Abstract

The present paper investigates the reflection and transmission properties of piles of nonscattering sheets. Using a spectral prediction model, we perform a detailed analysis of the spectral and color variations induced by variations of the number of superposed sheets, the absorbance of the sheet material, the refractive index of the medium between the sheets, and the reflectance of the background. The spectral prediction model accounts for the multiple reflections and transmissions of light between the interfaces bounding the layers. We describe in detail the procedure for deducing model parameters from measured data. Tests performed with nonscattering plastic sheets demonstrate the excellent accuracy of the predictions. A large set of predicted spectra illustrate the different evolutions of reflected and transmitted spectra as well as the corresponding colors for various types of piles.

© 2008 Optical Society of America

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References

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  1. J. L. Saunderson, “Calculation of the color pigmented plastics,” J. Opt. Soc. Am. 32, 727-736 (1942).
    [CrossRef]
  2. P. Kubelka, “New contributions to the optics of intensely light-scattering material, part I,” J. Opt. Soc. Am. 38, 448-457 (1948).
    [CrossRef] [PubMed]
  3. F. C. Williams and F. R. Clapper, “Multiple internal reflections in photographic color prints,” J. Opt. Soc. Am. 29, 595-599 (1953).
    [CrossRef]
  4. J. D. Shore and J. P. Spoonhower, “Reflection density in photographic color prints: Generalizations of the Williams-Clapper transform,” J. Imaging Sci. Technol. 45, 484-488 (2001).
  5. 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]
  6. M. Hébert and R. D. Hersch, “Deducing ink-transmittance spectra from reflectance and transmittance measurements of prints,” Proc. SPIE 6493, 649314-1-13 (2007).
  7. 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]
  8. H.-H. Perkampus, Encyclopedia of Spectroscopy (VCH, 1995).
  9. M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999).
  10. W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).
  11. M. Hébert and R. D. Hersch, “Classical print reflection models: A radiometric approach,” J. Imaging Sci. Technol. 48, 363-374 (2004).
  12. G. Sharma, “Color fundamentals for digital imaging,” in Digital Color Imaging Handbook, G. Sharma, ed. (CRC Press, 2003), pp. 30-36.
  13. M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete 2 flux models for reflectance and transmittance of diffusing layers,” J. Opt. A, Pure Appl. Opt. 10, 035006 (2008).
    [CrossRef]
  14. P. Kubelka, “New contributions to the optics of intensely light-scattering materials, part II: Nonhomogeneous layers,” J. Opt. Soc. Am. 44, 330-335 (1954).
    [CrossRef]
  15. M. Hébert, R. Hersch, and J.-M. Becker, “Compositional reflectance and transmittance model for multilayer specimens,” J. Opt. Soc. Am. A 24, 2628-2644 (2007).
    [CrossRef]
  16. D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329-332 (1942).
  17. M. Elias and L. Simonot, “Separation between the different fluxes scattered by art glazes: Explanation of the special color saturation,” Appl. Opt. 45, 3163-3172 (2006).
    [CrossRef] [PubMed]
  18. L. Simonot, M. Elias, and E. Charron, “Special visual effect of art-glazes explained by the radiative transfer equation,” Appl. Opt. 43, 2580-2587 (2004).
    [CrossRef] [PubMed]
  19. M. Hébert and R. D. Hersch, “A reflectance and transmittance model for recto-verso halftone prints,” J. Opt. Soc. Am. A 22, 1952-1967 (2006).
    [CrossRef]
  20. M. Hébert, “Compositional model for predicting multilayer reflectances and transmittances in color reproduction,” Ph.D. dissertation (Ecole Polytchnique Fédérale de Lausanne, 2006), p. 139.

2008

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete 2 flux models for reflectance and transmittance of diffusing layers,” J. Opt. A, Pure Appl. Opt. 10, 035006 (2008).
[CrossRef]

2007

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

M. Hébert and R. D. Hersch, “Deducing ink-transmittance spectra from reflectance and transmittance measurements of prints,” Proc. SPIE 6493, 649314-1-13 (2007).

2006

2004

L. Simonot, M. Elias, and E. Charron, “Special visual effect of art-glazes explained by the radiative transfer equation,” Appl. Opt. 43, 2580-2587 (2004).
[CrossRef] [PubMed]

M. Hébert and R. D. Hersch, “Classical print reflection models: A radiometric approach,” J. Imaging Sci. Technol. 48, 363-374 (2004).

2003

G. Sharma, “Color fundamentals for digital imaging,” in Digital Color Imaging Handbook, G. Sharma, ed. (CRC Press, 2003), pp. 30-36.

2001

J. D. Shore and J. P. Spoonhower, “Reflection density in photographic color prints: Generalizations of the Williams-Clapper transform,” J. Imaging Sci. Technol. 45, 484-488 (2001).

1999

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999).

1995

H.-H. Perkampus, Encyclopedia of Spectroscopy (VCH, 1995).

1994

W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).

1954

1953

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]

F. C. Williams and F. R. Clapper, “Multiple internal reflections in photographic color prints,” J. Opt. Soc. Am. 29, 595-599 (1953).
[CrossRef]

1948

1942

J. L. Saunderson, “Calculation of the color pigmented plastics,” J. Opt. Soc. Am. 32, 727-736 (1942).
[CrossRef]

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329-332 (1942).

Becker, J.-M.

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete 2 flux models for reflectance and transmittance of diffusing layers,” J. Opt. A, Pure Appl. Opt. 10, 035006 (2008).
[CrossRef]

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999).

Charron, E.

Clapper, F. R.

F. C. Williams and F. R. Clapper, “Multiple internal reflections in photographic color prints,” J. Opt. Soc. Am. 29, 595-599 (1953).
[CrossRef]

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]

Elias, M.

Hébert, M.

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete 2 flux models for reflectance and transmittance of diffusing layers,” J. Opt. A, Pure Appl. Opt. 10, 035006 (2008).
[CrossRef]

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

M. Hébert and R. D. Hersch, “Deducing ink-transmittance spectra from reflectance and transmittance measurements of prints,” Proc. SPIE 6493, 649314-1-13 (2007).

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]

M. Hébert and R. D. Hersch, “A reflectance and transmittance model for recto-verso halftone prints,” J. Opt. Soc. Am. A 22, 1952-1967 (2006).
[CrossRef]

M. Hébert, “Compositional model for predicting multilayer reflectances and transmittances in color reproduction,” Ph.D. dissertation (Ecole Polytchnique Fédérale de Lausanne, 2006), p. 139.

M. Hébert and R. D. Hersch, “Classical print reflection models: A radiometric approach,” J. Imaging Sci. Technol. 48, 363-374 (2004).

Hersch, R.

Hersch, R. D.

M. Hébert and R. D. Hersch, “Deducing ink-transmittance spectra from reflectance and transmittance measurements of prints,” Proc. SPIE 6493, 649314-1-13 (2007).

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]

M. Hébert and R. D. Hersch, “A reflectance and transmittance model for recto-verso halftone prints,” J. Opt. Soc. Am. A 22, 1952-1967 (2006).
[CrossRef]

M. Hébert and R. D. Hersch, “Classical print reflection models: A radiometric approach,” J. Imaging Sci. Technol. 48, 363-374 (2004).

Judd, D. B.

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329-332 (1942).

Kubelka, P.

McCluney, W. R.

W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).

Perkampus, H.-H.

H.-H. Perkampus, Encyclopedia of Spectroscopy (VCH, 1995).

Saunderson, J. L.

Sharma, G.

G. Sharma, “Color fundamentals for digital imaging,” in Digital Color Imaging Handbook, G. Sharma, ed. (CRC Press, 2003), pp. 30-36.

G. Sharma, “Color fundamentals for digital imaging,” in Digital Color Imaging Handbook, G. Sharma, ed. (CRC Press, 2003), pp. 30-36.

Shore, J. D.

J. D. Shore and J. P. Spoonhower, “Reflection density in photographic color prints: Generalizations of the Williams-Clapper transform,” J. Imaging Sci. Technol. 45, 484-488 (2001).

Simonot, L.

Spoonhower, J. P.

J. D. Shore and J. P. Spoonhower, “Reflection density in photographic color prints: Generalizations of the Williams-Clapper transform,” J. Imaging Sci. Technol. 45, 484-488 (2001).

Williams, F. C.

F. C. Williams and F. R. Clapper, “Multiple internal reflections in photographic color prints,” J. Opt. Soc. Am. 29, 595-599 (1953).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999).

Yule, J. A. C.

Appl. Opt.

J. Imaging Sci. Technol.

J. D. Shore and J. P. Spoonhower, “Reflection density in photographic color prints: Generalizations of the Williams-Clapper transform,” J. Imaging Sci. Technol. 45, 484-488 (2001).

M. Hébert and R. D. Hersch, “Classical print reflection models: A radiometric approach,” J. Imaging Sci. Technol. 48, 363-374 (2004).

J. Opt. A, Pure Appl. Opt.

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete 2 flux models for reflectance and transmittance of diffusing layers,” J. Opt. A, Pure Appl. Opt. 10, 035006 (2008).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Res. Natl. Bur. Stand.

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329-332 (1942).

Proc. SPIE

M. Hébert and R. D. Hersch, “Deducing ink-transmittance spectra from reflectance and transmittance measurements of prints,” Proc. SPIE 6493, 649314-1-13 (2007).

Other

H.-H. Perkampus, Encyclopedia of Spectroscopy (VCH, 1995).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999).

W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).

M. Hébert, “Compositional model for predicting multilayer reflectances and transmittances in color reproduction,” Ph.D. dissertation (Ecole Polytchnique Fédérale de Lausanne, 2006), p. 139.

G. Sharma, “Color fundamentals for digital imaging,” in Digital Color Imaging Handbook, G. Sharma, ed. (CRC Press, 2003), pp. 30-36.

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Figures (16)

Fig. 1
Fig. 1

Reflection and transmission of light by an interfaced layer.

Fig. 2
Fig. 2

Transmission and total reflection of light rays within a multilayer.

Fig. 3
Fig. 3

Pile composed of three nonscattering sheets.

Fig. 4
Fig. 4

Measured (solid curves) and predicted (dashed curves) reflectance spectra of air-bound piles of blue plastic (1 to 5 sheets, and infinity of sheets).

Fig. 5
Fig. 5

Measured (solid curves) and predicted (dashed curves) transmittance spectra of air-bound piles of blue plastic sheets (1 to 5 sheets).

Fig. 6
Fig. 6

Measured (solid curves) and predicted (dashed curves) transmittance spectra of liquid-bound piles of blue plastic sheets (1 to 5 sheets).

Fig. 7
Fig. 7

Reflectance of plastic-bound piles of blue plastic sheets.

Fig. 8
Fig. 8

( L * , C * ) coordinates (left) and ( C * , h * ) coordinates (right) of the colors reflected and transmitted by air-bound piles (solid curves), liquid-bound piles (dashed curves) and plastic-bound piles (dotted curves) as a function of the number of sheets.

Fig. 9
Fig. 9

Multiple reflections and transmissions of light between two identical nonscattering sheets, where the global reflectance R ( θ ) and the global transmittance T ( θ ) compose the multiple reflections within each sheet.

Fig. 10
Fig. 10

Diffusing support superposed with a pile of colored nonscattering sheets.

Fig. 11
Fig. 11

Multiple reflections between the diffusing support and the interfaced multilayer.

Fig. 12
Fig. 12

Measured (solid curves) and predicted (dashed curves) reflectance spectra of blue plastic sheets superposed with a white PVC support and bound by air. Dotted curve, support reflectance spectrum.

Fig. 13
Fig. 13

Measured (solid curves) and predicted (dashed curves) reflectance spectra of blue plastic sheets superposed with a white PVC support and bound by a liquid. Dotted curve, support reflectance spectrum.

Fig. 14
Fig. 14

Measured (solid curves) and predicted (dashed curves) reflectance spectra of blue plastic sheets superposed with a yellow-green paper support and bound by air. Dotted curve, reflectance of the support and of an infinite pile.

Fig. 15
Fig. 15

Predicted reflectance spectra of plastic-bound piles of blue plastic on white PVC.

Fig. 16
Fig. 16

( C * , L * ) coordinates (left) and ( C * , h * ) coordinates of the color reflected by piles of nonscattering sheets bound by air, liquid, or plastic superposed on a white PVC support.

Tables (1)

Tables Icon

Table 1 Specular Reflectance of an Interface According to the Light Source and the Capturing Device

Equations (52)

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T ij ( θ i ) = 1 R ij ( θ i ) .
R ji ( θ j ) = R ij ( θ i ) ,
T ji ( θ j ) = T ij ( θ i ) .
n 0 sin θ 0 = n 1 sin θ 1 = n 2 sin θ 2 .
R 012 ( θ 0 ) = R 01 ( θ 0 ) + T 01 2 ( θ 0 ) R 12 ( θ 1 ) t 2 cos θ 1 1 R 01 ( θ 0 ) R 12 ( θ 1 ) t 2 cos θ 1 ,
T 012 ( θ 0 ) = T 01 ( θ 0 ) T 12 ( θ 1 ) t 1 cos θ 1 1 R 01 ( θ 0 ) R 12 ( θ 1 ) t 2 cos θ 1 .
R 0 . . ij ( θ 0 ) = R 0 . . i ( θ 0 ) + T 0 . . i 2 ( θ 0 ) R ij ( θ i ) t i 2 cos θ i 1 R i . . 0 ( θ i ) R ij ( θ i ) t i 2 cos θ i ,
R ji . . 0 ( θ j ) = R ji ( θ j ) + T ij 2 ( θ i ) R i . . 0 ( θ i ) t i 2 cos θ i 1 R i . . 0 ( θ i ) R ij ( θ i ) t i 2 cos θ i ,
T 0 . . ij ( θ 0 ) = T ij ( θ i ) T 0 . . i ( θ 0 ) t i 1 cos θ i 1 R i . . 0 ( θ i ) R ij ( θ i ) t i 2 cos θ i ,
T ji . . 0 ( θ j ) = T ij ( θ i ) T i . . 0 ( θ i ) t i 1 cos θ i 1 R i . . 0 ( θ i ) R ij ( θ i ) t i 2 cos θ i ,
R ij ( θ i ) = { 1 n j n 0 sin θ 0 R ij ( θ i ) n j > n 0 sin θ 0 } ,
f = θ = 0 π 2 F ( θ ) sin 2 θ d θ ,
L t = d 2 Φ t d s cos θ 0 sin θ 0 d θ 0 d ϕ ,
L i = d 2 Φ i d s cos θ k sin θ k d θ k d ϕ ,
d 2 Φ t = T 0 . . k ( θ 0 ) d 2 Φ i .
sin θ 0 = ( n k n 0 ) sin θ k .
cos θ 0 d θ 0 = ( n k n 0 ) cos θ k d θ k ,
L t = ( n 0 n k ) 2 T 0 . . k ( θ 0 ) L i .
R ( λ ) = r 0 + ( 1 r 0 2 ) r 0 t 2 ( λ ) 1 r 0 2 t 2 ( λ ) .
t ( λ ) = R ( λ ) r 0 r 0 [ 1 2 r 0 2 + r 0 R ( λ ) ] .
T ( λ ) = ( 1 r 0 ) 2 t ( λ ) 1 r 0 2 t 2 ( λ ) .
t ( λ ) = ( 1 r 0 ) 4 + 4 r 0 2 T 2 ( λ ) ( 1 r 0 ) 2 2 r 0 2 T ( λ ) .
C * = a * 2 + b * 2
h * = arctan ( b * a * )
R 2 ( θ ) = R ( θ ) + T 2 ( θ ) R ( θ ) 1 R 2 ( θ ) .
T 2 ( θ ) = T 2 ( θ ) 1 R 2 ( θ ) .
R k ( θ ) = R ( θ ) + T 2 ( θ ) R k 1 ( θ ) 1 R ( θ ) R k 1 ( θ ) ,
T k ( θ ) = T ( θ ) T k 1 ( θ ) 1 R ( θ ) R k 1 ( θ ) ,
R ( θ ) = R ( θ ) + T 2 ( θ ) R ( θ ) 1 R ( θ ) R ( θ ) ,
R 2 ( θ ) 1 + R 2 ( θ ) T 2 ( θ ) R ( θ ) R ( θ ) + 1 = 0 .
R ( θ ) = a b ,
a = 1 + R 2 ( θ ) T 2 ( θ ) 2 R ( θ )
b = a 2 1 .
R g ( λ ) = r u + T in T ex ρ ( λ ) 1 r i ρ ( λ ) ,
r i = 0 π 2 R 30 ( θ 3 ) sin 2 θ 3 d θ 3 .
r u = R 03 ( 0 ) ,
T in = θ 0 = 0 π 2 T 03 ( θ 0 ) sin 2 θ 0 d θ 0 ,
T ex = ( n 0 n 3 ) 2 T 03 ( 0 ) ,
ρ ( λ ) = R g ( λ ) r u T in T ex + r i [ R g ( λ ) r u ] .
R 012 . . 123 ( θ 0 ) = R 012 . . 12 ( θ 0 ) + T 012 . . 12 2 ( θ 0 ) R 23 ( θ 2 ) 1 R 21 . . 210 ( θ 2 ) R 23 ( θ 2 ) ,
R 321 . . 210 ( θ 3 ) = R 32 ( θ 3 ) + T 23 2 ( θ 2 ) R 21 . . 210 ( θ 2 ) 1 R 21 . . 210 ( θ 2 ) R 23 ( θ 2 ) ,
T 012 . . 123 ( θ 0 ) = T 012 . . 12 ( θ 0 ) T 23 ( θ 2 ) 1 R 21 . . 210 ( θ 2 ) R 23 ( θ 2 ) ,
T 321 . . 210 ( θ 3 ) = T 21 . . 210 ( θ 2 ) T 23 ( θ 2 ) 1 R 21 . . 210 ( θ 2 ) R 23 ( θ 2 ) ,
R ( λ ) = r u ( λ ) + T in ( λ ) T ex ( λ ) ρ ( λ ) 1 r i ( λ ) ρ ( λ ) ,
r i ( λ ) = θ 3 = 0 π 2 R 321 . . 210 ( θ 3 ) sin 2 θ 3 d θ 3 ,
r u ( λ ) = R 012 . . 123 ( 0 ) ,
T in ( λ ) = θ 0 = 0 π 2 T 012 . . 123 ( θ 0 ) sin 2 θ 0 d θ 0 ,
T ex ( λ ) = ( n 0 n 3 ) 2 T 012 . . 123 ( 0 ) .
T in = T 03 ( θ ) .
T in = θ 0 = 0 π 2 T 03 ( θ 0 ) sin ( 2 θ 0 ) d θ 0 .
T ex = ( n 0 n 3 ) 2 T 03 ( θ ) .
T ex = θ 3 = 0 π 2 T 30 ( θ 3 ) sin ( 2 θ 3 ) d θ 3 .

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