Abstract

It is shown that the long-scale (smooth) component of the roughness spectrum of a slightly rough dielectric layer critically affects the angular distribution of radiation scattered from the surface. The interference pattern obtained from a sample with only small-scale roughness differs drastically from a sample with the same small-scale roughness but possessing slight (of the order λ/10) variation of the thickness of the dielectric layer. It is shown that when interference phenomena are significant and the dielectric film has long-scale roughness, conventional perturbation theory is invalid, even if the rms of roughness is much smaller than the wavelength. A model is presented that correctly predicts the measured angular intensity distributions in the scattered-light field for samples that possess arbitrary scales of roughness.

© 1999 Optical Society of America

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References

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  1. T. Young, “Lecture on the theory of light and colours,” Philos. Trans. R. Soc. London, Part 1, 41–48 (1802).
  2. G. Stokes, “On the colours of thick plates,” Trans. Cambridge Philos. Soc. 9, 147–176 (1851).
  3. C. V. Raman, G. L. Datta, “On Quetelet’s rings and other allied phenomena,” Philos. Mag. Ser. 6 42, 826–840 (1921).
    [CrossRef]
  4. A. J. de Witte, “Interference in scattered light,” Am. J. Phys. 35, 301–313 (1967).
    [CrossRef]
  5. M. Françon, Laser Speckle and Applications in Optics (Academic, New York, 1979).
  6. M. Françon, “Information processing using speckle pat-terns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1984).
  7. J. Q. Lu, J. A. Sánchez-Gil, E. R. Méndez, Z.-H. Gu, A. A. Maradudin, “Scattering of light from a rough dielectric film on a reflecting substrate: diffuse fringes,” J. Opt. Soc. Am. A 15, 185–195 (1998).
    [CrossRef]
  8. A. I. Kalmykov, I. Fuks, I. Sherbinin, V. Tsymbal, A. Matveev, V. Freilikher, M. Fix, “Radar observation of strong subsurface scatterers,” in Proceedings of the International Geoscience and Remote Sensing Symposium ’95 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), Vol. 3, p. 1702.
  9. J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
    [CrossRef]
  10. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  11. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  12. V. Freilikher, M. Pustilnik, I. Yurkevich, V. Tatarskii, “Polarization of light scattered from slightly rough dielectric film,” Opt. Lett. 19, 1382–1384 (1994).
    [CrossRef] [PubMed]
  13. F. G. Bass, I. M. Fuks, Wave Scattering from a Statistically Rough Surface (Pergamon, New York, 1979).
  14. Yu. Kaganovskii, V. D. Freilikher, V. V. Grischenko, I. M. Popova, “Estimate of micro roughness parameters on a solid surface by light scattering method,” Pribory i Tekh. Exp., No. 4, 258–260 (1976) (in Russian).
  15. Yu. S. Kaganovskii, V. D. Freilikher, E. Kanzieper, Y. Nafcha, M. Rosenbluh, “Interference in light scattering from slightly rough dielectric layers,” Opt. Lett. 23, 316–318 (1998).
    [CrossRef]
  16. V. Freilikher, M. Pustilnik, I. Yurkevich, “Wave scattering from a bounded medium with disorder,” Phys. Lett. A 193, 467–470 (1994).
    [CrossRef]
  17. J. Sánchez-Gil, A. Maradudin, J. Lu, V. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
    [CrossRef]
  18. M. Nieto Vesperinas, J. C. Dainty, eds., Scattering in Volumes and Surfaces (North-Holland, Amsterdam, 1990).
  19. V. Freilikher, E. Kanzieper, A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
    [CrossRef]

1998 (2)

1997 (1)

V. Freilikher, E. Kanzieper, A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[CrossRef]

1996 (1)

J. Sánchez-Gil, A. Maradudin, J. Lu, V. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

1995 (1)

J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
[CrossRef]

1994 (2)

V. Freilikher, M. Pustilnik, I. Yurkevich, “Wave scattering from a bounded medium with disorder,” Phys. Lett. A 193, 467–470 (1994).
[CrossRef]

V. Freilikher, M. Pustilnik, I. Yurkevich, V. Tatarskii, “Polarization of light scattered from slightly rough dielectric film,” Opt. Lett. 19, 1382–1384 (1994).
[CrossRef] [PubMed]

1976 (1)

Yu. Kaganovskii, V. D. Freilikher, V. V. Grischenko, I. M. Popova, “Estimate of micro roughness parameters on a solid surface by light scattering method,” Pribory i Tekh. Exp., No. 4, 258–260 (1976) (in Russian).

1967 (1)

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

1921 (1)

C. V. Raman, G. L. Datta, “On Quetelet’s rings and other allied phenomena,” Philos. Mag. Ser. 6 42, 826–840 (1921).
[CrossRef]

1851 (1)

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

1802 (1)

T. Young, “Lecture on the theory of light and colours,” Philos. Trans. R. Soc. London, Part 1, 41–48 (1802).

Bass, F. G.

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

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Datta, G. L.

C. V. Raman, G. L. Datta, “On Quetelet’s rings and other allied phenomena,” Philos. Mag. Ser. 6 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]

Fix, M.

A. I. Kalmykov, I. Fuks, I. Sherbinin, V. Tsymbal, A. Matveev, V. Freilikher, M. Fix, “Radar observation of strong subsurface scatterers,” in Proceedings of the International Geoscience and Remote Sensing Symposium ’95 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), Vol. 3, p. 1702.

Françon, M.

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

M. Françon, Laser Speckle and Applications in Optics (Academic, New York, 1979).

Freilikher, V.

V. Freilikher, E. Kanzieper, A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[CrossRef]

J. Sánchez-Gil, A. Maradudin, J. Lu, V. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

V. Freilikher, M. Pustilnik, I. Yurkevich, V. Tatarskii, “Polarization of light scattered from slightly rough dielectric film,” Opt. Lett. 19, 1382–1384 (1994).
[CrossRef] [PubMed]

V. Freilikher, M. Pustilnik, I. Yurkevich, “Wave scattering from a bounded medium with disorder,” Phys. Lett. A 193, 467–470 (1994).
[CrossRef]

A. I. Kalmykov, I. Fuks, I. Sherbinin, V. Tsymbal, A. Matveev, V. Freilikher, M. Fix, “Radar observation of strong subsurface scatterers,” in Proceedings of the International Geoscience and Remote Sensing Symposium ’95 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), Vol. 3, p. 1702.

Freilikher, V. D.

Yu. S. Kaganovskii, V. D. Freilikher, E. Kanzieper, Y. Nafcha, M. Rosenbluh, “Interference in light scattering from slightly rough dielectric layers,” Opt. Lett. 23, 316–318 (1998).
[CrossRef]

Yu. Kaganovskii, V. D. Freilikher, V. V. Grischenko, I. M. Popova, “Estimate of micro roughness parameters on a solid surface by light scattering method,” Pribory i Tekh. Exp., No. 4, 258–260 (1976) (in Russian).

Fuks, I.

A. I. Kalmykov, I. Fuks, I. Sherbinin, V. Tsymbal, A. Matveev, V. Freilikher, M. Fix, “Radar observation of strong subsurface scatterers,” in Proceedings of the International Geoscience and Remote Sensing Symposium ’95 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), Vol. 3, p. 1702.

Fuks, I. M.

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

Grischenko, V. V.

Yu. Kaganovskii, V. D. Freilikher, V. V. Grischenko, I. M. Popova, “Estimate of micro roughness parameters on a solid surface by light scattering method,” Pribory i Tekh. Exp., No. 4, 258–260 (1976) (in Russian).

Gu, Z.-H.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Kaganovskii, Yu.

Yu. Kaganovskii, V. D. Freilikher, V. V. Grischenko, I. M. Popova, “Estimate of micro roughness parameters on a solid surface by light scattering method,” Pribory i Tekh. Exp., No. 4, 258–260 (1976) (in Russian).

Kaganovskii, Yu. S.

Kalmykov, A. I.

A. I. Kalmykov, I. Fuks, I. Sherbinin, V. Tsymbal, A. Matveev, V. Freilikher, M. Fix, “Radar observation of strong subsurface scatterers,” in Proceedings of the International Geoscience and Remote Sensing Symposium ’95 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), Vol. 3, p. 1702.

Kanzieper, E.

Yu. S. Kaganovskii, V. D. Freilikher, E. Kanzieper, Y. Nafcha, M. Rosenbluh, “Interference in light scattering from slightly rough dielectric layers,” Opt. Lett. 23, 316–318 (1998).
[CrossRef]

V. Freilikher, E. Kanzieper, A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[CrossRef]

Lu, J.

J. Sánchez-Gil, A. Maradudin, J. Lu, V. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

Lu, J. Q.

Maradudin, A.

V. Freilikher, E. Kanzieper, A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[CrossRef]

J. Sánchez-Gil, A. Maradudin, J. Lu, V. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

Maradudin, A. A.

Matveev, A.

A. I. Kalmykov, I. Fuks, I. Sherbinin, V. Tsymbal, A. Matveev, V. Freilikher, M. Fix, “Radar observation of strong subsurface scatterers,” in Proceedings of the International Geoscience and Remote Sensing Symposium ’95 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), Vol. 3, p. 1702.

McGilp, J. F.

J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
[CrossRef]

Méndez, E. R.

Nafcha, Y.

Popova, I. M.

Yu. Kaganovskii, V. D. Freilikher, V. V. Grischenko, I. M. Popova, “Estimate of micro roughness parameters on a solid surface by light scattering method,” Pribory i Tekh. Exp., No. 4, 258–260 (1976) (in Russian).

Pustilnik, M.

J. Sánchez-Gil, A. Maradudin, J. Lu, V. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

V. Freilikher, M. Pustilnik, I. Yurkevich, “Wave scattering from a bounded medium with disorder,” Phys. Lett. A 193, 467–470 (1994).
[CrossRef]

V. Freilikher, M. Pustilnik, I. Yurkevich, V. Tatarskii, “Polarization of light scattered from slightly rough dielectric film,” Opt. Lett. 19, 1382–1384 (1994).
[CrossRef] [PubMed]

Raman, C. V.

C. V. Raman, G. L. Datta, “On Quetelet’s rings and other allied phenomena,” Philos. Mag. Ser. 6 42, 826–840 (1921).
[CrossRef]

Rosenbluh, M.

Sánchez-Gil, J.

J. Sánchez-Gil, A. Maradudin, J. Lu, V. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

Sánchez-Gil, J. A.

Sherbinin, I.

A. I. Kalmykov, I. Fuks, I. Sherbinin, V. Tsymbal, A. Matveev, V. Freilikher, M. Fix, “Radar observation of strong subsurface scatterers,” in Proceedings of the International Geoscience and Remote Sensing Symposium ’95 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), Vol. 3, p. 1702.

Stokes, G.

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

Tatarskii, V.

Tsymbal, V.

A. I. Kalmykov, I. Fuks, I. Sherbinin, V. Tsymbal, A. Matveev, V. Freilikher, M. Fix, “Radar observation of strong subsurface scatterers,” in Proceedings of the International Geoscience and Remote Sensing Symposium ’95 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), Vol. 3, p. 1702.

van de Hulst, C.

C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Young, T.

T. Young, “Lecture on the theory of light and colours,” Philos. Trans. R. Soc. London, Part 1, 41–48 (1802).

Yurkevich, I.

J. Sánchez-Gil, A. Maradudin, J. Lu, V. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

V. Freilikher, M. Pustilnik, I. Yurkevich, V. Tatarskii, “Polarization of light scattered from slightly rough dielectric film,” Opt. Lett. 19, 1382–1384 (1994).
[CrossRef] [PubMed]

V. Freilikher, M. Pustilnik, I. Yurkevich, “Wave scattering from a bounded medium with disorder,” Phys. Lett. A 193, 467–470 (1994).
[CrossRef]

Am. J. Phys. (1)

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

J. Mod. Opt. (1)

J. Sánchez-Gil, A. Maradudin, J. Lu, V. Freilikher, M. Pustilnik, I. Yurkevich, “Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1996).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Lett. (2)

Philos. Mag. Ser. 6 (1)

C. V. Raman, G. L. Datta, “On Quetelet’s rings and other allied phenomena,” Philos. Mag. Ser. 6 42, 826–840 (1921).
[CrossRef]

Philos. Trans. R. Soc. London (1)

T. Young, “Lecture on the theory of light and colours,” Philos. Trans. R. Soc. London, Part 1, 41–48 (1802).

Phys. Lett. A (1)

V. Freilikher, M. Pustilnik, I. Yurkevich, “Wave scattering from a bounded medium with disorder,” Phys. Lett. A 193, 467–470 (1994).
[CrossRef]

Phys. Rep. (1)

V. Freilikher, E. Kanzieper, A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[CrossRef]

Pribory i Tekh. Exp. (1)

Yu. Kaganovskii, V. D. Freilikher, V. V. Grischenko, I. M. Popova, “Estimate of micro roughness parameters on a solid surface by light scattering method,” Pribory i Tekh. Exp., No. 4, 258–260 (1976) (in Russian).

Prog. Surf. Sci. (1)

J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
[CrossRef]

Trans. Cambridge Philos. Soc. (1)

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

Other (7)

C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

M. Françon, Laser Speckle and Applications in Optics (Academic, New York, 1979).

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

A. I. Kalmykov, I. Fuks, I. Sherbinin, V. Tsymbal, A. Matveev, V. Freilikher, M. Fix, “Radar observation of strong subsurface scatterers,” in Proceedings of the International Geoscience and Remote Sensing Symposium ’95 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), Vol. 3, p. 1702.

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

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

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Figures (6)

Fig. 1
Fig. 1

Four beam trajectories for single scattering from a single surface scatterer, denoted by the point S. In all four trajectories the incident beam impinges on the sample at an incident angle θin and is scattered into an angle θs. (a) and (b) correspond to forward scattering; (c) and (d) correspond to backward scattering.

Fig. 2
Fig. 2

Interference patterns from a slightly rough Fabry–Perot plate under illumination by an Ar ion laser beam at 514 nm for various incident angles θin. The pattern is viewed on an observation screen in which a hole has been drilled to allow the incident beam to pass through on its way to the sample. The visible line along the incident beam path is produced by scattering of the beam by dust particles in the air. A neutral density filter is positioned at the reflected-beam position in order to decrease the beam’s brightness. Note that the reflected beam can be on a ring maximum [(b), (d)] or minimum [(c)] as the incident angle is varied. Ring intensity oscillates periodically with increase in θin: (a) θin=0, (b) θin=2.41°, (c) θin=2.98°, (d) θin=3.51°.

Fig. 3
Fig. 3

Interference pattern from a slightly rough ordinary glass plate at various incident angles θin. The plate was illuminated by two collinear laser beams from a He–Ne and Ar ion laser at the corresponding red and green colors. The reflected beam in this case is always positioned on an interference maximum for any incident angle. One of the interference maxima always coincides with the specular and retroreflected directions. (a) θin=4°, (b) θin=6°, (c) θin=8°.

Fig. 4
Fig. 4

Computer simulation of interference patterns for (a) one, (b) two, and (c), (d) twenty scatterers located randomly on a plane surface. The colors correspond to the intensity, with violet representing interference minima and red representing maxima. The wavelength was assumed to be 633 nm, H=150 µm, ϵ=2.25. (a), (b), (c) θin=4.9°; (d) θin=5.8°. Although the speckle size and positions depend on the number of scatterers and their distribution on the plane as well as on the angle of incidence, the positions of the rings stay constant. The scattered intensity oscillates periodically with increasing incidence angle.

Fig. 5
Fig. 5

Comparison of the measured (solid curves) and the calculated (dotted curves) scattering diagrams for two regular dielectric layers containing LSR. The wavelength in both cases was 633 nm. (a) H=2.8 µm, ϵ=3.24, θin=30°, (b) H=150 µm, ϵ=2.25, θin=4°.

Fig. 6
Fig. 6

Angular intensity distribution of green (thin curve) and red (thick curve) scattered light from a regular glass plate with LSR rms height σ=0.06 µm and SSR rms height h=0.05 µm. The amplitude of oscillations of the red light is almost two times higher than that of the green light. The positions of the interference maxima are different for the different wavelengths, in accordance with Eq. (1).

Equations (10)

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

ϕ(θs)-ϕ(θin)=2πn,
ϕ(θs)=2πn.
I(θin,θs)=I0|g(θin)g(θs)|2S(q).
g(θ)=14π{1-exp[iϕ(θ)]}[1+r(θ)]1-r(θ)exp[iϕ(θ)],
r(θ)=cos θ-(ϵ-sin2 θ)1/2cos θ+(ϵ-sin2 θ)1/2
4πg(θ)={1-exp[iνϕ(θ)]}[1+r(θ)]×ν=0rν(θ)exp[iνϕ(θ)]=1-exp[iϕ(θ)]+r(θ){1-exp[i2ϕ(θ)]}+O(r2)+,
(4π2)|g(θin)g(θs)|2=2[1-cos ϕ(θin)]×[1-cos ϕ(θs)].
I(θin, θs)=18π2I0{1-cos ϕ(θs)exp-σ22H2ϕ2(θs)-cos ϕ(θin)exp-σ22H2ϕ2(θin)+F+ cos[ϕ(θs)+ϕ(θin)]+F- cos[ϕ(θs)-ϕ(θin)]}S(q).
F±=12exp-σ22H2[ϕ2(θin)+ϕ2(θs)±2ϕ(θin)ϕ(θs)W(Λθin+Λθs)],
Λθ=2H|tan θ|,ϕ(θ)=2kH(ϵ-sin2 θ)1/2.

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