Abstract

A theoretical and numerical study is made of the scattering of light and other electromagnetic waves from rough surfaces separating vacuum from a dielectric. The extinction theorem, both above and below the surface, is used to obtain the boundary values of the field and its normal derivative. Then we calculate the angular distribution of the ensemble average of intensity of the reflected and transmitted fields. The scattering equations are solved numerically by generating one-dimensional surface profiles through a Monte Carlo method. The effect of roughness σ and correlation distance T on the aforementioned angular distribution, as well as on the reflectance, is analyzed. Enhanced backscattering and new transmission effects are observed, also depending on the permittivity. The ratio σ/T is large in all cases studied, and thus no analytical approximation, such as the Kirchhoff approximation (KA) and small perturbation methods, could a priori be expected to hold. We find, however, that the range of validity of the KA can be much broader than that previously found in perfect conductors.

© 1991 Optical Society of America

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  1. 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]
  2. P. Tran, V. Celli, “Monte Carlo calculation of backscattering enhancement for a randomly rough grating,” J. Opt. Soc. Am. A 5, 1635–1637 (1988).
    [Crossref]
  3. 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]
  4. K. A. O’Donnell, E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [Crossref]
  5. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett. 12, 979–981 (1987).
    [Crossref] [PubMed]
  6. J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
    [Crossref]
  7. A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
    [Crossref] [PubMed]
  8. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, Ann. Phys. (NY) 203, 255–307 (1990).
    [Crossref]
  9. A. J. Sant, J. C. Dainty, M. J. Kim, “Comparison of surface scattering between identical, randomly rough metal and dielectric diffusers,” Opt. Lett. 14, 1183–1185 (1989).
    [Crossref] [PubMed]
  10. 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]
  11. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Light-diffracted intensities from very deep gratings,” Phys. Rev. B 38, 7250–7259 (1988).
    [Crossref]
  12. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Connection between blazes from gratings and enhancements from random rough surfaces,” Phys. Rev. B 39, 8193–8197 (1989).
    [Crossref]
  13. J. M. Soto-Crespo, M. Nieto-Vesperinas, “Enhancement of all antispecular orders from deep gratings,” Opt. Commun. 69, 185–188 (1989).
    [Crossref]
  14. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 83, 78–92 (1987).
    [Crossref]
  15. E. I. Thorsos, D. R. Jackson, “The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 86, 261–277 (1989).
    [Crossref]
  16. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo calculations of speckle contrast from perfectly conductive random rough surfaces,” Opt. Commun. 75, 215–218 (1990).
    [Crossref]
  17. 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]
  18. C. Macaskill, “Geometric optics and enhanced backscatter from very rough surfaces,” J. Opt. Soc. Am. A 8, 88–96 (1991).
    [Crossref]
  19. M. Nieto-Vesperinas, J. A. Sánchez-Gil, A. J. Sant, J. C. Dainty, “Light transmission from a randomly rough dielectric diffuser: theoretical and experimental results,” Opt. Lett. 15, 1261–1263 (1990).
    [Crossref] [PubMed]
  20. M. Nieto-Vesperinas, J. C. Dainty, eds., Scattering in Volumes and Surfaces (North-Holland, Amsterdam, 1990).
  21. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).
  22. P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1961), pp. 55–69.
  23. F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).
  24. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 4808–4816 (1951).
    [Crossref]
  25. G. R. Valenzuela, “Depolarization of electromagnetic waves by slightly rough surfaces,”IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
    [Crossref]
  26. V. Celli, A. Marvin, F. Toigo, “Light scattering from rough surfaces,” Phys. Rev. B 11, 1779–1786 (1975).
    [Crossref]
  27. G. S. Agarwal, “Interaction of EM waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
    [Crossref]
  28. F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
    [Crossref]
  29. N. Garcia, V. Celli, M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
    [Crossref]
  30. M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of a scalar wave from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
    [Crossref]
  31. 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]
  32. D. N. Pattanayak, E. Wolf, “General form and a new interpretation of the Ewald-Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
    [Crossref]
  33. A. K. Fung, M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2, 2274–2284 (1985).
    [Crossref]
  34. J. D. Jackson, Classical Electodinamics, 2nd ed. (Wiley, New York, 1975), Sec. I.5.
  35. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. I.5.
  36. M. Cadilhac, in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), p. 54.
    [Crossref]
  37. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 364.
  38. M. Saillard, D. Maystre, “Scattering from metallic and dielectric surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990); see also M. Saillard, “Theoretical and numerical study of light scattering from dielectric and conducting rough surfaces,” Ph.D. dissertation (University of Aix-Marseille III, Aix-en-Provence, France, 1990).
    [Crossref]

1991 (1)

1990 (6)

1989 (6)

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

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Connection between blazes from gratings and enhancements from random rough surfaces,” Phys. Rev. B 39, 8193–8197 (1989).
[Crossref]

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Enhancement of all antispecular orders from deep gratings,” Opt. Commun. 69, 185–188 (1989).
[Crossref]

A. J. Sant, J. C. Dainty, M. J. Kim, “Comparison of surface scattering between identical, randomly rough metal and dielectric diffusers,” Opt. Lett. 14, 1183–1185 (1989).
[Crossref] [PubMed]

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
[Crossref]

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[Crossref] [PubMed]

1988 (2)

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

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Light-diffracted intensities from very deep gratings,” Phys. Rev. B 38, 7250–7259 (1988).
[Crossref]

1987 (4)

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 83, 78–92 (1987).
[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. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[Crossref]

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett. 12, 979–981 (1987).
[Crossref] [PubMed]

1985 (2)

1982 (1)

1981 (1)

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

1979 (1)

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

1977 (2)

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

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

1975 (1)

V. Celli, A. Marvin, F. Toigo, “Light scattering from rough surfaces,” Phys. Rev. B 11, 1779–1786 (1975).
[Crossref]

1972 (1)

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

1967 (1)

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

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 4808–4816 (1951).
[Crossref]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 364.

Agarwal, G. S.

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

Bass, F. G.

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

Beckmann, P.

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

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1961), pp. 55–69.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. I.5.

Cadilhac, M.

M. Cadilhac, in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), p. 54.
[Crossref]

Celli, V.

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

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

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

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

V. Celli, A. Marvin, F. Toigo, “Light scattering from rough surfaces,” Phys. Rev. B 11, 1779–1786 (1975).
[Crossref]

Chen, M. F.

Dainty, J. C.

Friberg, A. T.

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]

Fuks, I. M.

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

Fung, A. K.

Garcia, N.

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

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

Hill, N. R.

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

Jackson, D. R.

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

Jackson, J. D.

J. D. Jackson, Classical Electodinamics, 2nd ed. (Wiley, New York, 1975), Sec. I.5.

Kim, M. J.

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]

A. J. Sant, J. C. Dainty, M. J. Kim, “Comparison of surface scattering between identical, randomly rough metal and dielectric diffusers,” Opt. Lett. 14, 1183–1185 (1989).
[Crossref] [PubMed]

Macaskill, C.

Maradudin, A. A.

Marvin, A.

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

V. Celli, A. Marvin, F. Toigo, “Light scattering from rough surfaces,” Phys. Rev. B 11, 1779–1786 (1975).
[Crossref]

Marvin, A. M.

Maystre, D.

McGurn, A. R.

Mendez, E. R.

Méndez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, Ann. Phys. (NY) 203, 255–307 (1990).
[Crossref]

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[Crossref] [PubMed]

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]

Michel, T.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, J. A. Sánchez-Gil, A. J. Sant, J. C. Dainty, “Light transmission from a randomly rough dielectric diffuser: theoretical and experimental results,” Opt. Lett. 15, 1261–1263 (1990).
[Crossref] [PubMed]

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo calculations of speckle contrast from perfectly conductive random rough surfaces,” Opt. Commun. 75, 215–218 (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]

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
[Crossref]

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Enhancement of all antispecular orders from deep gratings,” Opt. Commun. 69, 185–188 (1989).
[Crossref]

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Connection between blazes from gratings and enhancements from random rough surfaces,” Phys. Rev. B 39, 8193–8197 (1989).
[Crossref]

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Light-diffracted intensities from very deep gratings,” Phys. Rev. B 38, 7250–7259 (1988).
[Crossref]

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett. 12, 979–981 (1987).
[Crossref] [PubMed]

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, N. Garcia, “A detailed study of the scattering of a scalar wave from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[Crossref]

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

O’Donnell, K. A.

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. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[Crossref]

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]

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 4808–4816 (1951).
[Crossref]

Saillard, M.

Sánchez-Gil, J. A.

Sant, A. J.

Soto-Crespo, J. 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, J. M. Soto-Crespo, “Monte Carlo calculations of speckle contrast from perfectly conductive random rough surfaces,” Opt. Commun. 75, 215–218 (1990).
[Crossref]

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Connection between blazes from gratings and enhancements from random rough surfaces,” Phys. Rev. B 39, 8193–8197 (1989).
[Crossref]

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Enhancement of all antispecular orders from deep gratings,” Opt. Commun. 69, 185–188 (1989).
[Crossref]

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
[Crossref]

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Light-diffracted intensities from very deep gratings,” Phys. Rev. B 38, 7250–7259 (1988).
[Crossref]

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett. 12, 979–981 (1987).
[Crossref] [PubMed]

Spizzichino, A.

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

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 364.

Thorsos, E. I.

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

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 83, 78–92 (1987).
[Crossref]

Toigo, F.

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

V. Celli, A. Marvin, F. Toigo, “Light scattering from rough surfaces,” Phys. Rev. B 11, 1779–1786 (1975).
[Crossref]

Tran, P.

Valenzuela, G. R.

G. R. Valenzuela, “Depolarization of electromagnetic waves by slightly rough surfaces,”IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[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, Oxford, 1980), Sec. I.5.

Ann. Phys. (NY) (1)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, Ann. Phys. (NY) 203, 255–307 (1990).
[Crossref]

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 4808–4816 (1951).
[Crossref]

IEEE Trans. Antennas Propag. (1)

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

J. Acoust. Soc. Am. (2)

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 83, 78–92 (1987).
[Crossref]

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

J. Opt. Soc. Am A (1)

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

J. Opt. Soc. Am. (1)

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

A. K. Fung, M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2, 2274–2284 (1985).
[Crossref]

M. Saillard, D. Maystre, “Scattering from metallic and dielectric surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990); see also M. Saillard, “Theoretical and numerical study of light scattering from dielectric and conducting rough surfaces,” Ph.D. dissertation (University of Aix-Marseille III, Aix-en-Provence, France, 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]

C. Macaskill, “Geometric optics and enhanced backscatter from very rough surfaces,” J. Opt. Soc. Am. A 8, 88–96 (1991).
[Crossref]

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
[Crossref]

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]

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

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

Opt. Acta (1)

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

Opt. Commun. (5)

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

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Enhancement of all antispecular orders from deep gratings,” Opt. Commun. 69, 185–188 (1989).
[Crossref]

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

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo calculations of speckle contrast from perfectly conductive random rough surfaces,” Opt. Commun. 75, 215–218 (1990).
[Crossref]

Opt. Lett. (4)

Phys. Rev. B (5)

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Light-diffracted intensities from very deep gratings,” Phys. Rev. B 38, 7250–7259 (1988).
[Crossref]

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Connection between blazes from gratings and enhancements from random rough surfaces,” Phys. Rev. B 39, 8193–8197 (1989).
[Crossref]

V. Celli, A. Marvin, F. Toigo, “Light scattering from rough surfaces,” Phys. Rev. B 11, 1779–1786 (1975).
[Crossref]

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

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

Other (8)

J. D. Jackson, Classical Electodinamics, 2nd ed. (Wiley, New York, 1975), Sec. I.5.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. I.5.

M. Cadilhac, in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), p. 54.
[Crossref]

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 364.

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

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

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1961), pp. 55–69.

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

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Figures (16)

Fig. 1
Fig. 1

Scattering geometry.

Fig. 2
Fig. 2

Illustration of the local angle of incidence ϑ(x) at the tangent plane.

Fig. 3
Fig. 3

Angular distribution of mean reflected intensity from a dielectric surface with σ = 1.86λ, T = 4.69λ, and = 1.991, at θo = 0°, 20°, and 40° (dashed curves, s polarization; solid curves, p polarization). The average is over 400 samples. The specular direction is shown by the mark at the upper right. The backscattering direction is marked by vertical lines. The unitarity is as shown.

Fig. 4
Fig. 4

Same as Fig. 3 for σ = 1.9λ and T = 3.16λ; θo = 0°, and 10°: (a) = 2.04, (b) = 7.5.

Fig. 5
Fig. 5

Angular distribution of mean transmitted intensity from a dielectric surface with σ = 1.9λ, T = 3.16λ, (dashed curves, s polarization; solid curves, p polarization). The average is over 400 samples. The straight-through direction is shown by the mark at the upper right. The specular direction of refraction, namely, that from Snell’s law for a plane, is marked by vertical lines: (a) = 2.04 at θo = 20°, 40°, (b) = 7.5 at θo = 20°.

Fig. 6
Fig. 6

Same as Fig. 5 for the diffuse component of mean transmitted (trans.) intensity: σ = 0.5λ, T = 3.16λ, = 2.04; θo = 0°, 20°, 40°.

Fig. 7
Fig. 7

Same as Fig. 3: σ = 0.5λ, T = 3.16λ, = 2.04; θo = 0°, 20°, 40°.

Fig. 8
Fig. 8

Same as Fig. 3 for T = 3.16λ, = 2.04, and θo = 55° (Brewster angle for a plane): (a) σ = 0.5λ, (b) σ = 1.9λ.

Fig. 9
Fig. 9

Average reflectance from 400 samples versus θo, for = 2.04, T = 3.16λ, σ = 0.5λ, and of σ = 1.9λ. The reflectance from a plane is also shown.

Fig. 10
Fig. 10

Same as Fig. 5 for the diffuse component of mean transmitted (trans.) intensity for σ = 0.2λ, T = 0.2λ, and = 2.04; θo = 10°, 40°.

Fig. 11
Fig. 11

Same as Fig. 3 for the diffuse component of mean reflected (ref.) intensity for σ = 0.2λ, T = 0.2λ, and = 2.04; θo = 10°, 40°.

Fig. 12
Fig. 12

Same as Fig. 9 from 240 samples versus θo, for σ = 0.2λ, T = 0.2λ, and = 2.04. The reflectance from a plane is also shown.

Fig. 13
Fig. 13

Same as Fig. 6 for mean transmitted (trans.) intensity KA.

Fig. 14
Fig. 14

Same as Fig. 7 with the KA.

Fig. 15
Fig. 15

Same as Fig. 5 for σ = 1.86λ, T = 4.69λ, and = 1.991 at θo, = 0°, 20°, 40° with the KA.

Fig. 16
Fig. 16

Same as Fig. 3 with the KA.

Equations (94)

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K o k o ( sin θ o , 0 , - cos θ o ) ,
K k o ( sin θ o , 0 , cos θ ) ,
K t [ ( ω ) ] 1 / 2 k o ( sin θ t , 0 , - cos θ t ) ,
K 2 = K o 2 = k o 2 = ω 2 c 2 = ( 2 π / λ ) 2 ,
K t 2 = ( ω ) k o 2 ,
E ( i ) ( r ) = j ^ E ( i ) exp ( i K o · r ) .
H ( i ) ( r ) = j ^ H ( i ) exp ( i K o · r ) ,
2 E ( out ) ( r ) + k o 2 E ( out ) ( r ) = 0 ,             z > D ( x ) , ( r V ˜ ) ,
2 E ( in ) ( r ) + ( ω ) k o 2 E ( in ) ( r ) = 0 ,             z > D ( x ) , ( r V ) .
[ E ( in ) - E ( out ) ] × n ^ = 0 , [ H ( in ) - H ( out ) ] × n ^ = 0 ,
E ( out ) ( r ) z = D ( + ) ( x ) = E ( in ) ( r ) z = D ( - ) ( x ) ,
[ E ( out ) ( r ) n ] z = D ( + ) ( x ) = [ E ( in ) ( r ) n ] z = D ( - ) ( x ) ,
/ n = ( n ^ · ) ,
n ^ ( 1 / γ ) { - d [ D ( x ) ] / d x , 1 } .
G o ( r , r ) r 2 E ( out ) ( r ) - E ( out ) ( r ) r 2 G o ( r , r ) = 4 π δ ( r - r ) E ( out ) ( r ) ,
G ( r , r ) r 2 E ( in ) ( r ) - E ( in ) ( r ) r 2 G ( r , r ) = 4 π δ ( r - r ) E ( in ) ( r ) .
G o ( r , r ) = π i H 0 ( 1 ) ( k o r - r ) ,
G ( r , r ) = π i H 0 ( 1 ) { [ ( ω ) ] 1 / 2 k o r - r } .
1 4 π V ˜ d r · [ G o ( r , r ) r E ( r ) - E ( r ) r G o ( r , r ) ] = E ( out ) ( r ) ,             r V ˜ ,
1 4 π V ˜ d r · [ G o ( r , r ) r E ( r ) - E ( r ) r G o ( r , r ) ] = 0 ,             r V ,
V ˜ d r = Σ ( ) d r + z = D ( x ) d S ( - ) ,
d S ( - ) = - d S ( + ) = ( - n ^ ) d S = - n ^ γ d x .
E ( out ) ( r ) = E ( i ) ( r ) + E ( r ) ( r ) ,
Σ ( ) [ G o E ( out ) - E ( out ) G o ] = 4 π E ( i ) .
E ( i ) ( r ) + 1 4 π - d x [ E ( out ) ( r ) G o ( r , r ) n - G o ( r , r ) E ( out ) ( r ) n ] γ = E ( out ) ( r ) ,             r V ˜ ,
E ( i ) ( r ) + 1 4 π - + d x [ E ( out ) ( r ) G o ( r , r ) n - G o ( r , r ) E ( out ) ( r ) n ] γ = 0 ,             r V ,
Σ ( - ) [ G E ( in ) - E ( in ) G ] = 0.
- 1 4 π - d x [ E ( in ) ( r ) G ( r , r ) n - G ( r , r ) E ( in ) ( r ) n ] γ = 0 ,             r V ˜ ,
- 1 4 π - d x [ E ( in ) ( r ) G ( r , r ) n - G ( r , r ) E ( in ) ( r ) n ] γ = E ( in ) ( r ) ,             r V .
E ( x ) = E ( out ) [ x , D ( x ) ] = E ( in ) [ x , D ( x ) ] ,
F ( x ) = γ [ E ( out ) ( r ) n ] z = D ( + ) ( x ) = γ [ E ( in ) ( r ) n ] z = D ( - ) ( x ) ,
E ( i ) ( r ) + 1 4 π - d x { E ( x ) [ G o z - D ( x ) G o x ] - G o F ( x ) } = E ( out ) ( r ) ,
- 1 4 π - d x { E ( x ) [ G z - D ( x ) G x ] - G F ( x ) } = 0.
E ( i ) [ x , D ( x ) ] + 1 4 π - d x { E ( x ) [ G o z - D ( x ) G o x ] - G o F ( x ) } = E ( x ) ,
- 1 4 π - d x { E ( x ) [ G z - D ( x ) G x ] - G F ( x ) } = 0 ,
r > , < - r r > , < - r > , < · r ,
E ( r ) ( r > , θ ) = exp [ i ( k o r > - π / 4 ) ] 2 ( 2 π k o r > ) 1 / 2 × - d x { k o [ cos θ - D ( x ) sin θ ] E ( x ) - i F ( x ) } exp ( - i K · r ) ,
E ( t ) ( r > , θ t ) = exp [ i ( k o r < - π / 4 ) ] 2 ( 2 π k o r < ) 1 / 2 × - d x { k o [ cos θ t - D ( x ) sin θ t ] E ( x ) + i F ( x ) } exp ( - i K t · r ) ,
I o E ( i ) 2 L cos θ o .
( 1 / I o ) I s ( r ) ( θ ) = ( r > / I o ) E ( r ) ( r > , θ ) 2 ,
( 1 / I o ) I s ( t ) ( θ t ) = ( r < / I o ) E ( t ) ( r < , θ t ) 2 ,
( 4 π / c ) S = ( / μ ) 1 / 2 E 2 = ( μ / ) 1 / 2 H 2 .
( 1 / I o ) Δ I s ( r ) ( θ ) = ( r > / I o ) [ E ( r ) ( r > , θ ) 2 - E ( r ) ( r > , θ ) 2 ] ,
( 1 / I o ) Δ I s ( t ) ( θ t ) = ( r < / I o ) [ E ( t ) ( r < , θ t ) 2 - E ( t ) ( r < , θ t ) 2 ] .
R = 1 I o - π / 2 π / 2 I ( r ) ( θ ) d θ
T = 1 I o - π / 2 π / 2 I ( t ) ( θ t ) d θ t .
R + T = 1.
2 H ( out ) ( r ) + k o 2 H ( out ) ( r ) = 0 ,             z > D ( x ) , ( r V ˜ ) ,
2 H ( in ) ( r ) + ( ω ) k o 2 H ( in ) ( r ) = 0 ,             z > D ( x ) , ( r V ) ,
H ( out ) ( r ) z = D ( + ) ( x ) = H ( in ) ( r ) z = D ( - ) ( x ) ,
[ H ( out ) ( r ) n ] z = D ( + ) ( x ) = 1 ( ω ) [ H ( in ) ( r ) n ] z = D ( - ) ( x ) .
H ( i ) ( r ) + 1 4 π - d x [ H ( out ) ( r ) G o ( r , r ) n - G o ( r , r ) H ( out ) ( r ) n ] γ = H ( out ) ( r ) ,             r V ˜ ,
H ( i ) ( r ) + 1 4 π - d x [ H ( out ) ( r ) G o ( r , r ) n - G o ( r , r ) H ( out ) ( r ) n ] γ = 0 ,             r V ,
- 1 4 π - d x [ H ( in ) ( r ) G ( r , r ) n - G ( r , r ) H ( in ) ( r ) n ] γ = 0 ,             r V ˜ ,
- 1 4 π - d x [ H ( in ) ( r ) G ( r , r ) n - G ( r , r ) H ( in ) ( r ) n ] γ = H ( in ) ( r ) ,             r V .
H ( x ) = H ( out ) [ x , D ( x ) ] = H ( in ) [ x , D ( x ) ] ,
L ( x ) = γ H ( out ) ( r ) n = γ ( ω ) H ( in ) ( r ) n .
H ( r ) ( r > , θ ) = exp [ i ( k o r > - π / 4 ) ] 2 ( 2 π k o r > ) 1 / 2 × - d x { k o [ cos θ - D ( x ) sin θ ] H ( x ) - i L ( x ) } exp ( - i K · r ) ,
H ( t ) ( r > , θ t ) = exp [ i ( k o r < - π / 4 ) ] 2 ( 2 π k o r < ) 1 / 2 × - d x { k o [ cos θ t - D ( x ) sin θ t ] H ( x ) - i L ( x ) } exp ( - i K t · r ) ,
H ( i ) [ x , D ( x ) ] + 1 4 π - d x × { H ( x ) [ G o z - D ( x ) G o x ] - G o L ( x ) } = H ( x ) ,
- 1 4 π - d x { H ( x ) [ G z - D ( x ) G x ] - ( ω ) G L ( x ) } = 0.
( 1 / I o ) I p ( r ) ( θ ) = ( r > / I o ) H ( r ) ( r > , θ 2 ,
( 1 / I o ) I p ( t ) ( θ t ) = ( 1 / ) ( r < / I o ) H ( t ) ( r < , θ t 2 ,
E ( out ) [ x , D ( x ) ] = [ 1 + R s ( x ) ] E ( i ) [ x , D ( x ) ] ,
[ E ( out ) ( r ) n ] z = D ( + ) ( x ) = i K o · n ^ [ 1 - R s ( x ) ] E ( i ) [ x , D ( x ) ] ,
H ( out ) [ x , D ( x ) ] = [ 1 + R p ( x ) ] H ( i ) [ x , D ( x ) ] ,
[ H ( out ) ( r ) n ] z = D ( + ) ( x ) = i K o · n ^ [ 1 - R p ( x ) ] H ( i ) [ x , D ( x ) ] ,
R s ( x ) = cos ϑ ( x ) - cos ϑ t ( x ) cos ϑ ( x ) + cos ϑ t ( x ) ,
R p ( x ) = cos ϑ ( x ) - cos ϑ t ( x ) cos ϑ ( x ) - cos ϑ t ( x ) .
ϑ ( x ) = θ o - α = θ o - arctan D ( x ) ,
sin ϑ t ( x ) = sin ϑ ( x ) .
E KA [ x , D ( x ) ] = [ 1 + R s ( x ) ] E ( i ) × exp { i k o [ x sin θ o - D ( x ) cos θ o ] } ,
F KA [ x , D ( x ) ] = i γ K o · n ^ [ 1 - R s ( x ) ] E ( i ) × exp { i k o [ x sin θ o - D ( x ) cos θ o ] } ,
H KA [ x , D ( x ) ] = [ 1 + R p ( x ) ] H ( i ) × exp { i k o [ x sin θ o - D ( x ) cos θ o ] } ,
L KA [ x , D ( x ) ] = i γ K o · n ^ [ 1 - R p ( x ) ] H ( i ) × exp { i k o [ x sin θ o - D ( x ) cos θ o ] } .
[ A ( o ) + I B ( o ) A - I B ] [ E F ] = 2 [ E ( i ) 0 ] ,
[ A ( o ) + I B ( o ) A - I ( ω ) B ] [ H L ] = 2 [ H ( i ) 0 ] .
E n ( i ) = E ( i ) ( x n ) ,             H n ( i ) = H ( i ) ( x n ) ,
E n = E ( x n ) ,             F n = F ( x n ) ,
H n = H ( x n ) ,             L n = L ( x n ) ,
A m n = { i k o Δ x 2 D ( x n ) ( x m - x n ) - [ D ( x m ) - D ( x n ) ] { ( x m - x n ) 2 + [ D ( x m ) - D ( x n ) ] 2 } 1 / 2 H 1 ( 1 ) ( k o { ( x m - x m ) 2 + [ D ( x m ) - D ( x n ) ] 2 } 1 / 2 ) , m n - Δ x 2 π γ - 2 D ( x n ) , m = n ,
B m n = { ( i Δ x / 2 ) H 0 ( 1 ) ( k o { ( x m - x n ) 2 + [ D ( x m ) - D ( x n ) ] 2 } 1 / 2 ) , m n ( i Δ x / 2 ) H 0 ( 1 ) ( k 0 γ Δ x / 2 e ) , m = n .
δ m n = { 0 , m n 1 , m = n .
1 I o I s ( r ) ( θ ) = 1 8 π k o L cos θ o × | Δ x n = 1 N { k o [ cos θ - D ( x n ) sin θ ] E n - i F n } × exp { - i k o [ x n sin θ + D ( x n ) cos θ ] } | 2 ,
1 I o I s ( t ) ( θ t ) = 1 8 π k o L cos θ o × | Δ x n = 1 N { k o [ cos θ t - D ( x n ) sin θ t ] E n + i F n } × exp { - i k o [ x n sin θ t - D ( x n ) cos θ t ] } | 2 ,
1 I o I p ( r ) ( θ ) = 1 8 π k o L cos θ o × | Δ x n = 1 N { k o [ cos θ - D ( x n ) sin θ ] H n - i L n } × exp { - i k o [ x n sin θ + D ( x n ) cos θ ] } | 2 ,
1 I o I p ( t ) ( θ t ) = - 1 8 π k o L cos θ o × | Δ x n = 1 N { k o [ cos θ t - D ( x n ) sin θ t ] H n + i L n } × exp { - i k o [ x n sin θ t - D ( x n ) cos θ t ] } | 2 ,
E n KA = [ 1 + R s ( x n ) ] exp { i k o [ x n sin θ o - D ( x n ) cos θ o ] } ,
F n KA = - i k o [ D ( x n ) sin θ o + cos θ o ] [ 1 - R s ( x n ) ] × exp { i k o [ x n sin θ o - D ( x n ) cos θ o ] } ,
H n KA = [ 1 + R p ( x n ) ] exp { i k o [ x n sin θ o - D ( x n ) cos θ o ] } ,
L n KA = - i k o [ D ( x n ) sin θ o + cos θ o ] [ 1 - R p ( x n ) ] × exp { i k o [ x n sin θ o - D ( x n ) cos θ o ] } ,
D ( x ) = 0.
σ = [ D ( x ) D ( x ) ] 1 / 2 .
c ( τ ) = ( 1 / σ 2 ) D ( x ) D ( x + τ ) = exp [ - ( τ 2 / T 2 ) ] .

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