Abstract

Computer codes that are based on Elson’s theory for light scattering by interfacial roughness in multilayer coatings were used to predict the bidirectional reflectance-distribution function (BRDF) of several opaque coatings from surface-roughness profiles measured by either a scanning–tunneling microscope or an atomic-force microscope. The predictions usually agreed with measured BRDF values to within a factor of 2. The coatings consisted of single layers of Ag or Ni and dielectric stacks with up to three layers.

© 1995 Optical Society of America

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References

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  1. J. M. Elson, “Low-efficiency diffraction grating theory,” Rep. AFWL–TR–75–210 (Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., March1976).
  2. J. M. Elson, “Infrared light scattering from surfaces covered with multiple dielectric overlayers,” Appl. Opt. 16, 2872–2881 (1977).
    [CrossRef] [PubMed]
  3. J. M. Elson, “Light scattering from semi-infinite media for nonnormal incidence,” Phys. Rev. B 12, 2541–2542 (1975).
    [CrossRef]
  4. P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
    [CrossRef]
  5. A. A. Maradudin, D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
    [CrossRef]
  6. A. Marvin, F. Toigo, V. Celli, “Light scattering from rough surfaces: general incidence angle and polarization,” Phys. Rev. B 11, 2777–2782 (1975).
    [CrossRef]
  7. C. Amra, P. Bousquet, “Scattering from surfaces and multilayer coatings: recent advances for a better investigation of experiment,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 82–97 (1988).
  8. D. Neal, TRW, One Space Park, Redondo Beach, Calif. 90278 (personal communication, 1991).
  9. 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]
  10. J. M. Elson, J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).
  11. J. M. Elson, J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. 69, 31–47 (1979).
    [CrossRef]
  12. R. J. Noll, P. Glenn, “Mirror surface autocovariance functions and their associated visible scattering,” Appl. Opt. 21, 1824–1838 (1982).
    [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. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).
  15. E. D. Palik, ed., Handbook of Optical Constants of Solids II (Academic, New York, 1991).

1992 (1)

1982 (1)

1981 (1)

1980 (1)

1979 (2)

1977 (1)

1975 (3)

J. M. Elson, “Light scattering from semi-infinite media for nonnormal incidence,” Phys. Rev. B 12, 2541–2542 (1975).
[CrossRef]

A. A. Maradudin, D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
[CrossRef]

A. Marvin, F. Toigo, V. Celli, “Light scattering from rough surfaces: general incidence angle and polarization,” Phys. Rev. B 11, 2777–2782 (1975).
[CrossRef]

Amra, C.

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, P. Bousquet, “Scattering from surfaces and multilayer coatings: recent advances for a better investigation of experiment,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 82–97 (1988).

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. 71, 1115–1123 (1981).
[CrossRef]

C. Amra, P. Bousquet, “Scattering from surfaces and multilayer coatings: recent advances for a better investigation of experiment,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 82–97 (1988).

Celli, V.

A. Marvin, F. Toigo, V. Celli, “Light scattering from rough surfaces: general incidence angle and polarization,” Phys. Rev. B 11, 2777–2782 (1975).
[CrossRef]

Elson, J. M.

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. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).

J. M. Elson, J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. 69, 31–47 (1979).
[CrossRef]

J. M. Elson, “Infrared light scattering from surfaces covered with multiple dielectric overlayers,” Appl. Opt. 16, 2872–2881 (1977).
[CrossRef] [PubMed]

J. M. Elson, “Light scattering from semi-infinite media for nonnormal incidence,” Phys. Rev. B 12, 2541–2542 (1975).
[CrossRef]

J. M. Elson, “Low-efficiency diffraction grating theory,” Rep. AFWL–TR–75–210 (Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., March1976).

Flory, F.

Glenn, P.

Maradudin, A. A.

A. A. Maradudin, D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
[CrossRef]

Marvin, A.

A. Marvin, F. Toigo, V. Celli, “Light scattering from rough surfaces: general incidence angle and polarization,” Phys. Rev. B 11, 2777–2782 (1975).
[CrossRef]

Mills, D. L.

A. A. Maradudin, D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
[CrossRef]

Neal, D.

D. Neal, TRW, One Space Park, Redondo Beach, Calif. 90278 (personal communication, 1991).

Noll, R. J.

Pelletier, E.

Rahn, J. P.

Roche, P.

Toigo, F.

A. Marvin, F. Toigo, V. Celli, “Light scattering from rough surfaces: general incidence angle and polarization,” Phys. Rev. B 11, 2777–2782 (1975).
[CrossRef]

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

Opt. Eng. (1)

J. M. Elson, J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).

Phys. Rev. B (3)

J. M. Elson, “Light scattering from semi-infinite media for nonnormal incidence,” Phys. Rev. B 12, 2541–2542 (1975).
[CrossRef]

A. A. Maradudin, D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
[CrossRef]

A. Marvin, F. Toigo, V. Celli, “Light scattering from rough surfaces: general incidence angle and polarization,” Phys. Rev. B 11, 2777–2782 (1975).
[CrossRef]

Other (5)

C. Amra, P. Bousquet, “Scattering from surfaces and multilayer coatings: recent advances for a better investigation of experiment,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 82–97 (1988).

D. Neal, TRW, One Space Park, Redondo Beach, Calif. 90278 (personal communication, 1991).

J. M. Elson, “Low-efficiency diffraction grating theory,” Rep. AFWL–TR–75–210 (Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., March1976).

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).

E. D. Palik, ed., Handbook of Optical Constants of Solids II (Academic, New York, 1991).

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Figures (17)

Fig. 1
Fig. 1

Coordinate system showing the polar θ and azimuthal ϕ scattering angles and the angle of incidence θ0.

Fig. 2
Fig. 2

Scheme for sectioning a 400 data point × 400 data point surface profile into nine 128 data point × 128 data point subprofiles.

Fig. 3
Fig. 3

Measured spectral response of the scatterometer for the Xe lamp with the SWP-800 filter.

Fig. 4
Fig. 4

Diagram of the scatterometer test geometry.

Fig. 5
Fig. 5

Predicted and measured BRDF for an 85.0-nm-thick Ag coating on a microsheet-glass substrate: 30° incidence, He–Ne laser, and circularly polarized incident light.

Fig. 6
Fig. 6

Predicted and measured BRDF for an 85.0-nm-thick Ag coating on a TP-2 substrate: 30° incidence, He–Ne laser, and circularly polarized incident light.

Fig. 7
Fig. 7

Predicted and measured BRDF in the plane of incidence for 326.0-nm layer of Si on a microsheet-glass substrate: 30° incidence, Xe lamp, 500-nm filter, and unpolarized incident light.

Fig. 8
Fig. 8

Predicted and measured BRDF out of the plane of incidence for a 326.0-nm layer of Si on a microsheet-glass substrate: 41° incidence, Xe lamp, 500-nm filter, and unpolarized incident light.

Fig. 9
Fig. 9

Typical AFM roughness profile for a 326.0-nm Si on a microsheet-glass substrate.

Fig. 10
Fig. 10

Predicted and measured BRDF in the plane of incidence for a type-4 coating on a microsheet-glass substrate: 30° incidence, Xe lamp, 500-nm filter, and unpolarized incident light.

Fig. 11
Fig. 11

Predicted and measured BRDF out of the plane of incidence for a type-4 coating on a microsheet-glass substrate: 41° incidence, Xe lamp, 500-nm filter, and unpolarized incident light.

Fig. 12
Fig. 12

Typical AFM roughness profile for the top surface of a type-4 coating on a microsheet-glass substrate.

Fig. 13
Fig. 13

Predicted and measured BRDF in the plane of incidence for a type-5 coating on a microsheet-glass substrate: 30° incidence, Xe lamp, 500-nm filter, unpolarized incident light.

Fig. 14
Fig. 14

Predicted and measured BRDF out of the plane of incidence for a type-5 coating on a microsheet-glass substrate: 41° incidence, Xe lamp, 500-nm filter, and unpolarized incident light.

Fig. 15
Fig. 15

Typical AFM roughness profile for the top surface of a type-5 coating on a microsheet glass substrate.

Fig. 16
Fig. 16

Predicted and measured BRDF in the plane of incidence for a 580-nm layer of Si on a microsheet-glass substrate: 0° incidence, Xe lamp, SWP = 800 filter, and unpolarized incident light.

Fig. 17
Fig. 17

Predicted and measured BRDF in the plane of incidence for a type-7 coating on a microsheet-glass substrate.

Tables (4)

Tables Icon

Table 1 Types of Coatings Investigateda

Tables Icon

Table 2 Types of Scanner Used for Measurements of Each Layer

Tables Icon

Table 3 rms Roughness Values of Surfaces of Top Layers

Tables Icon

Table 4 Scatterometer Test Conditions

Equations (2)

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( d P / d Ω ) / P = m = 1 L + 1 n = 1 L + 1 F m F n * P m n ,
BRDF = BRDF ( λ ) R ( λ ) d λ R ( λ ) d λ ,

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