Abstract

A critical evaluation of various theoretical techniques for calculating the reflectivity of one-dimensional metallic randomly rough surfaces is presented. We proceed by comparing experimental and rigorous numerical results with those obtained with three perturbation theories and the Kirchhoff approximation. The samples were fabricated in photoresist, and their metallized surface profiles constitute good approximations to Gaussian-correlated, Gaussian random processes. The correlation lengths of these surfaces range from approximately one third to approximately three times the infrared wavelengths employed. The results show that the phase-perturbation theory has wider applicability than the other perturbation theories and the results based on the Kirchhoff approximation.

© 1998 Optical Society of America

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    [CrossRef]
  2. Lord Rayleigh , The Theory of Sound, 2nd ed. (Macmillan, London, 1896), Vol. 2, pp. 89–96.
  3. Lord Rayleigh , “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
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  5. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), p. 245.
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    [CrossRef]
  15. J. C. Stover, S. A. Serati, C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
    [CrossRef]
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    [CrossRef]
  17. F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
    [CrossRef]
  18. 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]
  19. A. A. Maradudin, “Electromagnetic surface excitations on rough surfaces,” in Electromagnetic Surface Excitations, R. F. Wallis, G. I. Stegeman, eds. (Springer-Verlag, New York, 1986), pp. 57–131.
    [CrossRef]
  20. A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
    [CrossRef]
  21. J. Shen, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
    [CrossRef]
  22. D. P. Winebrenner, A. Ishimaru, “Investigation of a surface field phase perturbation techniques for scattering from rough surfaces,” Radio Sci. 20, 161–170 (1985).
    [CrossRef]
  23. D. P. Winebrenner, A. Ishimaru, “Application of the phase-perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2294 (1985).
    [CrossRef]
  24. M. Nieto-Vesperinas, N. García, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
    [CrossRef]
  25. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–307 (1990).
    [CrossRef]
  26. M. E. Veysoglu, H. A. Yuch, R. T. Shin, J. A. Kong, “Polarimetric passive remote sensing of periodic surfaces,” J. Electromagn. Waves Appl. 5, 267–280 (1991).
  27. R. García-Molina, A. A. Maradudin, T. A. Leskova, “The impedance boundary condition for a curved surface,” Phys. Rep. 194, 351–359 (1990).
    [CrossRef]
  28. A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A 207, 302–314 (1994).
    [CrossRef]
  29. I. N. Shkliarevskii, V. G. Padlaka, “Measurement of optical constants of copper, gold and nickel in the infrared spectral region,” Opt. Spektrosk. 6, 78–84 (1959).
  30. R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
    [CrossRef]
  31. P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
    [CrossRef]
  32. 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]
  33. 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]
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    [CrossRef] [PubMed]
  35. 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]
  36. K. A. O’Donnell, M. E. Knotts, “Polarization-dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 9, 585–596 (1992).
  37. G. S. Brown, “Scattering from a class of randomly rough surfaces,” Radio Sci. 17, 1274–1280 (1982).
    [CrossRef]

1995

J. A. Saénchez-Gil, A. A. Maradudin, E. R. Méndez, “Limits of validity of three perturbation theories of the coherent scattering of light from a one-dimensional randomly rough dielectric surface,” J. Opt. Soc. Am. A 12, 1547–1558 (1995).
[CrossRef]

R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

1994

A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A 207, 302–314 (1994).
[CrossRef]

M. E. Knotts, K. A. O’Donnell, “Measurement of light-scattering by a series of conducting surfaces with one-dimensional roughness,” J. Opt. Soc. Am. A 11, 697–710 (1994).
[CrossRef]

1993

A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
[CrossRef]

1992

1991

M. E. Veysoglu, H. A. Yuch, R. T. Shin, J. A. Kong, “Polarimetric passive remote sensing of periodic surfaces,” J. Electromagn. Waves Appl. 5, 267–280 (1991).

1990

R. García-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. (New York) 203, 255–307 (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]

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]

1989

1987

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]

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]

1985

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

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

1984

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

1982

G. S. Brown, “Scattering from a class of randomly rough surfaces,” Radio Sci. 17, 1274–1280 (1982).
[CrossRef]

1981

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

1980

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

1978

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

1977

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

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

1971

1963

1961

1959

I. N. Shkliarevskii, V. G. Padlaka, “Measurement of optical constants of copper, gold and nickel in the infrared spectral region,” Opt. Spektrosk. 6, 78–84 (1959).

1951

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

1907

Lord Rayleigh , “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1987), p. 72.

Bennett, H. E.

Birkebak, R. C.

Brown, G. S.

G. S. Brown, “Scattering from a class of randomly rough surfaces,” Radio Sci. 17, 1274–1280 (1982).
[CrossRef]

Celli, V.

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

Dainty, J. C.

Depew, C. A.

Friberg, A. T.

García, N.

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

García-Molina, R.

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

Gray, P. F.

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

Gu, Z. H.

R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

Hill, N. R.

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

Ishimaru, A.

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

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

Kim, M. J.

Knotts, M. E.

Kong, J. A.

M. E. Veysoglu, H. A. Yuch, R. T. Shin, J. A. Kong, “Polarimetric passive remote sensing of periodic surfaces,” J. Electromagn. Waves Appl. 5, 267–280 (1991).

Leskova, T. A.

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

Lu, J. Q.

R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

Luna, R. E.

R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
[CrossRef]

Maradudin, A. A.

J. A. Saénchez-Gil, A. A. Maradudin, E. R. Méndez, “Limits of validity of three perturbation theories of the coherent scattering of light from a one-dimensional randomly rough dielectric surface,” J. Opt. Soc. Am. A 12, 1547–1558 (1995).
[CrossRef]

A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A 207, 302–314 (1994).
[CrossRef]

A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
[CrossRef]

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

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

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

A. A. Maradudin, “Electromagnetic surface excitations on rough surfaces,” in Electromagnetic Surface Excitations, R. F. Wallis, G. I. Stegeman, eds. (Springer-Verlag, New York, 1986), pp. 57–131.
[CrossRef]

Marvin, A.

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

McGurn, A. R.

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

Méndez, E. R.

R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

J. A. Saénchez-Gil, A. A. Maradudin, E. R. Méndez, “Limits of validity of three perturbation theories of the coherent scattering of light from a one-dimensional randomly rough dielectric surface,” J. Opt. Soc. Am. A 12, 1547–1558 (1995).
[CrossRef]

A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A 207, 302–314 (1994).
[CrossRef]

A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
[CrossRef]

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

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]

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. (New York) 203, 255–307 (1990).
[CrossRef]

Nieto-Vesperinas, M.

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]

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

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), p. 245.

O’Donnell, K. A.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, 1991), p. 38.

Padlaka, V. G.

I. N. Shkliarevskii, V. G. Padlaka, “Measurement of optical constants of copper, gold and nickel in the infrared spectral region,” Opt. Spektrosk. 6, 78–84 (1959).

Porteus, J. O.

Rayleigh, Lord

Lord Rayleigh , “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).

Lord Rayleigh , The Theory of Sound, 2nd ed. (Macmillan, London, 1896), Vol. 2, pp. 89–96.

Rice, S. O.

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

Saénchez-Gil, J. A.

Sant, A. J.

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]

Shin, R. T.

M. E. Veysoglu, H. A. Yuch, R. T. Shin, J. A. Kong, “Polarimetric passive remote sensing of periodic surfaces,” J. Electromagn. Waves Appl. 5, 267–280 (1991).

Shkliarevskii, I. N.

I. N. Shkliarevskii, V. G. Padlaka, “Measurement of optical constants of copper, gold and nickel in the infrared spectral region,” Opt. Spektrosk. 6, 78–84 (1959).

Soto-Crespo, J. M.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1987), p. 72.

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]

Toigo, F.

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

Veysoglu, M. E.

M. E. Veysoglu, H. A. Yuch, R. T. Shin, J. A. Kong, “Polarimetric passive remote sensing of periodic surfaces,” J. Electromagn. Waves Appl. 5, 267–280 (1991).

Weir, R. D.

Welford, W. T.

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

Winebrenner, D. P.

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

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

Yuch, H. A.

M. E. Veysoglu, H. A. Yuch, R. T. Shin, J. A. Kong, “Polarimetric passive remote sensing of periodic surfaces,” J. Electromagn. Waves Appl. 5, 267–280 (1991).

Ann. Phys. (New York)

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

Appl. Opt.

Commun. Pure Appl. Math.

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

J. Electromagn. Waves Appl.

M. E. Veysoglu, H. A. Yuch, R. T. Shin, J. A. Kong, “Polarimetric passive remote sensing of periodic surfaces,” J. Electromagn. Waves Appl. 5, 267–280 (1991).

J. Mod. Opt.

R. E. Luna, E. R. Méndez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

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

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

Opt. Commun.

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. Eng.

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

Opt. Lett.

Opt. Quantum Electron.

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

Opt. Spektrosk.

I. N. Shkliarevskii, V. G. Padlaka, “Measurement of optical constants of copper, gold and nickel in the infrared spectral region,” Opt. Spektrosk. 6, 78–84 (1959).

Phys. Rep.

R. García-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

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

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

Physica A

A. A. Maradudin, E. R. Méndez, “Theoretical studies of the enhanced backscattering of light from one-dimensional randomly rough metal surfaces by the use of a nonlocal impedance boundary condition,” Physica A 207, 302–314 (1994).
[CrossRef]

Proc. R. Soc. London Ser. A

Lord Rayleigh , “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).

Radio Sci.

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

G. S. Brown, “Scattering from a class of randomly rough surfaces,” Radio Sci. 17, 1274–1280 (1982).
[CrossRef]

Waves Random Media

A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
[CrossRef]

Other

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, 1991), p. 38.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), p. 245.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1987), p. 72.

Ref. 4, p. 73.

Ref. 5, p. 238.

Lord Rayleigh , The Theory of Sound, 2nd ed. (Macmillan, London, 1896), Vol. 2, pp. 89–96.

A. A. Maradudin, “Electromagnetic surface excitations on rough surfaces,” in Electromagnetic Surface Excitations, R. F. Wallis, G. I. Stegeman, eds. (Springer-Verlag, New York, 1986), pp. 57–131.
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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the scattering geometry.

Fig. 2
Fig. 2

Statistics of sample D estimated from the profilometer data. (a) Height correlation function. The solid curve corresponds to the estimated correlation, and the dashed curve to a Gaussian-correlation function with the estimated correlation length. Probability density functions of (b) heights, (c) slopes, and (d) curvatures. The estimated densities are shown as histograms, and the dashed curves correspond to Gaussian densities with the estimated variances.

Fig. 3
Fig. 3

Schematic diagram of the scatterometer used for the measurements.

Fig. 4
Fig. 4

Reflectivity as a function of angle of incidence for s-polarized light of wavelength λ = 5.5 μm. (a) Reflectivity of sample A (δ = 0.38 μm, a = 2.8 μm). (b) Reflectivity of sample D (δ = 0.75 μm, a = 6.8 μm). (c) Reflectivity of sample F (δ = 0.41 μm, a = 15.0 μm). The curves were obtained experimentally (filled circles) by means of a numerical simulation of the problem (open squares), by use of the KA (filled triangles), with the SAPT (dash–dot curve), with the PPT (solid curve), and with the SEPT (dotted curve).

Fig. 5
Fig. 5

Same as Fig. 4, but for p-polarized light.

Fig. 6
Fig. 6

Reflectivity as a function of angle of incidence for s-polarized light of wavelength λ = 10.6 μm. (a) Reflectivity of sample A (δ = 0.38 μm, a = 2.8 μm). (b) Reflectivity of sample D (δ = 0.75 μm, a = 6.8 μm). (c) Reflectivity of sample F (δ = 0.41 μm, a = 15.0 μm). The curves were obtained experimentally (filled circles), by means of a numerical simulation of the problem (open squares), by use of the KA (filled triangles), with the SAPT (dash–dot curve), with the PPT (solid curve), and with the SEPT (dotted curve).

Fig. 7
Fig. 7

Same as Fig. 6, but for p-polarized light.

Fig. 8
Fig. 8

Ratio of the reflectivities ℛ s 0)/ℛ p 0) as a function of the wavelength. The curves represent calculations based on the PPT, assuming a dielectric constant of ∊(ω) = -713.34 + i510.4 and an angle of incidence of 5°. The symbols represent the ratio obtained from the experimental (filled circles) and numerical (open squares) reflectivities.

Tables (1)

Tables Icon

Table 1 Estimated Standard Deviation of Heights and Correlation Lengths for the Three Surfaces Employed

Equations (35)

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H2>x1, x3|ω=Hincx1, x3|ω+-dq2π Rpq|k×expiqx1+iα0q, ωx3,
α0(q, ω)=((ω2/c2)-q2)1/2q2<ω2/c2,
=i(q2-(ω2/c2))1/2q2>ω2/c2.
E2>x1, x3|ω=Eincx1, x3|ω+-dq2π Rsq|k×expiqx1+iα0q, ωx3.
Rp,sθscoh = 1L1ω2πccos2 θscos θ0 |Rp,sq|k|2,
k=ω/csin θ0, q=ω/csin θs.
Rp,sθscoh=δθs-θ0p,sθ0,
-dq2πIαp, ω-α0q, ω|p-qαp, ω-α0q, ω×αp, ωα0q, ω+pqRpq|k=Iαp, ω+α0k, ω|p-kαp, ω+α0k, ω×αp, ωα0k, ω-pk, 
α(q, ω)=[(ω)(ω2/c2)-q2]1/2,Re α(q, ω)>0,Im α(q, ω)>0,
Iγ|Q=-dx1exp-iγζx1exp-iQx1.
-dq2πIαp, ω-α0q, ω|p-qαp, ω-α0q, ω Rsq|k=-Iαp, ω+α0k, ω|p-kαq, ω+α0k, ω.
pθ0=Rpθ01-4 δ2a2×eωω-1cosθ0ωcos2 θ0-sin2 θ0 μpθ0,
Rp(θ0)=(ω)cos θ0- (ω)-sin2 θ01/2(ω)cos θ0+(ω)-sin2 θ01/22
sθ0=Rsθ01-4 δ2a2ecos θ01-(ω) μsθ0,
Rsθ0=cos θ0-(ω)-sin2 θ01/2cos θ0+(ω)-sin2 θ01/22
μpθ0=(ω)-1(ω)ωac2[(ω)-sin2 θ0]1/2×[(ω)-2 sin2 θ0]-12π(ω)-1(ω)2ωac3[(ω)-sin2 θ0]×-dx[(ω)-x2]1/2exp[-ωa/2c2×sin θ0-x2]+12π(ω)-1(ω)2ωac3×-dx {x(ω)sin θ0-[(ω)-x2]1/2[(ω)-sin2 θ0]1/2}2(ω)(1-x2)1/2+[(ω)-x2]1/2×exp[-ωa/2c2sin θ0-x2],
μsθ0=-(ω)-1ωac2[(ω)-sin2 θ0]1/2+12π(ω)-12ωac2×-dx exp[-(ωa/2c)2sin θ0-x2](1-x2)1/2+(ω)-x21/2.
Rp,s(q|k)=-2πδ(q-k)-2iGp,s(q|k)αo(k).
Rp,s(q|k)=-2πδ(q-k)-2iGp,s(q|k)αo(k).
Gp,s(q|k)=2πδ(q-k)Gp,s(o)(k)+Gp,s(o)(q)×-dp2π Mp,s(q|p)Gp,s(p|k),
Gp,s(q|k)=2πδ(q-k)Gp,s(k),
Mp,s(q|k)=2πδ(q-k)Mp,s(k),
Gp,sk=1Gok-1-Mk.
Gpok=i(ω)(ω)αok+αk,
Gsok=iαok+αk,
Rpq|k=2πδq-k(ω)αok-αk+i(ω)Mpk(ω)αok+αk-i(ω)Mpk,
Rsq|k=2πδq-kαok-αk+i(ω)Mskαok+αk-i(ω)Msk.
pθ0=(ω)cos θ0-(ω)-sin2 θ01/2-δ2/a2μpθ0(ω)cos θ0+(ω)-sin2 θ01/2+δ2/a2μpθ02,
sθ0=cos θ0-(ω)-sin2 θ01/2-δ2/a2μsθ0cos θ0+(ω)-sin2 θ01/2+δ2/a2μsθ02,
pθ0=Rpθ0exp-4 δ2a2×Re(ω)(ω)-1cos θ0(ω)cos2 θ0-sin2 θ0 μpθ0,
sθ0=Rsθ0exp-4 δ2a2Recos θ01-(ω) μsθ0,
p,sθ0=Rp,sθ0exp[-(4πδ/λ)2cos2 θ0],
p,sθ0=Iscθ0, θs-Iincθ0, θsdθsI0,
Rsθ01-4 cos θ0Re1(δ2/a2)μsθ0.
Rp(θ0)(ω)cos θ0-(δ2/a2)μpθ0(ω)cos θ0+(δ2/a2)μpθ02×1-4 Re(ω)[(ω)-sin2 θ0]1/2cos θ02(ω)cos2 θ0-(δ/a)4μp2θ0.

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