Abstract

We calculate the coherent and incoherent scattering of p- and s-polarized light incident from a dielectric medium characterized by a real, positive, dielectric constant 0 onto its one-dimensional, randomly rough interface with a dielectric medium characterized by a real, positive, dielectric constant . We use a perturbation theory with a new small parameter, namely, the dielectric contrast η=0- between the medium of incidence and the scattering medium. The proper self-energy entering the expression for the reflectivity is obtained as an expansion in powers of η through the second order in η, and the reducible vertex function in terms of which the scattered intensity is expressed is obtained as an expansion in powers of η through the fourth. The roughness-induced shifts of the Brewster angle (in p polarization) and of the critical angle for total internal reflection (0>) are obtained. The angular dependence of the intensity of the incoherent component of the scattered light displays an enhanced backscattering peak, which is due to the coherent interference of multiply scattered lateral waves supported by the interface and their reciprocal partners. Analogs of the Yoneda peaks observed in the scattering of x rays from solid surfaces are also present. The results obtained by our small-contrast perturbation theory are in good agreement with those obtained in computer simulation studies.

© 2000 Optical Society of America

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References

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  1. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [CrossRef]
  2. A. G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, New York, 1994).
  3. G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
    [CrossRef]
  4. 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]
  5. C. Baylard, J.-J. Greffet, A. A. Maradudin, “Coherent reflection factor of a random rough surface: applications,” J. Opt. Soc. Am. A 10, 2637–2647 (1993).
    [CrossRef]
  6. T. Kawanishi, H. Ogura, Z. L. Wang, “Scattering of electromagnetic wave from a slightly random dielectric surface. Yoneda peak and Brewster angle in incoherent scattering,” Waves Random Media 7, 351–384 (1997).
    [CrossRef]
  7. T. Tamir, “The lateral wave,” in Electromagnetic SurfaceModes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 521–548.
  8. Y. Yoneda, “Anomalous surface reflection of x rays,” Phys. Rev. 131, 2010–2013 (1963).
    [CrossRef]
  9. A. A. Maradudin, T. A. Leskova, “X-ray scattering from a randomly rough surface,” Waves Random Media 7, 395–434 (1997).
    [CrossRef]
  10. C. S. West, K. A. O’Donnell, “Observations of backscattering enhancement from polaritons on a rough metal surface,” J. Opt. Soc. Am. A 12, 390–397 (1995).
    [CrossRef]
  11. A. A. Maradudin, A. R. McGurn, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  12. G. C. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
    [CrossRef]
  13. R. Kubo, “Generalized cumulant expansion method,” Phys. Soc. Jpn. 17, 1100–1120 (1962).
    [CrossRef]
  14. M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
    [CrossRef]
  15. J.-J. Greffet, “Theoretical model of the shift of the Brewster angle on a rough surface,” Opt. Lett. 17, 238–240 (1992).
    [CrossRef] [PubMed]
  16. M. Saillard, “A characterization tool for dielectric random rough surfaces: Brewster’s phenomenon,” Waves Random Media 2, 67–79 (1992).
    [CrossRef]
  17. V. Freilikher, I. Fuks, M. Pustilnik, “Effect of fluctuations on the reflectivity near the angle of total internal reflection,” J. Electromagn. Waves Appl. 9, 1141–1148 (1995).
  18. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
    [CrossRef]
  19. T. Kawanishi, I. Iwata, M. Kitano, H. Ogura, Z. L. Wang, M. Izutsu, “Brewster’s scattering angle and quasi-anomalous scattering in random scattering from dielectric interfaces,” J. Opt. Soc. Am. A 16, 339–342 (1999).
    [CrossRef]
  20. A. A. Maradudin, E. R. Méndez, “Enhanced backscattering of light from weakly rough, random metal surfaces,” Appl. Opt. 32, 3335–3343 (1993).
    [CrossRef] [PubMed]
  21. A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.
  22. M. Nieto-Vesperinas, J. A. Sánchez-Gil, “Light scattering from a random rough interface with total internal reflection,” J. Opt. Soc. Am. A 9, 424–436 (1992).
    [CrossRef]
  23. D. De Roo, F. T. Ulaby, “Bistatic specular scattering from rough dielectric surfaces,” IEEE Trans. Antennas Propag. 42, 220–227 (1995).
    [CrossRef]
  24. A. Sentenac, J.-J. Greffet, “Mean-field theory of light scattering by one-dimensional rough surfaces,” J. Opt. Soc. Am. A 15, 528–532 (1998).
    [CrossRef]
  25. A. Sentenac, J. Toso, M. Saillard, “Study of coherent scattering from one-dimensional rough surfaces with a mean-field theory,” J. Opt. Soc. Am. A 15, 924–931 (1998).
    [CrossRef]
  26. O. Calvo-Perez, A. Sentenac, J.-J. Greffet, “Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory,” Radio Sci. 34, 311–335 (1999).
    [CrossRef]

1999 (2)

O. Calvo-Perez, A. Sentenac, J.-J. Greffet, “Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory,” Radio Sci. 34, 311–335 (1999).
[CrossRef]

T. Kawanishi, I. Iwata, M. Kitano, H. Ogura, Z. L. Wang, M. Izutsu, “Brewster’s scattering angle and quasi-anomalous scattering in random scattering from dielectric interfaces,” J. Opt. Soc. Am. A 16, 339–342 (1999).
[CrossRef]

1998 (2)

1997 (2)

T. Kawanishi, H. Ogura, Z. L. Wang, “Scattering of electromagnetic wave from a slightly random dielectric surface. Yoneda peak and Brewster angle in incoherent scattering,” Waves Random Media 7, 351–384 (1997).
[CrossRef]

A. A. Maradudin, T. A. Leskova, “X-ray scattering from a randomly rough surface,” Waves Random Media 7, 395–434 (1997).
[CrossRef]

1995 (3)

V. Freilikher, I. Fuks, M. Pustilnik, “Effect of fluctuations on the reflectivity near the angle of total internal reflection,” J. Electromagn. Waves Appl. 9, 1141–1148 (1995).

C. S. West, K. A. O’Donnell, “Observations of backscattering enhancement from polaritons on a rough metal surface,” J. Opt. Soc. Am. A 12, 390–397 (1995).
[CrossRef]

D. De Roo, F. T. Ulaby, “Bistatic specular scattering from rough dielectric surfaces,” IEEE Trans. Antennas Propag. 42, 220–227 (1995).
[CrossRef]

1993 (3)

1992 (3)

1990 (2)

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

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

1985 (2)

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

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

1983 (1)

G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
[CrossRef]

1963 (1)

Y. Yoneda, “Anomalous surface reflection of x rays,” Phys. Rev. 131, 2010–2013 (1963).
[CrossRef]

1962 (1)

R. Kubo, “Generalized cumulant expansion method,” Phys. Soc. Jpn. 17, 1100–1120 (1962).
[CrossRef]

1951 (1)

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

Baylard, C.

Brown, G. C.

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

G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
[CrossRef]

Calvo-Perez, O.

O. Calvo-Perez, A. Sentenac, J.-J. Greffet, “Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory,” Radio Sci. 34, 311–335 (1999).
[CrossRef]

Celli, V.

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

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

G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
[CrossRef]

Coopersmith, M.

G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
[CrossRef]

De Roo, D.

D. De Roo, F. T. Ulaby, “Bistatic specular scattering from rough dielectric surfaces,” IEEE Trans. Antennas Propag. 42, 220–227 (1995).
[CrossRef]

Freilikher, V.

V. Freilikher, I. Fuks, M. Pustilnik, “Effect of fluctuations on the reflectivity near the angle of total internal reflection,” J. Electromagn. Waves Appl. 9, 1141–1148 (1995).

Fuks, I.

V. Freilikher, I. Fuks, M. Pustilnik, “Effect of fluctuations on the reflectivity near the angle of total internal reflection,” J. Electromagn. Waves Appl. 9, 1141–1148 (1995).

Greffet, J.-J.

Haller, M.

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

G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
[CrossRef]

Iwata, I.

Izutsu, M.

Kawanishi, T.

T. Kawanishi, I. Iwata, M. Kitano, H. Ogura, Z. L. Wang, M. Izutsu, “Brewster’s scattering angle and quasi-anomalous scattering in random scattering from dielectric interfaces,” J. Opt. Soc. Am. A 16, 339–342 (1999).
[CrossRef]

T. Kawanishi, H. Ogura, Z. L. Wang, “Scattering of electromagnetic wave from a slightly random dielectric surface. Yoneda peak and Brewster angle in incoherent scattering,” Waves Random Media 7, 351–384 (1997).
[CrossRef]

Kitano, M.

Kubo, R.

R. Kubo, “Generalized cumulant expansion method,” Phys. Soc. Jpn. 17, 1100–1120 (1962).
[CrossRef]

Leskova, T. A.

A. A. Maradudin, T. A. Leskova, “X-ray scattering from a randomly rough surface,” Waves Random Media 7, 395–434 (1997).
[CrossRef]

Luna, R. E.

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.

A. A. Maradudin, T. A. Leskova, “X-ray scattering from a randomly rough surface,” Waves Random Media 7, 395–434 (1997).
[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]

C. Baylard, J.-J. Greffet, A. A. Maradudin, “Coherent reflection factor of a random rough surface: applications,” J. Opt. Soc. Am. A 10, 2637–2647 (1993).
[CrossRef]

A. A. Maradudin, E. R. Méndez, “Enhanced backscattering of light from weakly rough, random metal surfaces,” Appl. Opt. 32, 3335–3343 (1993).
[CrossRef] [PubMed]

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

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

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

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

Marvin, A.

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

Maystre, D.

McGurn, A. R.

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

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

Méndez, E. R.

A. A. Maradudin, 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, E. R. Méndez, “Enhanced backscattering of light from weakly rough, random metal surfaces,” Appl. Opt. 32, 3335–3343 (1993).
[CrossRef] [PubMed]

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 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 grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 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 grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

Nieto-Vesperinas, M.

O’Donnell, K. A.

Ogura, H.

T. Kawanishi, I. Iwata, M. Kitano, H. Ogura, Z. L. Wang, M. Izutsu, “Brewster’s scattering angle and quasi-anomalous scattering in random scattering from dielectric interfaces,” J. Opt. Soc. Am. A 16, 339–342 (1999).
[CrossRef]

T. Kawanishi, H. Ogura, Z. L. Wang, “Scattering of electromagnetic wave from a slightly random dielectric surface. Yoneda peak and Brewster angle in incoherent scattering,” Waves Random Media 7, 351–384 (1997).
[CrossRef]

Pustilnik, M.

V. Freilikher, I. Fuks, M. Pustilnik, “Effect of fluctuations on the reflectivity near the angle of total internal reflection,” J. Electromagn. Waves Appl. 9, 1141–1148 (1995).

Rice, S. O.

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

Saillard, M.

Sánchez-Gil, J. A.

Sentenac, A.

Tamir, T.

T. Tamir, “The lateral wave,” in Electromagnetic SurfaceModes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 521–548.

Toso, J.

Ulaby, F. T.

D. De Roo, F. T. Ulaby, “Bistatic specular scattering from rough dielectric surfaces,” IEEE Trans. Antennas Propag. 42, 220–227 (1995).
[CrossRef]

Voronovich, A. G.

A. G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, New York, 1994).

Wang, Z. L.

T. Kawanishi, I. Iwata, M. Kitano, H. Ogura, Z. L. Wang, M. Izutsu, “Brewster’s scattering angle and quasi-anomalous scattering in random scattering from dielectric interfaces,” J. Opt. Soc. Am. A 16, 339–342 (1999).
[CrossRef]

T. Kawanishi, H. Ogura, Z. L. Wang, “Scattering of electromagnetic wave from a slightly random dielectric surface. Yoneda peak and Brewster angle in incoherent scattering,” Waves Random Media 7, 351–384 (1997).
[CrossRef]

West, C. S.

Yoneda, Y.

Y. Yoneda, “Anomalous surface reflection of x rays,” Phys. Rev. 131, 2010–2013 (1963).
[CrossRef]

Ann. Phys. (N.Y.) (1)

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

Appl. Opt. (1)

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. (1)

D. De Roo, F. T. Ulaby, “Bistatic specular scattering from rough dielectric surfaces,” IEEE Trans. Antennas Propag. 42, 220–227 (1995).
[CrossRef]

J. Electromagn. Waves Appl. (1)

V. Freilikher, I. Fuks, M. Pustilnik, “Effect of fluctuations on the reflectivity near the angle of total internal reflection,” J. Electromagn. Waves Appl. 9, 1141–1148 (1995).

J. Opt. Soc. Am. A (7)

Opt. Lett. (1)

Phys. Rev. (1)

Y. Yoneda, “Anomalous surface reflection of x rays,” Phys. Rev. 131, 2010–2013 (1963).
[CrossRef]

Phys. Rev. B (2)

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

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

Phys. Soc. Jpn. (1)

R. Kubo, “Generalized cumulant expansion method,” Phys. Soc. Jpn. 17, 1100–1120 (1962).
[CrossRef]

Radio Sci. (1)

O. Calvo-Perez, A. Sentenac, J.-J. Greffet, “Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory,” Radio Sci. 34, 311–335 (1999).
[CrossRef]

Surf. Sci. (1)

G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
[CrossRef]

Waves Random Media (4)

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]

T. Kawanishi, H. Ogura, Z. L. Wang, “Scattering of electromagnetic wave from a slightly random dielectric surface. Yoneda peak and Brewster angle in incoherent scattering,” Waves Random Media 7, 351–384 (1997).
[CrossRef]

A. A. Maradudin, T. A. Leskova, “X-ray scattering from a randomly rough surface,” Waves Random Media 7, 395–434 (1997).
[CrossRef]

M. Saillard, “A characterization tool for dielectric random rough surfaces: Brewster’s phenomenon,” Waves Random Media 2, 67–79 (1992).
[CrossRef]

Other (3)

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

A. G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, New York, 1994).

T. Tamir, “The lateral wave,” in Electromagnetic SurfaceModes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 521–548.

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Figures (5)

Fig. 1
Fig. 1

Physical system studied in this paper.

Fig. 2
Fig. 2

Logarithm of the reflectivity of a one-dimensional randomly rough dielectric–dielectric interface (solid curves) and of a planar interface (dashed curves) as functions of the angle of incidence, θ0, when 0=2.4336 and =2.6569 for (a) p polarization and (b) s polarization, and when 0=2.6569 and =2.4336 for (c) p polarization and (d) s polarization.

Fig. 3
Fig. 3

Reflectivity of a one-dimensional randomly rough dielectric–dielectric interface characterized by the Gaussian power spectrum (solid curves) and of a planar interface (dashed curves) as functions of the angle of incidence, θ0, for θ0 in the immediate vicinity of the critical angle for total internal reflection θc, when 0=2.6569 and =2.4336 for (a) p polarization and (b) s polarization. The insets show plots of |Cp(θ0)|2 and |Cs(θ0)|2 as functions of θ0, for θ0 in the immediate vicinity of θc. The arrows indicate the critical angle in the presence of the random roughness.

Fig. 4
Fig. 4

Contribution to the mean differential reflection coefficient from the incoherent component of the scattered light as a function of the scattering angle θs when (a) p-polarized light and (b) s-polarized light are incident from the optically less dense medium and when (c) p-polarized light and (d) s-polarized light are incident from the optically more dense medium. The angle of incidence θ0=0°. The interface roughness is characterized by the Gaussian power spectrum Eq. (2.8) with δ=0.1λ and a=0.2λ. Dashed curves show the results of rigorous numerical simulations.

Fig. 5
Fig. 5

Contribution to the mean differential reflection coefficient from the incoherent component of the scattered light as a function of the scattering angles θs when (a) p-polarized light and (b) s-polarized light are incident from the optically less dense medium and when (c) p-polarized light and (d) s-polarized light are incident from the optically more dense medium. The interface roughness is characterized by the rectangular power spectrum Eq. (2.9) centered at (min{,0})1/2ω/c with δ=0.1λ and θmax=12°.

Equations (101)

Equations on this page are rendered with MathJax. Learn more.

ζ(x1)=0,
ζ(x1)ζ(x1)=δ2W(|x1-x1|),
δ2=ζ2(x1).
ζ(x1)=- dQ2π ζˆ(Q)exp(iQx1).
ζˆ(Q)=0,
ζˆ(Q)ζˆ(Q)=2πδ(Q+Q)δ2g(|Q|),
g(|Q|)=-dx1W(|x1|)exp(-iQx1).
g(|Q|)=πa exp(-Q2a2/4),
g(|Q|)=(π/Δk)(rect{[Q-(ω/c)]/Δk}+rect{[Q+(ω/c)]/Δk}),
Φ>(x1, x3|ω)=exp[ikx1-iα0(k)x3]+- dq2π R(q|k)×exp[iqx1+iα0(q)x3],
Φ<(x1, x3|ω)=- dq2π T(q|k)exp[iqx1-iα(q)x3],
α0(q)=[0(ω2/c2)-q2]1/2,Re α0(q)>0,Im α0(q)>0,α(q)=[(ω2/c2)-q2]1/2,Re α(q)>0,Im α(q)>0.
Rθscoh=1L1 0 ω2πc cos2 θscos θ0 |R(q|k)|2,
Rθsincoh=1L1 0 ω2πc cos2 θscos θ0[|R(q|k)|2-|R(q|k)|2].
- dq2π M(p|q)R(q|k)=-N(p|k),
Mp(p|q)=pq+α(p)α0(q)α(p)-α0(q) I[α(p)-α0(q)|p-q],
Np(p|k)=pk-α(p)α0(k)α(p)+α0(k) I[α(p)+α0(k)|p-k],
Ms(p|q)=I[α(p)-α0(q)|p-q]α(p)-α0(q),
Ns(p|k)=I[α(p)+α0(k)|p-k]α(p)+α0(k),
I(γ|Q)=-dx1 exp{-i[Qx1+γζ(x1)]}.
R(q|k)=2πδ(q-k)R(0)(k)-2iG(0)(q)T(q|k)G(0)(k)α0(k),
Rp(0)(k)=α0(k)-0α(k)α0(k)+0α(k)Δp(k)dp(k),
Rs(0)(k)=α0(k)-α(k)α0(k)+a(k)Δs(k)ds(k),
Gp(0)(k)=iα0(k)+0α(k),
Gs(0)(k)=iα0(k)+α(k).
T(q|k)=V(q|k)+- dp2π V(q|p)G(0)(p)T(p|k)
=V(q|k)+- dp2π T(q|p)G(0)(p)V(p|k),
- dq2π [N(p|q)-M(p|q)] V(q|k)α0(q)
=N(p|k)+M(p|k)R(0)(k)G(0)(k)α0(k).
G(q|k)=2πδ(q-k)G(0)(k)+G(0)(q)×- dp2π V(q|p)G(p|k)
=2πδ(q-k)G(0)(k)+G(0)(q)T(q|k)G(0)(k).
R(q|k)=-2πδ(q-k)-2iG(q|k)α0(k).
I(γ|Q)=2πδ(Q)+J(γ|Q)
J(γ|Q)=-dx1 exp(-iQx1){exp[-iγζ(x1)]-1}.
V(q|k)=-iv(q|k)/˜,
v(q|k)=ηA(q|k)+η- dp2π F(q|p)v(p|k).
A(q|k)=[n(q|k)d(k)+m(q|k)Δ(k)] 12˜α0(k),
F(q|k)=[m(q|k)-n(q|k)] 12˜α0(k),
mp(q|k)=qk+α(q)α0(k)α(q)-α0(k) J[α(q)-α0(k)|q-k],
np(q|k)=qk-α(q)α0(k)α(q)+α0(k) J[α(q)+α0(k)|q-k],
ms(q|k)=ω2/c2α(q)-α0(k) J[α(q)+α0(k)|q-k],
ns(q|k)=(ω2/c2)α(q)+α0(k) J[α(q)+α0(k)|q-k].
v(q|k)=ηA(q|k)+η2- dp12π F(q|p1)A(p1|k)+η3- dp12π - dp22π F(q|p1)F(p1|p2)×A(p2|k)+ .
R(q|k)=-2πδ(q-k)-2iG(q|k)α0(k).
G(q|k)=2πδ(q-k)G(k)=2πδ(q-k) 1G(0)(k)-1-M(k),
M(q|k)=2πδ(q-k)M(k),
M(q|k)=V(q|k)+- dp2π - dr2π M(q|p)×G(p|r)w(r|k),
w(q|k)=V(q|k)-M(q|k).
R(q|k)=2πδ(q-k)r(θ0),
rp(θ0)=0 cos θ0-0(0 cos2 θ0-η)1/2+i(c/ω)Mp[0(ω/c)sin θ0]0 cos θ0+0(0 cos2 θ0-η)1/2-i(c/ω)Mp[0(ω/c)sin θ0],
rs(θ0)=0 cos θ0-(0 cos2 θ0-η)1/2+i(c/ω)Ms[0(ω/c)sin θ0]0 cos θ0+(0 cos2 θ0-η)1/2-i(c/ω)Ms[0(ω/c)sin θ0].
Rp,s(θ0)=|rp,s(θ0)|2.
M(q|k)=n=1Mn(q|k),
M1(q|k)={V(q|k)},
M2(q|k)=- dp2π {V(q|p)V(p|k)}G(0)(p)-1-M(p),
J1J2Jn=i=1n{Ji}+θ(n-2)i, j=1(i<j)n{JiJj}k=1(ki, j)n{Jk}+θ(n-3)i, j,k=1(i<j<k)n{JiJjJk}l=1(li, j, k)n{Jl}+θ(n-4)i, j,k,l=1(i<j<k<l)n{JiJjJkJl}×m=1(mi, j, k, l)n{Jm}+θ(n-4)×i, j=1(i<j)nk,l=1(k<l)n{JiJj}{JkJl}m=1(mi, j, k, l)n{Jm}
(ik, l; jk, l)
+termswithfewerthann-4factorsof{Jp},
A(q|k)=2πδ(q-k) a(k)η,
F(q|k)=2πδ(q-k)-f(k)η,
a(k)=d(k)Δ(k)2˜α0(k) [Y(k)-X(k)],
f(k)=d(k)X(k)+Δ(k)Y(k)2˜α0(k)-1,
X(k)=exp-δ22 [α(k)-α0(k)]2,
Y(k)=exp-δ22 [α(k)+α0(k)]2.
M(k)=1i˜ a(k)1+f(k)+η21i˜2 1[1+f(k)]2×- dp2π {m(k|p)G(0)(p)X-1(p)A1(p|k)},
A1(q|k)=[n(q|k)d(k)X(k)+m(q|k)Δ(k)Y(k)]12˜α0(k).
{J(γ1|Q1)J(γ2|Q2)}
=2πδ(Q1+Q2)exp-12 (γ12+γ22)δ2×-du exp(-iQ1u){exp[-γ1γ2δ 2W(|u|)]-1}.
[|R(q|k)|2-|R(q|k)|2]
=4α02(k)[|G(q|k)|2-|G(q|k)|2].
G(q|k)=G(q)2πδ(q-k)+G(q)t(q|k)G(k),
- dp2π w(q|p)G(p|k)=t(q|k)G(k),
t(q|k)=w(q|k)+- dp2π w(q|p)G(p)t(p|k).
G(q|k)G*(q|k)=(2π)2δ2(q-k)|G(k)|2+|G(q)|2{τ(q, k|q, k)}|G(k)|2,
Rθsincoh=1L1 0 2π ωc3cos2 θs cos θ0|G(q)|2×τ(q, k|q, k)|G(k)|2.
t(q|k)=w(q|k)+- dp2π w(q|p)G(p)w(p|k)+- dp2π - dr2π w(q|p)G(p)w(p|r)×G(r)w(r|k)+ ,
{τ(q, k|q, k)}=η2˜2 1|1+f(q)|2 [τ˜2(q|k)+ητ˜3(q|k)+η2τ˜4(q|k)] 1|1+f(k)|2,
{J(γ1|Q1)J(γn|Qn)}
=2πδ(Q1+Q2++Qn){J(γ1|Q1)J(γn|Qn)}0,
τ˜2(q|k)=L1{A1(q|k)A1*(q|k)}0,
τ˜3(q|k)=2L1 Imm(q|p) G(p)1+f(p)×A1(p|k)A1*(q|k)0,
τ˜4(1)(q|k)=L1m(q|p) G(p)1+f(p) A1(p|k)m*(q|r)×G*(r)1+f*(r) A1*(r|k)0+2 Rem(q|p) G(p)1+f(p)×m(p|r) G(r)1+f(r) A1(r|k)A1*(q|k)0,
τ˜4(2)(q|k)=L1m(q|p) G(p)1+f(p){A1(p|k)×m*(q|q+k-p)}0 G*(q+k-p)1+f*(q+k-p)×A1*(q+k-p|k)0+m(q|p) G(p)1+f(p) m*(q|p)0×A1(p|k) G*(p)1+f*(p) A1*(p|k)0+2 Rem(q|p) G(p)1+f(p)×A1(p-q+k|k)0m(p|p-q+k)×G(p-q+k)1+f(p-q+k) A1*(q|k)0,
θ0=θB(0)=tan-101/2
0 cos θ0-0(0 cos2 θ0-η)1/2
=-Re{i(c/ω)Mp[0(ω/c)sin θ0]},
θ0-θB(0)=(c/ω)Im Mp[0(ω/c)sin θB(0)]η[(0/)(0+)sin θB(0)].
θ0=θc(0)=sin-101/2,
Cp(θ0)(-0 sin2 θ0)1/2-i(/0)(c/ω)×Mp[0(ω/c)sin θ0]=0
Cs(θ0)(-0 sin2 θ0)1/2-i(c/ω)×Ms[0(ω/c)sin θ0]=0
Cp(θ0)=0 cos θ0 1-rp(θ0)1+rp(θ0),
Cs(θ0)=0 cos θ0 1-rs(θ0)1+rs(θ0),
q=(ω/c)0 sin θs,k=(ω/c)0 sin θ0,
α(q)=(ω/c)(cos2 θs-η)1/2,
α(k)=(ω/c)(cos2 θ0-η)1/2.
{v(q|k)}=v(q|k)=ηA(q|k)+η2- dp12π F(q|p1)A(p1|k)+η3- dp12π - dp22π F(q|p1)F(p1|p2)A(p2|k)+η4- dp12π - dp22π- dp32π F(q|p1)×F(p1|p2)F(p2|p3)A(p3|k)+ .
{v(q|k)}(0)=2πδ(q-k)[a(k)-f(k)a(k)+f2(k)a(k)-]=2πδ(q-k) a(k)1+f(k).
{v(q|k)}(21)=η2[1-f(q)+f2(q)-]×- dp2π {F(q|p)[1-f(p)+f2(p)-]A(p|k)}=η2 11+f(q) - dp2π {F(q|p)A(p|k)}1+f(p).
{v(q|k)}(22)=η2[1-f(q)+f2(q)-]×- dp2π {F(q|p)[1-f(p)+f2(p)-]F(p|k)}×[1-f(k)+f2(k)-]a(k)=η2 11+f(q) - dp2π {F(q|p)F(p|k)}1+f(p)×a(k)1+f(k).
{v(q|p)v(p|k)}=v(q|p)v(p|k)-v(q|p)v(p|k),
{v(q|p)v(p|k)}=η2A(q|p)1+f(q)+F(q|p)1+f(q) a(p)1+f(p)×A(p|k)1+f(p)+F(p|k)1+f(p)×a(k)1+f(k).

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