Abstract

In this paper we suggest a method for calculating light gradients in scattering and absorbing media. The method is based on the Kubelka-Munk theory and involves computational modeling of light fluxes in a multilayered object, when every layer satisfies the prerequisites of the Kubelka-Munk theory. The model also includes specular reflection that may contribute strongly to internal photon fluence rates for diffuse light. To illustrate the possible effects of light gradients, a cotyledon of Cucurbita pepo is described in terms of this model. It is argued that a number of results in in vivo spectroscopy cannot be correctly interpreted unless light gradients or optics in general are included in the discussion, i.e., the light flux at the site of the pigment has to be known. To outline the difficulties involved some methods of measuring light gradients or internal photon fluence rates are critically considered.

© 1983 Optical Society of America

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References

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  1. P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).
  2. G. Kortüm, Reflectance Spectroscopy (Springer, Berlin, 1969).
    [CrossRef]
  3. L. Fukshansky, N. Kazarinova, J. Opt. Soc. Am. 70, 1101 (1980).
    [CrossRef]
  4. J. W. Ryde, B. S. Cooper, Proc. R. Soc. London Ser. A 131, 464 (1931).
    [CrossRef]
  5. S. Q. Duntley, J. Opt. Soc. Am. 32, 61 (1942).
    [CrossRef]
  6. H. G. Völz, Sixth FATIPEC Congress 1962 (Verlag Chemie, Weinheim/Bergstrasse, 1962), p. 62.
  7. H. G. Völz, Seventh FATIPEC Congress 1964 (Verlag Chemie, Weinheim/Bergstrasse, 1964), p. 194.
  8. P. S. Mudgett, L. W. Richards, Appl. Opt. 10, 1485 (1971).
    [CrossRef] [PubMed]
  9. A. L. Lathrop, J. Opt. Soc. Am. 55, 1097 (1965).
    [CrossRef]
  10. E. Schäfer, L. Fukshansky, W. Shropshire, in Encyclopedia of Plant PhysiologyW. Shropshire, H. Mohr, Eds. (Springer, Berlin, 1983).
  11. K. M. Hartmann, in Biophysik, W. Hoppe, W. Lohmann, H. Markl, H. Ziegler, Eds. (Springer, Berlin, 1982), p. 122.
  12. H. R. Gordon, I. B. Brown, Appl. Opt. 13, 2153 (1974).
    [CrossRef] [PubMed]
  13. H. D. L. Spence, R. M. Campell, J. Chrystal, J. Ecol. 61, 317 (1973).
    [CrossRef]
  14. J. Leverenz, P. G. Jarvis, J. Appl. Ecol. 16, 919 (1979).
    [CrossRef]
  15. J. B. Dawson, D. J. Barker, D. J. Ellis, E. Grassam, J. A. Cotterill, G. W. Fisher, J. W. Feather, Phys. Med. Biol. 25, 695 (1980).
    [CrossRef] [PubMed]
  16. G. S. Stokes, Proc. R. Soc. London 11, 545 (1862).
  17. P. Kubelka, J. Opt. Soc. Am. 44, 330 (1954).
    [CrossRef]
  18. A. Seyboldt, A. Weisweiler, Bot. Arch. 44, 456 (1943).
  19. H. W. Gausman, Photogramm. Eng. 40, 183 (1974).
  20. W. H. Foster, E. I. Stearns, J. Opt. Soc. Am. 60, 61 (1970).
  21. S. E. Orchard, J. Opt. Soc. Am. 59, 1584 (1969).
    [CrossRef]
  22. M. Seyfried, L. Fukshansky, E. Schäfer, Appl. Opt. 22, 492 (1983).
    [CrossRef] [PubMed]
  23. H. Smith, Phytochrome and Photomorphogenesis (McGraw-Hill, New York1975).
  24. H. W. Gausman, W. A. Allen, D. E. Escobar, Appl. Opt. 13, 109 (1974).
    [CrossRef] [PubMed]
  25. S. W. Thome, J. T. Duniec, J. A. Lee, Photobiochem. Photobiophys. 1, 161 (1980).
  26. W. A. Allen, H. W. Gausman, A. J. Richardson, J. Opt. Soc. Am. 60, 542 (1970).
    [CrossRef]
  27. R. Willstatter, A. Stoll, Untersuchungen ueber die Assimilation der Kohlensäure, Berlin (1918).
  28. W. Schmidt, D. Marmé, P. Quail, E. Schäfer, Planta 111, 329 (1973).
    [CrossRef]
  29. L. Fukshansky, J. Math. Biol. 6, 177 (1978).
    [CrossRef]
  30. W. L. Butler, J. Opt. Soc. Am. 52, 292 (1962).
    [CrossRef]

1983 (1)

1980 (3)

J. B. Dawson, D. J. Barker, D. J. Ellis, E. Grassam, J. A. Cotterill, G. W. Fisher, J. W. Feather, Phys. Med. Biol. 25, 695 (1980).
[CrossRef] [PubMed]

L. Fukshansky, N. Kazarinova, J. Opt. Soc. Am. 70, 1101 (1980).
[CrossRef]

S. W. Thome, J. T. Duniec, J. A. Lee, Photobiochem. Photobiophys. 1, 161 (1980).

1979 (1)

J. Leverenz, P. G. Jarvis, J. Appl. Ecol. 16, 919 (1979).
[CrossRef]

1978 (1)

L. Fukshansky, J. Math. Biol. 6, 177 (1978).
[CrossRef]

1974 (3)

1973 (2)

H. D. L. Spence, R. M. Campell, J. Chrystal, J. Ecol. 61, 317 (1973).
[CrossRef]

W. Schmidt, D. Marmé, P. Quail, E. Schäfer, Planta 111, 329 (1973).
[CrossRef]

1971 (1)

1970 (2)

W. H. Foster, E. I. Stearns, J. Opt. Soc. Am. 60, 61 (1970).

W. A. Allen, H. W. Gausman, A. J. Richardson, J. Opt. Soc. Am. 60, 542 (1970).
[CrossRef]

1969 (1)

1965 (1)

1962 (1)

1954 (1)

1943 (1)

A. Seyboldt, A. Weisweiler, Bot. Arch. 44, 456 (1943).

1942 (1)

1931 (2)

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

J. W. Ryde, B. S. Cooper, Proc. R. Soc. London Ser. A 131, 464 (1931).
[CrossRef]

1862 (1)

G. S. Stokes, Proc. R. Soc. London 11, 545 (1862).

Allen, W. A.

Barker, D. J.

J. B. Dawson, D. J. Barker, D. J. Ellis, E. Grassam, J. A. Cotterill, G. W. Fisher, J. W. Feather, Phys. Med. Biol. 25, 695 (1980).
[CrossRef] [PubMed]

Brown, I. B.

Butler, W. L.

Campell, R. M.

H. D. L. Spence, R. M. Campell, J. Chrystal, J. Ecol. 61, 317 (1973).
[CrossRef]

Chrystal, J.

H. D. L. Spence, R. M. Campell, J. Chrystal, J. Ecol. 61, 317 (1973).
[CrossRef]

Cooper, B. S.

J. W. Ryde, B. S. Cooper, Proc. R. Soc. London Ser. A 131, 464 (1931).
[CrossRef]

Cotterill, J. A.

J. B. Dawson, D. J. Barker, D. J. Ellis, E. Grassam, J. A. Cotterill, G. W. Fisher, J. W. Feather, Phys. Med. Biol. 25, 695 (1980).
[CrossRef] [PubMed]

Dawson, J. B.

J. B. Dawson, D. J. Barker, D. J. Ellis, E. Grassam, J. A. Cotterill, G. W. Fisher, J. W. Feather, Phys. Med. Biol. 25, 695 (1980).
[CrossRef] [PubMed]

Duniec, J. T.

S. W. Thome, J. T. Duniec, J. A. Lee, Photobiochem. Photobiophys. 1, 161 (1980).

Duntley, S. Q.

Ellis, D. J.

J. B. Dawson, D. J. Barker, D. J. Ellis, E. Grassam, J. A. Cotterill, G. W. Fisher, J. W. Feather, Phys. Med. Biol. 25, 695 (1980).
[CrossRef] [PubMed]

Escobar, D. E.

Feather, J. W.

J. B. Dawson, D. J. Barker, D. J. Ellis, E. Grassam, J. A. Cotterill, G. W. Fisher, J. W. Feather, Phys. Med. Biol. 25, 695 (1980).
[CrossRef] [PubMed]

Fisher, G. W.

J. B. Dawson, D. J. Barker, D. J. Ellis, E. Grassam, J. A. Cotterill, G. W. Fisher, J. W. Feather, Phys. Med. Biol. 25, 695 (1980).
[CrossRef] [PubMed]

Foster, W. H.

W. H. Foster, E. I. Stearns, J. Opt. Soc. Am. 60, 61 (1970).

Fukshansky, L.

M. Seyfried, L. Fukshansky, E. Schäfer, Appl. Opt. 22, 492 (1983).
[CrossRef] [PubMed]

L. Fukshansky, N. Kazarinova, J. Opt. Soc. Am. 70, 1101 (1980).
[CrossRef]

L. Fukshansky, J. Math. Biol. 6, 177 (1978).
[CrossRef]

E. Schäfer, L. Fukshansky, W. Shropshire, in Encyclopedia of Plant PhysiologyW. Shropshire, H. Mohr, Eds. (Springer, Berlin, 1983).

Gausman, H. W.

Gordon, H. R.

Grassam, E.

J. B. Dawson, D. J. Barker, D. J. Ellis, E. Grassam, J. A. Cotterill, G. W. Fisher, J. W. Feather, Phys. Med. Biol. 25, 695 (1980).
[CrossRef] [PubMed]

Hartmann, K. M.

K. M. Hartmann, in Biophysik, W. Hoppe, W. Lohmann, H. Markl, H. Ziegler, Eds. (Springer, Berlin, 1982), p. 122.

Jarvis, P. G.

J. Leverenz, P. G. Jarvis, J. Appl. Ecol. 16, 919 (1979).
[CrossRef]

Kazarinova, N.

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy (Springer, Berlin, 1969).
[CrossRef]

Kubelka, P.

P. Kubelka, J. Opt. Soc. Am. 44, 330 (1954).
[CrossRef]

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

Lathrop, A. L.

Lee, J. A.

S. W. Thome, J. T. Duniec, J. A. Lee, Photobiochem. Photobiophys. 1, 161 (1980).

Leverenz, J.

J. Leverenz, P. G. Jarvis, J. Appl. Ecol. 16, 919 (1979).
[CrossRef]

Marmé, D.

W. Schmidt, D. Marmé, P. Quail, E. Schäfer, Planta 111, 329 (1973).
[CrossRef]

Mudgett, P. S.

Munk, F.

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

Orchard, S. E.

Quail, P.

W. Schmidt, D. Marmé, P. Quail, E. Schäfer, Planta 111, 329 (1973).
[CrossRef]

Richards, L. W.

Richardson, A. J.

Ryde, J. W.

J. W. Ryde, B. S. Cooper, Proc. R. Soc. London Ser. A 131, 464 (1931).
[CrossRef]

Schäfer, E.

M. Seyfried, L. Fukshansky, E. Schäfer, Appl. Opt. 22, 492 (1983).
[CrossRef] [PubMed]

W. Schmidt, D. Marmé, P. Quail, E. Schäfer, Planta 111, 329 (1973).
[CrossRef]

E. Schäfer, L. Fukshansky, W. Shropshire, in Encyclopedia of Plant PhysiologyW. Shropshire, H. Mohr, Eds. (Springer, Berlin, 1983).

Schmidt, W.

W. Schmidt, D. Marmé, P. Quail, E. Schäfer, Planta 111, 329 (1973).
[CrossRef]

Seyboldt, A.

A. Seyboldt, A. Weisweiler, Bot. Arch. 44, 456 (1943).

Seyfried, M.

Shropshire, W.

E. Schäfer, L. Fukshansky, W. Shropshire, in Encyclopedia of Plant PhysiologyW. Shropshire, H. Mohr, Eds. (Springer, Berlin, 1983).

Smith, H.

H. Smith, Phytochrome and Photomorphogenesis (McGraw-Hill, New York1975).

Spence, H. D. L.

H. D. L. Spence, R. M. Campell, J. Chrystal, J. Ecol. 61, 317 (1973).
[CrossRef]

Stearns, E. I.

W. H. Foster, E. I. Stearns, J. Opt. Soc. Am. 60, 61 (1970).

Stokes, G. S.

G. S. Stokes, Proc. R. Soc. London 11, 545 (1862).

Stoll, A.

R. Willstatter, A. Stoll, Untersuchungen ueber die Assimilation der Kohlensäure, Berlin (1918).

Thome, S. W.

S. W. Thome, J. T. Duniec, J. A. Lee, Photobiochem. Photobiophys. 1, 161 (1980).

Völz, H. G.

H. G. Völz, Sixth FATIPEC Congress 1962 (Verlag Chemie, Weinheim/Bergstrasse, 1962), p. 62.

H. G. Völz, Seventh FATIPEC Congress 1964 (Verlag Chemie, Weinheim/Bergstrasse, 1964), p. 194.

Weisweiler, A.

A. Seyboldt, A. Weisweiler, Bot. Arch. 44, 456 (1943).

Willstatter, R.

R. Willstatter, A. Stoll, Untersuchungen ueber die Assimilation der Kohlensäure, Berlin (1918).

Appl. Opt. (4)

Bot. Arch. (1)

A. Seyboldt, A. Weisweiler, Bot. Arch. 44, 456 (1943).

J. Appl. Ecol. (1)

J. Leverenz, P. G. Jarvis, J. Appl. Ecol. 16, 919 (1979).
[CrossRef]

J. Ecol. (1)

H. D. L. Spence, R. M. Campell, J. Chrystal, J. Ecol. 61, 317 (1973).
[CrossRef]

J. Math. Biol. (1)

L. Fukshansky, J. Math. Biol. 6, 177 (1978).
[CrossRef]

J. Opt. Soc. Am. (8)

Photobiochem. Photobiophys. (1)

S. W. Thome, J. T. Duniec, J. A. Lee, Photobiochem. Photobiophys. 1, 161 (1980).

Photogramm. Eng. (1)

H. W. Gausman, Photogramm. Eng. 40, 183 (1974).

Phys. Med. Biol. (1)

J. B. Dawson, D. J. Barker, D. J. Ellis, E. Grassam, J. A. Cotterill, G. W. Fisher, J. W. Feather, Phys. Med. Biol. 25, 695 (1980).
[CrossRef] [PubMed]

Planta (1)

W. Schmidt, D. Marmé, P. Quail, E. Schäfer, Planta 111, 329 (1973).
[CrossRef]

Proc. R. Soc. London (1)

G. S. Stokes, Proc. R. Soc. London 11, 545 (1862).

Proc. R. Soc. London Ser. A (1)

J. W. Ryde, B. S. Cooper, Proc. R. Soc. London Ser. A 131, 464 (1931).
[CrossRef]

Z. Tech. Phys. (1)

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

Other (7)

G. Kortüm, Reflectance Spectroscopy (Springer, Berlin, 1969).
[CrossRef]

E. Schäfer, L. Fukshansky, W. Shropshire, in Encyclopedia of Plant PhysiologyW. Shropshire, H. Mohr, Eds. (Springer, Berlin, 1983).

K. M. Hartmann, in Biophysik, W. Hoppe, W. Lohmann, H. Markl, H. Ziegler, Eds. (Springer, Berlin, 1982), p. 122.

H. G. Völz, Sixth FATIPEC Congress 1962 (Verlag Chemie, Weinheim/Bergstrasse, 1962), p. 62.

H. G. Völz, Seventh FATIPEC Congress 1964 (Verlag Chemie, Weinheim/Bergstrasse, 1964), p. 194.

R. Willstatter, A. Stoll, Untersuchungen ueber die Assimilation der Kohlensäure, Berlin (1918).

H. Smith, Phytochrome and Photomorphogenesis (McGraw-Hill, New York1975).

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Figures (9)

Fig. 1
Fig. 1

Notation used in our application of the Kubelka-Munk theory for two incident light fluxes. The scattering layer is of thickness d.

Fig. 2
Fig. 2

Light fluxes incident from above on layer j. Both incident light fluxes contribute to this. Adding up the contribution leads to a geometrical progression (see text). The layers j and j − 1 have been separated in this drawing to better illustrate the situation: u = upper layer, l = lower layer.

Fig. 3
Fig. 3

Whole model. It consists of reflecting boundaries, any number of scattering layers, and two possible incident fluxes. All components are facultative, as required from the original object and measuring conditions, e.g., if the object is embedded in a proper immersion fluid, the reflecting boundaries are omitted. T is transmittance, R reflectance, the indices denote the layers involved.

Fig. 4
Fig. 4

(a) Reflection and transmission spectra of a moderately green four-day old cotyledon of Cucurbita pepo, grown under continuous white light (16.000 lux). The originally measured values have been processed so as to obey the prerequisites of the KMT (i.e., reflectance does not include specular reflection). For details of the measuring procedure see Ref. 22. - - -, reflectance from above (Ru); -·-, reflectance from below (Rl); —, transmission, (b) Absorption coefficient K (—) and scattering coefficient S (- - -). Calculated from (a) using the KMT. For these calculations we used the reflectance values from the upper surface.

Fig. 5
Fig. 5

Light gradients obtained from the scattering and absorption values of Fig. 4(b) at 660(—) and 730 nm (- - -), and the light gradients calculated from these transmissions using Beer’s law (—, - - -). Symbols at depth = 0 represent 1 + R; symbols at depth = d give the actual transmittance values (Δ = 660 nm, O = 730 nm).

Fig. 6
Fig. 6

Light gradients as in Fig. 5 but this time taking into account specularly reflecting boundaries (n1 = 1.425). The lowest curve is the light gradient at 450 nm (□ = 450 nm).

Fig. 7
Fig. 7

Same model as for λ = 660 nm in the previous figure (replotted as - - -) but with an additional flux incident from the rear (—). Light intensity is 20% of the light flux applied from the front side, thus simulating canopy light. — and - - - are again the corresponding Beer fits.

Fig. 8
Fig. 8

Structured cotyledon. The four layers are upper epidermis, palisade, spongy, and lower epidermis (compare Table I). O is light with top incidence, and Δ is light from the back side (same intensities). Note that the gradient for back incidence is much higher for the same depths measured from the illuminated surface.

Fig. 9
Fig. 9

Procedure for calculating internal light intensities from the measurements of reflectances and transmittance of the two parts of a cut object. Specular reflection must be accounted for on each separately measured slice. Upwelling and downwelling fluxes contribute to the local light intensities.

Tables (1)

Tables Icon

Table I Four Layers of a Cotyledon of Cucurbita pepo and the Scattering and Absorption Coefficients Chosen to Describe These Layers; the Layers are Considered Macrohomogeneous

Equations (8)

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

I i tot ( x ) = I 0 u · KM ( x ) + I 0 l · KM ( d x ) ,
a = ( S + K ) / S , b = a 2 1 , for convenience , KM ( x ) = T u [ a + 1 b sinh ( b S x ) + cosh ( b S x ) ] .
R = 1 a + b coth ( b S d ) , T = b a sinh ( b S d ) + b cosh ( b S d ) .
I 0 u [ T u + T u R u R l + T u ( R u R l ) 2 + ] = I 0 u T u 1 R u R l ,
I 0 l [ T l R u + T l R u R l R u + T l R u ( R l R u ) 2 + ] = I 0 l T l R u 1 R l R u .
T 1,2 = T 1 T 2 1 R 1 R 2 .
R 1,2 = R 1 + T 1 2 R 2 1 R 1 R 2 .
T 1,2,3 = T 1,2 T 3 1 R 2,1 R 3 .

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