Abstract

We derive expressions for the intensity and polarization of light singly scattered by flake pigments or a rough surface beneath a smooth transparent coating using the ray or facet model. The distribution of local surface normals is used to calculate the bidirectional reflectance distribution function (BRDF). We discuss the different distribution functions that can be used to characterize the distribution of local surface normals. The light-scattering model is validated by measurements of the BRDF and polarization by a metallic flake pigmented coating. The results enable the extraction of a slope distribution function from the data, which is shown to be consistent over a variety of scattering geometries. These models are appropriate to estimate or predict the appearance of flake pigment automotive paints.

© 2004 Optical Society of America

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References

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  1. R. Glausch, M. Kieser, R. Maisch, G. Pfaff, J. Weitzel, Special Effect Pigments (Vincentz-Verlag, Hannover, Germany, 1998).
  2. G. Pfaff, P. Reynders, “Angle-dependent optical effects deriving from submicron structures of films and pigments,” Chem. Rev. 99, 1963–1981 (1999).
    [CrossRef]
  3. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for ReflectanceNBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).
  4. J. C. Stover, Optical Scattering: Measurement and Analysis, Vol. PM24 of the SPIE Press Monographs (SPIE Press, Bellingham, Wash., 1995).
    [CrossRef]
  5. D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
    [CrossRef]
  6. D. O. Muhleman, “Radar scattering from Venus and the Moon,” Astron. J. 69, 34–41 (1964).
    [CrossRef]
  7. K. E. Torrance, E. M. Sparrow, “Theory of off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1105–1114 (1967).
    [CrossRef]
  8. L. Sung, M. E. Nadal, M. E. McKnight, E. Marx, B. Laurenti, “Optical reflectance of metallic coatings: effect of aluminum flake orientation,” J. Coatings Technol. 74, 55–63 (2002).
    [CrossRef]
  9. M. E. McKnight, T. V. Vorburger, E. Marx, M. E. Nadal, P. Y. Barnes, M. A. Galler, “Measurements and predictions of light scattering from clear coatings,” Appl. Opt. 40, 2159–2168 (2001).
    [CrossRef]
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).
  11. T. A. Germer, C. C. Asmail, “A goniometric optical scatter instrument for bidirectional reflectance distribution function measurements with out-of-plane and polarimetry capabilities,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 220–231 (1997).
    [CrossRef]
  12. T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
    [CrossRef]
  13. T. A. Germer, “Polarized light scattering by microroughness and small defects in dielectric layers,” J. Opt. Soc. Am. A 18, 1279–1288 (2001).
    [CrossRef]
  14. E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).
  15. T. A. Germer, M. E. Nadal, “Modeling the appearance of special effect pigment coatings,” in Surface Scattering and Diffraction for Advanced Metrology, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE4447, 77–86 (2001).
    [CrossRef]
  16. B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
    [CrossRef]

2002 (1)

L. Sung, M. E. Nadal, M. E. McKnight, E. Marx, B. Laurenti, “Optical reflectance of metallic coatings: effect of aluminum flake orientation,” J. Coatings Technol. 74, 55–63 (2002).
[CrossRef]

2001 (2)

1999 (2)

T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

G. Pfaff, P. Reynders, “Angle-dependent optical effects deriving from submicron structures of films and pigments,” Chem. Rev. 99, 1963–1981 (1999).
[CrossRef]

1968 (1)

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
[CrossRef]

1967 (2)

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
[CrossRef]

K. E. Torrance, E. M. Sparrow, “Theory of off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1105–1114 (1967).
[CrossRef]

1964 (1)

D. O. Muhleman, “Radar scattering from Venus and the Moon,” Astron. J. 69, 34–41 (1964).
[CrossRef]

Asmail, C. C.

T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

T. A. Germer, C. C. Asmail, “A goniometric optical scatter instrument for bidirectional reflectance distribution function measurements with out-of-plane and polarimetry capabilities,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 220–231 (1997).
[CrossRef]

Barnes, P. Y.

Barrick, D. E.

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

Galler, M. A.

Germer, T. A.

T. A. Germer, “Polarized light scattering by microroughness and small defects in dielectric layers,” J. Opt. Soc. Am. A 18, 1279–1288 (2001).
[CrossRef]

T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

T. A. Germer, C. C. Asmail, “A goniometric optical scatter instrument for bidirectional reflectance distribution function measurements with out-of-plane and polarimetry capabilities,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 220–231 (1997).
[CrossRef]

T. A. Germer, M. E. Nadal, “Modeling the appearance of special effect pigment coatings,” in Surface Scattering and Diffraction for Advanced Metrology, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE4447, 77–86 (2001).
[CrossRef]

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for ReflectanceNBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).

Glausch, R.

R. Glausch, M. Kieser, R. Maisch, G. Pfaff, J. Weitzel, Special Effect Pigments (Vincentz-Verlag, Hannover, Germany, 1998).

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for ReflectanceNBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).

Kieser, M.

R. Glausch, M. Kieser, R. Maisch, G. Pfaff, J. Weitzel, Special Effect Pigments (Vincentz-Verlag, Hannover, Germany, 1998).

Laurenti, B.

L. Sung, M. E. Nadal, M. E. McKnight, E. Marx, B. Laurenti, “Optical reflectance of metallic coatings: effect of aluminum flake orientation,” J. Coatings Technol. 74, 55–63 (2002).
[CrossRef]

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for ReflectanceNBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).

Maisch, R.

R. Glausch, M. Kieser, R. Maisch, G. Pfaff, J. Weitzel, Special Effect Pigments (Vincentz-Verlag, Hannover, Germany, 1998).

Marx, E.

L. Sung, M. E. Nadal, M. E. McKnight, E. Marx, B. Laurenti, “Optical reflectance of metallic coatings: effect of aluminum flake orientation,” J. Coatings Technol. 74, 55–63 (2002).
[CrossRef]

M. E. McKnight, T. V. Vorburger, E. Marx, M. E. Nadal, P. Y. Barnes, M. A. Galler, “Measurements and predictions of light scattering from clear coatings,” Appl. Opt. 40, 2159–2168 (2001).
[CrossRef]

McKnight, M. E.

L. Sung, M. E. Nadal, M. E. McKnight, E. Marx, B. Laurenti, “Optical reflectance of metallic coatings: effect of aluminum flake orientation,” J. Coatings Technol. 74, 55–63 (2002).
[CrossRef]

M. E. McKnight, T. V. Vorburger, E. Marx, M. E. Nadal, P. Y. Barnes, M. A. Galler, “Measurements and predictions of light scattering from clear coatings,” Appl. Opt. 40, 2159–2168 (2001).
[CrossRef]

Muhleman, D. O.

D. O. Muhleman, “Radar scattering from Venus and the Moon,” Astron. J. 69, 34–41 (1964).
[CrossRef]

Nadal, M. E.

L. Sung, M. E. Nadal, M. E. McKnight, E. Marx, B. Laurenti, “Optical reflectance of metallic coatings: effect of aluminum flake orientation,” J. Coatings Technol. 74, 55–63 (2002).
[CrossRef]

M. E. McKnight, T. V. Vorburger, E. Marx, M. E. Nadal, P. Y. Barnes, M. A. Galler, “Measurements and predictions of light scattering from clear coatings,” Appl. Opt. 40, 2159–2168 (2001).
[CrossRef]

T. A. Germer, M. E. Nadal, “Modeling the appearance of special effect pigment coatings,” in Surface Scattering and Diffraction for Advanced Metrology, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE4447, 77–86 (2001).
[CrossRef]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for ReflectanceNBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).

Pfaff, G.

G. Pfaff, P. Reynders, “Angle-dependent optical effects deriving from submicron structures of films and pigments,” Chem. Rev. 99, 1963–1981 (1999).
[CrossRef]

R. Glausch, M. Kieser, R. Maisch, G. Pfaff, J. Weitzel, Special Effect Pigments (Vincentz-Verlag, Hannover, Germany, 1998).

Reynders, P.

G. Pfaff, P. Reynders, “Angle-dependent optical effects deriving from submicron structures of films and pigments,” Chem. Rev. 99, 1963–1981 (1999).
[CrossRef]

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for ReflectanceNBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).

Smith, B. G.

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
[CrossRef]

Sparrow, E. M.

Stover, J. C.

J. C. Stover, Optical Scattering: Measurement and Analysis, Vol. PM24 of the SPIE Press Monographs (SPIE Press, Bellingham, Wash., 1995).
[CrossRef]

Sung, L.

L. Sung, M. E. Nadal, M. E. McKnight, E. Marx, B. Laurenti, “Optical reflectance of metallic coatings: effect of aluminum flake orientation,” J. Coatings Technol. 74, 55–63 (2002).
[CrossRef]

Torrance, K. E.

Vorburger, T. V.

Weitzel, J.

R. Glausch, M. Kieser, R. Maisch, G. Pfaff, J. Weitzel, Special Effect Pigments (Vincentz-Verlag, Hannover, Germany, 1998).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

Appl. Opt. (1)

Astron. J. (1)

D. O. Muhleman, “Radar scattering from Venus and the Moon,” Astron. J. 69, 34–41 (1964).
[CrossRef]

Chem. Rev. (1)

G. Pfaff, P. Reynders, “Angle-dependent optical effects deriving from submicron structures of films and pigments,” Chem. Rev. 99, 1963–1981 (1999).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
[CrossRef]

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
[CrossRef]

J. Coatings Technol. (1)

L. Sung, M. E. Nadal, M. E. McKnight, E. Marx, B. Laurenti, “Optical reflectance of metallic coatings: effect of aluminum flake orientation,” J. Coatings Technol. 74, 55–63 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Rev. Sci. Instrum. (1)

T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

Other (7)

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).

T. A. Germer, M. E. Nadal, “Modeling the appearance of special effect pigment coatings,” in Surface Scattering and Diffraction for Advanced Metrology, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE4447, 77–86 (2001).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

T. A. Germer, C. C. Asmail, “A goniometric optical scatter instrument for bidirectional reflectance distribution function measurements with out-of-plane and polarimetry capabilities,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 220–231 (1997).
[CrossRef]

R. Glausch, M. Kieser, R. Maisch, G. Pfaff, J. Weitzel, Special Effect Pigments (Vincentz-Verlag, Hannover, Germany, 1998).

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for ReflectanceNBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).

J. C. Stover, Optical Scattering: Measurement and Analysis, Vol. PM24 of the SPIE Press Monographs (SPIE Press, Bellingham, Wash., 1995).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Ray trajectory for an oblique surface element. Azimuthal angles associated with directions are shown in parentheses.

Fig. 2
Fig. 2

Calculated BRDF for different exponential slope distribution functions, characterized by rms slope σ and coating index n. The incident angle was θ i = 60°, the substrate was aluminum, and the wavelength was 633 nm.

Fig. 3
Fig. 3

Data for an aluminum flake pigment under a smooth coating. The light was incident at an angle of θ i = 60° and polarized 45° from the plane of incidence, and the wavelength was 633 nm. The top frame shows the BRDF, the middle frame shows the degree of polarization and degree of circular polarization, and the bottom frame shows the principal angle of polarization.

Fig. 4
Fig. 4

Slope distribution function derived from the data shown in Fig. 3.

Tables (1)

Tables Icon

Table 1 Total Integrated Reflectance of the Coating on Rough Perfectly Conducting, Aluminum, and Silicon Surfaces for Different Exponential Slope Distribution Functions and Coating Indicesa

Equations (64)

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frθi, θr, ϕr=Pθi, θr, ϕrRnet/cos θr.
n sin θi=sin θi,
n sin θr=sin θr,
ϕr=ϕr.
kˆi=xˆsin θi-zˆ cos θi,
kˆr=xˆ sin θr cos ϕr+ŷ sin θr sin ϕr+zˆ cos θi.
nˆ=kˆr-kˆi/|kˆr-kˆi|.
cos α=1-sin θi sin θr cos ϕr+cos θi cos θr/21/2,
cos θn=cos θi+cos θr/2 cos α.
ϕn=arctancos θr cos ϕr-cos θi, cos θr sin ϕr,
ŝr=-xˆ cos ϕr+ŷ cos ϕr,
pˆr=xˆ cos θr cos ϕr+ŷ cos θr sin ϕr+zˆ sin θr.
Ei=tiθi; 1, n·Ei,
tiθi; 1, n=tsθi; 1, nŝiŝi+tpθi; 1, npˆipˆi,
rα; n, nf=rsα; n, nfŝrŝi+rpα; n, nfpˆrpˆi,
trθr; n, 1=tsθr; n, 1ŝrŝr+tpθr; n, 1pˆrpˆr.
rnet=trθr; n, 1·rα; n, nf·tiθi; 1, n,
ts,pθr; n, 1=n cos θrcos θr ts,pθr; 1, n
trθr; n, 1=n cos θrcos θrtsθr; 1, nŝrŝr+tpθr; 1, npˆrpˆr.
rnet=n cos θrcos θrq.
q=qssŝrŝi+qspŝrpˆi+qpspˆrŝi+qpppˆrpˆi,
qss=tsθi; 1, ntsθr; 1, n×rpα; n, nf×sin θi sin θr sin2 ϕr+a2a3rsα; n, nf/a1,
qps=-tsθi; 1, ntpθr; 1, n×sin ϕra2rsα; n, nfsin θr-a3rpα; n, nfsin θi/a1,
qsp=-tpθi; 1, ntsθr; 1, n×sin ϕra3rsα; n, nfsin θi-a2rpα; n, nfsin θr/a1,
qpp=tpθi; 1, ntpθr; 1, n×rsα; n, nfsin θi sin θr sin2 ϕr+a2a3rpα; n, nf/a1.
a1=sin2 2α,
a2=cos θi sin θr+sin θi cos θr cos ϕr,
a3=sin θi cos θr+cos θi sin θr cos ϕr.
Rnet=cos θi cos θrcos θi cos θr|rnet·Ei|2|Ei|2=n2 cos θi cos θrcos θi cos θr|q·Ei|2|Ei|2.
ζx=tan θn cos ϕn,
ζy=tan θn sin ϕn.
P1xyθn, ϕndΩn.
P2xyθndθn=sin θndθn  dϕnP1xyθn, ϕn.
P2xyθndθn=2πP1xyθn, 0sin θndθn.
P3xyζx, ζydζxdζy.
P4xyζdζ=ζdζ  dϕnP3xyζx, ζy.
P4xyζdζ=2πζP3xyζ, 0dζ.
ζx, ζyθn, ϕn=sin θn sec3 θn,
P3xyζx, ζy=cos3 θnP1xyθn, ϕn,
P4xyζ=cos2 θnP2xyθn.
 dθn sin θn  dϕnP1xyθn, ϕn=C,
 dθnP2xyθn=C,
 dζx  dζyP3xyζx, ζy=C,
 dζP4xyζ=C.
P1Aθn, ϕn=sec θnP1xyθn, ϕn/Csec θn,
sec θn=1C  dϕn  dθn sin θn sec θnP1xyθn, ϕn.
 dθn sin θn  dϕnP1Aθn, ϕn=1,
 dθnP2Aθn=1,
 dζx  dζyP3Aζx, ζy=1,
 dζP4Aζ=1.
P1Aθn, ϕn=sec4 θnP3xyζx, ζy/Csec θn.
P3xyζx, ζy=3Cπσ2exp-6ζ/σ,
P1xyθn, ϕncos αdΩn/cos θn cos θi.
θr, ϕr/θr, ϕr=n cos θr/cos θr.
θr, ϕr/θn, ϕn=4sin θn/sin θrcos α.
dΩr/dΩn=sin θr/sin θnθr, ϕr/θn, ϕn=4n2 cos θr cos α/cos θr.
Pθi, θr, ϕrdΩr=P1xyθn, ϕncos θrdΩr4n2 cos θr cos θi cos θn.
fr=P1xyθn, ϕn4 cos θi cos θr cos θn|q·Ej|2|Ei|2.
fr=P3xyζx, ζy4 cos θi cos θr cos4 θn|q·Ei|2|Ei|2.
fr=Csec θnP1Aθn, ϕn4 cos θi cos θr|q·Ei|2|Ei|2.
rsθ; n1, n2=cos θ-n2/n12-sin2 θ1/2cos θ+n2/n12-sin2 θ1/2,
rpθ; n1, n2=n2/n1cos θ-n2/n12-sin2 θ1/2n2/n1cos θ+n2/n12-sin2 θ1/2
tsθ; n1, n2=2n1 cos θcos θ+n2/n12-sin2 θ1/2,
tpθ; n1, n2=2n1/n2cos θn2/n1cos θ+n2/n12-sin2 θ1/2.

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