Abstract

In this paper simple analytical equations for the reflection and transmission coefficients of fluorescent turbid media are given. The case of weakly absorbing optically thick media is considered (e.g., paper, textiles, tissues). The calculations are performed in the framework of the two-flux approximation for finite layers under monochromatic illumination conditions. The relationships of Kubelka–Munk parameters to the true absorption and transport extinction coefficients of fluorescent turbid media are derived. The results can be used for the development of various optimization procedures in the paper and textile industries and also in the area of fluorescence spectroscopy of turbid media.

© 2009 Optical Society of America

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References

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  4. T. Shakespeare, “Colorant modeling for on-line paper coloring: evaluations of models and an extension to Kubelka-Munk model,” Ph.D. thesis (Tampere University of Technology, 2000).
  5. J. Wu, M. S. Feld, and R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585-3595 (1993).
    [CrossRef] [PubMed]
  6. A. D. Klose and A. H. Hielscher, “Fluorescence tomography with simulated data based on the equation of radiative transfer,” Opt. Lett. 28, 1019-1021 (2003).
    [CrossRef] [PubMed]
  7. H. R. Gordon, “Diffuse reflectance of the ocean: the theory of its augmentation by chlorophyll a fluorescence at 685 nm,” Appl. Opt. 18, 5356-5367 (1979).
  8. A. D. Klose, “Radiative transfer of luminescence light in biological tissue,” in Light Scattering Reviews, A.A.Kokhanovsky, ed. (Springer-Praxis, 2009), Vol. 4, 347-405.
  9. M. G. I. Muller, I. Georgakoudi, Q. Zhang, J. Wu, and M. S. Feld, “Intrinsic fluorescence spectroscopy in turbid media: disentangling effects of scattering and absorption,” Appl. Opt. 40, 4633-4646 (2001).
    [CrossRef]
  10. Q. Zhang, M. G. Muller, J. Wu, and M. S. Feld, “Turbidity-free fluorescence spectroscopy of biological tissue,” Opt. Lett. 25, 1451-1453 (2000).
    [CrossRef]
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    [CrossRef]
  12. P. Kubelka and F. Munk, “Ein Beitrag Zur Optik der Farbanstriche,” Z. fur Techn. Physik 12, 593-601 (1931).
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  16. T. Shakespeare and J. Shakespeare, “A fluorescent extension to the Kubelka-Munk model,” Color Res. Appl. 28, 4-14 (2002).
    [CrossRef]
  17. S. D. Howison and R. J. Lawrence, “Fluorescent transfer of light in dyed materials,” SIAM J. Appl. Math. 53, 447-458 (1993).
    [CrossRef]
  18. L. E. Elsgolts, Differential Equations (Gordon & Breach, 1961).
  19. A. A. Kokhanovsky, Cloud Optics (Springer, 2006).
    [CrossRef]
  20. A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka-Munk theory,” J. Phys. D 40, 2210-2216 (2007).
    [CrossRef]
  21. A. A. Kokhanovsky, Light Scattering Media Optics (Springer-Praxis, 2004).
  22. A. A. Kokhanovsky, “Asymptotic radiative transfer,” in Light Scattering Reviews, A.A.Kokhanovsky, ed. (Springer-Praxis, 2006), Vol. 1, pp. 253-290.
    [CrossRef]
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2007 (1)

A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka-Munk theory,” J. Phys. D 40, 2210-2216 (2007).
[CrossRef]

2003 (2)

J. Swartling, A. Pifferi, A. M. K. Enejder, and S. Anderson-Engels, “Accelerated Monte Carlo models to simulate fluorescence spectra from layered tissues,” J. Opt. Soc. Am. 20, 714-727 (2003).
[CrossRef]

A. D. Klose and A. H. Hielscher, “Fluorescence tomography with simulated data based on the equation of radiative transfer,” Opt. Lett. 28, 1019-1021 (2003).
[CrossRef] [PubMed]

2002 (1)

T. Shakespeare and J. Shakespeare, “A fluorescent extension to the Kubelka-Munk model,” Color Res. Appl. 28, 4-14 (2002).
[CrossRef]

2001 (1)

2000 (1)

1993 (2)

S. D. Howison and R. J. Lawrence, “Fluorescent transfer of light in dyed materials,” SIAM J. Appl. Math. 53, 447-458 (1993).
[CrossRef]

J. Wu, M. S. Feld, and R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585-3595 (1993).
[CrossRef] [PubMed]

1980 (1)

1979 (1)

H. R. Gordon, “Diffuse reflectance of the ocean: the theory of its augmentation by chlorophyll a fluorescence at 685 nm,” Appl. Opt. 18, 5356-5367 (1979).

1971 (1)

1964 (1)

1959 (1)

T. H. Morton, “Fluorescent brightening agents on textiles: elementary optical theory and its practical applications,” J. Soc. Dyers Colour. 79, 238-242 (1959).
[CrossRef]

1954 (1)

1948 (1)

1931 (1)

P. Kubelka and F. Munk, “Ein Beitrag Zur Optik der Farbanstriche,” Z. fur Techn. Physik 12, 593-601 (1931).

Allen, E.

Anderson-Engels, S.

J. Swartling, A. Pifferi, A. M. K. Enejder, and S. Anderson-Engels, “Accelerated Monte Carlo models to simulate fluorescence spectra from layered tissues,” J. Opt. Soc. Am. 20, 714-727 (2003).
[CrossRef]

Elsgolts, L. E.

L. E. Elsgolts, Differential Equations (Gordon & Breach, 1961).

Enejder, A. M. K.

J. Swartling, A. Pifferi, A. M. K. Enejder, and S. Anderson-Engels, “Accelerated Monte Carlo models to simulate fluorescence spectra from layered tissues,” J. Opt. Soc. Am. 20, 714-727 (2003).
[CrossRef]

Feld, M. S.

Fukshansky, L.

Georgakoudi, I.

Gordon, H. R.

H. R. Gordon, “Diffuse reflectance of the ocean: the theory of its augmentation by chlorophyll a fluorescence at 685 nm,” Appl. Opt. 18, 5356-5367 (1979).

Hielscher, A. H.

Howison, S. D.

S. D. Howison and R. J. Lawrence, “Fluorescent transfer of light in dyed materials,” SIAM J. Appl. Math. 53, 447-458 (1993).
[CrossRef]

Kazarinova, N.

Klose, A. D.

A. D. Klose and A. H. Hielscher, “Fluorescence tomography with simulated data based on the equation of radiative transfer,” Opt. Lett. 28, 1019-1021 (2003).
[CrossRef] [PubMed]

A. D. Klose, “Radiative transfer of luminescence light in biological tissue,” in Light Scattering Reviews, A.A.Kokhanovsky, ed. (Springer-Praxis, 2009), Vol. 4, 347-405.

Kokhanovsky, A. A.

A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka-Munk theory,” J. Phys. D 40, 2210-2216 (2007).
[CrossRef]

A. A. Kokhanovsky, Cloud Optics (Springer, 2006).
[CrossRef]

A. A. Kokhanovsky, “Asymptotic radiative transfer,” in Light Scattering Reviews, A.A.Kokhanovsky, ed. (Springer-Praxis, 2006), Vol. 1, pp. 253-290.
[CrossRef]

A. A. Kokhanovsky, Light Scattering Media Optics (Springer-Praxis, 2004).

Kubelka, P.

Lawrence, R. J.

S. D. Howison and R. J. Lawrence, “Fluorescent transfer of light in dyed materials,” SIAM J. Appl. Math. 53, 447-458 (1993).
[CrossRef]

Morton, T. H.

T. H. Morton, “Fluorescent brightening agents on textiles: elementary optical theory and its practical applications,” J. Soc. Dyers Colour. 79, 238-242 (1959).
[CrossRef]

Mudgett, P. S.

Muller, M. G.

Muller, M. G. I.

Munk, F.

P. Kubelka and F. Munk, “Ein Beitrag Zur Optik der Farbanstriche,” Z. fur Techn. Physik 12, 593-601 (1931).

Pifferi, A.

J. Swartling, A. Pifferi, A. M. K. Enejder, and S. Anderson-Engels, “Accelerated Monte Carlo models to simulate fluorescence spectra from layered tissues,” J. Opt. Soc. Am. 20, 714-727 (2003).
[CrossRef]

Rava, R. P.

Richards, L. W.

Shakespeare, J.

T. Shakespeare and J. Shakespeare, “A fluorescent extension to the Kubelka-Munk model,” Color Res. Appl. 28, 4-14 (2002).
[CrossRef]

Shakespeare, T.

T. Shakespeare and J. Shakespeare, “A fluorescent extension to the Kubelka-Munk model,” Color Res. Appl. 28, 4-14 (2002).
[CrossRef]

T. Shakespeare, “Colorant modeling for on-line paper coloring: evaluations of models and an extension to Kubelka-Munk model,” Ph.D. thesis (Tampere University of Technology, 2000).

Swartling, J.

J. Swartling, A. Pifferi, A. M. K. Enejder, and S. Anderson-Engels, “Accelerated Monte Carlo models to simulate fluorescence spectra from layered tissues,” J. Opt. Soc. Am. 20, 714-727 (2003).
[CrossRef]

van de Hulst, H. C.

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

Wu, J.

Zhang, Q.

Appl. Opt. (4)

Color Res. Appl. (1)

T. Shakespeare and J. Shakespeare, “A fluorescent extension to the Kubelka-Munk model,” Color Res. Appl. 28, 4-14 (2002).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Phys. D (1)

A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka-Munk theory,” J. Phys. D 40, 2210-2216 (2007).
[CrossRef]

J. Soc. Dyers Colour. (1)

T. H. Morton, “Fluorescent brightening agents on textiles: elementary optical theory and its practical applications,” J. Soc. Dyers Colour. 79, 238-242 (1959).
[CrossRef]

Opt. Lett. (2)

SIAM J. Appl. Math. (1)

S. D. Howison and R. J. Lawrence, “Fluorescent transfer of light in dyed materials,” SIAM J. Appl. Math. 53, 447-458 (1993).
[CrossRef]

Z. fur Techn. Physik (1)

P. Kubelka and F. Munk, “Ein Beitrag Zur Optik der Farbanstriche,” Z. fur Techn. Physik 12, 593-601 (1931).

Other (7)

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

T. Shakespeare, “Colorant modeling for on-line paper coloring: evaluations of models and an extension to Kubelka-Munk model,” Ph.D. thesis (Tampere University of Technology, 2000).

L. E. Elsgolts, Differential Equations (Gordon & Breach, 1961).

A. A. Kokhanovsky, Cloud Optics (Springer, 2006).
[CrossRef]

A. A. Kokhanovsky, Light Scattering Media Optics (Springer-Praxis, 2004).

A. A. Kokhanovsky, “Asymptotic radiative transfer,” in Light Scattering Reviews, A.A.Kokhanovsky, ed. (Springer-Praxis, 2006), Vol. 1, pp. 253-290.
[CrossRef]

A. D. Klose, “Radiative transfer of luminescence light in biological tissue,” in Light Scattering Reviews, A.A.Kokhanovsky, ed. (Springer-Praxis, 2009), Vol. 4, 347-405.

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Tables (1)

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Table 1 Radiative Characteristics of Thick Strongly Light-Scattering Layers in the Framework of KMT a

Equations (48)

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

I ( x = 0 ) = I 0 , J ( x = L ) = 0 ,
r = J ( 0 ) I 0 , t = I ( L ) I 0 .
I = ɛ I + s J + F ( x ) ,
J = ɛ J + s I + F ( x ) .
J = ɛ J + s I + F ( x ) ,
J μ 2 J = Φ ( x ) ,
μ = ɛ 2 s 2 ,
Φ ( x ) = ( ɛ + s ) F ( x ) F ( x ) .
J = J h + J p .
J h = c 1 exp ( μ x ) + c 2 exp ( μ x ) ,
F ( x ) = 1 2 k 1 i 1 ( x ) φ 12 .
i 1 ( x ) = a e ξ x + b e ξ x ,
a = ( 1 + r 1 ) I 0 1 r 1 2 e 2 ξ L , b = ( 1 + r 1 ) r 1 e 2 ξ L I 0 1 r 1 2 e 2 ξ L ,
r 1 = s 1 ɛ 1 + ξ .
Φ ( x ) = A e ξ x + B e ξ x ,
A = a 2 k 1 φ 12 ( ɛ + s ξ ) ,
B = b 2 k 1 φ 12 ( ɛ + s + ξ ) .
J p μ 2 J p = A e ξ x + B e ξ x .
J p = 1 Ψ ( D 2 μ 2 ) [ A e ξ x + B e ξ x ] ,
1 Ψ ( D 2 μ 2 ) [ A e ξ x ] = A e ξ x ξ 2 μ 2 .
J p = 1 ξ 2 μ 2 [ A e ξ x + B e ξ x ]
J ( x ) = c 1 exp ( μ x ) + c 2 exp ( μ x ) + A e ξ x + B e ξ x ξ 2 μ 2
I = s 1 ( ɛ J J F ) ,
I ( x ) = D 1 e μ x + D 2 e μ x + D 3 e ξ x + D 4 e ξ x ,
D 1 = ( ɛ + μ ) c 1 s , D 2 = ( ɛ μ ) c 2 s ,
D 3 = ( ɛ + ξ ) A ( ξ 2 μ 2 ) s k 1 φ 12 a 2 s , D 4 = ( ɛ ξ ) B ( ξ 2 μ 2 ) s k 1 φ 12 b 2 s .
c 1 = q 1 + p 2 c 2 p 1 , c 2 = q 1 p 1 1 e 2 μ L q 2 e 2 μ L 1 p 2 p 1 1 e 2 μ L ,
q 1 = A ( ɛ + ξ ) + B ( ɛ ξ ) ξ 2 μ 2 1 2 k 1 φ 12 ( a + b ) s I 0 ,
q 2 = A e ξ L + B e ξ L ξ 2 μ 2 ,
p 1 = ɛ + μ , p 2 = ɛ μ .
r = c 1 + c 2 + A + B ξ 2 μ 2 ,
t = D 1 e μ L + D 2 e μ L + D 3 e ξ L + D 4 e ξ L ,
b = 0 , B = 0 , c 2 = 0 , a = 1 + r 1 ,
A = 1 2 ( 1 + r 1 ) ( ɛ + s ξ ) k 1 φ 2 ,
c 1 = r 2 + 1 + r 1 2 ( ɛ + μ ) k 1 φ 2 ( ɛ + ξ ) A ( ɛ + μ ) ( ξ 2 μ 2 ) ,
r = c 1 + A ξ 2 μ 2
r = r 2 + 1 + r 1 2 ( ɛ + μ ) k 1 φ 12 A ( ɛ + μ ) ( ξ μ ) .
r = r 2 + k 1 φ 12 ( 1 + r 1 ) ( 1 + r 2 ) 2 ( ξ + μ ) ,
r as = r as ( 1 β e 2 p ) 1 l 2 e 2 p , t as = α e p 1 l 2 e 2 p ,
r KM = r KM ( 1 e 2 p ¯ ) 1 r KM 2 e 2 p ¯ , t KM = ( 1 r KM 2 ) e p ¯ 1 r KM 2 e 2 p ¯ .
γ = 3 ( 1 ω 0 ) ( 1 g ) , r as = 1 4 γ ( 3 ( 1 g ) ) ,
l = r as , α = 1 r as 2 , β = 1 .
r KM = 1 + k s 2 k s + k 2 s 2 , p ¯ = k 2 + 2 k s L .
1 + k s 2 k s + k 2 s 2 = 1 4 γ 3 ( 1 g ) , k 2 + 2 k s = γ k ext .
2 k s = 4 γ 3 ( 1 g ) , 2 k s = γ k ext ,
k s = 8 ( 1 ω 0 ) 3 ( 1 g ) , k s = 3 ( 1 ω 0 ) ( 1 g ) 2 k ext 2
k = 2 κ abs , s = 0.75 ( 1 g ) κ ext .
k = 2 κ abs , s = 0.75 ( 1 g ) κ sca ,

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