Abstract

Calculations of radiative transfer require knowledge of the absorption and scattering coefficients and the asymmetry factor of scattering in the medium. A method is presented for estimating these coefficients in living plant leaves from fiber-optic measurements. We consider the plant leaf as consisting of two layers of different refractive indices and with reflecting surfaces. Light intensities at the boundaries of these layers in several irradiated plant leaves have been measured using a thin (70-μm) glass fiber connected to a photomultiplier. The diffuse reflection and transmission were measured with an integrating sphere. From these values we derive an estimation of the scattering and absorption coefficients and the asymmetry factor of scattering applying an inversion of the multiflux theory of light propagation in turbid media. In addition, we compare these coefficients with those obtained by using the Kubelka–Munk theory.

© 1991 Optical Society of America

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References

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  1. L. Fukshansky, “Optical properties of plant tissue,” in Plants and the Daylight Spectrum, H. Smith, ed. (Academic, London, 1981), pp. 21–40.
  2. C. J. Gomer, guest ed., special issue on photodynamic therapy, Photochem. Photobiol.46 (1987).
  3. E. Schafer, L. Fukshansky, W. Shropshire, “Action spectroscopy of photoreversible pigment systems,” in Photo-morphogenesis Encyclopedia of Plant Physiology, N.S. 16A, W. Shropshire, H. Mohr, eds. (Springer-Verlag, Berlin, 1883), pp. 31–68.
  4. L. Fukshansky, “Absorption statistics in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 38, 389–406 (1987).
    [CrossRef]
  5. R. A. J. Groenhuis, H. A. Ferwerda, J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements,” Appl. Opt. 22, 2456–2467 (1983).
    [CrossRef] [PubMed]
  6. S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–317 (1987).
  7. L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the gallaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  8. G. Yoon, A. J. Welch, M. Motamedi, M. C. J. van Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1732 (1987).
    [CrossRef]
  9. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  10. A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1 (1905); reprinted in D. H. Menzel, Selected Papers on the Transfer of Radiation (Dover, New York, 1966).
    [CrossRef]
  11. L. Silberstein, “The transparency of turbid media,” Philos. Mag. 4, 1291–1302 (1927).
  12. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 11a, 593–601 (1931).
  13. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–656 (1948).
    [CrossRef] [PubMed]
  14. J. W. Ryde, “The scattering of light by turbid media. Part I,” Proc. R. Soc. London Ser. A 131, 451–463 (1931).
    [CrossRef]
  15. J. W. Ryde, B. S. Cooper, “The Scattering of light by turbid media. Part II” Proc. R. Soc. London Ser. A 131, 464–476 (1931).
    [CrossRef]
  16. S. Q. Duntley, “The optical properties of diffusing materials,” J. Opt. Soc. Am. 32, 61–70 (1942).
    [CrossRef]
  17. H. G. Volz, “Ein Beitrag zur phaenomenologischen Theorie lichtstreuender und absorbierender Medien,” in Proceedings of the Seventh FATIPEC Congress (Verlag Chemie, Weinheim/Bergstrasse, 1964), pp. 194–201.
  18. P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
    [CrossRef] [PubMed]
  19. M. Seyfried, L. Fukshansky, “Light gradients in plant tissue,” Appl. Opt. 22, 1402–1408 (1983).
    [CrossRef] [PubMed]
  20. M. Seyfried, L. Fukshansky, E. Schafer, “Correcting remission and transmission spectra of plant tissue measured in glass cuvettes: a technique,” Appl. Opt. 22, 492–696 (1983).
    [CrossRef] [PubMed]
  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. R. G. Giovanelly, “Reflection by Semi-infinite diffusors,” Opt. Acta 2, 153–162 (1955).
    [CrossRef]
  23. H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Datzell, “Multiple scattering: comparison of theory with experiment,” J. Heat Transfer 92, 285–292 (1970).
    [CrossRef]
  24. L. Fukshansky, N. Fukshansky-Kazarinova, “Extension of the Kubelka–Munk theory of light propagation in intensely scattering materials to fluorescent media,” J. Opt. Soc. Am. 70, 1101–1111 (1980).
    [CrossRef]
  25. T. C. Vogelmann, L. O. Bjorn, “Measurement of light gradients and spectral regime in plant tissue with fiber optic probe,” Physiol. Plant 60, 361–368 (1984), Sec. II, para. 2, line 9.
    [CrossRef]
  26. N. Fukshansky-Kazarinova, W. Lork, E. Schafer, L. Fukshansky, “Photon flux gradients in layered turbid media: application to biological tissues,” Appl. Opt. 25, 780–788 (1986).
    [CrossRef] [PubMed]
  27. W. T. Walsh, “The reflection factor of a polished glass surface for diffused light,” Dep. Sci. Ind. Res. Illum. Res. Tech. Pap. 2, 10–76 (1926).
  28. D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329–332 (1942).
    [CrossRef]
  29. G. Kortum, Reflectance Spectroscopy: Principles, Methods, Applications (Springer-Verlag, Berlin, 1966), p. 366.
  30. W. F. Kaufmann, K. H. Hartmann, “Internal brightness of disk-shaped samples,” J. Photochem. Photobiol. 1, 337–360 (1988).
    [CrossRef]
  31. J. McClendon, L. Fukshansky, “On the interpretation of absorption spectra of leaves—I. Introduction and the correction of leaf spectra for surface reflection,” J. Photochem. Photobiol. 51, 203–270 (1990).
    [CrossRef]
  32. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, London, 1988).

1990 (1)

J. McClendon, L. Fukshansky, “On the interpretation of absorption spectra of leaves—I. Introduction and the correction of leaf spectra for surface reflection,” J. Photochem. Photobiol. 51, 203–270 (1990).
[CrossRef]

1988 (1)

W. F. Kaufmann, K. H. Hartmann, “Internal brightness of disk-shaped samples,” J. Photochem. Photobiol. 1, 337–360 (1988).
[CrossRef]

1987 (3)

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–317 (1987).

G. Yoon, A. J. Welch, M. Motamedi, M. C. J. van Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1732 (1987).
[CrossRef]

L. Fukshansky, “Absorption statistics in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 38, 389–406 (1987).
[CrossRef]

1986 (1)

1984 (2)

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]

T. C. Vogelmann, L. O. Bjorn, “Measurement of light gradients and spectral regime in plant tissue with fiber optic probe,” Physiol. Plant 60, 361–368 (1984), Sec. II, para. 2, line 9.
[CrossRef]

1983 (3)

1980 (1)

1971 (1)

1970 (1)

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Datzell, “Multiple scattering: comparison of theory with experiment,” J. Heat Transfer 92, 285–292 (1970).
[CrossRef]

1955 (1)

R. G. Giovanelly, “Reflection by Semi-infinite diffusors,” Opt. Acta 2, 153–162 (1955).
[CrossRef]

1948 (1)

1942 (2)

S. Q. Duntley, “The optical properties of diffusing materials,” J. Opt. Soc. Am. 32, 61–70 (1942).
[CrossRef]

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

1941 (1)

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the gallaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

1931 (3)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 11a, 593–601 (1931).

J. W. Ryde, “The scattering of light by turbid media. Part I,” Proc. R. Soc. London Ser. A 131, 451–463 (1931).
[CrossRef]

J. W. Ryde, B. S. Cooper, “The Scattering of light by turbid media. Part II” Proc. R. Soc. London Ser. A 131, 464–476 (1931).
[CrossRef]

1927 (1)

L. Silberstein, “The transparency of turbid media,” Philos. Mag. 4, 1291–1302 (1927).

1926 (1)

W. T. Walsh, “The reflection factor of a polished glass surface for diffused light,” Dep. Sci. Ind. Res. Illum. Res. Tech. Pap. 2, 10–76 (1926).

1905 (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1 (1905); reprinted in D. H. Menzel, Selected Papers on the Transfer of Radiation (Dover, New York, 1966).
[CrossRef]

Alter, C. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–317 (1987).

Bjorn, L. O.

T. C. Vogelmann, L. O. Bjorn, “Measurement of light gradients and spectral regime in plant tissue with fiber optic probe,” Physiol. Plant 60, 361–368 (1984), Sec. II, para. 2, line 9.
[CrossRef]

Cooper, B. S.

J. W. Ryde, B. S. Cooper, “The Scattering of light by turbid media. Part II” Proc. R. Soc. London Ser. A 131, 464–476 (1931).
[CrossRef]

Datzell, W. H.

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Datzell, “Multiple scattering: comparison of theory with experiment,” J. Heat Transfer 92, 285–292 (1970).
[CrossRef]

Duntley, S. Q.

Ferwerda, H. A.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, London, 1988).

Fukshansky, L.

J. McClendon, L. Fukshansky, “On the interpretation of absorption spectra of leaves—I. Introduction and the correction of leaf spectra for surface reflection,” J. Photochem. Photobiol. 51, 203–270 (1990).
[CrossRef]

L. Fukshansky, “Absorption statistics in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 38, 389–406 (1987).
[CrossRef]

N. Fukshansky-Kazarinova, W. Lork, E. Schafer, L. Fukshansky, “Photon flux gradients in layered turbid media: application to biological tissues,” Appl. Opt. 25, 780–788 (1986).
[CrossRef] [PubMed]

M. Seyfried, L. Fukshansky, E. Schafer, “Correcting remission and transmission spectra of plant tissue measured in glass cuvettes: a technique,” Appl. Opt. 22, 492–696 (1983).
[CrossRef] [PubMed]

M. Seyfried, L. Fukshansky, “Light gradients in plant tissue,” Appl. Opt. 22, 1402–1408 (1983).
[CrossRef] [PubMed]

L. Fukshansky, N. Fukshansky-Kazarinova, “Extension of the Kubelka–Munk theory of light propagation in intensely scattering materials to fluorescent media,” J. Opt. Soc. Am. 70, 1101–1111 (1980).
[CrossRef]

L. Fukshansky, “Optical properties of plant tissue,” in Plants and the Daylight Spectrum, H. Smith, ed. (Academic, London, 1981), pp. 21–40.

E. Schafer, L. Fukshansky, W. Shropshire, “Action spectroscopy of photoreversible pigment systems,” in Photo-morphogenesis Encyclopedia of Plant Physiology, N.S. 16A, W. Shropshire, H. Mohr, eds. (Springer-Verlag, Berlin, 1883), pp. 31–68.

Fukshansky-Kazarinova, N.

Giovanelly, R. G.

R. G. Giovanelly, “Reflection by Semi-infinite diffusors,” Opt. Acta 2, 153–162 (1955).
[CrossRef]

Gouesbet, G.

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the gallaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Groenhuis, R. A. J.

Hartmann, K. H.

W. F. Kaufmann, K. H. Hartmann, “Internal brightness of disk-shaped samples,” J. Photochem. Photobiol. 1, 337–360 (1988).
[CrossRef]

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the gallaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hottel, H. C.

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Datzell, “Multiple scattering: comparison of theory with experiment,” J. Heat Transfer 92, 285–292 (1970).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Jacques, S. L.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–317 (1987).

Judd, D. B.

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

Kaufmann, W. F.

W. F. Kaufmann, K. H. Hartmann, “Internal brightness of disk-shaped samples,” J. Photochem. Photobiol. 1, 337–360 (1988).
[CrossRef]

Kortum, G.

G. Kortum, Reflectance Spectroscopy: Principles, Methods, Applications (Springer-Verlag, Berlin, 1966), p. 366.

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–656 (1948).
[CrossRef] [PubMed]

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 11a, 593–601 (1931).

Letoulouzan, J. N.

Lork, W.

Maheu, B.

McClendon, J.

J. McClendon, L. Fukshansky, “On the interpretation of absorption spectra of leaves—I. Introduction and the correction of leaf spectra for surface reflection,” J. Photochem. Photobiol. 51, 203–270 (1990).
[CrossRef]

Motamedi, M.

G. Yoon, A. J. Welch, M. Motamedi, M. C. J. van Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1732 (1987).
[CrossRef]

Mudgett, P. S.

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 11a, 593–601 (1931).

Prahl, S. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–317 (1987).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, London, 1988).

Richards, L. W.

Ryde, J. W.

J. W. Ryde, “The scattering of light by turbid media. Part I,” Proc. R. Soc. London Ser. A 131, 451–463 (1931).
[CrossRef]

J. W. Ryde, B. S. Cooper, “The Scattering of light by turbid media. Part II” Proc. R. Soc. London Ser. A 131, 464–476 (1931).
[CrossRef]

Sarofim, A. F.

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Datzell, “Multiple scattering: comparison of theory with experiment,” J. Heat Transfer 92, 285–292 (1970).
[CrossRef]

Schafer, E.

Schuster, A.

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1 (1905); reprinted in D. H. Menzel, Selected Papers on the Transfer of Radiation (Dover, New York, 1966).
[CrossRef]

Seyfried, M.

Shropshire, W.

E. Schafer, L. Fukshansky, W. Shropshire, “Action spectroscopy of photoreversible pigment systems,” in Photo-morphogenesis Encyclopedia of Plant Physiology, N.S. 16A, W. Shropshire, H. Mohr, eds. (Springer-Verlag, Berlin, 1883), pp. 31–68.

Silberstein, L.

L. Silberstein, “The transparency of turbid media,” Philos. Mag. 4, 1291–1302 (1927).

Ten Bosch, J. J.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, London, 1988).

van Gemert, M. C. J.

G. Yoon, A. J. Welch, M. Motamedi, M. C. J. van Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1732 (1987).
[CrossRef]

Vasalos, I. A.

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Datzell, “Multiple scattering: comparison of theory with experiment,” J. Heat Transfer 92, 285–292 (1970).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, London, 1988).

Vogelmann, T. C.

T. C. Vogelmann, L. O. Bjorn, “Measurement of light gradients and spectral regime in plant tissue with fiber optic probe,” Physiol. Plant 60, 361–368 (1984), Sec. II, para. 2, line 9.
[CrossRef]

Volz, H. G.

H. G. Volz, “Ein Beitrag zur phaenomenologischen Theorie lichtstreuender und absorbierender Medien,” in Proceedings of the Seventh FATIPEC Congress (Verlag Chemie, Weinheim/Bergstrasse, 1964), pp. 194–201.

Walsh, W. T.

W. T. Walsh, “The reflection factor of a polished glass surface for diffused light,” Dep. Sci. Ind. Res. Illum. Res. Tech. Pap. 2, 10–76 (1926).

Welch, A. J.

G. Yoon, A. J. Welch, M. Motamedi, M. C. J. van Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1732 (1987).
[CrossRef]

Yoon, G.

G. Yoon, A. J. Welch, M. Motamedi, M. C. J. van Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1732 (1987).
[CrossRef]

Appl. Opt. (6)

Astrophys. J. (2)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1 (1905); reprinted in D. H. Menzel, Selected Papers on the Transfer of Radiation (Dover, New York, 1966).
[CrossRef]

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the gallaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Dep. Sci. Ind. Res. Illum. Res. Tech. Pap. (1)

W. T. Walsh, “The reflection factor of a polished glass surface for diffused light,” Dep. Sci. Ind. Res. Illum. Res. Tech. Pap. 2, 10–76 (1926).

IEEE J. Quantum Electron. (1)

G. Yoon, A. J. Welch, M. Motamedi, M. C. J. van Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1732 (1987).
[CrossRef]

J. Heat Transfer (1)

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Datzell, “Multiple scattering: comparison of theory with experiment,” J. Heat Transfer 92, 285–292 (1970).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Photochem. Photobiol. (2)

W. F. Kaufmann, K. H. Hartmann, “Internal brightness of disk-shaped samples,” J. Photochem. Photobiol. 1, 337–360 (1988).
[CrossRef]

J. McClendon, L. Fukshansky, “On the interpretation of absorption spectra of leaves—I. Introduction and the correction of leaf spectra for surface reflection,” J. Photochem. Photobiol. 51, 203–270 (1990).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

L. Fukshansky, “Absorption statistics in turbid media,” J. Quant. Spectrosc. Radiat. Transfer 38, 389–406 (1987).
[CrossRef]

J. Res. Natl. Bur. Stand. (1)

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

Lasers Life Sci. (1)

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–317 (1987).

Opt. Acta (1)

R. G. Giovanelly, “Reflection by Semi-infinite diffusors,” Opt. Acta 2, 153–162 (1955).
[CrossRef]

Philos. Mag. (1)

L. Silberstein, “The transparency of turbid media,” Philos. Mag. 4, 1291–1302 (1927).

Physiol. Plant (1)

T. C. Vogelmann, L. O. Bjorn, “Measurement of light gradients and spectral regime in plant tissue with fiber optic probe,” Physiol. Plant 60, 361–368 (1984), Sec. II, para. 2, line 9.
[CrossRef]

Proc. R. Soc. London Ser. A (2)

J. W. Ryde, “The scattering of light by turbid media. Part I,” Proc. R. Soc. London Ser. A 131, 451–463 (1931).
[CrossRef]

J. W. Ryde, B. S. Cooper, “The Scattering of light by turbid media. Part II” Proc. R. Soc. London Ser. A 131, 464–476 (1931).
[CrossRef]

Z. Tech. Phys. (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 11a, 593–601 (1931).

Other (7)

L. Fukshansky, “Optical properties of plant tissue,” in Plants and the Daylight Spectrum, H. Smith, ed. (Academic, London, 1981), pp. 21–40.

C. J. Gomer, guest ed., special issue on photodynamic therapy, Photochem. Photobiol.46 (1987).

E. Schafer, L. Fukshansky, W. Shropshire, “Action spectroscopy of photoreversible pigment systems,” in Photo-morphogenesis Encyclopedia of Plant Physiology, N.S. 16A, W. Shropshire, H. Mohr, eds. (Springer-Verlag, Berlin, 1883), pp. 31–68.

H. G. Volz, “Ein Beitrag zur phaenomenologischen Theorie lichtstreuender und absorbierender Medien,” in Proceedings of the Seventh FATIPEC Congress (Verlag Chemie, Weinheim/Bergstrasse, 1964), pp. 194–201.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

G. Kortum, Reflectance Spectroscopy: Principles, Methods, Applications (Springer-Verlag, Berlin, 1966), p. 366.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, London, 1988).

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Figures (4)

Fig. 1
Fig. 1

Geometry of the measurements of internal fluxes in a plane-parallel sample with a fiber-optical probe: F0, photon flux incident at the boundary of the sample; ξ, spatial coordinate perpendicular to the sample surface; 1, …, i, …, 5, numbers of channels (partial fluxes having different angles θ1, …, θi, …, θ5 with respect to the coordinate ξ); a, fiber-optic probe with acceptance angle β measuring the partial flux in channel 3 at depth ξ.

Fig. 2
Fig. 2

External fluxes measured with the integrating sphere and internal boundary fluxes in a two-layer sample irradiated with a collimated flux Fi: ξ, the spatial coordinate directed along the internal normal to the upper surface; L, ξ ¯, thicknesses of the sample and the upper layer, respectively; θ, polar angle. Absorption k and scattering s coefficients and the fraction of forward scattering f are specified for the upper and lower layers by subscripts 1 and 2, respectively. For definitions of the transformation coefficients, see Subsection III.A. Subscripts 0 and L assigned to the fluxes specify values at the upper and lower boundaries, respectively; subscripts c and d specify collimated and diffuse Lambertian fluxes, respectively. Subscripts R and T specify remitted fluxes, respectively, as measured with the integrating sphere; RM and TM are the formal remission and transmission coefficients of the sample: (a) the scheme of measured external fluxes; (b) the scheme of relationships between partial external and internal boundary fluxes; (c) the scheme of internal boundary fluxes.

Fig. 3
Fig. 3

(a) Parameters k1, s1, and asymmetry factor f1 for the upper layer of a leaf of Catalpa bignonioides calculated with α1 from our 1 measurement (see Table I); k1 and s1 are in inverse millimeters, f1 is dimensionless. (b) Parameters k2, s2, and asymmetry factor f2 for the lower layer of a leaf of Catalpa bignonioides calculated with α2 from our 1 measurement (see Table I); k2 and s2 are in inverse millimeters, f2 is dimensionless. (c) Light gradients for a leaf of Catalpa bignoniodes at 480 and 720 nm obtained from the Kubelka–Munk theory and from multiflux theory calculated with α1 and α2 from our 1 measurement (see Table I).

Fig. 4
Fig. 4

(a) Parameters k1, s1, and asymmetry factor f1 for the upper layer of a leaf of Catalpa bignonioides calculated with a from our 2 measurement (see Table I); k1 and s1 are in inverse millimeters, f1 is dimensionless. (b) Parameters k2, s2, and asymmetry factor f2 for the lower layer of a leaf of Catalpa bignonioides calculated with α2 from our 2 measurement (see Table I); k2 and s2 are in inverse millimeters, f2 is dimensionless. (c) Light gradients for a leaf of Catalpa bignonioides at 480 and 720 nm obtained from the Kubelka–Munk theory and from multiflux theory calculated with α1 and α2 from our 2 measurement (see Table I).

Tables (1)

Tables Icon

Table I Values of α1 and α2 Used for the Results Given In Figs. 3 and 4

Equations (30)

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

d F d ξ = - ( k + s ) F + s G , d G d ξ = - s F + ( k + s ) G ,
I i = F i δ ( Ω - Ω 0 ) ,
4 π δ ( Ω - Ω 0 ) d Ω = 1
δ ( Ω - Ω 0 ) = δ ( θ - θ 0 ) δ ( φ - φ 0 ) sin θ ,             d Ω = sin θ d θ d φ ,
0 π δ ( θ - θ 0 ) d θ = 1 ,             0 π δ ( φ - φ 0 ) d φ = 1.
F d 0 = F i d ( 1 - r r ) + G d 0 r d ,             G d L = F c L d r d + F d L r d ,
F R = F i ( 1 - d ) r c r + F i d r r + G d 0 ( 1 - r d ) , F T = F c L ( 1 - d ) ( 1 - r c d ) + F c L d ( 1 - r d ) + F d L ( 1 - r d )
r r = 1 - t r ,             r d = 1 - t d = 1 - ( n r n d ) 2 t r = 1 - ( n r n d ) 2 ( 1 - r r ) ,
F i [ ( 1 - d ) r c r + d r r ] = F measured
d F c d ξ = - ( k + s ) F c , d F d d ξ = - 2 [ k + s ( 1 - f ) ] F d + 2 s ( 1 - f ) G d + s f F c , d G d d ξ = 2 [ k + s ( 1 - f ) ] G d - 2 s ( 1 - f ) F d - s ( 1 - f ) F c ,
W = F d + G d + F c , U = F d - G d + F c ,
d W d ξ = - 2 [ k + 2 s ( 1 - f ) ] U + [ k + 2 s ( 1 - f ) ] F c , d U d ξ = - 2 k W + k F c ,
d 2 W d ξ 2 = 4 k [ k + 2 s ( 1 - f ) ] W - 2 k { k + 2 s ( 1 - f ) } F c - [ k + 2 s ( 1 - f ) ] ( k + s ) F c .
x = ξ / L ,
α = ( k + s ) L ,
μ 2 = 4 k [ k + 2 s ( 1 - f ) ] L 2 , ν 2 = k + 2 s ( 1 - f ) k ,
1 - f = μ ( ν 2 - 1 ) 2 ( 2 α ν - μ ) ,
d 2 W d x 2 = μ 2 W - μ 2 + ν μ α 2 F c , d W d x = - μ ν ( U - F c 2 ) .
W ( 0 ) = F c 0 ( 1 - R ) , d W d x ( 0 ) = - μ ν [ F c 0 ( 1 - R ) - F c 0 2 ] , W ( 1 ) = F c 0 T ( 1 + R g ) , d W d x ( 1 ) = - μ ν [ F c 0 T ( 1 - R g ) - F c 1 2 ] ,
R = G d 0 F c 0 + F d 0 ,             R g = G d L F c L + F d L ,             T = F c L + F d L F c 0 + F d 0 .
F c 1 = F c 0 exp [ - ( α 1 ¯ + α 2 ¯ ) ] ,
α 1 ¯ = α 1 x ¯ , α 2 ¯ = α 2 ( 1 - x ¯ )
d 2 W 1 d x 2 = μ 1 2 W 1 - μ 1 2 + ν 1 μ 1 α 1 2 F c 0 exp ( - α 1 x ) , U 1 ( x ) = - 1 μ 1 ν 1 d W 1 d x + 1 2 F c 0 exp ( - α 1 x )
W 1 ( 0 ) = ( F c 0 + F d 0 ) ( 1 + R ) , d W 1 d x ( 0 ) = - μ 1 ν 1 ( F c 0 + F d 0 ) [ ( 1 - R ) - F c 0 2 ( F c 0 + F d 0 ) ] .
W 1 ( x ) ( F c 0 + F d 0 ) = ( 1 + R ) cosh μ 1 x - ν 1 ( 1 - R ) sinh μ 1 x + F c 0 F c 0 + F d 0 { ν 1 2 sinh μ 1 x - μ 1 + ν 1 α 1 2 ( α 1 2 - μ 1 2 ) × [ μ 1 exp ( - α 1 x ) - μ 1 cosh μ 1 x + α 1 sinh μ 1 x ] } , U 1 ( x ) ( F c 0 + F d 0 ) = - 1 ν 1 [ ( 1 + R ) sinh μ 1 x - ν 1 ( 1 - R ) cosh μ 1 x ] - F c 0 ν 1 ( F c 0 + F d 0 ) { ν 1 2 cosh μ 1 x - μ 1 + ν 1 α 1 2 ( α 1 2 - μ 1 2 ) × [ - α 1 exp ( - α 1 x ) - μ 1 sinh μ 1 x + α 1 cosh μ 1 x ] - ν 1 2 exp ( - α 1 x ) } .
d 2 W 2 d x 2 = μ 2 2 W 2 - μ 2 2 + ν 2 μ 2 α 2 2 F c 0 exp [ - ( α 1 ¯ + α 2 ¯ ) exp [ α 2 ( 1 - x ) ] , U 2 ( x ) = - 1 μ 2 ν 2 d W 2 d x + 1 2 F c 0 exp [ - ( α 1 ¯ + α 2 ¯ ) ] exp [ α 2 ( 1 - x ) ] ,
W 2 ( 1 ) = ( F c 0 + F d 0 ) T ( 1 + R g ) , d W 2 d x ( 1 ) = - μ 2 ν 2 { ( F c 0 + F d 0 ) T ( 1 - R g ) - ½ F c 0 exp [ - ( α 1 ¯ + α 2 ¯ ) ] }
W 2 ( x ) ( F c 0 + F d 0 ) = T [ ( 1 + R g ) cosh μ 2 ( 1 - x ) + ν 2 ( 1 - R ) sinh μ 2 ( 1 - x ) ] + 1 2 ( F c 0 + F d 0 ) exp [ - ( α 1 ¯ + α 2 ¯ ) ] { - ν 2 sinh μ 2 ( 1 - x ) - μ 2 + ν 2 α 2 α 2 2 - μ 2 2 × [ μ 2 exp [ α 2 ( 1 - x ) ] - μ 2 cosh μ 2 ( 1 - x ) - α 2 sinh μ 2 ( 1 - x ) } , U 2 ( x ) ( F c 0 + F d 0 ) = T ν 2 [ ( 1 + R g ) sinh μ 2 ( 1 - x ) + ν 2 ( 1 - R g ) cosh μ 2 ( 1 - x ) ] - 1 2 ν 2 F c 0 F c 0 + F d 0 exp [ - ( α 1 ¯ + α 2 ¯ ) ] ( ν 2 cosh μ 2 ( 1 - x ) - μ 2 + ν 2 α 2 α 2 2 - μ 2 2 { - α 2 exp [ α 2 ( 1 - x ) ] + μ 2 sinh μ 2 ( 1 - x ) + α 2 cosh μ 2 ( 1 - x ) } - ν 2 exp [ α 2 ( 1 - x ) ] ) .
φ 1 = W 1 above ( x ¯ ) - W 2 above ( x ¯ ) = 0 , φ 2 = U 1 above ( x ¯ ) - U 2 above ( x ¯ ) = 0 , φ 3 = W 1 below ( x ¯ ) - W 2 below ( x ¯ ) = 0 , φ 4 = U 1 below ( x ¯ ) - U 2 below ( x ¯ ) = 0 ,
K = j = 1 4 φ j 2 ,

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