Abstract

A quantitative treatment of the mechanism of phototropism in plants (directed growth caused by assymetrical illumination) implies the description of light propagation in both nonscattering and intensely scattering finite cylinders. We solve this problem for cylinders illuminated unilaterally by parallel and diffuse light applying the diffusion theory. As a first step, the complicated spatial distribution of the coherent intensity as caused by the jump of the refractive index and the nonzero curvature of the cylinder has been derived by means of differential geometry. This distribution (which is also the final solution for nonscattering objects) provides the source term in the diffusion equation (for isotropic scattering) for the turbid cylinder, which is then solved using a Green function.

© 1985 Optical Society of America

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References

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  1. H. Senger, The Blue Light Syndrome (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  2. D. S. Dennison, “Phototropism,” in Encyclopedia of Plant Physiology, A. Pirson, M. H. Zimmermann, eds. (Springer-Verlag, Berlin, 1979), Vol. VII, pp. 506–560.
  3. F. Lenci, G. Colombetti, eds., Photoreception and Sensory Transduction in Aneural Organisms (Plenum, New York, 1980).
    [CrossRef]
  4. W. Shropshire, “The lens effect and phototropism of phycomyces,” J. Gen. Physiol. 45, 949–958 (1962).
  5. A. J. Jesaitis, “Linear dichroism and orientation of the phycomyces photopigment,” J. Gen. Physiol. 63, 1–21 (1974).
  6. L. Jaffe, H. Etzold, “Orientation and locus of tropic photoreceptor molecules in spores of Botrytis and Osmunda,” J. Cell Biol. 13, 13–31 (1962).
  7. K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).
  8. E. Schäfer, L. Fukshansky, W. Shropshire, “Action spectroscopy of photoreversible pigment systems,” in Encyclopedia of Plant Physiology, W. Shropshire, H. Mohr, eds. (Springer-Verlag, Berlin, 1983), Vol. 16, pp. 39–68.
  9. L. Fukshansky, “Optical properties of plants,” in Plants and the Daylight Spectrum, H. Smith, ed. (Academic, New York, 1981), p. 21.
  10. L. Fukshansky, N. Kazarinova, “Extension of the Kubelka–Munk theory of light propagation in intensity scattering materials to fluorescent media,” J. Opt. Soc. Am. 70, 1101–1011 (1980).
    [CrossRef]
  11. M. G. Holmes, L. Fukshansky, “Phytochrome photoequilibria in green leaves under polychromatic radiation: a theoretical approach,” Plant Cell Environ. 2, 59–65 (1979).
    [CrossRef]
  12. M. Seyfried, E. Schäfer, L. Fukshansky, “Biological aspects of light distribution in scattering media,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 574–580 (1983).
  13. K. M. Hartmann, “Wirkungsspektroskopie,” in Biophysik, W. Hoppe, W. Lohmann, H. Mark, H. Ziegler, eds. (Springer-Verlag, Berlin, 1982), p. 122.
  14. E. D. Lipson, D. Presti, “Light induced absorbance changes in Phycomyces photomutants,” Photochem. Photobiol. 25, 203–208 (1977).
    [CrossRef]
  15. R. A. J. Groenhuis, H. A. Ferwerda, J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory,” Appl. Opt. 22, 2456–2462 (1983).
    [CrossRef] [PubMed]
  16. S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
    [PubMed]
  17. L. Reynolds, C. Johnson, A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt. 15, 2059–2067 (1976).
    [CrossRef] [PubMed]
  18. K. G. Privoznik, K. J. Daniel, F. P. Incropera, “Absorption, extinction and phase function measurements for algal suspensions of Chlorella pyrenoidosa,” J. Quant. Spectrosc. Radiat. Transfer 20, 345–352 (1978).
    [CrossRef]
  19. A. L. Crosbie, R. L. Dougherty, “Two-dimensional isotropic scattering in a semi-infinite cylindrical medium,” J. Quant. Spectrosc. Radiat. Transfer 20, 155–173 (1978), and references cited therein.
    [CrossRef]
  20. A. L. Crosbie, R. L. Dougherty, “Two-dimensional radiative transfer in a cylindrical geometry with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 25, 551–569 (1980).
    [CrossRef]
  21. M. Seyfried, doctoral dissertation (University of Freiburg, Freiburg, 1980) (unpublished).
  22. M. Seyfried, E. Schäfer, L. Fukshansky, “Correcting remission and transmission spectra of plant tissue measured in glass cuvettes: a technique,” Appl. Opt. 22, 492–496 (1983).
    [CrossRef] [PubMed]
  23. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, Chap. 9.
  24. A. Ishimaru, Y. Kuga, R. L.-T. Cheung, K. Shimizu, “Scattering and diffusion of a beam wave in randomly distributed scatterers,” J. Opt. Soc. Am. 73, 131–136 (1983).
    [CrossRef]
  25. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 41.
  26. The concept of wave fronts and caustics is discussed at length in O. N. Stavroudis, The Optics of Rays, Wave Fronts and Caustics (Academic, New York, 1972);M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).
  27. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 170.
  28. K. Strubecker, Differentialgeometrie I, Kurυentheorie (Gruyter, Berlin, 1964), p. 69.
  29. K. Strubecker, Differentialgeometrie I, Kurυentheorie (Gruyter, Berlin, 1964), p. 18.
  30. d/dx∫A(x)B(x)F(x,ξ)dξ=∫A(x)B(x)∂/∂xF(x,ξ)dξ+F[x,B(x)]dB(x)/dx−F[x,A(x)]dA(x)/dx.
  31. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 17.

1983 (4)

1981 (1)

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

1980 (2)

A. L. Crosbie, R. L. Dougherty, “Two-dimensional radiative transfer in a cylindrical geometry with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 25, 551–569 (1980).
[CrossRef]

L. Fukshansky, N. Kazarinova, “Extension of the Kubelka–Munk theory of light propagation in intensity scattering materials to fluorescent media,” J. Opt. Soc. Am. 70, 1101–1011 (1980).
[CrossRef]

1979 (1)

M. G. Holmes, L. Fukshansky, “Phytochrome photoequilibria in green leaves under polychromatic radiation: a theoretical approach,” Plant Cell Environ. 2, 59–65 (1979).
[CrossRef]

1978 (2)

K. G. Privoznik, K. J. Daniel, F. P. Incropera, “Absorption, extinction and phase function measurements for algal suspensions of Chlorella pyrenoidosa,” J. Quant. Spectrosc. Radiat. Transfer 20, 345–352 (1978).
[CrossRef]

A. L. Crosbie, R. L. Dougherty, “Two-dimensional isotropic scattering in a semi-infinite cylindrical medium,” J. Quant. Spectrosc. Radiat. Transfer 20, 155–173 (1978), and references cited therein.
[CrossRef]

1977 (1)

E. D. Lipson, D. Presti, “Light induced absorbance changes in Phycomyces photomutants,” Photochem. Photobiol. 25, 203–208 (1977).
[CrossRef]

1976 (1)

1974 (1)

A. J. Jesaitis, “Linear dichroism and orientation of the phycomyces photopigment,” J. Gen. Physiol. 63, 1–21 (1974).

1969 (1)

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

1962 (2)

W. Shropshire, “The lens effect and phototropism of phycomyces,” J. Gen. Physiol. 45, 949–958 (1962).

L. Jaffe, H. Etzold, “Orientation and locus of tropic photoreceptor molecules in spores of Botrytis and Osmunda,” J. Cell Biol. 13, 13–31 (1962).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 17.

Anderson, R. R.

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

Bergmann, K.

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 41.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 170.

Burke, P. V.

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

Cerdá-Olmedo, E.

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

Cheung, R. L.-T.

Crosbie, A. L.

A. L. Crosbie, R. L. Dougherty, “Two-dimensional radiative transfer in a cylindrical geometry with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 25, 551–569 (1980).
[CrossRef]

A. L. Crosbie, R. L. Dougherty, “Two-dimensional isotropic scattering in a semi-infinite cylindrical medium,” J. Quant. Spectrosc. Radiat. Transfer 20, 155–173 (1978), and references cited therein.
[CrossRef]

Daniel, K. J.

K. G. Privoznik, K. J. Daniel, F. P. Incropera, “Absorption, extinction and phase function measurements for algal suspensions of Chlorella pyrenoidosa,” J. Quant. Spectrosc. Radiat. Transfer 20, 345–352 (1978).
[CrossRef]

David, C. N.

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

Delbrück, M.

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

Dennison, D. S.

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

D. S. Dennison, “Phototropism,” in Encyclopedia of Plant Physiology, A. Pirson, M. H. Zimmermann, eds. (Springer-Verlag, Berlin, 1979), Vol. VII, pp. 506–560.

Dougherty, R. L.

A. L. Crosbie, R. L. Dougherty, “Two-dimensional radiative transfer in a cylindrical geometry with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 25, 551–569 (1980).
[CrossRef]

A. L. Crosbie, R. L. Dougherty, “Two-dimensional isotropic scattering in a semi-infinite cylindrical medium,” J. Quant. Spectrosc. Radiat. Transfer 20, 155–173 (1978), and references cited therein.
[CrossRef]

Etzold, H.

L. Jaffe, H. Etzold, “Orientation and locus of tropic photoreceptor molecules in spores of Botrytis and Osmunda,” J. Cell Biol. 13, 13–31 (1962).

Ferwerda, H. A.

Foster, K. W.

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

Fukshansky, L.

M. Seyfried, E. Schäfer, L. Fukshansky, “Biological aspects of light distribution in scattering media,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 574–580 (1983).

M. Seyfried, E. Schäfer, L. Fukshansky, “Correcting remission and transmission spectra of plant tissue measured in glass cuvettes: a technique,” Appl. Opt. 22, 492–496 (1983).
[CrossRef] [PubMed]

L. Fukshansky, N. Kazarinova, “Extension of the Kubelka–Munk theory of light propagation in intensity scattering materials to fluorescent media,” J. Opt. Soc. Am. 70, 1101–1011 (1980).
[CrossRef]

M. G. Holmes, L. Fukshansky, “Phytochrome photoequilibria in green leaves under polychromatic radiation: a theoretical approach,” Plant Cell Environ. 2, 59–65 (1979).
[CrossRef]

E. Schäfer, L. Fukshansky, W. Shropshire, “Action spectroscopy of photoreversible pigment systems,” in Encyclopedia of Plant Physiology, W. Shropshire, H. Mohr, eds. (Springer-Verlag, Berlin, 1983), Vol. 16, pp. 39–68.

L. Fukshansky, “Optical properties of plants,” in Plants and the Daylight Spectrum, H. Smith, ed. (Academic, New York, 1981), p. 21.

Goodell, E. W.

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

Groenhuis, R. A. J.

Hartmann, K. M.

K. M. Hartmann, “Wirkungsspektroskopie,” in Biophysik, W. Hoppe, W. Lohmann, H. Mark, H. Ziegler, eds. (Springer-Verlag, Berlin, 1982), p. 122.

Heisenberg, M.

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

Holmes, M. G.

M. G. Holmes, L. Fukshansky, “Phytochrome photoequilibria in green leaves under polychromatic radiation: a theoretical approach,” Plant Cell Environ. 2, 59–65 (1979).
[CrossRef]

Incropera, F. P.

K. G. Privoznik, K. J. Daniel, F. P. Incropera, “Absorption, extinction and phase function measurements for algal suspensions of Chlorella pyrenoidosa,” J. Quant. Spectrosc. Radiat. Transfer 20, 345–352 (1978).
[CrossRef]

Ishimaru, A.

Jaffe, L.

L. Jaffe, H. Etzold, “Orientation and locus of tropic photoreceptor molecules in spores of Botrytis and Osmunda,” J. Cell Biol. 13, 13–31 (1962).

Jesaitis, A. J.

A. J. Jesaitis, “Linear dichroism and orientation of the phycomyces photopigment,” J. Gen. Physiol. 63, 1–21 (1974).

Johnson, C.

Kazarinova, N.

Kuga, Y.

Lipson, E. D.

E. D. Lipson, D. Presti, “Light induced absorbance changes in Phycomyces photomutants,” Photochem. Photobiol. 25, 203–208 (1977).
[CrossRef]

Meissner, G.

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

Parrish, J. A.

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

Presti, D.

E. D. Lipson, D. Presti, “Light induced absorbance changes in Phycomyces photomutants,” Photochem. Photobiol. 25, 203–208 (1977).
[CrossRef]

Privoznik, K. G.

K. G. Privoznik, K. J. Daniel, F. P. Incropera, “Absorption, extinction and phase function measurements for algal suspensions of Chlorella pyrenoidosa,” J. Quant. Spectrosc. Radiat. Transfer 20, 345–352 (1978).
[CrossRef]

Reynolds, L.

Schäfer, E.

M. Seyfried, E. Schäfer, L. Fukshansky, “Correcting remission and transmission spectra of plant tissue measured in glass cuvettes: a technique,” Appl. Opt. 22, 492–496 (1983).
[CrossRef] [PubMed]

M. Seyfried, E. Schäfer, L. Fukshansky, “Biological aspects of light distribution in scattering media,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 574–580 (1983).

E. Schäfer, L. Fukshansky, W. Shropshire, “Action spectroscopy of photoreversible pigment systems,” in Encyclopedia of Plant Physiology, W. Shropshire, H. Mohr, eds. (Springer-Verlag, Berlin, 1983), Vol. 16, pp. 39–68.

Senger, H.

H. Senger, The Blue Light Syndrome (Springer-Verlag, Berlin, 1980).
[CrossRef]

Seyfried, M.

M. Seyfried, E. Schäfer, L. Fukshansky, “Biological aspects of light distribution in scattering media,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 574–580 (1983).

M. Seyfried, E. Schäfer, L. Fukshansky, “Correcting remission and transmission spectra of plant tissue measured in glass cuvettes: a technique,” Appl. Opt. 22, 492–496 (1983).
[CrossRef] [PubMed]

M. Seyfried, doctoral dissertation (University of Freiburg, Freiburg, 1980) (unpublished).

Shimizu, K.

Shropshire, W.

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

W. Shropshire, “The lens effect and phototropism of phycomyces,” J. Gen. Physiol. 45, 949–958 (1962).

E. Schäfer, L. Fukshansky, W. Shropshire, “Action spectroscopy of photoreversible pigment systems,” in Encyclopedia of Plant Physiology, W. Shropshire, H. Mohr, eds. (Springer-Verlag, Berlin, 1983), Vol. 16, pp. 39–68.

Stavroudis, O. N.

The concept of wave fronts and caustics is discussed at length in O. N. Stavroudis, The Optics of Rays, Wave Fronts and Caustics (Academic, New York, 1972);M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 17.

Strubecker, K.

K. Strubecker, Differentialgeometrie I, Kurυentheorie (Gruyter, Berlin, 1964), p. 69.

K. Strubecker, Differentialgeometrie I, Kurυentheorie (Gruyter, Berlin, 1964), p. 18.

Ten Bosch, J. J.

Wan, S.

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 170.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 41.

Zalokar, M.

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

Appl. Opt. (3)

Bact. Rev. (1)

K. Bergmann, P. V. Burke, E. Cerdá-Olmedo, C. N. David, M. Delbrück, K. W. Foster, E. W. Goodell, M. Heisenberg, G. Meissner, M. Zalokar, D. S. Dennison, W. Shropshire, “Phycomyces,” Bact. Rev. 33, 99–157 (1969).

J. Cell Biol. (1)

L. Jaffe, H. Etzold, “Orientation and locus of tropic photoreceptor molecules in spores of Botrytis and Osmunda,” J. Cell Biol. 13, 13–31 (1962).

J. Gen. Physiol. (2)

W. Shropshire, “The lens effect and phototropism of phycomyces,” J. Gen. Physiol. 45, 949–958 (1962).

A. J. Jesaitis, “Linear dichroism and orientation of the phycomyces photopigment,” J. Gen. Physiol. 63, 1–21 (1974).

J. Opt. Soc. Am. (2)

J. Quant. Spectrosc. Radiat. Transfer (3)

K. G. Privoznik, K. J. Daniel, F. P. Incropera, “Absorption, extinction and phase function measurements for algal suspensions of Chlorella pyrenoidosa,” J. Quant. Spectrosc. Radiat. Transfer 20, 345–352 (1978).
[CrossRef]

A. L. Crosbie, R. L. Dougherty, “Two-dimensional isotropic scattering in a semi-infinite cylindrical medium,” J. Quant. Spectrosc. Radiat. Transfer 20, 155–173 (1978), and references cited therein.
[CrossRef]

A. L. Crosbie, R. L. Dougherty, “Two-dimensional radiative transfer in a cylindrical geometry with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 25, 551–569 (1980).
[CrossRef]

Photochem. Photobiol. (2)

E. D. Lipson, D. Presti, “Light induced absorbance changes in Phycomyces photomutants,” Photochem. Photobiol. 25, 203–208 (1977).
[CrossRef]

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

Plant Cell Environ. (1)

M. G. Holmes, L. Fukshansky, “Phytochrome photoequilibria in green leaves under polychromatic radiation: a theoretical approach,” Plant Cell Environ. 2, 59–65 (1979).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

M. Seyfried, E. Schäfer, L. Fukshansky, “Biological aspects of light distribution in scattering media,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 574–580 (1983).

Other (15)

K. M. Hartmann, “Wirkungsspektroskopie,” in Biophysik, W. Hoppe, W. Lohmann, H. Mark, H. Ziegler, eds. (Springer-Verlag, Berlin, 1982), p. 122.

H. Senger, The Blue Light Syndrome (Springer-Verlag, Berlin, 1980).
[CrossRef]

D. S. Dennison, “Phototropism,” in Encyclopedia of Plant Physiology, A. Pirson, M. H. Zimmermann, eds. (Springer-Verlag, Berlin, 1979), Vol. VII, pp. 506–560.

F. Lenci, G. Colombetti, eds., Photoreception and Sensory Transduction in Aneural Organisms (Plenum, New York, 1980).
[CrossRef]

E. Schäfer, L. Fukshansky, W. Shropshire, “Action spectroscopy of photoreversible pigment systems,” in Encyclopedia of Plant Physiology, W. Shropshire, H. Mohr, eds. (Springer-Verlag, Berlin, 1983), Vol. 16, pp. 39–68.

L. Fukshansky, “Optical properties of plants,” in Plants and the Daylight Spectrum, H. Smith, ed. (Academic, New York, 1981), p. 21.

M. Seyfried, doctoral dissertation (University of Freiburg, Freiburg, 1980) (unpublished).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, Chap. 9.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 41.

The concept of wave fronts and caustics is discussed at length in O. N. Stavroudis, The Optics of Rays, Wave Fronts and Caustics (Academic, New York, 1972);M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 170.

K. Strubecker, Differentialgeometrie I, Kurυentheorie (Gruyter, Berlin, 1964), p. 69.

K. Strubecker, Differentialgeometrie I, Kurυentheorie (Gruyter, Berlin, 1964), p. 18.

d/dx∫A(x)B(x)F(x,ξ)dξ=∫A(x)B(x)∂/∂xF(x,ξ)dξ+F[x,B(x)]dB(x)/dx−F[x,A(x)]dA(x)/dx.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 17.

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Figures (4)

Fig. 1
Fig. 1

Coordinate system and angles that are used in the derivation of the equations for the caustic and the wave fronts.

Fig. 2
Fig. 2

Normal section of the caustic for a cylindrical medium with mc = 1.42 and the geometrical meaning of the parameters ξ and η.

Fig. 3
Fig. 3

The wave fronts corresponding to some important values of the parameter s that define the integration limits in Eq. (38): 1, s = −a[m + (1 − m2)1/2]; 2, s = −a[(l − m2)1/2], 3, s = a ( 1 m 2 ) 1 / 2 ( 3 1 ); 4, s = a[2 − m − (1 − m2)1/2]; 5, s = a(1 − m2)1/2.

Fig. 4
Fig. 4

Coordinate system and angles that are used in the derivation of the source term in the case of diffuse illumination.

Equations (95)

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I ( r , ŝ ) = I c ( r , ŝ ) + I d ( r , ŝ ) .
I d ( r , ŝ ) = U d ( r ) + 3 / 4 π F d ( r ) ŝ
U d ( r ) = 1 / 4 π 4 π I d ( r , ŝ ) d ω
( 2 ϰ 2 ) U d ( r ) = α U c ( r ) ,
U d ( r s ) μ / n U d ( r s ) = 0 .
ϰ 2 3 ρ 2 σ a σ t , α 3 ρ 2 σ s σ t , μ 2 / ( 3 ρ σ t ) .
d I c ( r , ŝ ) / d s = ρ σ t I c ( r , ŝ ) .
U d ( r ) = 3 / 4 π F d ( r )
I d ( r , ŝ ) = U d ( r ) + 3 / 4 π F d ( r ) ŝ .
F z + + F z = 0 ,
U d ( r ) = α U c ( r ) G ( r , r ) d A ,
G ( r , φ ; r , φ ) = ( 1 / 2 π ) m = 0 cos m ( φ φ ) { Y m ( ϰ r ) I m ( ϰ r ) , r < r Y m ( ϰ r ) I m ( ϰ r ) , r > r .
Y m ( ϰ r ) = Z m ( ϰ a ) I m ( ϰ r ) + K m ( ϰ r ) ,
Z m ( ϰ a ) = [ K m ( ϰ r ) + μ K m ( ϰ r ) ] / [ I m ( ϰ r ) + μ I m ( ϰ r ) ] r = a ,
x ( φ ¯ ) = a cos φ ¯ , y ( φ ) = a sin φ ¯ .
m 0 sin γ i = m c sin γ 0 ,
γ i = π / 2 φ ¯ , 0 φ ¯ π / 2 , γ i = φ ¯ π / 2 , π / 2 φ ¯ π ,
| m cos φ ¯ | = sin γ 0 , 0 φ ¯ π ,
ŝ I c ( r , ŝ ) = ρ σ t I c ( r , ŝ ) ,
{ cos ( Ω φ ) / r + [ sin ( Ω φ ) ] / r / φ } I c ( r , ŝ ) = ρ σ t I c ( r , ŝ ) ,
I c ( r , φ ; φ ¯ ) = I t exp [ τ ( r , φ ; φ ¯ ) ] ,
τ ( r , φ ; φ ¯ ) = ρ σ t [ a 2 + r 2 2 arccos ( φ ¯ φ ) ] 1 / 2 .
I t = A ( φ ¯ ) I 0 ,
A ( φ ¯ ) = cos γ 0 / m sin φ ¯ | T ( φ ¯ ) | 2 .
T = 2 m cos φ ¯ sin φ ¯ / sin Ω cos Ω ¯
T = 2 m cos φ ¯ sin φ ¯ / cos Ω ¯
| T ( φ ¯ ) | 2 = 1 / 2 ( | T ( φ ¯ ) | 2 + | T ( φ ¯ ) | 2 ) .
U c ( r ) = 1 / 4 π 4 π I c ( r , ŝ ) d ω ,
x / ξ + y / η = 1 ,
ξ = a sin γ 0 / sin Ω , η = a sin γ 0 / cos Ω ,
F ( x , y ; φ ¯ ) = x sin Ω y cos Ω a sin γ 0 = 0 .
F ( x , y ; φ ¯ ) = 0 , F ( x , y ; φ ¯ ) / φ ¯ = 0 .
x c ( φ ¯ ) = a m 2 cos 2 φ ¯ ,
y c ( φ ¯ ) = a m ( m cos 2 φ ¯ sin Ω 1 ) / Ψ ( φ ¯ ) ,
φ ¯ 1 / 2 = arccos [ ± ( 4 / 3 1 / 3 m 2 ) 1 / 2 ] ,
φ ¯ 3 / 4 = arccos ( ± 1 / m ) ,
0 φ ¯ arccos [ ± ( 4 / 3 1 / 3 m 2 ) 1 / 2 ] .
x c ( π φ ¯ ) = x c ( φ ¯ ) , y c ( π φ ¯ ) = y c ( φ ¯ ) ,
x w ( φ ¯ ) = x c ( φ ¯ ) [ s c ( φ ¯ ) s ] x c ( φ ¯ ) / ( x c 2 + y c 2 ) 1 / 2 ,
y w ( φ ¯ ) = y c ( φ ¯ ) [ s c ( φ ¯ ) s ] y c ( φ ¯ ) / ( x c 2 + y c 2 ) 1 / 2 ,
s c ( φ ¯ ) = 0 φ ¯ [ x c 2 ( ) + y c 2 ( ) ] 1 / 2 d ,
s c ( φ ¯ ) = 3 a m 2 0 φ ¯ cos 2 sin / cos Ω ( ) d ,
s c ( φ ¯ ) = a / 4 [ ( cos 2 Ω ¯ + 3 cos 2 Ω ) / cos Ω cos φ ¯ ] a ( 1 m 2 ) 1 / 2
r w ( φ ¯ , s ) = a ( m 2 cos 2 φ ¯ + Φ 2 ) 1 / 2 ,
φ w ( φ ¯ , s ) = arctan { tan Ω 2 m / [ ( Γ + Φ 2 ) cos Ω ] } ,
Φ Ψ η / a ,
η s + a ( 1 m 2 ) 1 / 2 ,
Γ 1 + m 2 ( η / a ) 2 .
s min = a [ m + ( 1 m 2 ) 1 / 2 ] , s max = a ( 1 m 2 ) 1 / 2 .
a ( 1 m 2 ) 1 / 2 s a ( 1 m 2 ) 1 / 2 ( 3 1 )
φ ¯ ± = arcsin ( { η / a ± 2 [ ( η / a ) 2 3 ( 1 m 2 ) ] 1 / 2 } / 3 m ) ,
φ ¯ 0 = arcsin ( η / a m ) .
s min s s 2 a ( 1 m 2 ) 1 / 2 , φ ¯ 0 φ ¯ π / 2 , s 2 s 0 , 0 φ ¯ π / 2 , 0 s s 3 a ( 1 m 2 ) 1 / 2 ( 3 1 ) , 0 φ ¯ φ ¯ s ( branch a ) , φ ¯ s φ ¯ π / 2 ( branch b ) , s 3 s s 4 a [ 2 m ( 1 m 2 ) 1 / 2 ] , 0 φ ¯ φ ¯ ( branch a ) , φ ¯ + φ ¯ π / 2 ( branch b ) , s 4 s s max , 0 φ ¯ φ ¯ ( branch a ) .
a [ m sin φ ¯ + ( 1 m 2 ) 1 / 2 ] s r a [ 2 cos γ 0 m sin φ ¯ ( 1 m 2 ) 1 / 2 ] .
U c ( r , φ ¯ ) = s min s max φ ¯ min ( s ) φ ¯ max ( s ) I t ( φ ¯ ) r 1 × exp [ τ ( r , φ ; φ ¯ ) ] δ ( r r w ) δ ( φ φ w ) d s d φ ¯ ,
U d ( r , φ ) = α A s min s max φ ¯ min ( s ) φ ¯ max ( s ) I t ( φ ¯ ) r 1 × exp [ τ ( r , φ ; φ ¯ ) ] G ( r , φ ; r , φ ) × δ ( r r w ) δ ( φ φ w ) r d r d φ d φ ¯ d s .
U d ( r , φ ) = α s min s max φ ¯ min ( s ) φ ¯ max ( s ) × H ( r w , φ w ; φ ¯ ) G ( r , φ ; r w φ w ) d s d φ ¯ ,
H ( r w , φ w ; φ ¯ ) I 0 A ( φ ¯ ) exp [ τ ( r w , φ w ; φ ¯ ) ] .
U d ( r , φ ) = α / 2 π m = 0 { I m ( ϰ r ) × ( s min s ( r ) φ ¯ 0 π / 2 + s ( r ) s ( r ) φ ¯ min ( s ) φ ¯ ( r , s ) + s ( r ) s max φ ¯ min ( s ) φ ¯ max ( s ) × [ H ( r w , φ w ; φ ¯ ) cos m ( φ φ w ) Y m ( ϰ r w ) d s d φ ¯ ] ) + Y m ( ϰ r ) s ( r ) s ( r ) φ ¯ ( r , s ) φ ¯ max ( s ) H ( r w , φ w ; φ ¯ ) × cos m ( φ φ w ) I m ( ϰ r ) d s d φ ¯ } .
U d ( r , φ ) = α / 2 π m = 0 [ I m ( ϰ r ) × H ( r w , φ w ; φ ¯ ) f m ( φ , φ w ) Y m ( ϰ r w ) d s d φ ¯ + Y m ( ϰ r ) × s ( r ) s ( r ) φ ¯ ( r , s ) φ ¯ max ( s ) H ( r w , φ w ; φ ) f m ( φ , φ w ) I m ( ϰ r w ) d s d φ ¯ ] ,
f m ( φ , φ w ) = 2 cos [ m ( 2 φ π ) / 2 ] cos [ m ( π 2 φ w ) / 2 ] .
F r ( r , φ ) = 3 α ρ σ t / 8 π 2 m = 0 [ I m ( ϰ r ) / r × H ( r w , φ w ; φ ¯ ) f m ( φ , φ w ) Y m ( ϰ r w ) d s d φ ¯ + Y m ( ϰ r ) / r × s ( r ) s ( r ) φ ¯ ( r , s ) φ ¯ max ( s ) f m ( φ , φ w ) H ( r w , φ w ; φ ¯ ) I m ( ϰ r w ) d s d φ ¯ ] F φ ( r , φ ) = 3 α ρ σ t / 8 π 2 r m = 0 m [ I m ( ϰ r ) × H ( r w , φ w ; φ ¯ ) f ¯ m ( φ , φ w ) Y m ( ϰ r w ) d s d φ ¯ + Y m ( ϰ r ) × s ( r ) s ( r ) φ ¯ ( r , s ) φ ¯ max ( s ) f ¯ m ( φ , φ w ) H ( r w , φ w ; φ ¯ ) I m ( ϰ r ) d s d φ ¯ ] ,
f ¯ m ( φ , φ w ) = 2 sin [ m ( 2 φ π ) / 2 ] cos [ m ( π 2 φ w ) / 2 ] .
γ 0 ( φ ¯ ) = arctan { [ r sin ( φ φ ¯ ) ] / a r cos ( φ φ ¯ ) } ,
γ i ( φ ¯ ) = arcsin { [ r sin ( φ φ ¯ ) / m s } ,
s = [ a 2 + r 2 2 arccos ( φ ¯ φ ) ] 1 / 2 = τ ( r , φ : φ ¯ ) / ρ σ t .
I [ β ( φ ¯ ) ] = I 0 h [ β ( φ ¯ ) ] .
β = φ ¯ γ i ,
U c ( r , φ ) = I 0 0 2 π A ( φ ¯ ) h [ β ( φ ¯ ) ] exp [ τ ( r , φ ; φ ¯ ) ] d φ ¯ .
U c ( r , φ ) = I 0 0 2 π A ( φ ¯ ) exp [ τ ( r , φ ; φ ¯ ) ] d φ ¯ .
U d ( r , φ ) = α 0 2 π 0 2 π 0 a A ( φ ¯ ) h [ β ( φ ¯ ) ] × exp [ τ ( r , φ ; φ ¯ ) ] G ( r , φ ; r φ ) r d r d φ d φ ¯ ,
U d ( r , φ ) = α I 0 / 2 π m = 0 { Y m ( ϰ r ) 0 2 π 0 2 π 0 r A ( φ ¯ ) h [ β ( φ ¯ ) ] × exp [ τ ( r , φ ; φ ¯ ) ] × cos m ( φ φ ) I m ( ϰ r ) r d r d φ d φ ¯ + I m ( ϰ r ) 0 2 π 0 2 π r a A ( φ ¯ ) h [ β ( φ ¯ ) ] × exp [ τ ( r , φ ; φ ¯ ) ] cos m ( φ φ ) × Y m ( ϰ r ) r d r d φ d φ ¯ } .
F r ( r , φ ) = 3 α ρ σ t I 0 / 8 π 2 m = 0 { Y m ( ρ ) / r 0 2 π 0 2 π 0 r × A ( φ ¯ ) h [ β ( φ ¯ ) ] exp [ τ ( r , φ ; φ ¯ ) ] × cos m ( φ φ ) I m ( ϰ r ) r d r d φ d φ ¯ + I m ( ϰ r ) / r 0 2 π 0 2 π r a A ( φ ¯ ) h [ β ( φ ¯ ) ] × exp [ τ ( r , φ ; φ ¯ ) ] × cos m ( φ φ ) Y m ( ϰ r ) r d r d φ d φ ¯ } , F φ ( r , φ ) = 3 α ρ σ t I 0 / 8 π 2 r m = 0 m { Y m ( ϰ r ) 0 2 π 0 2 π r a × A ( φ ¯ ) h [ β ( φ ¯ ) ] exp [ τ ( r , φ ; φ ¯ ) ] × sin m ( φ φ ) I m ( ϰ r ) r d r d φ d φ ¯ + I m ( ϰ r ) 0 2 π 0 2 π 0 r A ( φ ¯ ) h [ β ( φ ¯ ) ] × exp [ τ ( r , φ ; φ ¯ ) ] × sin m ( φ φ ) Y m ( ϰ r ) r d r d φ d φ ¯ } ,
( 2 ϰ 2 ) G ( r , r ) = δ ( r r )
G ( r , r s ) μ / n G ( r , r s ) = 0 ,
G ( r , φ ; r , φ ) = 1 / 2 π m = exp [ Im ( φ φ ) ] g m ( r , r ) ,
L ( r ) g m ( r , r ) = δ ( r r ) ,
L ( r ) = d / d r ( r d / d r ) r ( m 2 / r 2 + ϰ 2 ) .
[ g m ( r , r ) + μ / r g m ( r , r ) ] r = a = 0 ,
L ( r ) g m ( r , r ) = 0
u 1 ( ϰ r ) = A I m ( ϰ r ) + B K m ( ϰ r ) , r < r , u 2 ( ϰ r ) = C I m ( ϰ r ) + D K m ( ϰ r ) , r > r ,
C = Z m ( ϰ a ) D ,
Z m ( ϰ a ) = [ K m ( ϰ r ) + μ K m ( ϰ r ) ] / [ I m ( ϰ r ) + μ I m ( ϰ r ) ] r = a
D = I m ( ϰ r ) A = Z m ( ϰ a ) I m ( ϰ r ) + K m ( ϰ r )
G ( r , φ ; r , φ ) = 1 / 2 π m = exp [ Im ( φ φ ) ] { Y m ( ϰ r ) I m ( ϰ r ) , r < r I m ( ϰ r ) Y m ( ϰ r ) , r > r ,
Y m ( ϰ r ) = Z m ( ϰ a ) I m ( ϰ r ) + K m ( ϰ r ) .
G ( r , φ ; r , φ ) = 1 / 2 π m = 0 cos m ( φ φ ) { Y m ( ϰ r ) I m ( ϰ r ) , r < r I m ( ϰ r ) Y m ( ϰ r ) , r > r .
s ( r ) = a [ 1 m ( 1 m 2 ) 1 / 2 ] r ,
s ( r ) = a [ 1 m ( 1 m 2 ) 1 / 2 ] + r .
m 2 cos φ ¯ 2 + ( cos γ 0 m sin φ ¯ ) 2 2 ( η / a ) ( cos γ 0 m sin φ ¯ ) + ( η / a ) 2 = δ 2 ,
μ = m sin φ ¯ + η ( s ) / a
μ 4 ( 8 / 3 ) ( η / a ) μ 3 + ( 2 / 3 ) μ 2 [ 2 ( 1 m 2 ) + ( η / a ) 2 + k 2 ] k 4 / 3 = 0 ,
φ ¯ ( r , s ) = arcsin { 1 / m [ μ η ( s ) / a ] } .
0 1 / m [ μ η ( s ) / a ] 1 .
d/dxA(x)B(x)F(x,ξ)dξ=A(x)B(x)/xF(x,ξ)dξ+F[x,B(x)]dB(x)/dxF[x,A(x)]dA(x)/dx.

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