Abstract

Using a statistical analysis of light propagation in media, we propose a revision to Kubelka–Munk (K–M) theory by taking into account the effect of scattering on the path length of light propagation (path variation). This leads to new relationships between the K–M scattering S and absorbing K coefficients and the intrinsic scattering s and absorbing a coefficients of a material that indicate that the S and K coefficients depend nonlinearly on both a and s. The additivity law that bridges K–M S and K coefficients of a composite medium, such as dye-dispersed paper (dyed paper) and those of its material components (dye and paper), is also revised. It is further shown that experimental findings on dyed paper that the original K–M theory failed to explain can be clearly understood and accommodated by the new K–M theoretical framework (two-flux approach). Numerical simulations with the revised theory on model ink, paper, and dyed paper have been carried out.

© 2004 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 contribution to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [CrossRef] [PubMed]
  3. W. R. Blevin, W. J. Brown, “Light-scattering properties of pigment suspensions,” J. Opt. Soc. Am. 51, 1250–1256 (1962).
    [CrossRef]
  4. E. Allen, “Basic equations used in computer color matching. II. Tristimulus match, two-constant theory,” J. Opt. Soc. Am. 64, 991–993 (1974).
    [CrossRef]
  5. D. B. Judd, G. Wyszecki, “Physics and psychophysics of colorant layers,” in Color in Business, Science, and Industry (Wiley, New York, 1975).
  6. 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]
  7. H. Granberg, M. Rundlöf, L. Mattsson, “Influence of surface-induced nonuniform reflectance on the diffuse reflectance factor. Part II: experimental verification on coated substrates,” J. Pulp Pap. Sci. 29, 254–259 (2003).
  8. E. Allen, “Colorant formation and shading,” in Optical Radiation Measurements, Vol. 2, Color Measurements, F. Grum, C. J. Bartleson, eds. (Academic, New York, 1980), pp. 290–336.
  9. B. Philips-Invernizzi, D. Dupont, C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. 40, 1082–1092 (2001).
    [CrossRef]
  10. W. J. Foote, “An investigation of the fundamental scattering and absorption coefficients of dye handsheets,” Pap. Trade J. 109, 333–340 (1939).
  11. L. Nordman, P. Aaltonen, T. Makkonen, “Relationships between mechanical and optical properties of paper affected by web consolidation,” in Transactions of the Symposium on Consolidation of the Paper Web, F. Bolam, ed. (British Paper and Board Makers’ Association, London, 1966), pp. 909–927.
  12. H. Granberg, P. Edström, “Quantification of the intrinsic error of the Kubelka–Munk model caused by strong light absorption,” J. Pulp Pap. Sci. 29, 386–390 (2003).
  13. 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).
  14. J. A. Van den Akker, “Scattering and absorption of light in paper and other diffusing media,” Tappi J. 32, 498–501 (1940).
  15. J. A. Van den Akker, Theory of Some of the Discrepancies Observed in Application of the Kubelka–Munk Equations to Particulate Systems (Plenum, New York, 1968).
  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, New York, 1998).
  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: Sivy’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]
  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. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, London, 1965), pp. 32–34.
  24. W. Wendlandt, H. Hecht, Reflectance Spectroscopy (Wiley Interscience, New York, 1966), Chap. 3.
  25. P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology II,” J. Colloid Interface Sci. 39, 551–567 (1972).
    [CrossRef]
  26. L. Yang, “Characterization of inks and ink application for inkjet printing: model and simulation,” J. Opt. Soc. Am. A 20, 1149–1154 (2003).
    [CrossRef]
  27. P. Emmel, R. D. Hersch, “A one-channel spectral colour prediction model for transparent fluorescent inks on a transparent support,” Proceedings of the IS&T/SID Color Imaging Conference C: Color Science, Systems, and Appli-cations (Society for Imaging Science and TechnologySpringfield, Va., 1997), pp. 70–77.
  28. L. Yang, “Determining depth of ink penetration in ink-jet printing,” in Proceedings of the Technical Association of Graphic Arts (TAGA) (TAGA, Rochester, N.Y., 2003), pp. 466–480.
  29. W. Wendlandt, H. Hecht, Reflectance Spectroscopy (Wiley Interscience, New York, 1966), Chap. 9.
  30. L. Yang, B. Kruse, S. J. Miklavcic, “Revised Kubelka–Munk theory. II. Unified framework for homogeneous and inhomogeneous optical media,” J. Opt. Soc. Am. A 21, 1942–1952 (2004).
    [CrossRef]

2004 (1)

2003 (4)

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

H. Granberg, M. Rundlöf, L. Mattsson, “Influence of surface-induced nonuniform reflectance on the diffuse reflectance factor. Part II: experimental verification on coated substrates,” J. Pulp Pap. Sci. 29, 254–259 (2003).

H. Granberg, P. Edström, “Quantification of the intrinsic error of the Kubelka–Munk model caused by strong light absorption,” J. Pulp Pap. Sci. 29, 386–390 (2003).

J. F. Bloch, R. Sève, “About the theoretical aspect of multiple light scattering: Sivy’s theory,” Color Res. Appl. 28, 227–228 (2003).
[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. 40, 1082–1092 (2001).
[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)

1984 (1)

1974 (1)

1972 (1)

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology II,” J. Colloid Interface Sci. 39, 551–567 (1972).
[CrossRef]

1962 (1)

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

1948 (1)

1940 (1)

J. A. Van den Akker, “Scattering and absorption of light in paper and other diffusing media,” Tappi J. 32, 498–501 (1940).

1939 (1)

W. J. Foote, “An investigation of the fundamental scattering and absorption coefficients of dye 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,” in Transactions of the Symposium on Consolidation of the Paper Web, F. Bolam, ed. (British Paper and Board Makers’ Association, London, 1966), pp. 909–927.

Allen, E.

E. Allen, “Basic equations used in computer color matching. II. Tristimulus match, two-constant theory,” J. Opt. Soc. Am. 64, 991–993 (1974).
[CrossRef]

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

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: Sivy’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, New York, 1998).

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. 40, 1082–1092 (2001).
[CrossRef]

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. 40, 1082–1092 (2001).
[CrossRef]

Edström, P.

H. Granberg, P. Edström, “Quantification of the intrinsic error of the Kubelka–Munk model caused by strong light absorption,” J. Pulp Pap. Sci. 29, 386–390 (2003).

Emmel, P.

P. Emmel, R. D. Hersch, “A one-channel spectral colour prediction model for transparent fluorescent inks on a transparent support,” Proceedings of the IS&T/SID Color Imaging Conference C: Color Science, Systems, and Appli-cations (Society for Imaging Science and TechnologySpringfield, Va., 1997), pp. 70–77.

Foote, W. J.

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

Gouesbet, G.

Granberg, H.

H. Granberg, P. Edström, “Quantification of the intrinsic error of the Kubelka–Munk model caused by strong light absorption,” J. Pulp Pap. Sci. 29, 386–390 (2003).

H. Granberg, M. Rundlöf, L. Mattsson, “Influence of surface-induced nonuniform reflectance on the diffuse reflectance factor. Part II: experimental verification on coated substrates,” J. Pulp Pap. Sci. 29, 254–259 (2003).

Hecht, H.

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

W. Wendlandt, H. Hecht, Reflectance Spectroscopy (Wiley Interscience, New York, 1966), Chap. 9.

Hersch, R. D.

P. Emmel, R. D. Hersch, “A one-channel spectral colour prediction model for transparent fluorescent inks on a transparent support,” Proceedings of the IS&T/SID Color Imaging Conference C: Color Science, Systems, and Appli-cations (Society for Imaging Science and TechnologySpringfield, Va., 1997), pp. 70–77.

Huffman, D. R.

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

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 (Wiley, New York, 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 contribution 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,” in Transactions of the Symposium on Consolidation of the Paper Web, F. Bolam, ed. (British Paper and Board Makers’ Association, London, 1966), pp. 909–927.

Mattsson, L.

H. Granberg, M. Rundlöf, L. Mattsson, “Influence of surface-induced nonuniform reflectance on the diffuse reflectance factor. Part II: experimental verification on coated substrates,” J. Pulp Pap. Sci. 29, 254–259 (2003).

Miklavcic, S. J.

Mudgett, P. S.

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology II,” J. Colloid Interface Sci. 39, 551–567 (1972).
[CrossRef]

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Niklasson, G. A.

Nordman, L.

L. Nordman, P. Aaltonen, T. Makkonen, “Relationships between mechanical and optical properties of paper affected by web consolidation,” in Transactions of the Symposium on Consolidation of the Paper Web, F. Bolam, ed. (British Paper and Board Makers’ Association, London, 1966), pp. 909–927.

Philips-Invernizzi, B.

B. Philips-Invernizzi, D. Dupont, C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. 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]

Reif, F.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, London, 1965), pp. 32–34.

Richards, L. W.

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology II,” J. Colloid Interface Sci. 39, 551–567 (1972).
[CrossRef]

Rundlöf, M.

H. Granberg, M. Rundlöf, L. Mattsson, “Influence of surface-induced nonuniform reflectance on the diffuse reflectance factor. Part II: experimental verification on coated substrates,” J. Pulp Pap. Sci. 29, 254–259 (2003).

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: Sivy’s theory,” Color Res. Appl. 28, 227–228 (2003).
[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 (1940).

J. A. Van den Akker, Theory of Some of the Discrepancies Observed in Application of the Kubelka–Munk Equations to Particulate Systems (Plenum, New York, 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, New York, 1966), Chap. 3.

W. Wendlandt, H. Hecht, Reflectance Spectroscopy (Wiley Interscience, New York, 1966), Chap. 9.

Wyszecki, G.

D. B. Judd, G. Wyszecki, “Physics and psychophysics of colorant layers,” in Color in Business, Science, and Industry (Wiley, New York, 1975).

Yang, L.

Appl. Opt. (3)

Color Res. Appl. (1)

J. F. Bloch, R. Sève, “About the theoretical aspect of multiple light scattering: Sivy’s theory,” Color Res. Appl. 28, 227–228 (2003).
[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. Colloid Interface Sci. (1)

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology II,” J. Colloid Interface Sci. 39, 551–567 (1972).
[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. (3)

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

J. Pulp Pap. Sci. (4)

H. Granberg, M. Rundlöf, L. Mattsson, “Influence of surface-induced nonuniform reflectance on the diffuse reflectance factor. Part II: experimental verification on coated substrates,” J. Pulp Pap. Sci. 29, 254–259 (2003).

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

H. Granberg, P. Edström, “Quantification of the intrinsic error of the Kubelka–Munk model caused by strong light absorption,” J. Pulp Pap. Sci. 29, 386–390 (2003).

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. (1)

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

Pap. Trade J. (1)

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

Tappi J. (1)

J. A. Van den Akker, “Scattering and absorption of light in paper and other diffusing media,” Tappi J. 32, 498–501 (1940).

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, New York, 1998).

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

L. Nordman, P. Aaltonen, T. Makkonen, “Relationships between mechanical and optical properties of paper affected by web consolidation,” in Transactions of the Symposium on Consolidation of the Paper Web, F. Bolam, ed. (British Paper and Board Makers’ Association, London, 1966), pp. 909–927.

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

D. B. Judd, G. Wyszecki, “Physics and psychophysics of colorant layers,” in Color in Business, Science, and Industry (Wiley, New York, 1975).

P. Emmel, R. D. Hersch, “A one-channel spectral colour prediction model for transparent fluorescent inks on a transparent support,” Proceedings of the IS&T/SID Color Imaging Conference C: Color Science, Systems, and Appli-cations (Society for Imaging Science and TechnologySpringfield, Va., 1997), pp. 70–77.

L. Yang, “Determining depth of ink penetration in ink-jet printing,” in Proceedings of the Technical Association of Graphic Arts (TAGA) (TAGA, Rochester, N.Y., 2003), pp. 466–480.

W. Wendlandt, H. Hecht, Reflectance Spectroscopy (Wiley Interscience, New York, 1966), Chap. 9.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, London, 1965), pp. 32–34.

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

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the studied medium system.

Fig. 2
Fig. 2

Schematic representation of light propagation in a light absorbing and scattering medium.

Fig. 3
Fig. 3

Schematic representation of a layer of absorbing and light scattering medium.

Fig. 4
Fig. 4

KM absorbing and scattering powers of the magenta ink.26

Fig. 5
Fig. 5

Spectral dependence of the SIPV factor μi of magenta ink.

Fig. 6
Fig. 6

Computed absorbing and scattering powers of magenta ink.

Fig. 7
Fig. 7

Computed KM absorbing (upper panel) and scattering powers (lower panel) of a single sheet of paper.

Fig. 8
Fig. 8

Variations of the SIPV-factors for paper μp.

Fig. 9
Fig. 9

Computed absorbing (upper panel) and scattering (lower panel) powers of the paper.

Fig. 10
Fig. 10

Spectral variations of the KM absorption (upper panel) and scattering (lower panel) powers of dyed paper (magenta) with dye percentage ρ=0.5, 1, 2.5, 5, 10, 20% (% of zi). ρ=0 corresponds to white paper and is denoted by dashed curves.

Equations (60)

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

dI=-(K+S)Idz+SJdz,
dJ=(K+S)Jdz-SIdz.
S=s,K=2a,
Kipzp=ρKizi+Kpzp,
Sipzp=ρSizi+Spzp,
la=1/a.
ls=1/s.
R=n=1Nln.
l=n=1N|ln|=Nls,
N=la/ls.
R2=RR=n=1Nm=1Nrnrm=n=1Nrn2=Nr2,
R=(|R|2)1/2(N)1/2ls(lals)1/2.
μla/R(la/ls)1/2=(s/a)1/2.
μ(λ)=(s(λ)/a(λ))1/2,s(λ)a(λ)1,otherwise.
dlI=μdz0π/21IIϕdϕcos ϕ=μαIdz,
αI=0π/21IIϕdϕcos ϕ.
I/ϕ=I sin 2ϕ,
αI=2.
I/ϕ=Iδ(ϕ),
dlJ=μ0π/21JJϕdϕcos ϕdz=μαJdz,
αJ=0π/21JJϕdϕcos ϕ.
(a+s)IdlI=μ(a+s)IαIdz,
(a+s)JdlJ=μ(a+s)JαJdz.
dI=-μ(a+s)IαIdz+μsJαJdz,
dJ=μ(a+s)JαJdz-μsIαIdz.
dI=-(KI+SI)Idz+SJJdz,
dJ=(KJ+SJ)Jdz-SIIdz,
KI=μαIa,SI=μαIs,
KJ=μαJa,SJ=μαJs.
dI=-(K+S)Idz+SJdz,
dJ=(K+S)Jdz-SIdz,
K=μαa,S=μαs.
K=μαa,S=μαs/2.
K=2μa,S=μs.
K=μαa=α(sa)1/2,(sa)αa,otherwise,
S=12 μαs=12α(s3/a)1/2,(sa)12αs,otherwise.
si/ai=2Si/Ki.
μi(λ)=(2Si(λ)/Ki(λ))1/2,2Si(λ)Ki(λ)1,otherwise.
sizi=2Sizi/αiμi,aizi=Kizi/αiμi.
aipzp=ρaizi+apzp,
sipzp=ρsizi+spzp,
μip(λ)=ρsi(λ)zi+sp(λ)zpρai(λ)zi+ap(λ)zp1/2,sip(λ)aip(λ)1,otherwise.
Kipzp=μipαipaipzp=μipαip(ρaizi+apzp),
Sipzp=μipαipsipzp2=μipαip(ρsizi+spzp)2.
Kipzp=ρ μipμiαipαi Kizi+μipμpαipαp Kpzp,
Sipzp=ρ μipμiαipαi Sizi+μipμpαipαp Spzp.
Kipzp=ρ(μip/μi)Kizi+(μip/μp)Kpzp,
Sipzp=ρ(μip/μi)Sizi+(μip/μp)Spzp.
Sipzp(μip/μp)Spzp.
μip(λ)sp(λ)zpρai(λ)zi+ap(λ)zp1/2=μp1+ρ ai(λ)ziap(λ)zp1/2,sip(λ)aip(λ).
SipzpSpzp/1+ρ ai(λ)ziap(λ)zp1/2.
Kipzpρ(μip/μi)Kizi.
μip(λ)sp(λ)zpρai(λ)zi+ap(λ)zp1/2=spzpsizi1/2μiρ+ap(λ)zpai(λ)zi1/2sip(λ)aip(λ).
Kipzpspzpsizi1/21ρ+ap(λ)zpai(λ)zi1/2ρKizi.
f(x)=k(x/γ)2exp(-x2/γ2),
1=k0(x/γ)2exp[-(x2/γ2)]dx.
k=4/γπ.
ls=|x|=k0x(x/γ)2exp(-x2/γ2)dx,
γ=(π1/2/2)ls.
|x|2=k0x2(x/γ)2exp(-x2/γ2)dx=(3π/8)ls2.

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