Abstract

The scattering of light from a one-dimensional randomly rough dielectric film deposited on a flat reflecting substrate is studied by means of a small-amplitude perturbation theory and rigorous numerical simulations. In particular, we study the appearance of well-defined fringes in the angular distribution of the diffusely scattered intensity and their dependence on the angle of incidence, the roughness of the film, and the film’s mean thickness. We find that, for slightly rough films, the angle of incidence modulates the intensity of the fringes but has no effect on their angular position. For rougher films the contrast of the pattern decreases, and the fringes move with the angle of incidence in such a way that there are always bright fringes in the specular and backscattering directions. Eventually, for very rough films, the fringe pattern disappears, and a well-defined backscattering peak emerges in the retroreflection direction.

© 1998 Optical Society of America

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References

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  1. Sir Isaac Newton, Opticks (originally published in London, 1704; new edition by Dover, New York, 1952), pp. 289 ff.
  2. T. Young, “The Bakerian lecture. On the theory of light and colours,” Philos. Trans. R. Soc. London Part I 12–48, (1802).
  3. Sir John Herschel, On the Theory of Light (Encyclopedia Metropolitana, London, 1828), Art. 676.
  4. L. Schläfli, “Ueber eine durch zerstreutes Licht bewirkte Interferenzerscheinung,” Mitt. Natureforsch. Ges. (Bern) Nos. 131–132, 177–183 (1848).
  5. Sir George Stokes, “On the colours of thick plates,” Trans. Cambridge Philos. Soc. 9, Part 2, 147–176 (1851).
  6. C. Exner, “Ueber die durch zahlreiche, unregelmaessig vertheilte Koerperchen hervorgebrachten Beugungserscheinung,” Sitzungsber. K. Preuss. Akad. Wiss. 90, Abt. II, 827–879 (1884).
  7. E. Lommel, “Ueber die Newtow’schen Staubringe,” Ann. Phys. Chem. N. F. 8, No. 10, 193–244 (1879).
  8. P. Selényi, “Über Lichtzerstreuung im Raume Wienerscher Interferenzen und neue, diesen reziproke Interferenzerscheinungen,” Ann. Phys. Chem. 35, 440–460 (1911).
  9. C. V. Raman and G. L. Datta, “On Quételet’s rings and other allied phenomena,” Philos. Mag. 42, 826–840 (1921).
    [CrossRef]
  10. C. F. Meyer, The Diffraction of Light, X-rays, and Material Particles (Edwards, Ann Arbor, Mich., 1949), p. 244.
  11. R. W. Pohl, Optik und Atomphysik (Springer-Verlag, Berlin, 1954), p. 82.
  12. A. J. de Witte, “Interference in scattered light,” Am. J. Phys. 35, 301–313 (1967).
    [CrossRef]
  13. M. Françon, “Information processing using speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1984).
  14. J. M. Elson, J. P. Rahn, and J. M. Bennett, “Light scattering from multilayer optics: comparisons of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
    [CrossRef] [PubMed]
  15. A. Duparré and S. Kassam, “Relation between light scattering and the microstructure of optical thin films,” Appl. Opt. 32, 5475–5480 (1993).
    [CrossRef] [PubMed]
  16. C. Amra, “From light scattering to the microstructure of thin film multilayers,” Appl. Opt. 32, 5481–5491 (1993).
    [CrossRef] [PubMed]
  17. R. García-Llamas and L. E. Regalado, “Transmitted scattered light from a thin film with shallow rough interfaces,” Appl. Opt. 35, 5595–5599 (1996).
    [CrossRef]
  18. J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D Freilikher, M. Pustilnik, and I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
    [CrossRef]
  19. J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, and V. D. Freilikher, “Satellite peaks in the p-polarized light scattered from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
    [CrossRef]
  20. J. Q. Lu, A. A. Maradudin, and T. R. Michel, “Enhanced backscattering from a rough dielectric film on a reflecting substrate,” J. Opt. Soc. Am. B 8, 311–318 (1991).
    [CrossRef]
  21. Y. Kuga and A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
    [CrossRef]
  22. L. Tsang and A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. A 1, 836–839 (1984).
    [CrossRef]
  23. A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  24. E. R. Méndez and K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
    [CrossRef]
  25. M. Nieto-Vesperinas and 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]
  26. E. Jakeman, “Enhanced backscattering through a deep random phase screen,” J. Opt. Soc. Am. A 5, 1638–1648 (1988).
    [CrossRef]
  27. A. A. Maradudin, E. R. Méndez, and 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]
  28. See, for example, E. Hecht and A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), p. 294.

1996

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, and V. D. Freilikher, “Satellite peaks in the p-polarized light scattered from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

R. García-Llamas and L. E. Regalado, “Transmitted scattered light from a thin film with shallow rough interfaces,” Appl. Opt. 35, 5595–5599 (1996).
[CrossRef]

1994

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D Freilikher, M. Pustilnik, and I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

1993

1991

1989

1988

1987

M. Nieto-Vesperinas and 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]

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

1985

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

1984

1980

1967

A. J. de Witte, “Interference in scattered light,” Am. J. Phys. 35, 301–313 (1967).
[CrossRef]

1921

C. V. Raman and G. L. Datta, “On Quételet’s rings and other allied phenomena,” Philos. Mag. 42, 826–840 (1921).
[CrossRef]

1884

C. Exner, “Ueber die durch zahlreiche, unregelmaessig vertheilte Koerperchen hervorgebrachten Beugungserscheinung,” Sitzungsber. K. Preuss. Akad. Wiss. 90, Abt. II, 827–879 (1884).

1802

T. Young, “The Bakerian lecture. On the theory of light and colours,” Philos. Trans. R. Soc. London Part I 12–48, (1802).

Amra, C.

Bennett, J. M.

Celli, V.

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

Datta, G. L.

C. V. Raman and G. L. Datta, “On Quételet’s rings and other allied phenomena,” Philos. Mag. 42, 826–840 (1921).
[CrossRef]

de Witte, A. J.

A. J. de Witte, “Interference in scattered light,” Am. J. Phys. 35, 301–313 (1967).
[CrossRef]

Duparré, A.

Elson, J. M.

Exner, C.

C. Exner, “Ueber die durch zahlreiche, unregelmaessig vertheilte Koerperchen hervorgebrachten Beugungserscheinung,” Sitzungsber. K. Preuss. Akad. Wiss. 90, Abt. II, 827–879 (1884).

Freilikher, V. D

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D Freilikher, M. Pustilnik, and I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

Freilikher, V. D.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, and V. D. Freilikher, “Satellite peaks in the p-polarized light scattered from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

García-Llamas, R.

Ishimaru, A.

Jakeman, E.

Kassam, S.

Kuga, Y.

Lu, J. Q.

Lu, Jun Q.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, and V. D. Freilikher, “Satellite peaks in the p-polarized light scattered from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D Freilikher, M. Pustilnik, and I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

Maradudin, A. A.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, and V. D. Freilikher, “Satellite peaks in the p-polarized light scattered from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D Freilikher, M. Pustilnik, and I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

J. Q. Lu, A. A. Maradudin, and T. R. Michel, “Enhanced backscattering from a rough dielectric film on a reflecting substrate,” J. Opt. Soc. Am. B 8, 311–318 (1991).
[CrossRef]

A. A. Maradudin, E. R. Méndez, and 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]

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

McGurn, A. R.

A. R. McGurn, A. A. Maradudin, and 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, E. R. Méndez, and 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]

E. R. Méndez and 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.

Michel, T. R.

Nieto-Vesperinas, M.

O’Donnell, K. A.

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

Pustilnik, M.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D Freilikher, M. Pustilnik, and I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

Rahn, J. P.

Raman, C. V.

C. V. Raman and G. L. Datta, “On Quételet’s rings and other allied phenomena,” Philos. Mag. 42, 826–840 (1921).
[CrossRef]

Regalado, L. E.

Sánchez-Gil, J. A.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, and V. D. Freilikher, “Satellite peaks in the p-polarized light scattered from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D Freilikher, M. Pustilnik, and I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

Soto-Crespo, J. M.

Tsang, L.

Young, T.

T. Young, “The Bakerian lecture. On the theory of light and colours,” Philos. Trans. R. Soc. London Part I 12–48, (1802).

Yurkevich, I.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D Freilikher, M. Pustilnik, and I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

Am. J. Phys.

A. J. de Witte, “Interference in scattered light,” Am. J. Phys. 35, 301–313 (1967).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, and V. D. Freilikher, “Satellite peaks in the p-polarized light scattered from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

E. R. Méndez and 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. Lett.

Philos. Mag.

C. V. Raman and G. L. Datta, “On Quételet’s rings and other allied phenomena,” Philos. Mag. 42, 826–840 (1921).
[CrossRef]

Philos. Trans. R. Soc. London Part I

T. Young, “The Bakerian lecture. On the theory of light and colours,” Philos. Trans. R. Soc. London Part I 12–48, (1802).

Phys. Rev. B

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

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D Freilikher, M. Pustilnik, and I. Yurkevich, “Scattering of electromagnetic waves from a bounded medium with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

Sitzungsber. K. Preuss. Akad. Wiss.

C. Exner, “Ueber die durch zahlreiche, unregelmaessig vertheilte Koerperchen hervorgebrachten Beugungserscheinung,” Sitzungsber. K. Preuss. Akad. Wiss. 90, Abt. II, 827–879 (1884).

Other

E. Lommel, “Ueber die Newtow’schen Staubringe,” Ann. Phys. Chem. N. F. 8, No. 10, 193–244 (1879).

P. Selényi, “Über Lichtzerstreuung im Raume Wienerscher Interferenzen und neue, diesen reziproke Interferenzerscheinungen,” Ann. Phys. Chem. 35, 440–460 (1911).

C. F. Meyer, The Diffraction of Light, X-rays, and Material Particles (Edwards, Ann Arbor, Mich., 1949), p. 244.

R. W. Pohl, Optik und Atomphysik (Springer-Verlag, Berlin, 1954), p. 82.

Sir John Herschel, On the Theory of Light (Encyclopedia Metropolitana, London, 1828), Art. 676.

L. Schläfli, “Ueber eine durch zerstreutes Licht bewirkte Interferenzerscheinung,” Mitt. Natureforsch. Ges. (Bern) Nos. 131–132, 177–183 (1848).

Sir George Stokes, “On the colours of thick plates,” Trans. Cambridge Philos. Soc. 9, Part 2, 147–176 (1851).

Sir Isaac Newton, Opticks (originally published in London, 1704; new edition by Dover, New York, 1952), pp. 289 ff.

M. Françon, “Information processing using speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1984).

See, for example, E. Hecht and A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), p. 294.

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Figures (11)

Fig. 1
Fig. 1

Scattering system studied in this work.

Fig. 2
Fig. 2

Three main kinds of scattering trajectories reflected from a weakly rough dielectric film on a reflecting substrate that give rise to (a) Selényi fringes, (b) Quételet fringes, (c) enhanced backscattering.

Fig. 3
Fig. 3

Angular distributions of the incoherent component of the mean DRC for s polarization, with λ=0.6328 µm, d=2.6896 +i0.0075, a=0.26 µm, and δ=0.01 µm. Solid curve, numerical simulation; dashed curve, perturbation theory. (a) θ0=18° and d=6 µm; (b) θ0=10° and d=3 µm. The vertical dotted lines indicate the positions of the extrema predicted by Eq. (1).

Fig. 4
Fig. 4

Same as Fig. 3 but for p polarization.

Fig. 5
Fig. 5

Gray-level intensity plots of the perturbation-theoretic results for the incoherent component of the mean DRC as a function of the angles of incidence and scattering for λ=0.6328 µm, d=6µm, d=2.6896+i0.0075, a=0.26 µm, and δ=0.01 µm. (a) s polarization, (b) p polarization.

Fig. 6
Fig. 6

Positions of the extrema predicted by (a) Eqs. (1) and (4), and (b) Eq. (9). The conditions for constructive interference are denoted by the solid lines and those for destructive interference by the dashed lines. An additional phase factor of π reverses the contrast of the fringes in (a).

Fig. 7
Fig. 7

Same as Fig. 5 but for d=3 µm.

Fig. 8
Fig. 8

Gray-level intensity plot of the numerical simulation results for the incoherent component of the mean DRC as a function of the angles of incidence and scattering for the following parameters: λ=0.6328 µm, d=6 µm, d=2.6896+i0.0075, a=0.26 µm, δ=0.1 µm, L=25.6 µm, N=400, and Np=500. (a) s polarization, (b) p polarization.

Fig. 9
Fig. 9

Angular distribution of the incoherent component of the mean DRC from Fig. 8. Solid curve, θ0=8°; dashed curve, θ0 =18°. (a) s polarization, (b) p polarization. The vertical dotted lines indicate the positions of the extrema predicted by Eq. (9).

Fig. 10
Fig. 10

Same as Fig. 8 but for rms height δ=0.25 µm.

Fig. 11
Fig. 11

Same as Fig. 9 but for rms height δ=0.25 µm. The vertical dotted lines indicate the backscattering directions.

Equations (11)

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

Δ=2nd cos θds,
maxima:2πλnd cos θds=m+12π;
minima:2πλnd cos θds=mπ,
Δ=2nd cos θd0,
θsmax=18.7°, 31.3°, 41°, 50°, 59°, 69.8°,
θsmin=7.5°, 25.6°, 36.3°, 45.5°, 54.5°, 64°, 77.3°.
θsmax=7.5°, 36.3°, 54.5°, 77.3°,
θsmin=25.6°, 45.5°, 64°.
Δ=2nd(cos θd0-cos θds),
θ0=8°:θs=±8°, ±25.8°, ±36.4°, ±45.7°, ±54.6°, ±64.3°, ±77.5°,
θ0=18°:θs=±18°, ±30°, ±40°, ±49.6°, ±58.7°, ±69.4°.

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