Abstract

Comparisons are made between calculated and measured angle-resolved light-scattering distributions from clear dielectric isotropic epoxy coatings over a range of rms roughness conditions, resulting in strongly specular scattering to diffuse scattering characteristics. Calculated distributions are derived from topography measurements performed with interferometric microscopes. Two methods of calculation are used. One determines the intensity of scattered light waves with a phase integral in the Kirchhoff approximation. The other is based on the reflection of light rays by locally flat surfaces. The angle-resolved scattering distributions for the coatings are measured with the spectral trifunction automated reference reflectometer (STARR) developed by the National Institute of Standards and Technology. Comparisons between measured and calculated results are shown for three surfaces with rms roughness values of approximately 3, 150, and 800 nm for an angle of incidence of 20°.

© 2001 Optical Society of America

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References

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  1. M. Maynard, “Automakers’ color guard not spinning its wheels,” USA Today, 26October1998, p. 12B.
  2. M. E. McKnight, J. W. Martin, “Advanced methods and models for describing coating appearance,” Prog. Org. Coat. 34, 152–159 (1998).
    [CrossRef]
  3. P. Y. Barnes, E. A. Early, A. C. Parr, “Spectral reflectance,” NIST (Natl. Inst. Stand. Technol.) Spec.Publ. 250–48 (1998).
  4. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 67 ff.
  5. R. Brodmann, M. Algäuer, “Comparison of light scattering from rough surfaces with optical and mechanical profilometry,” in Proceedings on Surface Measurement and Characterization, J. M. Bennett, ed., Proc. SPIE1009, 111–118 (1988).
    [CrossRef]
  6. J. T. Bendler, S. F. Feldman, H. Hatti, S. Y. Hobbs, “Approximate model of diffuse reflectance from rough polymer surfaces,” J. Appl. Phys. 83, 998–1004 (1998).
    [CrossRef]
  7. F. Y. Hunt, E. Marx, G. W. Meyer, T. V. Vorburger, P. A. Walker, H. B. Westlund, “A first step towards photorealistic rendering of coated surfaces and computer based standards of appearance,” in Service Life Methodology and Metrology, J. W. Martin, D. R. Bauer, eds., ACS Symposium Series (Oxford U. Press, New York, 2001).
    [CrossRef]
  8. Certain commercial equipment is identified in this paper to describe adequately an experimental procedure. Such identification does not imply that the equipment identified is necessarily the best available for the purpose nor does it imply any recommendation or endorsement by NIST.
  9. ASME Surface Quality Standard B46.1, Surface Texture (American Society of Mechanical Engineers, New York, 1995).
  10. D. B. Hall, P. Underhill, J. M. Torkelson, “Spin coating of thin and ultrathin polymer films,” Polym. Eng. Sci. 38, 2039–2045 (1998).
    [CrossRef]
  11. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (1977).
  12. J. E. Proctor, P. Y. Barnes, “NIST high accuracy reference reflectometer-spectrophotometer,” J. Res. Natl. Inst. Stand. Technol. 101, 619–627 (1996).
    [CrossRef]
  13. P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  16. WYKO NT-2000, Veeco Metrology Group, Tucson, Ariz. 85706.
  17. Zygo New View 5030, Zygo Corporation, Middlefield, Conn. 06455–0448.
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    [CrossRef]
  19. K. Creath, “Comparison of phase-measurement algorithms,” K. Creath, ed., Proc. SPIE680, 19–28 (1986).
  20. E. Marx, T. V. Vorburger, “Windowing effects on light scattered by sinusoidal surfaces” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 2–14 (1993).
    [CrossRef]
  21. E. Marx, T. R. Lettieri, T. V. Vorburger, “Light scattering by sinusoidal surfaces: illumination windows and harmonics in standards,” Appl. Opt. 34, 1269–1277 (1995).
    [CrossRef] [PubMed]
  22. D. E. Barrick, “Grazing behavior of scatter and propagation above any rough surface,” IEEE Trans. Antennas Propag. 46, 73–83 (1998).
    [CrossRef]
  23. E. Marx, “Integral equations for scattering by a dielectric,” IEEE Trans. Antennas Propag. 32, 166–172 (1984).
    [CrossRef]
  24. J. M. Elson, J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).
  25. B. N. Taylor, C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” (National Institute of Standards and Technology, Washington, D.C., 1994).

1999

T. Doi, T. V. Vorburger, P. Sullivan, “Effects of defocus and algorithm on optical step height calibration,” Proc. Eng. 23, 135–143 (1999).
[CrossRef]

1998

D. E. Barrick, “Grazing behavior of scatter and propagation above any rough surface,” IEEE Trans. Antennas Propag. 46, 73–83 (1998).
[CrossRef]

M. E. McKnight, J. W. Martin, “Advanced methods and models for describing coating appearance,” Prog. Org. Coat. 34, 152–159 (1998).
[CrossRef]

P. Y. Barnes, E. A. Early, A. C. Parr, “Spectral reflectance,” NIST (Natl. Inst. Stand. Technol.) Spec.Publ. 250–48 (1998).

J. T. Bendler, S. F. Feldman, H. Hatti, S. Y. Hobbs, “Approximate model of diffuse reflectance from rough polymer surfaces,” J. Appl. Phys. 83, 998–1004 (1998).
[CrossRef]

D. B. Hall, P. Underhill, J. M. Torkelson, “Spin coating of thin and ultrathin polymer films,” Polym. Eng. Sci. 38, 2039–2045 (1998).
[CrossRef]

1996

J. E. Proctor, P. Y. Barnes, “NIST high accuracy reference reflectometer-spectrophotometer,” J. Res. Natl. Inst. Stand. Technol. 101, 619–627 (1996).
[CrossRef]

1995

1994

1993

1985

1984

E. Marx, “Integral equations for scattering by a dielectric,” IEEE Trans. Antennas Propag. 32, 166–172 (1984).
[CrossRef]

1979

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

1977

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (1977).

Algäuer, M.

R. Brodmann, M. Algäuer, “Comparison of light scattering from rough surfaces with optical and mechanical profilometry,” in Proceedings on Surface Measurement and Characterization, J. M. Bennett, ed., Proc. SPIE1009, 111–118 (1988).
[CrossRef]

Barnes, P. Y.

P. Y. Barnes, E. A. Early, A. C. Parr, “Spectral reflectance,” NIST (Natl. Inst. Stand. Technol.) Spec.Publ. 250–48 (1998).

J. E. Proctor, P. Y. Barnes, “NIST high accuracy reference reflectometer-spectrophotometer,” J. Res. Natl. Inst. Stand. Technol. 101, 619–627 (1996).
[CrossRef]

Barrick, D. E.

D. E. Barrick, “Grazing behavior of scatter and propagation above any rough surface,” IEEE Trans. Antennas Propag. 46, 73–83 (1998).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 67 ff.

Bendler, J. T.

J. T. Bendler, S. F. Feldman, H. Hatti, S. Y. Hobbs, “Approximate model of diffuse reflectance from rough polymer surfaces,” J. Appl. Phys. 83, 998–1004 (1998).
[CrossRef]

Bennett, J. M.

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

Bhushan, B.

Brodmann, R.

R. Brodmann, M. Algäuer, “Comparison of light scattering from rough surfaces with optical and mechanical profilometry,” in Proceedings on Surface Measurement and Characterization, J. M. Bennett, ed., Proc. SPIE1009, 111–118 (1988).
[CrossRef]

Caber, P. J.

Creath, K.

K. Creath, “Comparison of phase-measurement algorithms,” K. Creath, ed., Proc. SPIE680, 19–28 (1986).

de Groot, P.

Deck, L.

Doi, T.

T. Doi, T. V. Vorburger, P. Sullivan, “Effects of defocus and algorithm on optical step height calibration,” Proc. Eng. 23, 135–143 (1999).
[CrossRef]

Early, E. A.

P. Y. Barnes, E. A. Early, A. C. Parr, “Spectral reflectance,” NIST (Natl. Inst. Stand. Technol.) Spec.Publ. 250–48 (1998).

Elson, J. M.

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

Feldman, S. F.

J. T. Bendler, S. F. Feldman, H. Hatti, S. Y. Hobbs, “Approximate model of diffuse reflectance from rough polymer surfaces,” J. Appl. Phys. 83, 998–1004 (1998).
[CrossRef]

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (1977).

Hall, D. B.

D. B. Hall, P. Underhill, J. M. Torkelson, “Spin coating of thin and ultrathin polymer films,” Polym. Eng. Sci. 38, 2039–2045 (1998).
[CrossRef]

Hatti, H.

J. T. Bendler, S. F. Feldman, H. Hatti, S. Y. Hobbs, “Approximate model of diffuse reflectance from rough polymer surfaces,” J. Appl. Phys. 83, 998–1004 (1998).
[CrossRef]

Hobbs, S. Y.

J. T. Bendler, S. F. Feldman, H. Hatti, S. Y. Hobbs, “Approximate model of diffuse reflectance from rough polymer surfaces,” J. Appl. Phys. 83, 998–1004 (1998).
[CrossRef]

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (1977).

Hunt, F. Y.

F. Y. Hunt, E. Marx, G. W. Meyer, T. V. Vorburger, P. A. Walker, H. B. Westlund, “A first step towards photorealistic rendering of coated surfaces and computer based standards of appearance,” in Service Life Methodology and Metrology, J. W. Martin, D. R. Bauer, eds., ACS Symposium Series (Oxford U. Press, New York, 2001).
[CrossRef]

Koliopoulos, C. L.

Kuyatt, C. E.

B. N. Taylor, C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” (National Institute of Standards and Technology, Washington, D.C., 1994).

Lettieri, T. R.

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (1977).

Martin, J. W.

M. E. McKnight, J. W. Martin, “Advanced methods and models for describing coating appearance,” Prog. Org. Coat. 34, 152–159 (1998).
[CrossRef]

Marx, E.

E. Marx, T. R. Lettieri, T. V. Vorburger, “Light scattering by sinusoidal surfaces: illumination windows and harmonics in standards,” Appl. Opt. 34, 1269–1277 (1995).
[CrossRef] [PubMed]

E. Marx, “Integral equations for scattering by a dielectric,” IEEE Trans. Antennas Propag. 32, 166–172 (1984).
[CrossRef]

F. Y. Hunt, E. Marx, G. W. Meyer, T. V. Vorburger, P. A. Walker, H. B. Westlund, “A first step towards photorealistic rendering of coated surfaces and computer based standards of appearance,” in Service Life Methodology and Metrology, J. W. Martin, D. R. Bauer, eds., ACS Symposium Series (Oxford U. Press, New York, 2001).
[CrossRef]

E. Marx, T. V. Vorburger, “Windowing effects on light scattered by sinusoidal surfaces” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 2–14 (1993).
[CrossRef]

Maynard, M.

M. Maynard, “Automakers’ color guard not spinning its wheels,” USA Today, 26October1998, p. 12B.

McKnight, M. E.

M. E. McKnight, J. W. Martin, “Advanced methods and models for describing coating appearance,” Prog. Org. Coat. 34, 152–159 (1998).
[CrossRef]

Meyer, G. W.

F. Y. Hunt, E. Marx, G. W. Meyer, T. V. Vorburger, P. A. Walker, H. B. Westlund, “A first step towards photorealistic rendering of coated surfaces and computer based standards of appearance,” in Service Life Methodology and Metrology, J. W. Martin, D. R. Bauer, eds., ACS Symposium Series (Oxford U. Press, New York, 2001).
[CrossRef]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (1977).

Parr, A. C.

P. Y. Barnes, E. A. Early, A. C. Parr, “Spectral reflectance,” NIST (Natl. Inst. Stand. Technol.) Spec.Publ. 250–48 (1998).

Proctor, J. E.

J. E. Proctor, P. Y. Barnes, “NIST high accuracy reference reflectometer-spectrophotometer,” J. Res. Natl. Inst. Stand. Technol. 101, 619–627 (1996).
[CrossRef]

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (1977).

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 67 ff.

Sullivan, P.

T. Doi, T. V. Vorburger, P. Sullivan, “Effects of defocus and algorithm on optical step height calibration,” Proc. Eng. 23, 135–143 (1999).
[CrossRef]

Taylor, B. N.

B. N. Taylor, C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” (National Institute of Standards and Technology, Washington, D.C., 1994).

Torkelson, J. M.

D. B. Hall, P. Underhill, J. M. Torkelson, “Spin coating of thin and ultrathin polymer films,” Polym. Eng. Sci. 38, 2039–2045 (1998).
[CrossRef]

Underhill, P.

D. B. Hall, P. Underhill, J. M. Torkelson, “Spin coating of thin and ultrathin polymer films,” Polym. Eng. Sci. 38, 2039–2045 (1998).
[CrossRef]

Vorburger, T. V.

T. Doi, T. V. Vorburger, P. Sullivan, “Effects of defocus and algorithm on optical step height calibration,” Proc. Eng. 23, 135–143 (1999).
[CrossRef]

E. Marx, T. R. Lettieri, T. V. Vorburger, “Light scattering by sinusoidal surfaces: illumination windows and harmonics in standards,” Appl. Opt. 34, 1269–1277 (1995).
[CrossRef] [PubMed]

F. Y. Hunt, E. Marx, G. W. Meyer, T. V. Vorburger, P. A. Walker, H. B. Westlund, “A first step towards photorealistic rendering of coated surfaces and computer based standards of appearance,” in Service Life Methodology and Metrology, J. W. Martin, D. R. Bauer, eds., ACS Symposium Series (Oxford U. Press, New York, 2001).
[CrossRef]

E. Marx, T. V. Vorburger, “Windowing effects on light scattered by sinusoidal surfaces” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 2–14 (1993).
[CrossRef]

Walker, P. A.

F. Y. Hunt, E. Marx, G. W. Meyer, T. V. Vorburger, P. A. Walker, H. B. Westlund, “A first step towards photorealistic rendering of coated surfaces and computer based standards of appearance,” in Service Life Methodology and Metrology, J. W. Martin, D. R. Bauer, eds., ACS Symposium Series (Oxford U. Press, New York, 2001).
[CrossRef]

Westlund, H. B.

F. Y. Hunt, E. Marx, G. W. Meyer, T. V. Vorburger, P. A. Walker, H. B. Westlund, “A first step towards photorealistic rendering of coated surfaces and computer based standards of appearance,” in Service Life Methodology and Metrology, J. W. Martin, D. R. Bauer, eds., ACS Symposium Series (Oxford U. Press, New York, 2001).
[CrossRef]

Wyant, J. C.

Appl. Opt.

IEEE Trans. Antennas Propag.

D. E. Barrick, “Grazing behavior of scatter and propagation above any rough surface,” IEEE Trans. Antennas Propag. 46, 73–83 (1998).
[CrossRef]

E. Marx, “Integral equations for scattering by a dielectric,” IEEE Trans. Antennas Propag. 32, 166–172 (1984).
[CrossRef]

J. Appl. Phys.

J. T. Bendler, S. F. Feldman, H. Hatti, S. Y. Hobbs, “Approximate model of diffuse reflectance from rough polymer surfaces,” J. Appl. Phys. 83, 998–1004 (1998).
[CrossRef]

J. Res. Natl. Inst. Stand. Technol.

J. E. Proctor, P. Y. Barnes, “NIST high accuracy reference reflectometer-spectrophotometer,” J. Res. Natl. Inst. Stand. Technol. 101, 619–627 (1996).
[CrossRef]

Natl. Bur. Stand. (U.S.) Monogr.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (1977).

NIST (Natl. Inst. Stand. Technol.) Spec.

P. Y. Barnes, E. A. Early, A. C. Parr, “Spectral reflectance,” NIST (Natl. Inst. Stand. Technol.) Spec.Publ. 250–48 (1998).

Opt. Eng.

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

Polym. Eng. Sci.

D. B. Hall, P. Underhill, J. M. Torkelson, “Spin coating of thin and ultrathin polymer films,” Polym. Eng. Sci. 38, 2039–2045 (1998).
[CrossRef]

Proc. Eng.

T. Doi, T. V. Vorburger, P. Sullivan, “Effects of defocus and algorithm on optical step height calibration,” Proc. Eng. 23, 135–143 (1999).
[CrossRef]

Prog. Org. Coat.

M. E. McKnight, J. W. Martin, “Advanced methods and models for describing coating appearance,” Prog. Org. Coat. 34, 152–159 (1998).
[CrossRef]

Other

M. Maynard, “Automakers’ color guard not spinning its wheels,” USA Today, 26October1998, p. 12B.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 67 ff.

R. Brodmann, M. Algäuer, “Comparison of light scattering from rough surfaces with optical and mechanical profilometry,” in Proceedings on Surface Measurement and Characterization, J. M. Bennett, ed., Proc. SPIE1009, 111–118 (1988).
[CrossRef]

F. Y. Hunt, E. Marx, G. W. Meyer, T. V. Vorburger, P. A. Walker, H. B. Westlund, “A first step towards photorealistic rendering of coated surfaces and computer based standards of appearance,” in Service Life Methodology and Metrology, J. W. Martin, D. R. Bauer, eds., ACS Symposium Series (Oxford U. Press, New York, 2001).
[CrossRef]

Certain commercial equipment is identified in this paper to describe adequately an experimental procedure. Such identification does not imply that the equipment identified is necessarily the best available for the purpose nor does it imply any recommendation or endorsement by NIST.

ASME Surface Quality Standard B46.1, Surface Texture (American Society of Mechanical Engineers, New York, 1995).

WYKO NT-2000, Veeco Metrology Group, Tucson, Ariz. 85706.

Zygo New View 5030, Zygo Corporation, Middlefield, Conn. 06455–0448.

B. N. Taylor, C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” (National Institute of Standards and Technology, Washington, D.C., 1994).

K. Creath, “Comparison of phase-measurement algorithms,” K. Creath, ed., Proc. SPIE680, 19–28 (1986).

E. Marx, T. V. Vorburger, “Windowing effects on light scattered by sinusoidal surfaces” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 2–14 (1993).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Schematic diagram illustrating fabrication of clear-epoxy samples with controlled roughness.

Fig. 2
Fig. 2

Schematic diagram of STARR with all the major components labeled in part (a). (a) Incident-flux measurement, (b) reflected-flux measurement.

Fig. 3
Fig. 3

Convention for incidence and scattering angles.

Fig. 4
Fig. 4

Schematic diagram of a typical interferometric microscope, which may be used either in the phase-shifting mode or the scanning white-light mode. PZT, piezoelectric transducer

Fig. 5
Fig. 5

Measured reflectance as a function of observation (scattering) angle for samples A, B, and C, and for a highly polished black-glass specimen at 550 nm and 60° angle of incidence.

Fig. 6
Fig. 6

Measured reflectance of sample B as a function of observation angle for incident angles of 20°, 45°, and 60°.

Fig. 7
Fig. 7

Surface topography map of sample A as measured by SWLI microscope 1; rms roughness, 805 nm.

Fig. 8
Fig. 8

Surface topography map of sample A as measured by SWLI microscope 2; rms roughness, 871 nm.

Fig. 9
Fig. 9

Surface topography map of sample B as measured by SWLI 1; rms roughness, 201 nm.

Fig. 10
Fig. 10

Surface topography map of sample B as measured by SWLI 2; rms roughness, 124 nm.

Fig. 11
Fig. 11

Typical surface topography map of sample C as measured by PSI microscope of type 1; rms roughness, 5.0 nm.

Fig. 12
Fig. 12

Comparison between relative reflectance curves as a function of scattering angle, measured and computed with a Schwartz window in the phase integral calculation.

Fig. 13
Fig. 13

Comparison of measured and computed reflectance curves, showing the difference between the effects of Schwartz and rectangular windows for data taken with SWLI 1. For sample B, calculated results are shown for ϕ i = 0° and ϕ i = 90°.

Fig. 14
Fig. 14

Comparison of measured and computed reflectance curves, showing the difference between the phase integral and ray approximations with SWLI 1 topographical data and the Schwartz window.

Tables (1)

Tables Icon

Table 1 Measurement Parameters for Scanning White-Light Interferometric Microscopes

Equations (12)

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

BRDFPs/Ωs cos θs/Pi
ρ=Ps/Pi=BRDF·Ωs cos θs.
ψθi, ϕi; θs, ϕs=F3θi, ϕi; θs, ϕs/A×expivθi, ϕi; θs, ϕs·rx, ydxdy,
F3θi, ϕi; θs, ϕs=1+cos θi cos θs+sin θi sin θs cosϕi-ϕscos θicos θi+cos θs,
v=-ksin θi cos ϕi+sin θs cos ϕseˆx+sin θi sin ϕi+sin θs sin ϕseˆy+cos θi+cos θseˆz,
v·r=-ksin θi cos ϕi+sin θs cos ϕsx+sin θi sin ϕi+sin θs sin ϕsy+cos θi+cos θsζx, y.
ψθi, ϕi; θs, ϕs=F3θi, ϕi; θs, ϕssincvxL1/2×sincvyL2/2,
ψθi, ϕi; θs, ϕs=F3θi, ϕi; θs, ϕs/AWx, y×expivθi, ϕi; θs, ϕs·rx, ydxdy.
Wx=exp{α1-Li2/Li2-x2}|x|Li0elsewhere,
I¯θ=-1/2π1/2πIθIrθ-θdθ.
kˆ=kˆi-2kˆi·nˆnˆ.
cos Δ=kˆ·rˆj=sin θ sin θj cosϕ-ϕj+cos θ cos θj,

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