Abstract

A generally applicable theoretical model describing light propagating through turbid media is proposed. The theory is a generalization of the revised Kubelka–Munk theory, extending its applicability to accommodate a wider range of absorption influences. A general expression for a factor taking into account the effect of scattering on the total photon path traversed in a turbid medium is derived. The extended model is applied to systems of ink-dyed paper sheets—mixtures of wood fibers with dyes—which represent examples of systems that have thus far eluded the original Kubelka–Munk model. The results of simulations of the spectral dependence of Kubelka–Munk coefficients of absorption and scattering show that they compare very well with those derived from experimental results.

© 2005 Optical Society of America

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  1. P. Kubelka, F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).
  2. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [CrossRef] [PubMed]
  3. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  4. W. R. Blevin, W. J. Brown, “Light-scattering properties of pigment suspensions,” J. Opt. Soc. Am. 51, 1250–1256 (1962).
    [CrossRef]
  5. E. Allen, “Calculations for colorant formulations,” in Industrial Color Technology: Advances in Chemistry Series 107 (American Chemical Society, 1971), pp. 87–119.
  6. D. B. Judd, G. Wyszecki, “Physics and Psychophysics of colorant layers,” in Color in Business, Science and Industry, 3rd ed. (Wiley, 1975).
  7. W. F. Cheong, A. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990), and references therein.
    [CrossRef]
  8. J. H. Nobbs, “Kubelka–Munk theory and the prediction of reflectance,” Rev. Prog. Color. Relat. Top. 15, 66–75 (1985).
    [CrossRef]
  9. B. Philips-Invernizzi, D. Dupont, C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082–1092 (2001).
    [CrossRef]
  10. W. J. Foote, “An investigation of the fundamental scattering and absorption coefficients of dyed handsheets,” Pap. Trade J. 109, 333–340 (1939).
  11. J. A. Van den Akker, “Scattering and absorption of light in paper and other diffusing media,” Tappi J. 32, 498–501 (1949).
  12. L. Nordman, P. Aaltonen, T. Makkonen, Relationships between Mechanical and Optical Properties of Paper Affected by Web Consolidation (British Paper and Board Maker’s Association, 1966).
  13. J. A. Van den Akker, Theory of Some of the Discrepancies Observed in Application of the Kubelka–Munk Equations to Particulate Systems (Plenum, 1968).
  14. M. Rundlöf, J. A. Bristow, “A note concerning the interaction between light scattering and light absorption in the application of the Kubelka–Munk equations,” J. Pulp Pap. Sci. 23, J220–223 (1997).
  15. E. Allen, “Colorant formation and shading,” in Optical Radiation Measurements, Vol. 2, Color Measurements, F. Grum and C. J. Bartleson, eds. (Academic, 1980), pp. 290–336.
  16. W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
    [CrossRef] [PubMed]
  17. A. A. Koukoulas, B. D. Jordan, “Effect of strong absorption on the Kubelka–Munk scattering coefficient,” J. Pulp Pap. Sci. 23, J224–232 (1997).
  18. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
    [CrossRef]
  19. W. E. Vargas, “Inversion methods from Kubelka–Munk analysis,” J. Opt. A, Pure Appl. Opt. 4, 452–456 (2002).
    [CrossRef]
  20. J. F. Bloch, R. Sève, “About the theoretical aspect of multiple light scattering: Silvy’s theory,” Color Res. Appl. 28, 227–228 (2003).
    [CrossRef]
  21. B. Maheu, J. N. Letoulouzan, G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz-Mie parameters,” Appl. Opt. 23, 3353–3362 (1984).
    [CrossRef] [PubMed]
  22. B. Maheu, J. P. Briton, G. Gouesbet, “Four-flux model and a Monte Carlo code: comparisons between two simple, complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24 (1989).
    [CrossRef] [PubMed]
  23. L. Yang, B. Kruse, “Revised Kubelka–Munk theory. I. Theory and applications,” J. Opt. Soc. Am. A 21, 1933–1941 (2004).
    [CrossRef]
  24. L. Yang, B. Kruse, S. J. Miklavcic, “Revised Kubelka–Munk theory. II. A study of ink penetration in ink-jet printing,” J. Opt. Soc. Am. A 21, 1942–1952 (2004).
    [CrossRef]
  25. W. Wendlandt, H. Hecht, Reflectance Spectroscopy (Wiley Interscience, 1966), Chap. 3.
  26. L. Yang, S. J. Miklavcic, “A theory of light propagation incorporating scattering and absorption in turbid media,” Opt. Lett. 30, 792–794 (2005).
    [CrossRef] [PubMed]
  27. P. Emmel, “Modèles de Prédiction Couleur Appliqués à L’impression Jet D’encre,” Thèse No. 1857 (École Polytechnique Fédérale de Lausanne, Switzerland, 1998).
  28. L. Yang, B. Kruse, “Ink penetration and its effect on printing,” Proc. SPIE 3963, 365–375 (2000).
    [CrossRef]
  29. L. Yang, “Characterization of inks and ink application for inkjet printing: Model and simulation,” J. Opt. Soc. Am. A 20, 1149–1154 (2003).
    [CrossRef]
  30. N. Pauler, private communication. Similar experimental observations may be found in Ref. [14].
  31. T. Shakespear, J. Shakespear, “A fluorescent extension to the Kubelka–Munk model,” Color Res. Appl. 28, 4–14 (2003), Fig. 4.
    [CrossRef]

2005 (1)

2004 (2)

2003 (3)

J. F. Bloch, R. Sève, “About the theoretical aspect of multiple light scattering: Silvy’s theory,” Color Res. Appl. 28, 227–228 (2003).
[CrossRef]

L. Yang, “Characterization of inks and ink application for inkjet printing: Model and simulation,” J. Opt. Soc. Am. A 20, 1149–1154 (2003).
[CrossRef]

T. Shakespear, J. Shakespear, “A fluorescent extension to the Kubelka–Munk model,” Color Res. Appl. 28, 4–14 (2003), Fig. 4.
[CrossRef]

2002 (1)

W. E. Vargas, “Inversion methods from Kubelka–Munk analysis,” J. Opt. A, Pure Appl. Opt. 4, 452–456 (2002).
[CrossRef]

2001 (1)

B. Philips-Invernizzi, D. Dupont, C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082–1092 (2001).
[CrossRef]

2000 (1)

L. Yang, B. Kruse, “Ink penetration and its effect on printing,” Proc. SPIE 3963, 365–375 (2000).
[CrossRef]

1997 (3)

M. Rundlöf, J. A. Bristow, “A note concerning the interaction between light scattering and light absorption in the application of the Kubelka–Munk equations,” J. Pulp Pap. Sci. 23, J220–223 (1997).

W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
[CrossRef] [PubMed]

A. A. Koukoulas, B. D. Jordan, “Effect of strong absorption on the Kubelka–Munk scattering coefficient,” J. Pulp Pap. Sci. 23, J224–232 (1997).

1990 (1)

W. F. Cheong, A. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990), and references therein.
[CrossRef]

1989 (1)

1985 (1)

J. H. Nobbs, “Kubelka–Munk theory and the prediction of reflectance,” Rev. Prog. Color. Relat. Top. 15, 66–75 (1985).
[CrossRef]

1984 (1)

1962 (1)

W. R. Blevin, W. J. Brown, “Light-scattering properties of pigment suspensions,” J. Opt. Soc. Am. 51, 1250–1256 (1962).
[CrossRef]

1949 (1)

J. A. Van den Akker, “Scattering and absorption of light in paper and other diffusing media,” Tappi J. 32, 498–501 (1949).

1948 (1)

1939 (1)

W. J. Foote, “An investigation of the fundamental scattering and absorption coefficients of dyed handsheets,” Pap. Trade J. 109, 333–340 (1939).

1931 (1)

P. Kubelka, F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Aaltonen, P.

L. Nordman, P. Aaltonen, T. Makkonen, Relationships between Mechanical and Optical Properties of Paper Affected by Web Consolidation (British Paper and Board Maker’s Association, 1966).

Allen, E.

E. Allen, “Colorant formation and shading,” in Optical Radiation Measurements, Vol. 2, Color Measurements, F. Grum and C. J. Bartleson, eds. (Academic, 1980), pp. 290–336.

E. Allen, “Calculations for colorant formulations,” in Industrial Color Technology: Advances in Chemistry Series 107 (American Chemical Society, 1971), pp. 87–119.

Blevin, W. R.

W. R. Blevin, W. J. Brown, “Light-scattering properties of pigment suspensions,” J. Opt. Soc. Am. 51, 1250–1256 (1962).
[CrossRef]

Bloch, J. F.

J. F. Bloch, R. Sève, “About the theoretical aspect of multiple light scattering: Silvy’s theory,” Color Res. Appl. 28, 227–228 (2003).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Bristow, J. A.

M. Rundlöf, J. A. Bristow, “A note concerning the interaction between light scattering and light absorption in the application of the Kubelka–Munk equations,” J. Pulp Pap. Sci. 23, J220–223 (1997).

Briton, J. P.

Brown, W. J.

W. R. Blevin, W. J. Brown, “Light-scattering properties of pigment suspensions,” J. Opt. Soc. Am. 51, 1250–1256 (1962).
[CrossRef]

Caze, C.

B. Philips-Invernizzi, D. Dupont, C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082–1092 (2001).
[CrossRef]

Chandrasekhar, S.

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

Cheong, W. F.

W. F. Cheong, A. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990), and references therein.
[CrossRef]

Dupont, D.

B. Philips-Invernizzi, D. Dupont, C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082–1092 (2001).
[CrossRef]

Emmel, P.

P. Emmel, “Modèles de Prédiction Couleur Appliqués à L’impression Jet D’encre,” Thèse No. 1857 (École Polytechnique Fédérale de Lausanne, Switzerland, 1998).

Foote, W. J.

W. J. Foote, “An investigation of the fundamental scattering and absorption coefficients of dyed handsheets,” Pap. Trade J. 109, 333–340 (1939).

Gouesbet, G.

Hecht, H.

W. Wendlandt, H. Hecht, Reflectance Spectroscopy (Wiley Interscience, 1966), Chap. 3.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Jordan, B. D.

A. A. Koukoulas, B. D. Jordan, “Effect of strong absorption on the Kubelka–Munk scattering coefficient,” J. Pulp Pap. Sci. 23, J224–232 (1997).

Judd, D. B.

D. B. Judd, G. Wyszecki, “Physics and Psychophysics of colorant layers,” in Color in Business, Science and Industry, 3rd ed. (Wiley, 1975).

Koukoulas, A. A.

A. A. Koukoulas, B. D. Jordan, “Effect of strong absorption on the Kubelka–Munk scattering coefficient,” J. Pulp Pap. Sci. 23, J224–232 (1997).

Kruse, B.

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
[CrossRef] [PubMed]

P. Kubelka, F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Letoulouzan, J. N.

Maheu, B.

Makkonen, T.

L. Nordman, P. Aaltonen, T. Makkonen, Relationships between Mechanical and Optical Properties of Paper Affected by Web Consolidation (British Paper and Board Maker’s Association, 1966).

Miklavcic, S. J.

Munk, F.

P. Kubelka, F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Niklasson, G. A.

Nobbs, J. H.

J. H. Nobbs, “Kubelka–Munk theory and the prediction of reflectance,” Rev. Prog. Color. Relat. Top. 15, 66–75 (1985).
[CrossRef]

Nordman, L.

L. Nordman, P. Aaltonen, T. Makkonen, Relationships between Mechanical and Optical Properties of Paper Affected by Web Consolidation (British Paper and Board Maker’s Association, 1966).

Pauler, N.

N. Pauler, private communication. Similar experimental observations may be found in Ref. [14].

Philips-Invernizzi, B.

B. Philips-Invernizzi, D. Dupont, C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082–1092 (2001).
[CrossRef]

Prahl, A. A.

W. F. Cheong, A. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990), and references therein.
[CrossRef]

Rundlöf, M.

M. Rundlöf, J. A. Bristow, “A note concerning the interaction between light scattering and light absorption in the application of the Kubelka–Munk equations,” J. Pulp Pap. Sci. 23, J220–223 (1997).

Sève, R.

J. F. Bloch, R. Sève, “About the theoretical aspect of multiple light scattering: Silvy’s theory,” Color Res. Appl. 28, 227–228 (2003).
[CrossRef]

Shakespear, J.

T. Shakespear, J. Shakespear, “A fluorescent extension to the Kubelka–Munk model,” Color Res. Appl. 28, 4–14 (2003), Fig. 4.
[CrossRef]

Shakespear, T.

T. Shakespear, J. Shakespear, “A fluorescent extension to the Kubelka–Munk model,” Color Res. Appl. 28, 4–14 (2003), Fig. 4.
[CrossRef]

Van den Akker, J. A.

J. A. Van den Akker, “Scattering and absorption of light in paper and other diffusing media,” Tappi J. 32, 498–501 (1949).

J. A. Van den Akker, Theory of Some of the Discrepancies Observed in Application of the Kubelka–Munk Equations to Particulate Systems (Plenum, 1968).

Vargas, W. E.

W. E. Vargas, “Inversion methods from Kubelka–Munk analysis,” J. Opt. A, Pure Appl. Opt. 4, 452–456 (2002).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
[CrossRef] [PubMed]

Welch, A. J.

W. F. Cheong, A. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990), and references therein.
[CrossRef]

Wendlandt, W.

W. Wendlandt, H. Hecht, Reflectance Spectroscopy (Wiley Interscience, 1966), Chap. 3.

Wyszecki, G.

D. B. Judd, G. Wyszecki, “Physics and Psychophysics of colorant layers,” in Color in Business, Science and Industry, 3rd ed. (Wiley, 1975).

Yang, L.

Appl. Opt. (3)

Color Res. Appl. (2)

J. F. Bloch, R. Sève, “About the theoretical aspect of multiple light scattering: Silvy’s theory,” Color Res. Appl. 28, 227–228 (2003).
[CrossRef]

T. Shakespear, J. Shakespear, “A fluorescent extension to the Kubelka–Munk model,” Color Res. Appl. 28, 4–14 (2003), Fig. 4.
[CrossRef]

IEEE J. Quantum Electron. (1)

W. F. Cheong, A. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990), and references therein.
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

W. E. Vargas, “Inversion methods from Kubelka–Munk analysis,” J. Opt. A, Pure Appl. Opt. 4, 452–456 (2002).
[CrossRef]

J. Opt. Soc. Am. (2)

P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
[CrossRef] [PubMed]

W. R. Blevin, W. J. Brown, “Light-scattering properties of pigment suspensions,” J. Opt. Soc. Am. 51, 1250–1256 (1962).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Pulp Pap. Sci. (2)

A. A. Koukoulas, B. D. Jordan, “Effect of strong absorption on the Kubelka–Munk scattering coefficient,” J. Pulp Pap. Sci. 23, J224–232 (1997).

M. Rundlöf, J. A. Bristow, “A note concerning the interaction between light scattering and light absorption in the application of the Kubelka–Munk equations,” J. Pulp Pap. Sci. 23, J220–223 (1997).

Opt. Eng. (Bellingham) (1)

B. Philips-Invernizzi, D. Dupont, C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082–1092 (2001).
[CrossRef]

Opt. Lett. (1)

Pap. Trade J. (1)

W. J. Foote, “An investigation of the fundamental scattering and absorption coefficients of dyed handsheets,” Pap. Trade J. 109, 333–340 (1939).

Proc. SPIE (1)

L. Yang, B. Kruse, “Ink penetration and its effect on printing,” Proc. SPIE 3963, 365–375 (2000).
[CrossRef]

Rev. Prog. Color. Relat. Top. (1)

J. H. Nobbs, “Kubelka–Munk theory and the prediction of reflectance,” Rev. Prog. Color. Relat. Top. 15, 66–75 (1985).
[CrossRef]

Tappi J. (1)

J. A. Van den Akker, “Scattering and absorption of light in paper and other diffusing media,” Tappi J. 32, 498–501 (1949).

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

P. Kubelka, F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Other (10)

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

E. Allen, “Colorant formation and shading,” in Optical Radiation Measurements, Vol. 2, Color Measurements, F. Grum and C. J. Bartleson, eds. (Academic, 1980), pp. 290–336.

L. Nordman, P. Aaltonen, T. Makkonen, Relationships between Mechanical and Optical Properties of Paper Affected by Web Consolidation (British Paper and Board Maker’s Association, 1966).

J. A. Van den Akker, Theory of Some of the Discrepancies Observed in Application of the Kubelka–Munk Equations to Particulate Systems (Plenum, 1968).

E. Allen, “Calculations for colorant formulations,” in Industrial Color Technology: Advances in Chemistry Series 107 (American Chemical Society, 1971), pp. 87–119.

D. B. Judd, G. Wyszecki, “Physics and Psychophysics of colorant layers,” in Color in Business, Science and Industry, 3rd ed. (Wiley, 1975).

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

P. Emmel, “Modèles de Prédiction Couleur Appliqués à L’impression Jet D’encre,” Thèse No. 1857 (École Polytechnique Fédérale de Lausanne, Switzerland, 1998).

W. Wendlandt, H. Hecht, Reflectance Spectroscopy (Wiley Interscience, 1966), Chap. 3.

N. Pauler, private communication. Similar experimental observations may be found in Ref. [14].

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of light scattering in a medium layer.

Fig. 2
Fig. 2

Average depth of the reflected light and the SIPV factor of white paper, with dashed lines representing a single paper sheet ( w p = 40 g m 2 ) and solid curves an infinite paper thickness ( w p = kg m 2 ) .

Fig. 3
Fig. 3

KM coefficients of scattering and absorption S p , K p (solid curves) computed from experimental spectra and their physical correspondences s p , a p (dashed curves) of paper.

Fig. 4
Fig. 4

Graphic evaluation of the uniqueness of the SIPV factor μ i p of dyed sheets, with w p = 40 g m 2 and w i = [ 0.0 , 0.05 , 0.2 ] g m 2 . The upper panel shows curves for a range between μ i p = 1 and μ i p , , while the lower panel is a finer view in the neighborhood around Δ ( μ i p ) = 0 .

Fig. 5
Fig. 5

KM coefficients of absorption and scattering for dyed paper sheets having nominal grammages w i p = 40.16 41.73 g m 2 computed from spectrophotometric measurements.[30] The amount of dye w i increases in the direction of the arrows. The results in the case of white paper are denoted by the dots.

Fig. 6
Fig. 6

Intrinsic absorption and scattering coefficients of cyan-dyed sheets, a i p and s i p , computed using Eqs. (1, 21, 22) with w p = 40 g m 2 and w i = [ 0 , 0.005 , 0.01 , 0.02 , 0.05 , 0.1 , 0.2 ] g m 2 . The results in the case of white paper ( w i = 0.0 g m 2 ) are denoted by the dots.

Fig. 7
Fig. 7

Predictions of the KM coefficients of absorption and scattering of the (a) cyan-, (b) magneta-, and (c) yellow-dyed sheets of paper. In these curves w p = 40 g m 2 and w i = [ 0.0 , 0.005 , 0.01 , 0.02 , 0.05 , 0.1 , 0.2 ] g m 2 . The results in the case of white paper ( w i = 0.0 g m 2 ) are denoted by the dots.

Equations (30)

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

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

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