Abstract

We show that a scattering–reduction effect is obtained by coating a rough surface with an antireflection layer. This research is a generalization of Amra’s [ J. Opt. Soc. Am. A 10, 365– 374 ( 1993)] study of smooth surfaces conducted with a first-order theory to the case of rough surfaces. We show that the differential method with the R matrix algorithm can be used to study scattering from multilayered rough surfaces. A comparison between numerical and experimental results is given.

© 1997 Optical Society of America

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References

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  1. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).
  2. J. M. Elson, J. P. Rahn, J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
    [CrossRef] [PubMed]
  3. P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 9, 1115–1123 (1981).
    [CrossRef]
  4. J. M. Elson, J. P. Rahn, J. M. Bennett, “Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation-length, and roughness cross-correlation properties,” Appl. Opt. 22, 3207–3219 (1983).
    [CrossRef]
  5. C. Amra, “Light scattering from multilayer optics. Part A: investigation tools,” J. Opt. Soc. Am. A 11, 197–210 (1994).
    [CrossRef]
  6. C. Amra, “Light scattering from multilayer optics. Part B: application to experiment,” J. Opt. Soc. Am. A 11, 211–226 (1994).
    [CrossRef]
  7. A. Duparré, S. Kassam, “Relation between light scattering and microstructure of optical thin films,” Appl. Opt. 32, 5475–5480 (1992).
    [CrossRef]
  8. S. Kassam, A. Duparré, K. Helm, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
    [CrossRef] [PubMed]
  9. C. Amra, “From light scattering to the microstructure of thin film multilayers,” Appl. Opt. 32, 5481–5491 (1993).
    [CrossRef] [PubMed]
  10. C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. 10, 365–374 (1993).
    [CrossRef]
  11. C. Amra, C. Grèzes–Besset, L. Bruel, “Comparison of surface and bulk scattering in optical coatings,” Appl. Opt. 32, 5492–5503 (1993).
    [CrossRef] [PubMed]
  12. C. Amra, D. Torricini, P. Roche, “Multiwavelength (0.45–10.6 mm) angle-resolved scatterometer or how to extend the optical window,” Appl. Opt. 32, 5462–5474 (1993).
    [CrossRef] [PubMed]
  13. C. Amra, J. H. Apfel, E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31, 3134–3151 (1992).
    [CrossRef] [PubMed]
  14. S. Maure, G. Albrand, C. Amra, “Low-level scattering and localized defects,” Appl. Opt. 35, 5573–5582 (1996).
    [CrossRef] [PubMed]
  15. C. Deumié, R. Richier, P. Dumas, C. Amra, “Multiscale roughness in optical multilayers: atomic force microscopy and light scattering,” Appl. Opt. 35, 5583–5594 (1996).
    [CrossRef] [PubMed]
  16. C. Deumié, H. Giovannini, C. Amra, “Ellipsometry of light scattering from multilayer coatings,” Appl. Opt. 35, 5600–5608 (1996).
    [CrossRef] [PubMed]
  17. P. Vincent, “Differential methods,” in Progress in Optics, E. Wolf, ed. (Springer–Verlag, Berlin, 1980), Vol. 22, pp. 101–121.
  18. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
    [CrossRef]
  19. M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
    [CrossRef]
  20. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–276 (1990).
    [CrossRef]
  21. M. Nieto–Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), pp. 232–289.
  22. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
    [CrossRef]
  23. J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
    [CrossRef]
  24. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  25. C. Amra, G. Albrand, P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 16, 2695–2702 (1986).
    [CrossRef]

1996 (4)

1994 (3)

1993 (4)

1992 (3)

1990 (2)

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

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–276 (1990).
[CrossRef]

1986 (2)

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
[CrossRef]

C. Amra, G. Albrand, P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 16, 2695–2702 (1986).
[CrossRef]

1983 (1)

1981 (1)

P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 9, 1115–1123 (1981).
[CrossRef]

1980 (1)

1977 (1)

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
[CrossRef]

Albrand, G.

S. Maure, G. Albrand, C. Amra, “Low-level scattering and localized defects,” Appl. Opt. 35, 5573–5582 (1996).
[CrossRef] [PubMed]

C. Amra, G. Albrand, P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 16, 2695–2702 (1986).
[CrossRef]

Amra, C.

C. Deumié, R. Richier, P. Dumas, C. Amra, “Multiscale roughness in optical multilayers: atomic force microscopy and light scattering,” Appl. Opt. 35, 5583–5594 (1996).
[CrossRef] [PubMed]

C. Deumié, H. Giovannini, C. Amra, “Ellipsometry of light scattering from multilayer coatings,” Appl. Opt. 35, 5600–5608 (1996).
[CrossRef] [PubMed]

S. Maure, G. Albrand, C. Amra, “Low-level scattering and localized defects,” Appl. Opt. 35, 5573–5582 (1996).
[CrossRef] [PubMed]

C. Amra, “Light scattering from multilayer optics. Part A: investigation tools,” J. Opt. Soc. Am. A 11, 197–210 (1994).
[CrossRef]

C. Amra, “Light scattering from multilayer optics. Part B: application to experiment,” J. Opt. Soc. Am. A 11, 211–226 (1994).
[CrossRef]

C. Amra, D. Torricini, P. Roche, “Multiwavelength (0.45–10.6 mm) angle-resolved scatterometer or how to extend the optical window,” Appl. Opt. 32, 5462–5474 (1993).
[CrossRef] [PubMed]

C. Amra, “From light scattering to the microstructure of thin film multilayers,” Appl. Opt. 32, 5481–5491 (1993).
[CrossRef] [PubMed]

C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. 10, 365–374 (1993).
[CrossRef]

C. Amra, C. Grèzes–Besset, L. Bruel, “Comparison of surface and bulk scattering in optical coatings,” Appl. Opt. 32, 5492–5503 (1993).
[CrossRef] [PubMed]

C. Amra, J. H. Apfel, E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31, 3134–3151 (1992).
[CrossRef] [PubMed]

C. Amra, G. Albrand, P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 16, 2695–2702 (1986).
[CrossRef]

Apfel, J. H.

Bennett, J. M.

Bousquet, P.

P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 9, 1115–1123 (1981).
[CrossRef]

Bruel, L.

Bussemer, P.

Deumié, C.

Dumas, P.

Duparré, A.

Elson, J. M.

Flory, F.

P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 9, 1115–1123 (1981).
[CrossRef]

Gaylord, T. K.

Giovannini, H.

Grèzes–Besset, C.

Helm, K.

Hugonin, J. P.

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
[CrossRef]

Kassam, S.

Li, L.

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–276 (1990).
[CrossRef]

Mattsson, L.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

Maure, S.

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. 203, 255–276 (1990).
[CrossRef]

Méndez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–276 (1990).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–276 (1990).
[CrossRef]

Moharam, M. G.

Montiel, F.

Neubert, J.

Nevière, M.

Nieto–Vesperinas, M.

M. Nieto–Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), pp. 232–289.

Pelletier, E.

Petit, R.

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
[CrossRef]

Rahn, J. P.

Richier, R.

Roche, P.

C. Amra, D. Torricini, P. Roche, “Multiwavelength (0.45–10.6 mm) angle-resolved scatterometer or how to extend the optical window,” Appl. Opt. 32, 5462–5474 (1993).
[CrossRef] [PubMed]

C. Amra, G. Albrand, P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 16, 2695–2702 (1986).
[CrossRef]

P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 9, 1115–1123 (1981).
[CrossRef]

Saillard, M.

Torricini, D.

Vincent, P.

P. Vincent, “Differential methods,” in Progress in Optics, E. Wolf, ed. (Springer–Verlag, Berlin, 1980), Vol. 22, pp. 101–121.

Ann. Phys. (1)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–276 (1990).
[CrossRef]

Appl. Opt. (12)

C. Amra, C. Grèzes–Besset, L. Bruel, “Comparison of surface and bulk scattering in optical coatings,” Appl. Opt. 32, 5492–5503 (1993).
[CrossRef] [PubMed]

C. Amra, D. Torricini, P. Roche, “Multiwavelength (0.45–10.6 mm) angle-resolved scatterometer or how to extend the optical window,” Appl. Opt. 32, 5462–5474 (1993).
[CrossRef] [PubMed]

C. Amra, J. H. Apfel, E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31, 3134–3151 (1992).
[CrossRef] [PubMed]

S. Maure, G. Albrand, C. Amra, “Low-level scattering and localized defects,” Appl. Opt. 35, 5573–5582 (1996).
[CrossRef] [PubMed]

C. Deumié, R. Richier, P. Dumas, C. Amra, “Multiscale roughness in optical multilayers: atomic force microscopy and light scattering,” Appl. Opt. 35, 5583–5594 (1996).
[CrossRef] [PubMed]

C. Deumié, H. Giovannini, C. Amra, “Ellipsometry of light scattering from multilayer coatings,” Appl. Opt. 35, 5600–5608 (1996).
[CrossRef] [PubMed]

J. M. Elson, J. P. Rahn, J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
[CrossRef] [PubMed]

J. M. Elson, J. P. Rahn, J. M. Bennett, “Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation-length, and roughness cross-correlation properties,” Appl. Opt. 22, 3207–3219 (1983).
[CrossRef]

A. Duparré, S. Kassam, “Relation between light scattering and microstructure of optical thin films,” Appl. Opt. 32, 5475–5480 (1992).
[CrossRef]

S. Kassam, A. Duparré, K. Helm, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
[CrossRef] [PubMed]

C. Amra, “From light scattering to the microstructure of thin film multilayers,” Appl. Opt. 32, 5481–5491 (1993).
[CrossRef] [PubMed]

C. Amra, G. Albrand, P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 16, 2695–2702 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. 10, 365–374 (1993).
[CrossRef]

P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 9, 1115–1123 (1981).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Commun. (1)

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
[CrossRef]

Other (3)

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

M. Nieto–Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), pp. 232–289.

P. Vincent, “Differential methods,” in Progress in Optics, E. Wolf, ed. (Springer–Verlag, Berlin, 1980), Vol. 22, pp. 101–121.

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Figures (6)

Fig. 1
Fig. 1

Decomposition of the rough region into Q different slices.

Fig. 2
Fig. 2

Comparison between differential and integral surface models: n1 = 1.52, n2 = 1, and λ = 1 µm. Normal incidence: (a) TE polarization δ = 0.1 µm, a = 1 µm, and L = 30 µm; (b) TM polarization δ = 0.1 µm, a = 0.3 µm, and L = 20 µm.

Fig. 3
Fig. 3

Scattering-reduction and antiscattering effects obtained when a smooth surface is coated. L = 25.32 µm and λ = 0.633 µm. Computation with the differential method. The specular reflection is not represented. The coating is a single layer of Cryolite [Na3AlF6, refractive index n = 1.3, and thickness e = λ/(4.n)]; n1 = 1.52 and n2 = 1. Normal incidence.

Fig. 4
Fig. 4

Scattering-reduction effect obtained when a rough surface is coated. Numerical result.

Fig. 5
Fig. 5

Angular pattern of the bare surface and of the coated surface for different values of r. Case of a rough surface: δ = 0.2 µm, a = 0.3 µm, and L = 30 µm. TE polarization.

Fig. 6
Fig. 6

Scattering-reduction effect on a rough surface. Experimental results. The wavelength of operation is λ = 0.633 µm. Normal incidence. Spot size ≈6 mm2.

Equations (4)

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ε m ( x ) = + ε ˆ m ( σ ) exp ( 2 π i σ x ) d σ ,
ε m ( x ) n = N / 2 N / 2 1 ε ˆ m n   exp 2 π i n L x ,
α n = α + ( 2 π / L ) n ,
α = 2 π λ n 2   sin   θ i ,

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