Abstract

A general theoretical approach to the description of light propagating through turbid media is proposed. The theory is a modification of the two-flux model of Kubelka–Munk (KM), extending its applicability to media systems containing an absorptive component. The modified KM model takes into account the influence of internal scattering on the total path length and accommodates a wide range of absorption influences. Experimental results obtained for dyed-paper systems illuminated by diffuse light are demonstrated to be qualitatively and quantitatively reproduced by the theory.

© 2005 Optical Society of America

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References

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  1. P. Kubelka and F. Munk, Z. Tech. Phys. (Leipzig) 12, 593 (1931).
  2. M. Rundlöf and J. A. Bristow, J. Pulp Pap. Sci. 23, J220 (1997).
  3. W. J. Foote, Pap. Trade J. 109, 333 (1939).
  4. J. A. Van den Akker, in Modern Aspects of Reflectance Spectroscopy, W. W. Wendlandt, ed. (Plenum, New York, 1968), pp. 24–26.
  5. N. Pauler, Digital Printing Center, Mid Sweden University (personal communication, 2004). Similar experimental results may be found in Ref. [2].
  6. B. J. Brinkworth, Appl. Opt. 11, 1434 (1972).
    [CrossRef] [PubMed]
  7. B. Philips-Invernizzi, D. Dupont, and C. Caze, Opt. Eng. 40, 1082 (2001).
    [CrossRef]
  8. J. F. Bloch and R. Sève, Color Res. Appl. 28, 227 (2003).
    [CrossRef]
  9. L. Yang and B. Kruse, J. Opt. Soc. Am. A 21, 1933 (2004).
    [CrossRef]
  10. L. Yang, J. Opt. Soc. Am. A 20, 1149 (2003).
    [CrossRef]

2004 (1)

2003 (2)

J. F. Bloch and R. Sève, Color Res. Appl. 28, 227 (2003).
[CrossRef]

L. Yang, J. Opt. Soc. Am. A 20, 1149 (2003).
[CrossRef]

2001 (1)

B. Philips-Invernizzi, D. Dupont, and C. Caze, Opt. Eng. 40, 1082 (2001).
[CrossRef]

1997 (1)

M. Rundlöf and J. A. Bristow, J. Pulp Pap. Sci. 23, J220 (1997).

1972 (1)

1939 (1)

W. J. Foote, Pap. Trade J. 109, 333 (1939).

1931 (1)

P. Kubelka and F. Munk, Z. Tech. Phys. (Leipzig) 12, 593 (1931).

Bloch, J. F.

J. F. Bloch and R. Sève, Color Res. Appl. 28, 227 (2003).
[CrossRef]

Brinkworth, B. J.

Bristow, J. A.

M. Rundlöf and J. A. Bristow, J. Pulp Pap. Sci. 23, J220 (1997).

Caze, C.

B. Philips-Invernizzi, D. Dupont, and C. Caze, Opt. Eng. 40, 1082 (2001).
[CrossRef]

Dupont, D.

B. Philips-Invernizzi, D. Dupont, and C. Caze, Opt. Eng. 40, 1082 (2001).
[CrossRef]

Foote, W. J.

W. J. Foote, Pap. Trade J. 109, 333 (1939).

Kruse, B.

Kubelka, P.

P. Kubelka and F. Munk, Z. Tech. Phys. (Leipzig) 12, 593 (1931).

Munk, F.

P. Kubelka and F. Munk, Z. Tech. Phys. (Leipzig) 12, 593 (1931).

Pauler, N.

N. Pauler, Digital Printing Center, Mid Sweden University (personal communication, 2004). Similar experimental results may be found in Ref. [2].

Philips-Invernizzi, B.

B. Philips-Invernizzi, D. Dupont, and C. Caze, Opt. Eng. 40, 1082 (2001).
[CrossRef]

Rundlöf, M.

M. Rundlöf and J. A. Bristow, J. Pulp Pap. Sci. 23, J220 (1997).

Sève, R.

J. F. Bloch and R. Sève, Color Res. Appl. 28, 227 (2003).
[CrossRef]

Van den Akker, J. A.

J. A. Van den Akker, in Modern Aspects of Reflectance Spectroscopy, W. W. Wendlandt, ed. (Plenum, New York, 1968), pp. 24–26.

Yang, L.

Appl. Opt. (1)

Color Res. Appl. (1)

J. F. Bloch and R. Sève, Color Res. Appl. 28, 227 (2003).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Pulp Pap. Sci. (1)

M. Rundlöf and J. A. Bristow, J. Pulp Pap. Sci. 23, J220 (1997).

Opt. Eng. (1)

B. Philips-Invernizzi, D. Dupont, and C. Caze, Opt. Eng. 40, 1082 (2001).
[CrossRef]

Pap. Trade J. (1)

W. J. Foote, Pap. Trade J. 109, 333 (1939).

Z. Tech. Phys. (Leipzig) (1)

P. Kubelka and F. Munk, Z. Tech. Phys. (Leipzig) 12, 593 (1931).

Other (2)

J. A. Van den Akker, in Modern Aspects of Reflectance Spectroscopy, W. W. Wendlandt, ed. (Plenum, New York, 1968), pp. 24–26.

N. Pauler, Digital Printing Center, Mid Sweden University (personal communication, 2004). Similar experimental results may be found in Ref. [2].

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Figures (3)

Fig. 1
Fig. 1

Experimental KMT absorption and scattering coefficients of dyed-paper sheets with w p = 40.16 41.73 g m 2 , provided by Pauler.[5] The values of the white paper are denoted by dots.

Fig. 2
Fig. 2

Schematic diagram of light scattering in a medium layer.

Fig. 3
Fig. 3

Predictions of the KMT absorption and scattering coefficients of dyed paper, with w p = 40 g m 2 and w i = ( 0.005 , 0.01 , 0.02 , 0.05 , 0.1 , 0.2 ) g m 2 . The values of white paper are denoted by dots.

Equations (16)

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d I d z = ( S + K ) I + S J , d J d z = ( S + K ) J + S I .
K = 2 a , S = s .
a i p = w p a p + w i a i w p + w i , s i p = w p s p + w i s i w p + w i ,
K = α μ a , S = α μ s 2 .
μ L R .
μ ( s a ) 1 2 ,
L N l s .
R R 2 1 2 = ( Σ n = 1 N r n 2 ) 1 2 N l s .
R R = 1 I 0 I ϕ D cos ϕ d ϕ = α D .
N = α 2 D 2 l s 2 .
μ = α s D .
I ( z ) = b 1 exp ( A z ) R + R b 2 exp ( A z ) ,
J ( z ) = b 1 exp ( A z ) + b 2 exp ( A z ) ,
b 1 = I 0 R 1 R 2 exp ( 2 A w p ) , b 2 = I 0 R exp ( 2 A w p ) 1 R 2 exp ( 2 A w p ) .
D = w p 0 J ( z ) z d z w p 0 J ( z ) d z = 1 2 A w p exp ( A w p ) exp ( 2 A w p ) A [ 1 2 exp ( A w p ) + exp ( 2 A w p ) ] .
μ = [ s 2 ( a 2 + a s ) ] 1 4

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