Abstract

The color of a surface structured at the mesoscopic scale differs from the one of a flat surface of the same material because of the light inter-reflections taking place in the concavities of the surface, as well as shadowing effects. The color variation arises not only in scattering materials, but also in the absence of scattering, e.g., in metals and clear dielectrics, just as a consequence of multiple specular reflections between neighboring flat facets of the surface. In this paper, we investigate such color variation in the case of an infinitely long V-shaped groove, having in mind the visual appearance of a surface composed of many structures of that sort, all parallel and identical. We develop a full model of multiple specular reflections, accounting for the ray position and orientation and the polarization effects occurring at each reflection. We compare that situation with two approximate models, more usual and easier to compute, where light is assumed to remain unpolarized all along, or where the $p$- and $s$-polarized components are treated separately. Spectral reflectances were predicted for various materials and angles of cavities, under diffuse illumination. In most cases, the three models predict very similar bi-hemispherical reflectances, but the hemispherical-directional reflectances can vary noticeably in certain observation directions. This study might help achieve a more physically realistic rendering of dielectric or metallic ridged surfaces in computer graphics.

© 2019 Optical Society of America

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References

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  1. S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71, 076401 (2008).
    [Crossref]
  2. R. Charrière, G. Lacaille, M. Pedeferri, J. Faucheu, and D. Delafosse, “Characterization of the gonioapparent character of colored anodized titanium surfaces,” Color Res. Appl. 40, 483–490 (2015).
    [Crossref]
  3. P. Kubelka, “New contributions to the optics of intensely light-scattering material, part I,” J. Opt. Soc. Am. A 38, 448–457 (1948).
    [Crossref]
  4. N. T. Melamed, “Optical properties of powders: part I. Optical absorption coefficients and the absolute value of the diffuse reflectance,” J. Appl. Phys. 34, 560–570 (1963).
    [Crossref]
  5. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  6. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 200–227.
  7. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (MA Artech House Inc., 1963), pp. 70–98.
  8. K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1105–1114 (1967).
    [Crossref]
  9. E. Heitz, J. Hanika, E. d’Eon, and C. Dachsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. (Proc. SIGGRAPH) 35, 58 (2016).
    [Crossref]
  10. F. Xie and P. Hanrahan, “Multiple scattering from distributions of specular v-grooves,” in ACM SIGGRAPH Asia 2018 Technical Papers (2018), paper 276.
  11. J. H. Lee, A. Jarabo, D. S. Jeon, D. Gutierrez, and M. H. Kim, “Practical multiple scattering for rough surfaces,” in ACM SIGGRAPH Asia 2018 Technical Papers (2018), paper 275.
  12. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part II: nonhomogeneous layers,” J. Opt. Soc. Am. A 44, 330–335 (1954).
    [Crossref]
  13. L. Simonot, M. Hébert, and R. D. Hersch, “Extension of the Williams–Clapper model to stacked nondiffusing colored coatings with different refractive indices,” J. Opt. Soc. Am. A 23, 1432–1441 (2006).
    [Crossref]
  14. M. Hébert, R. D. Hersch, and J.-M. Becker, “Compositional reflectance and transmittance model for multilayer specimens,” J. Opt. Soc. Am. A 24, 2628–2644 (2007).
    [Crossref]
  15. P. Emmel and R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).
  16. F. R. Clapper and J. A. C. Yule, “The effect of multiple internal reflections on the densities of halftone prints on paper,” J. Opt. Soc. Am. 43, 600–603 (1953).
    [Crossref]
  17. J. A. C. Yule and W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproductions,” in Proc. TAGA (1951), Vol. 3, pp. 65–76.
  18. G. Rogers, “Effect of light scatter on halftone color,” J. Opt. Soc. Am. A 15, 1813–1821 (1998).
    [Crossref]
  19. G. Rogers, “A generalized Clapper–Yule model of halftone reflectance,” Color Res. Appl. 25, 402–407 (2000).
    [Crossref]
  20. D. Saint-Pierre, R. Deeb, D. Muselet, L. Simonot, and M. Hébert, “Light interreflections and shadowing effects in a Lambertian V-cavity under diffuse illumination,” in IS&T Electronic Imaging Symposium, Material Appearance, Burlingame, California, USA, January29–February 2, 2018.
  21. R. Deeb, D. Muselet, M. Hebert, and A. Trémeau, “Spectral reflectance estimation from one RGB image using self-interreflections in a concave object,” Appl. Opt. 57, 4918–4929 (2018).
    [Crossref]
  22. D. Saint-Pierre, L. Simonot, and M. Hébert, “Reflectance computation for a specular only V-cavity,” in International Workshop on Computational Color Imaging (Springer, 2019), pp. 289–303.
  23. M. Born and E. Wolf, “Reflection and refraction of a plane wave,” in Principle of Optics, 7th ed. (Pergamon, 1999), pp. 43–48.
  24. F. E. Nicomedus, J. C. Richmond, and J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160, NBS, 52 (1977).
  25. J. P. Snyder, Map Projections—A Working Manual, U.S. Geological Survey Professional Paper (U.S. Government Printing Office, 1987), Vol. 1395, pp. 182–190.
  26. B. J. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
    [Crossref]

2018 (1)

2016 (1)

E. Heitz, J. Hanika, E. d’Eon, and C. Dachsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. (Proc. SIGGRAPH) 35, 58 (2016).
[Crossref]

2015 (1)

R. Charrière, G. Lacaille, M. Pedeferri, J. Faucheu, and D. Delafosse, “Characterization of the gonioapparent character of colored anodized titanium surfaces,” Color Res. Appl. 40, 483–490 (2015).
[Crossref]

2008 (1)

S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71, 076401 (2008).
[Crossref]

2007 (1)

2006 (1)

2000 (2)

P. Emmel and R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

G. Rogers, “A generalized Clapper–Yule model of halftone reflectance,” Color Res. Appl. 25, 402–407 (2000).
[Crossref]

1998 (1)

1967 (2)

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

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

1963 (1)

N. T. Melamed, “Optical properties of powders: part I. Optical absorption coefficients and the absolute value of the diffuse reflectance,” J. Appl. Phys. 34, 560–570 (1963).
[Crossref]

1954 (1)

P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part II: nonhomogeneous layers,” J. Opt. Soc. Am. A 44, 330–335 (1954).
[Crossref]

1953 (1)

1948 (1)

P. Kubelka, “New contributions to the optics of intensely light-scattering material, part I,” J. Opt. Soc. Am. A 38, 448–457 (1948).
[Crossref]

Becker, J.-M.

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (MA Artech House Inc., 1963), pp. 70–98.

Born, M.

M. Born and E. Wolf, “Reflection and refraction of a plane wave,” in Principle of Optics, 7th ed. (Pergamon, 1999), pp. 43–48.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Charrière, R.

R. Charrière, G. Lacaille, M. Pedeferri, J. Faucheu, and D. Delafosse, “Characterization of the gonioapparent character of colored anodized titanium surfaces,” Color Res. Appl. 40, 483–490 (2015).
[Crossref]

Clapper, F. R.

d’Eon, E.

E. Heitz, J. Hanika, E. d’Eon, and C. Dachsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. (Proc. SIGGRAPH) 35, 58 (2016).
[Crossref]

Dachsbacher, C.

E. Heitz, J. Hanika, E. d’Eon, and C. Dachsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. (Proc. SIGGRAPH) 35, 58 (2016).
[Crossref]

Deeb, R.

R. Deeb, D. Muselet, M. Hebert, and A. Trémeau, “Spectral reflectance estimation from one RGB image using self-interreflections in a concave object,” Appl. Opt. 57, 4918–4929 (2018).
[Crossref]

D. Saint-Pierre, R. Deeb, D. Muselet, L. Simonot, and M. Hébert, “Light interreflections and shadowing effects in a Lambertian V-cavity under diffuse illumination,” in IS&T Electronic Imaging Symposium, Material Appearance, Burlingame, California, USA, January29–February 2, 2018.

Delafosse, D.

R. Charrière, G. Lacaille, M. Pedeferri, J. Faucheu, and D. Delafosse, “Characterization of the gonioapparent character of colored anodized titanium surfaces,” Color Res. Appl. 40, 483–490 (2015).
[Crossref]

Emmel, P.

P. Emmel and R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

Faucheu, J.

R. Charrière, G. Lacaille, M. Pedeferri, J. Faucheu, and D. Delafosse, “Characterization of the gonioapparent character of colored anodized titanium surfaces,” Color Res. Appl. 40, 483–490 (2015).
[Crossref]

Gutierrez, D.

J. H. Lee, A. Jarabo, D. S. Jeon, D. Gutierrez, and M. H. Kim, “Practical multiple scattering for rough surfaces,” in ACM SIGGRAPH Asia 2018 Technical Papers (2018), paper 275.

Hanika, J.

E. Heitz, J. Hanika, E. d’Eon, and C. Dachsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. (Proc. SIGGRAPH) 35, 58 (2016).
[Crossref]

Hanrahan, P.

F. Xie and P. Hanrahan, “Multiple scattering from distributions of specular v-grooves,” in ACM SIGGRAPH Asia 2018 Technical Papers (2018), paper 276.

Hebert, M.

Hébert, M.

M. Hébert, R. D. Hersch, and J.-M. Becker, “Compositional reflectance and transmittance model for multilayer specimens,” J. Opt. Soc. Am. A 24, 2628–2644 (2007).
[Crossref]

L. Simonot, M. Hébert, and R. D. Hersch, “Extension of the Williams–Clapper model to stacked nondiffusing colored coatings with different refractive indices,” J. Opt. Soc. Am. A 23, 1432–1441 (2006).
[Crossref]

D. Saint-Pierre, L. Simonot, and M. Hébert, “Reflectance computation for a specular only V-cavity,” in International Workshop on Computational Color Imaging (Springer, 2019), pp. 289–303.

D. Saint-Pierre, R. Deeb, D. Muselet, L. Simonot, and M. Hébert, “Light interreflections and shadowing effects in a Lambertian V-cavity under diffuse illumination,” in IS&T Electronic Imaging Symposium, Material Appearance, Burlingame, California, USA, January29–February 2, 2018.

Heitz, E.

E. Heitz, J. Hanika, E. d’Eon, and C. Dachsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. (Proc. SIGGRAPH) 35, 58 (2016).
[Crossref]

Hersch, R. D.

Hsia, J. J.

F. E. Nicomedus, J. C. Richmond, and J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160, NBS, 52 (1977).

Jarabo, A.

J. H. Lee, A. Jarabo, D. S. Jeon, D. Gutierrez, and M. H. Kim, “Practical multiple scattering for rough surfaces,” in ACM SIGGRAPH Asia 2018 Technical Papers (2018), paper 275.

Jeon, D. S.

J. H. Lee, A. Jarabo, D. S. Jeon, D. Gutierrez, and M. H. Kim, “Practical multiple scattering for rough surfaces,” in ACM SIGGRAPH Asia 2018 Technical Papers (2018), paper 275.

Kim, M. H.

J. H. Lee, A. Jarabo, D. S. Jeon, D. Gutierrez, and M. H. Kim, “Practical multiple scattering for rough surfaces,” in ACM SIGGRAPH Asia 2018 Technical Papers (2018), paper 275.

Kinoshita, S.

S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71, 076401 (2008).
[Crossref]

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part II: nonhomogeneous layers,” J. Opt. Soc. Am. A 44, 330–335 (1954).
[Crossref]

P. Kubelka, “New contributions to the optics of intensely light-scattering material, part I,” J. Opt. Soc. Am. A 38, 448–457 (1948).
[Crossref]

Lacaille, G.

R. Charrière, G. Lacaille, M. Pedeferri, J. Faucheu, and D. Delafosse, “Characterization of the gonioapparent character of colored anodized titanium surfaces,” Color Res. Appl. 40, 483–490 (2015).
[Crossref]

Lee, J. H.

J. H. Lee, A. Jarabo, D. S. Jeon, D. Gutierrez, and M. H. Kim, “Practical multiple scattering for rough surfaces,” in ACM SIGGRAPH Asia 2018 Technical Papers (2018), paper 275.

Melamed, N. T.

N. T. Melamed, “Optical properties of powders: part I. Optical absorption coefficients and the absolute value of the diffuse reflectance,” J. Appl. Phys. 34, 560–570 (1963).
[Crossref]

Miyazaki, J.

S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71, 076401 (2008).
[Crossref]

Muselet, D.

R. Deeb, D. Muselet, M. Hebert, and A. Trémeau, “Spectral reflectance estimation from one RGB image using self-interreflections in a concave object,” Appl. Opt. 57, 4918–4929 (2018).
[Crossref]

D. Saint-Pierre, R. Deeb, D. Muselet, L. Simonot, and M. Hébert, “Light interreflections and shadowing effects in a Lambertian V-cavity under diffuse illumination,” in IS&T Electronic Imaging Symposium, Material Appearance, Burlingame, California, USA, January29–February 2, 2018.

Nicomedus, F. E.

F. E. Nicomedus, J. C. Richmond, and J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160, NBS, 52 (1977).

Nielsen, W. J.

J. A. C. Yule and W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproductions,” in Proc. TAGA (1951), Vol. 3, pp. 65–76.

Pedeferri, M.

R. Charrière, G. Lacaille, M. Pedeferri, J. Faucheu, and D. Delafosse, “Characterization of the gonioapparent character of colored anodized titanium surfaces,” Color Res. Appl. 40, 483–490 (2015).
[Crossref]

Richmond, J. C.

F. E. Nicomedus, J. C. Richmond, and J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160, NBS, 52 (1977).

Rogers, G.

G. Rogers, “A generalized Clapper–Yule model of halftone reflectance,” Color Res. Appl. 25, 402–407 (2000).
[Crossref]

G. Rogers, “Effect of light scatter on halftone color,” J. Opt. Soc. Am. A 15, 1813–1821 (1998).
[Crossref]

Saint-Pierre, D.

D. Saint-Pierre, R. Deeb, D. Muselet, L. Simonot, and M. Hébert, “Light interreflections and shadowing effects in a Lambertian V-cavity under diffuse illumination,” in IS&T Electronic Imaging Symposium, Material Appearance, Burlingame, California, USA, January29–February 2, 2018.

D. Saint-Pierre, L. Simonot, and M. Hébert, “Reflectance computation for a specular only V-cavity,” in International Workshop on Computational Color Imaging (Springer, 2019), pp. 289–303.

Simonot, L.

L. Simonot, M. Hébert, and R. D. Hersch, “Extension of the Williams–Clapper model to stacked nondiffusing colored coatings with different refractive indices,” J. Opt. Soc. Am. A 23, 1432–1441 (2006).
[Crossref]

D. Saint-Pierre, L. Simonot, and M. Hébert, “Reflectance computation for a specular only V-cavity,” in International Workshop on Computational Color Imaging (Springer, 2019), pp. 289–303.

D. Saint-Pierre, R. Deeb, D. Muselet, L. Simonot, and M. Hébert, “Light interreflections and shadowing effects in a Lambertian V-cavity under diffuse illumination,” in IS&T Electronic Imaging Symposium, Material Appearance, Burlingame, California, USA, January29–February 2, 2018.

Smith, B. J.

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

Snyder, J. P.

J. P. Snyder, Map Projections—A Working Manual, U.S. Geological Survey Professional Paper (U.S. Government Printing Office, 1987), Vol. 1395, pp. 182–190.

Sparrow, E. M.

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (MA Artech House Inc., 1963), pp. 70–98.

Torrance, K. E.

Trémeau, A.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 200–227.

Wolf, E.

M. Born and E. Wolf, “Reflection and refraction of a plane wave,” in Principle of Optics, 7th ed. (Pergamon, 1999), pp. 43–48.

Xie, F.

F. Xie and P. Hanrahan, “Multiple scattering from distributions of specular v-grooves,” in ACM SIGGRAPH Asia 2018 Technical Papers (2018), paper 276.

Yoshioka, S.

S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71, 076401 (2008).
[Crossref]

Yule, J. A. C.

F. R. Clapper and J. A. C. Yule, “The effect of multiple internal reflections on the densities of halftone prints on paper,” J. Opt. Soc. Am. 43, 600–603 (1953).
[Crossref]

J. A. C. Yule and W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproductions,” in Proc. TAGA (1951), Vol. 3, pp. 65–76.

ACM Trans. Graph. (Proc. SIGGRAPH) (1)

E. Heitz, J. Hanika, E. d’Eon, and C. Dachsbacher, “Multiple-scattering microfacet BSDFs with the Smith model,” ACM Trans. Graph. (Proc. SIGGRAPH) 35, 58 (2016).
[Crossref]

Appl. Opt. (1)

Color Res. Appl. (2)

G. Rogers, “A generalized Clapper–Yule model of halftone reflectance,” Color Res. Appl. 25, 402–407 (2000).
[Crossref]

R. Charrière, G. Lacaille, M. Pedeferri, J. Faucheu, and D. Delafosse, “Characterization of the gonioapparent character of colored anodized titanium surfaces,” Color Res. Appl. 40, 483–490 (2015).
[Crossref]

IEEE Trans. Antennas Propag. (1)

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

J. Appl. Phys. (1)

N. T. Melamed, “Optical properties of powders: part I. Optical absorption coefficients and the absolute value of the diffuse reflectance,” J. Appl. Phys. 34, 560–570 (1963).
[Crossref]

J. Imaging Sci. Technol. (1)

P. Emmel and R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Rep. Prog. Phys. (1)

S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71, 076401 (2008).
[Crossref]

Other (11)

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 200–227.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (MA Artech House Inc., 1963), pp. 70–98.

F. Xie and P. Hanrahan, “Multiple scattering from distributions of specular v-grooves,” in ACM SIGGRAPH Asia 2018 Technical Papers (2018), paper 276.

J. H. Lee, A. Jarabo, D. S. Jeon, D. Gutierrez, and M. H. Kim, “Practical multiple scattering for rough surfaces,” in ACM SIGGRAPH Asia 2018 Technical Papers (2018), paper 275.

J. A. C. Yule and W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproductions,” in Proc. TAGA (1951), Vol. 3, pp. 65–76.

D. Saint-Pierre, R. Deeb, D. Muselet, L. Simonot, and M. Hébert, “Light interreflections and shadowing effects in a Lambertian V-cavity under diffuse illumination,” in IS&T Electronic Imaging Symposium, Material Appearance, Burlingame, California, USA, January29–February 2, 2018.

D. Saint-Pierre, L. Simonot, and M. Hébert, “Reflectance computation for a specular only V-cavity,” in International Workshop on Computational Color Imaging (Springer, 2019), pp. 289–303.

M. Born and E. Wolf, “Reflection and refraction of a plane wave,” in Principle of Optics, 7th ed. (Pergamon, 1999), pp. 43–48.

F. E. Nicomedus, J. C. Richmond, and J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160, NBS, 52 (1977).

J. P. Snyder, Map Projections—A Working Manual, U.S. Geological Survey Professional Paper (U.S. Government Printing Office, 1987), Vol. 1395, pp. 182–190.

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Figures (9)

Fig. 1.
Fig. 1. Structured surface with parallel, periodical, and identical V-shaped ridges of dihedral angle α.
Fig. 2.
Fig. 2. 3D geometry of one cavity, and vector ${\bf e}$ representing the direction of illumination.
Fig. 3.
Fig. 3. (a–b) Representations of two light rays parallel to the unit vector ${\bf e}$ , striking the cavity on facet 2 in different positions. (b) 2D representation of the same two light rays projected onto the ( $yOz$ ) vertical plane. The light path can be represented by a straight line joining the successive images of the facets. The projection of the real light paths in broken straight lines is also represented. Geometry for the calculation of the number of reflections, for the same position ${y_P}$ of the ray, and two different orientations. (c–d) Geometry for the calculation of the number of reflections, for the same position ${y_P}$ of the ray, and two different orientations.
Fig. 4.
Fig. 4. RGB picture of a V-cavity made of gold with dihedral angle of 45°, placed in an integrating sphere and observed from a direction ( $\theta \approx {30}^\circ $ , $\varphi \approx {90}^\circ $ ). The red rectangle features the area where the cavity can be considered as a cavity of infinite length.
Fig. 5.
Fig. 5. When the (a) hemisphere is mapped onto a (b) disk according to the Lambert azimuthal equal area projection, any portion of the hemisphere with area $A$ is mapped into a portion of the disk with the same area $A$ .
Fig. 6.
Fig. 6. Color maps of hemispherical-directional reflectance for various materials, obtained with different dihedral angles of a cavity, represented with the Lambert azimuthal equal area projection.
Fig. 7.
Fig. 7. (a) Spectral bi-hemispherical reflectances of cavities with different dihedral angles made of gold, and (b) corresponding color components represented in the ( $L^*$ , $C^*$ ) diagram of the CIE 1976 $\text{L}^{*}\text{a}^{*}\text{b}^{*}$ color space, predicted by the rigorous model for various of cavity.
Fig. 8.
Fig. 8. Color maps of the hemispherical-directional reflectance thanks to the Lambert azimuthal equal area projection, generated for silicon, with a dihedral angle of cavity of 45°, by using (a) the rigorous model taking into account the polarization of light, (b) the first approximate model assuming that light remains unpolarized after each reflection, and (c) the second approximate model where the $p$ and $s$ components are assumed to be multiply reflected in parallel, independently from each other.
Fig. 9.
Fig. 9. Lightness $L^*$ in the CIE 1976 $\text{L}^{*}\text{a}^{*}\text{b}^{*}$ color space computed from the spectral reflectance of silicon according to the rigorous model and the two approximate ones for a dihedral angle of cavity of 45°, as a function of the polar observation angle $\theta $ (a) when the observation direction is perpendicular to the ridges ( $\varphi = {0}$ or $\pi $ ) and (b) when it is parallel to the ridges ( $\varphi = \pi /{2}$ or ${3}\pi \text{/2}$ ). These curves correspond to lightness profiles of the (a) horizontal diameter and (b) vertical diameter of the maps shown in Fig. 8.

Tables (3)

Tables Icon

Table 1. Bi-Hemispherical Reflectance at 550  nm (in %)

Tables Icon

Table 2. CIE 1994 $\Delta \text{E}$ Values between Colors Corresponding to Spectral Reflectances Predicted by Different Models

Tables Icon

Table 3. Bi-Hemispherical Reflectance (in %) at 550  nm of a V-Cavity Made of Silver

Equations (44)

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

N 1 = ( 0 , cos ( α / 2 ) , sin ( α / 2 ) ) ,
N 2 = ( 0 , cos ( α / 2 ) , sin ( α / 2 ) ) .
e = ( sin θ sin φ , sin θ cos φ , cos θ ) .
e = ( sin θ , cos θ ) ,
θ = arctan ( tan θ cos φ ) .
y Q = sin ( β H β G ) cos β G cos β H ,
P G = ( sin β G y P , cos β G cos ( α 2 ) )
det ( sin β G y P sin θ cos β G cos ( α 2 ) cos θ ) = 0.
sin ( β G θ ) = y P cos θ cos ( α 2 ) sin θ ,
β G = θ + arcsin [ y P cos θ cos ( α 2 ) sin θ ] .
sin ( β H θ ) = y P cos θ cos ( α 2 ) sin θ .
β H = θ + π arcsin [ y P cos θ cos ( α 2 ) sin θ ] .
γ H = { 2 π β H when y Q < 0 β H when y Q > 0 ,
m = floor [ γ H α 1 2 ] + 1 ,
R ( θ i ) = 1 2 ( | r P ( θ i ) | 2 + | r S ( θ i ) | 2 ) .
θ i ( 1 ) = { arccos ( e N 1 ) if y Q < 0 arccos ( e N 2 ) if y Q > 0 ,
N 1 ( j ) = ( 0 cos ( α / 2 + ( j 1 ) α ) sin ( α / 2 + ( j 1 ) α ) )
N 2 ( j ) = ( 0 cos ( α / 2 + ( j 1 ) α ) sin ( α / 2 + ( j 1 ) α ) ) ,
θ i ( j ) = { arccos ( e N 1 ( j ) ) if y Q < 0 arccos ( e N 2 ( j ) ) if y Q > 0 .
R ( θ , φ , y P ) = j = 1 m R [ θ i ( j ) ] .
R ( θ , φ , y P ) = 1 2 R p ( θ , φ , y P ) + R s ( θ , φ , y P ) .
{ E 1 p = r 1 p E 0 p E 1 s = r 1 s E 0 s ,
( E 1 p E 1 s ) = R 1 ( E 0 p E 0 s ) ,
R 1 = ( r 1 p 0 0 r 1 s ) .
{ E 1 p = E 1 p cos ψ 12 + E 1 s sin ψ 12 E 1 s = E 1 p sin ψ 12 + E 1 s cos ψ 12 .
M ( ψ 12 ) = ( cos ψ 12 sin ψ 12 sin ψ 12 cos ψ 12 ) ,
R 2 = ( r 2 p 0 0 r 2 s ) ,
( E 2 p E 2 s ) = R 2 M ( ψ 12 ) ( E 1 p E 1 s ) = R 2 M ( ψ 12 ) R 1 ( E 0 p E 0 s ) .
( E m p E m s ) = R m M ( ψ m 1 , m ) R 3 M ( ψ 23 ) R 2 M ( ψ 12 ) × R 1 ( E 0 p E 0 s ) .
{ E m p = a E 0 p + b E 0 s E m s = c E 0 p + d E 0 s ,
F m p = a 2 F 0 p + b 2 F 0 s = ( a 2 + b 2 ) F 0 / 2 , F m s = c 2 F 0 p + d 2 F 0 s = ( c 2 + d 2 ) F 0 / 2 ,
F m = F m p + F m s = ( a 2 + b 2 + c 2 + d 2 ) F 0 2 .
S j = e × N k ( j ) e × N k ( j ) .
cos ψ j , j + 1 = | S j S j + 1 | .
E i = L i cos θ Δ ω ,
F r = Δ x E i y p = sin ( α / 2 ) sin ( α / 2 ) R ( θ , φ , y P ) d y P .
R ( θ , φ ) = F r F i = 1 2 sin ( α / 2 ) y p = sin ( α / 2 ) sin ( α / 2 ) R ( θ , φ , y P ) d y P .
L r ( θ , φ ) = R ( θ , φ ) L i .
{ u = 2 sin ( θ / 2 ) cos φ v = 2 sin ( θ / 2 ) sin φ .
E i = θ = 0 π / 2 φ = 0 2 π L i cos θ sin θ d θ d φ = π L i ,
M = θ r = 0 π / 2 φ r = 0 2 π L r ( θ r , φ r ) cos θ r sin θ r d θ r d φ r .
R ¯ = M E i = 1 π L i θ r = 0 π / 2 φ r = 0 2 π L r ( θ r , φ r ) cos θ r sin θ r d θ r d φ r ,
R ¯ = 1 2 π sin ( α / 2 ) θ r = 0 π / 2 φ r = 0 2 π y p = sin ( α / 2 ) sin ( α / 2 ) R ( θ r , φ r , y ) d y P × cos θ r sin θ r d θ r d φ r .
C = a 2 + b 2 ,

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