divided by 3.0.

Fig. 17
Fig. 17

Same as in Fig. 16 but for wavelength 3.392 μm. Coherent components are present for surfaces A, B, and possibly C.

Fig. 18
Fig. 18

Angular dependence of the matrix element s12 for surfaces A–G for θi = 30° and wavelengths 1.152 μm (top) and 3.392 μm (bottom). Tick marks on the y axis are spaced by 0.05.

Fig. 19
Fig. 19

For surfaces A–G, the +45°-polarized intensity I+ for θi = 30° and λ = 1.152 μm (top) and λ = 3.392 μm (bottom).

Fig. 20
Fig. 20

Angular dependence of the matrix element for s34 surfaces A–G for θi = 30° and wavelengths 1.152 μm (top) and 3.392 μm (bottom). Tick marks on they axis are spaced by 0.05. Coherent components are present for surfaces A and B in the lower figure.

Fig. 21
Fig. 21

Angular dependence of the intensities Ip (diamonds) and Is (circles) for surfaces A–G for θi = 70° and wavelength 1.152 μm. Tick marks on the y axis are spaced by 0.05. Coherent components are seen for surfaces A–C; s11 has been divided by 2.0 for surface A.

Fig. 22
Fig. 22

Same as in Fig. 21 but for wavelength 3.392 μm. Tick marks on the y axis are spaced by 0.05. Coherent components are clearly seen in all cases except for Ip for surface G.

Fig. 23
Fig. 23

For θi = 86.5°, the diffuse intensities Ip (upper curves) and Is (lower curves) for wavelength 3.392 μm for surfaces A–G. Tick marks on the y-axis are spaced by 0.05. The diffusely scattered powers contained in Ip and Is range from 0.107 and 0.0066 for surface A to 0.276 and 0.147 for surface G.

Tables (1)

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Table 1 Statistical Properties of Rough Surfaces

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 = [ 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 ,
I ˜ p = r p 2 ,             I ˜ s = r s 2 , s ˜ 33 = 2 r p r s cos ( ϕ p - ϕ s ) , s ˜ 34 = - 2 r p r s sin ( ϕ p - ϕ s ) ,
s ˜ 11 = ( s ˜ 12 2 + s ˜ 33 2 + s ˜ 34 2 ) 1 / 2 .
I ˜ p + I ˜ s = [ ( I ˜ p - I ˜ s ) 2 + s ˜ 33 2 + s ˜ 34 2 ] 1 / 2 ,
I ˜ α = 1 2 exp [ - 4 k 2 σ 2 cos 2 ( θ i ) ] ,
I Pol = ( s 12 2 + s 33 2 + s 34 2 ) 1 / 2 ,
I RPol = s 11 - ( s 12 2 + s 33 2 + s 34 2 ) 1 / 2 ,

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