Abstract

An experimental investigation of light scattering from random rough surfaces is described. The surfaces, whose height fluctuations approximately follow Gaussian statistics, are fabricated in photoresist with a metal overcoating. When the lateral correlation length is larger than a wavelength and the surface slopes are mild, measurements of diffuse scattering are found to agree with the Beckmann theory, as long as the angle of incidence is not too large. For other surfaces that have stronger slopes, depolarization and enhanced backscattering may be observed in the diffuse scattering. Though we are unaware of theoretical calculations that compare with the measurements, the effects of multiple scattering are shown to be consistent with the major features of the observations.

© 1987 Optical Society of America

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  1. G. S. Brown, “A comparison of approximate theories for scattering from random rough surfaces,” Wave Motion 7, 195–205 (1985).
    [CrossRef]
  2. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  3. M. I. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,”IEEE Trans. Antennas Propag. AP-17, 577–585 (1969).
    [CrossRef]
  4. A. K. Fung, H. J. Eom, “Multiple scattering and depolarization by a randomly rough Kirchhoff surface,”IEEE Trans. Antennas Propag. AP-29, 463–471 (1981).
    [CrossRef]
  5. Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.
  6. U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),”J. Opt. Soc. Am. 31, 213–222 (1941).
    [CrossRef]
  7. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [CrossRef]
  8. 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]
  9. D. N. Pattanayak, E. Wolf, “General form and a new interpretation of the Ewald–Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
    [CrossRef]
  10. G. S. Agarwal, “Scattering from rough surfaces,” Opt. Commun. 14, 161–166 (1975).
    [CrossRef]
  11. G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
    [CrossRef]
  12. P. C. Waterman, “Scattering from periodic surfaces,”J. Acoust. Soc. Am. 57, 791–802 (1975).
    [CrossRef]
  13. M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
    [CrossRef]
  14. J. Shen, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
    [CrossRef]
  15. M. Nieto-Vesperinas, “Depolarization of electromagnetic waves scattered from slightly rough random surfaces: a study by means of the extinction theorem,”J. Opt. Soc. Am. 72, 539–547 (1982).
    [CrossRef]
  16. D. Winebrenner, A. Ishimaru, “Investigation of the surface field phase-perturbation technique for scattering from rough surfaces,” Radio Sci. 20, 161–170 (1985).
    [CrossRef]
  17. D. Winebrenner, A. Ishimaru, “Application of the phase-perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2294 (1985).
    [CrossRef]
  18. H. E. Bennett, J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,”J. Opt. Soc. Am. 51, 123–129 (1963).
    [CrossRef]
  19. H. E. Bennett, “Specular reflectance of aluminized ground glass and the height distribution of surface irregularities,”J. Opt. Soc. Am. 53, 1389–1394 (1963).
    [CrossRef]
  20. H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. 101, 209–214 (1954).
  21. A. F. Houchens, R. G. Herring, “Bidirectional reflectance of rough metal surfaces,” Prog. Astronaut. Aeronaut. 20, 65–89 (1967).
  22. D. H. Hensler, “Light scattering from fused polycrystalline aluminum oxide surfaces,” Appl. Opt. 11, 2522–2528 (1972).
    [CrossRef] [PubMed]
  23. D. Beaglehole, O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
    [CrossRef]
  24. 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]
  25. O. Hunderi, D. Beaglehole, “Study of the interaction of light with rough metal surfaces. II. Theory,” Phys. Rev. B 2, 321–329 (1970).
    [CrossRef]
  26. 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]
  27. J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from optical surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1979), Vol. VII.
    [CrossRef]
  28. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Measurement of the finish of diamond-turned metal surfaces by differential light scattering,” Opt. Eng. 16, 360–374 (1977).
    [CrossRef]
  29. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
    [CrossRef]
  30. J. C. Stover, S. A. Serati, C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
    [CrossRef]
  31. P. Roche, E. Pelletier, “Characterization of optical surfaces by measurement of scattering distribution,” Appl. Opt. 23, 3561–3566 (1984).
    [CrossRef] [PubMed]
  32. J. Renau, P. K. Cheo, H. G. Cooper, “Depolarization of linearly polarized electromagnetic waves backscattered from rough metals and inhomogeneous dielectrics,”J. Opt. Soc. Am. 57, 459–467 (1967).
    [CrossRef] [PubMed]
  33. See Refs. 2, 13, 14, 15, and 26.
  34. P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
    [CrossRef]
  35. J. W. Goodman, “Statistical properties of speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984).
  36. See Sec. 5 of Ref. 2.
  37. M. Nieto-Vesperinas, “Radiometry of rough surfaces,” Opt. Acta 29, 961–971 (1982).
    [CrossRef]
  38. Equation (35) in Sec. 5.3 of Ref. 2.
  39. F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).
  40. Equation (48) in Sec. 5.3 of Ref. 2.
  41. See Chap. 7 of Ref. 39.
  42. G. S. Brown, “The validity of shadowing corrections in rough surface scattering,” Radio Sci. 19, 1461–1468 (1984).
    [CrossRef]
  43. E. R. Mendez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
    [CrossRef]
  44. See Chap. 2 of Ref. 39.
  45. Somewhat similar patterns have also been attributed to multiple scattering by S. R. Pal, A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from atmospheric clouds,” Appl. Opt. 24, 3464–3471 (1985).
    [CrossRef] [PubMed]
  46. P. Oetking, “Photometric studies of diffusely reflecting surfaces with applications to the brightness of the moon,”J. Geophys. Res. 71, 2505–2513 (1966).
    [CrossRef]
  47. W. G. Egan, T. Hilgeman, “Retroreflectance measurements of photometric standards and coatings,” Appl. Opt. 15, 1845–1849 (1976).
    [CrossRef] [PubMed]
  48. F. Becker, P. Ramanantsizehena, M. Stoll, “Angular variation of the bidirectional reflectance of bare soils in the thermal infrared band,” Appl. Opt. 24, 365–375 (1985).
    [CrossRef] [PubMed]
  49. D. A. de Wolf, “Electromagnetic reflection from an extended turbulent medium: cumulative forward-scatter single-backscatter approximation,”IEEE Trans. Antennas Propag. AP-19, 254–262 (1971).
    [CrossRef]
  50. Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
    [CrossRef]
  51. L. Tsang, A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. A 1, 836–839 (1984).
    [CrossRef]
  52. M. P. van Albada, A. Langendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
    [CrossRef] [PubMed]
  53. P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
    [CrossRef] [PubMed]
  54. E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
    [CrossRef] [PubMed]
  55. S. Etemad, R. Thompson, M. J. Andrejeco, “Weak localization of photons: universal fluctuations and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
    [CrossRef] [PubMed]
  56. S. Chakravarty, A. Schmid, “Weak localization: the quasi-classical theory of electrons in a random potential,” Phys. Rep. 140, 193–236 (1986).
    [CrossRef]
  57. 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]
  58. D. E. Khmel’nitskii, “Localization and coherent scattering of electrons,” Physica 126B, 235–241 (1984).
  59. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

1987 (1)

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

1986 (3)

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

S. Etemad, R. Thompson, M. J. Andrejeco, “Weak localization of photons: universal fluctuations and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

S. Chakravarty, A. Schmid, “Weak localization: the quasi-classical theory of electrons in a random potential,” Phys. Rep. 140, 193–236 (1986).
[CrossRef]

1985 (9)

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]

F. Becker, P. Ramanantsizehena, M. Stoll, “Angular variation of the bidirectional reflectance of bare soils in the thermal infrared band,” Appl. Opt. 24, 365–375 (1985).
[CrossRef] [PubMed]

M. P. van Albada, A. Langendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Somewhat similar patterns have also been attributed to multiple scattering by S. R. Pal, A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from atmospheric clouds,” Appl. Opt. 24, 3464–3471 (1985).
[CrossRef] [PubMed]

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 random rough surfaces,” Wave Motion 7, 195–205 (1985).
[CrossRef]

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

D. Winebrenner, A. Ishimaru, “Application of the phase-perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2294 (1985).
[CrossRef]

1984 (6)

G. S. Brown, “The validity of shadowing corrections in rough surface scattering,” Radio Sci. 19, 1461–1468 (1984).
[CrossRef]

J. C. Stover, S. A. Serati, C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
[CrossRef]

P. Roche, E. Pelletier, “Characterization of optical surfaces by measurement of scattering distribution,” Appl. Opt. 23, 3561–3566 (1984).
[CrossRef] [PubMed]

Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
[CrossRef]

L. Tsang, A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. A 1, 836–839 (1984).
[CrossRef]

D. E. Khmel’nitskii, “Localization and coherent scattering of electrons,” Physica 126B, 235–241 (1984).

1982 (2)

1981 (2)

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

A. K. Fung, H. J. Eom, “Multiple scattering and depolarization by a randomly rough Kirchhoff surface,”IEEE Trans. Antennas Propag. AP-29, 463–471 (1981).
[CrossRef]

1980 (2)

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

1979 (1)

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

1978 (1)

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

1977 (2)

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Measurement of the finish of diamond-turned metal surfaces by differential light scattering,” Opt. Eng. 16, 360–374 (1977).
[CrossRef]

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

1976 (1)

1975 (2)

P. C. Waterman, “Scattering from periodic surfaces,”J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

G. S. Agarwal, “Scattering from rough surfaces,” Opt. Commun. 14, 161–166 (1975).
[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 (2)

D. N. Pattanayak, E. Wolf, “General form and a new interpretation of the Ewald–Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
[CrossRef]

D. H. Hensler, “Light scattering from fused polycrystalline aluminum oxide surfaces,” Appl. Opt. 11, 2522–2528 (1972).
[CrossRef] [PubMed]

1971 (1)

D. A. de Wolf, “Electromagnetic reflection from an extended turbulent medium: cumulative forward-scatter single-backscatter approximation,”IEEE Trans. Antennas Propag. AP-19, 254–262 (1971).
[CrossRef]

1970 (2)

D. Beaglehole, O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
[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]

1969 (1)

M. I. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,”IEEE Trans. Antennas Propag. AP-17, 577–585 (1969).
[CrossRef]

1967 (2)

1966 (1)

P. Oetking, “Photometric studies of diffusely reflecting surfaces with applications to the brightness of the moon,”J. Geophys. Res. 71, 2505–2513 (1966).
[CrossRef]

1963 (2)

1954 (1)

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. 101, 209–214 (1954).

1951 (1)

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

1941 (1)

Agarwal, G. S.

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

G. S. Agarwal, “Scattering from rough surfaces,” Opt. Commun. 14, 161–166 (1975).
[CrossRef]

Akkermans, E.

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

Andrejeco, M. J.

S. Etemad, R. Thompson, M. J. Andrejeco, “Weak localization of photons: universal fluctuations and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

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]

D. Beaglehole, O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
[CrossRef]

Becker, F.

Beckmann, P.

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

Bennett, H. E.

Bennett, J. M.

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from optical surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1979), Vol. VII.
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Brown, G. S.

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

G. S. Brown, “The validity of shadowing corrections in rough surface scattering,” Radio Sci. 19, 1461–1468 (1984).
[CrossRef]

Carswell, A. I.

Celli, V.

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]

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]

Chakravarty, S.

S. Chakravarty, A. Schmid, “Weak localization: the quasi-classical theory of electrons in a random potential,” Phys. Rep. 140, 193–236 (1986).
[CrossRef]

Cheo, P. K.

Church, E. L.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Measurement of the finish of diamond-turned metal surfaces by differential light scattering,” Opt. Eng. 16, 360–374 (1977).
[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]

Cooper, H. G.

Davies, H.

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. 101, 209–214 (1954).

de Wolf, D. A.

D. A. de Wolf, “Electromagnetic reflection from an extended turbulent medium: cumulative forward-scatter single-backscatter approximation,”IEEE Trans. Antennas Propag. AP-19, 254–262 (1971).
[CrossRef]

Egan, W. G.

Elson, J. M.

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from optical surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1979), Vol. VII.
[CrossRef]

Eom, H. J.

A. K. Fung, H. J. Eom, “Multiple scattering and depolarization by a randomly rough Kirchhoff surface,”IEEE Trans. Antennas Propag. AP-29, 463–471 (1981).
[CrossRef]

Etemad, S.

S. Etemad, R. Thompson, M. J. Andrejeco, “Weak localization of photons: universal fluctuations and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

Fano, U.

Fuks, I. M.

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

Fung, A. K.

A. K. Fung, H. J. Eom, “Multiple scattering and depolarization by a randomly rough Kirchhoff surface,”IEEE Trans. Antennas Propag. AP-29, 463–471 (1981).
[CrossRef]

Garcia, N.

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

Gillespie, C. H.

J. C. Stover, S. A. Serati, C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
[CrossRef]

Goodman, J. W.

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

Gray, P. F.

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

Hensler, D. H.

Herring, R. G.

A. F. Houchens, R. G. Herring, “Bidirectional reflectance of rough metal surfaces,” Prog. Astronaut. Aeronaut. 20, 65–89 (1967).

Hilgeman, T.

Houchens, A. F.

A. F. Houchens, R. G. Herring, “Bidirectional reflectance of rough metal surfaces,” Prog. Astronaut. Aeronaut. 20, 65–89 (1967).

Hunderi, O.

D. Beaglehole, O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
[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]

Ishimaru, A.

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Measurement of the finish of diamond-turned metal surfaces by differential light scattering,” Opt. Eng. 16, 360–374 (1977).
[CrossRef]

Khmel’nitskii, D. E.

D. E. Khmel’nitskii, “Localization and coherent scattering of electrons,” Physica 126B, 235–241 (1984).

Kuga, Y.

Langendijk, A.

M. P. van Albada, A. Langendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

Maradudin, A. A.

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]

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]

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

Maret, G.

P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Marvin, A. M.

Maynard, R.

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

McGurn, A. R.

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]

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]

Mendez, E. R.

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

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]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, “Depolarization of electromagnetic waves scattered from slightly rough random surfaces: a study by means of the extinction theorem,”J. Opt. Soc. Am. 72, 539–547 (1982).
[CrossRef]

M. Nieto-Vesperinas, “Radiometry of rough surfaces,” Opt. Acta 29, 961–971 (1982).
[CrossRef]

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

O’Donnell, K. A.

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

Oetking, P.

P. Oetking, “Photometric studies of diffusely reflecting surfaces with applications to the brightness of the moon,”J. Geophys. Res. 71, 2505–2513 (1966).
[CrossRef]

Pal, S. R.

Pattanayak, D. N.

D. N. Pattanayak, E. Wolf, “General form and a new interpretation of the Ewald–Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
[CrossRef]

Pelletier, E.

Porteus, J. O.

Ramanantsizehena, P.

Rayleigh,

Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.

Renau, J.

Rice, S. O.

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

Roche, P.

Sancer, M. I.

M. I. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,”IEEE Trans. Antennas Propag. AP-17, 577–585 (1969).
[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]

Schmid, A.

S. Chakravarty, A. Schmid, “Weak localization: the quasi-classical theory of electrons in a random potential,” Phys. Rep. 140, 193–236 (1986).
[CrossRef]

Serati, S. A.

J. C. Stover, S. A. Serati, C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
[CrossRef]

Shen, J.

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

Spizzichino, A.

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

Stoll, M.

Stover, J. C.

J. C. Stover, S. A. Serati, C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
[CrossRef]

Thompson, R.

S. Etemad, R. Thompson, M. J. Andrejeco, “Weak localization of photons: universal fluctuations and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

Tsang, L.

van Albada, M. P.

M. P. van Albada, A. Langendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

Waterman, P. C.

P. C. Waterman, “Scattering from periodic surfaces,”J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

Winebrenner, D.

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

D. Winebrenner, A. Ishimaru, “Application of the phase-perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2294 (1985).
[CrossRef]

Wolf, E.

D. N. Pattanayak, E. Wolf, “General form and a new interpretation of the Ewald–Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Wolf, P. E.

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Zavada, J. M.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Measurement of the finish of diamond-turned metal surfaces by differential light scattering,” Opt. Eng. 16, 360–374 (1977).
[CrossRef]

Appl. Opt. (5)

Commun. Pure Appl. Math. (1)

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

IEEE Trans. Antennas Propag. (3)

M. I. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,”IEEE Trans. Antennas Propag. AP-17, 577–585 (1969).
[CrossRef]

A. K. Fung, H. J. Eom, “Multiple scattering and depolarization by a randomly rough Kirchhoff surface,”IEEE Trans. Antennas Propag. AP-29, 463–471 (1981).
[CrossRef]

D. A. de Wolf, “Electromagnetic reflection from an extended turbulent medium: cumulative forward-scatter single-backscatter approximation,”IEEE Trans. Antennas Propag. AP-19, 254–262 (1971).
[CrossRef]

J. Acoust. Soc. Am. (1)

P. C. Waterman, “Scattering from periodic surfaces,”J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

J. Geophys. Res. (1)

P. Oetking, “Photometric studies of diffusely reflecting surfaces with applications to the brightness of the moon,”J. Geophys. Res. 71, 2505–2513 (1966).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (4)

Opt. Acta (3)

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

M. Nieto-Vesperinas, “Radiometry of rough surfaces,” Opt. Acta 29, 961–971 (1982).
[CrossRef]

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

Opt. Commun. (3)

D. N. Pattanayak, E. Wolf, “General form and a new interpretation of the Ewald–Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
[CrossRef]

G. S. Agarwal, “Scattering from rough surfaces,” Opt. Commun. 14, 161–166 (1975).
[CrossRef]

E. R. Mendez, 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. Eng. (3)

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Measurement of the finish of diamond-turned metal surfaces by differential light scattering,” Opt. Eng. 16, 360–374 (1977).
[CrossRef]

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

J. C. Stover, S. A. Serati, C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
[CrossRef]

Phys. Rep. (1)

S. Chakravarty, A. Schmid, “Weak localization: the quasi-classical theory of electrons in a random potential,” Phys. Rep. 140, 193–236 (1986).
[CrossRef]

Phys. Rev. B (6)

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]

D. Beaglehole, O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (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]

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

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

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

Phys. Rev. Lett. (4)

M. P. van Albada, A. Langendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

S. Etemad, R. Thompson, M. J. Andrejeco, “Weak localization of photons: universal fluctuations and ensemble averaging,” Phys. Rev. Lett. 57, 575–578 (1986).
[CrossRef] [PubMed]

Physica (1)

D. E. Khmel’nitskii, “Localization and coherent scattering of electrons,” Physica 126B, 235–241 (1984).

Proc. Inst. Electr. Eng. (1)

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. 101, 209–214 (1954).

Prog. Astronaut. Aeronaut. (1)

A. F. Houchens, R. G. Herring, “Bidirectional reflectance of rough metal surfaces,” Prog. Astronaut. Aeronaut. 20, 65–89 (1967).

Radio Sci. (3)

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

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]

G. S. Brown, “The validity of shadowing corrections in rough surface scattering,” Radio Sci. 19, 1461–1468 (1984).
[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]

Other (12)

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

Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from optical surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1979), Vol. VII.
[CrossRef]

See Refs. 2, 13, 14, 15, and 26.

Equation (35) in Sec. 5.3 of Ref. 2.

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

Equation (48) in Sec. 5.3 of Ref. 2.

See Chap. 7 of Ref. 39.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

See Chap. 2 of Ref. 39.

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

See Sec. 5 of Ref. 2.

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Figures (22)

Fig. 1
Fig. 1

Electron micrographs comparing one of the photoresist surfaces (diffuser #45, a = 4.8 μm, top) with a ground-glass surface (bottom). The length of the white lines is 10 μm.

Fig. 2
Fig. 2

A simplified diagram of the scattering instrument. The sample is rotated about the vertical axis to a fixed angle, and the detector arm then pivots about the same axis and records the angular dependence of the diffusely scattered intensity.

Fig. 3
Fig. 3

Histogram of surface height data of diffuser #80 in comparison with a Gaussian distribution of the same variance. This was produced from 1000 measurements taken with a surface profilometer.

Fig. 4
Fig. 4

Autocorrelation function (solid line) of profilometer surface height data of diffuser #80 as compared with a Gaussian function (dashed line).

Fig. 5
Fig. 5

Measured diffuse scattering from diffuser #80 with angle of incidence θi = 20°, λ = 0.633 μm, and s incident polarization (no depolarization observed). The solid curve shows the Beckmann theory.

Fig. 6
Fig. 6

Measured diffuse scattering from diffuser #80 with angle of incidence θi = 20°, λ = 10.6 μm, and s incident polarization (no depolarization observed). The Beckmann theory is shown by the solid curve, and the specular component that was observed is not shown.

Fig. 7
Fig. 7

Measured diffuse scattering from diffuser #80 with θi = 70°, λ = 0.633 μm, and s incident polarization (no depolarization observed). The Beckmann theory (solid line) predicts scattering at −90° and behind the diffuser.

Fig. 8
Fig. 8

Measured diffuse scattering from diffuser #80 with θi = 70°, λ = 10.6 μm, and s incident polarization (no depolarization observed). The data and the Beckmann theory (solid line) are both skewed with respect to the direction of specular reflection (dashed line). The specular component is not shown.

Fig. 9
Fig. 9

Diffuse-scattering measurements from diffuser #83 with incidence angle θi = 0°, λ = 0.633 μm, and s incident polarization. The ○’s denote the ss- (parallel-) and the +’s denote the sp- (orthogonal-) polarized scattering components.

Fig. 10
Fig. 10

Diffuse-scattering measurements from diffuser #83 with θi = 200, λ = 0.633 μm, and s incident polarization (the ○’s denote ss data and +’s denote sp data).

Fig. 11
Fig. 11

Diffuse-scattering measurements from diffuser #83 with θI = 40°, λ = 0.633 μm, and s incident polarization (the ○’s denote ss data and +’s denote sp data).

Fig. 12
Fig. 12

Photographs of far-field scattering pattern in normal incidence from a fine-scale diffuser at λ = 0.633 μm. The incident polarization is vertical; A, parallel and B, orthogonally polarized scattering. The dark spot is a hole in the screen that lets the laser source through. Figures 9 and 17 correspond to scans of these patterns along the x and y axes, respectively.

Fig. 13
Fig. 13

A possible multiple-scattering path in a valley of the surface (see text).

Fig. 14
Fig. 14

Multiple scattering of polarized light from a valley in a two-dimensional surface, as seen from above in Fig. 13. For paths A and B there is no depolarization. In the case of a perfect conductor, paths C and D cross polarize the light that is returned. In the transverse paths of C and D the polarization vector is at 45° with respect to the plane of the figure.

Fig. 15
Fig. 15

Optical micrographs (N.A. =0.80, 50×) of the same area of diffuser #45 with linearly polarized illumination and viewed with a parallel (left) and perpendicular (right) analyzer. The areas where depolarization is observed (see circles) exhibit patterns analogous to Fig. 14.

Fig. 16
Fig. 16

Diffuse scattering from diffuser #83, incidence angle θI = 0°, λ = 0.633 μm, and RHCP incident polarization. The ○’s denote RHCP and the +’s denote LHCP-detected radiation.

Fig. 17
Fig. 17

Diffuse-scattering measurements from diffuser #83 with incidence angle θi = 0°, λ = 0.633 μm, and p incident polarization (the ○’s denote pp and +’s denote ps scattering).

Fig. 18
Fig. 18

Diffuse-scattering measurements from diffuser #45 with θi = 0°, λ = 0.633 μm, and s incident polarization. The ○’s denote ss scattering and +’s denote sp scattering.

Fig. 19
Fig. 19

Diffuse-scattering measurements from diffuser #45 with θi = 20°, λ = 0.633 μm, and s incident polarization. The ○’s denote ss and the +’s denote sp data.

Fig. 20
Fig. 20

Measured diffuse scattering from diffuser #45 in normal incidence, λ = 0.514 μm (G), and λ = 0.633 μm (R), normalized to show relative strengths as discussed in the text. The upper two curves are the ss components, and the lower pair are the sp data.

Fig. 21
Fig. 21

Diffuse-scattering measurements from diffuser #83 with θi = 0°, λ = 10.6 μm, and s incident polarization (○’s denote ss data and +’s denote sp data).

Fig. 22
Fig. 22

Diffuse scattering from diffuser #83 with θi = 60°, λ = 10.6 μm, and s incident polarization (the ○’s denote ss and +’s denote sp data).

Equations (2)

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Δ ϕ = ( k i + k s ) · Δ r .
Δ θ s λ / w ,

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