Abstract

The widely accepted Willstätter-Stoll (W-S) theory of leaf reflectance has been investigated by extensive ray tracing through a model (W-S model) in which the leaf cellular structure is approximated by circular arcs. Calculations were performed on an IBM 1800 computer. The W-S model is treated as a two-dimensional, uncentered optical system consisting of a single medium and air. Optical properties of the medium are specified by a complex index of refraction. Given an incident ray, new reflected and transmitted rays are produced at each interface with properties determined by the laws of Snell, Fresnel, and Lambert. Calculations indicate that the W-S model, as exemplified by their artist’s conception, is too transparent, that is, the magnitude predicted for transmittance is too high. Transmittance is still too high if each interface is treated as a diffusive instead of a smooth surface. The W-S model can be easily improved, however, by introduction of more intercellular air spaces. The modified W-S model promises to be an excellent representation of physical reality. Accurate predictions, however, require an inordinate amount of computer time.

© 1973 Optical Society of America

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References

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  1. R. Willstätter, A. Stoll, Untersuchunger üaber die Assimilation der Kohlensäure (Springer-Verlag, Berlin, 1918).
  2. R. W. Ditchburn, Light (Interscience-Wiley, New York, 1963), Vol. 2, p. 551.
  3. Ref. 2, p. 551.
  4. L. Silberstein, Simplified Method of Tracing Rays Through Any Optical System of Lenses, Prisms, and Mirrors (Longmans, Green and Company, New York, 1918).
  5. D. P. Feder, J. Opt. Soc. Am. 41, 630 (1951).
    [Crossref]
  6. W. A. Allen, J. R. Snyder, J. Opt. Soc. Am. 42, 243 (1952).
    [Crossref]
  7. Ref. 2, p. 531.
  8. T. R. Sinclair, “Pathway of Solar Radiation Through Leaves,” M.S. Thesis, Purdue University Library, Lafayette, Indiana (1968), 179 pp.
  9. J. T. Woolley, Plant Physiol. 47, 656 (1971).
    [Crossref] [PubMed]
  10. W. A. Allen, H. W. Gausman, A. J. Richardson, J. Opt. Soc. Am. 60, 542 (1970).
    [Crossref]
  11. R. Kumar, L. F. Silva, Appl. Opt., in press (1973).

1971 (1)

J. T. Woolley, Plant Physiol. 47, 656 (1971).
[Crossref] [PubMed]

1970 (1)

1952 (1)

1951 (1)

Allen, W. A.

Ditchburn, R. W.

R. W. Ditchburn, Light (Interscience-Wiley, New York, 1963), Vol. 2, p. 551.

Feder, D. P.

Gausman, H. W.

Kumar, R.

R. Kumar, L. F. Silva, Appl. Opt., in press (1973).

Richardson, A. J.

Silberstein, L.

L. Silberstein, Simplified Method of Tracing Rays Through Any Optical System of Lenses, Prisms, and Mirrors (Longmans, Green and Company, New York, 1918).

Silva, L. F.

R. Kumar, L. F. Silva, Appl. Opt., in press (1973).

Sinclair, T. R.

T. R. Sinclair, “Pathway of Solar Radiation Through Leaves,” M.S. Thesis, Purdue University Library, Lafayette, Indiana (1968), 179 pp.

Snyder, J. R.

Stoll, A.

R. Willstätter, A. Stoll, Untersuchunger üaber die Assimilation der Kohlensäure (Springer-Verlag, Berlin, 1918).

Willstätter, R.

R. Willstätter, A. Stoll, Untersuchunger üaber die Assimilation der Kohlensäure (Springer-Verlag, Berlin, 1918).

Woolley, J. T.

J. T. Woolley, Plant Physiol. 47, 656 (1971).
[Crossref] [PubMed]

J. Opt. Soc. Am. (3)

Plant Physiol. (1)

J. T. Woolley, Plant Physiol. 47, 656 (1971).
[Crossref] [PubMed]

Other (7)

R. Kumar, L. F. Silva, Appl. Opt., in press (1973).

Ref. 2, p. 531.

T. R. Sinclair, “Pathway of Solar Radiation Through Leaves,” M.S. Thesis, Purdue University Library, Lafayette, Indiana (1968), 179 pp.

R. Willstätter, A. Stoll, Untersuchunger üaber die Assimilation der Kohlensäure (Springer-Verlag, Berlin, 1918).

R. W. Ditchburn, Light (Interscience-Wiley, New York, 1963), Vol. 2, p. 551.

Ref. 2, p. 551.

L. Silberstein, Simplified Method of Tracing Rays Through Any Optical System of Lenses, Prisms, and Mirrors (Longmans, Green and Company, New York, 1918).

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Figures (5)

Fig. 1
Fig. 1

(a) Artist’s conception of cross section of a maple leaf. (b) Optical system (100 circular arcs) used to simulate (a).

Fig. 2
Fig. 2

Intersection of a ray with a circle determined by three points. The minimum arc (solid line) is real, and the maximum arc (dashed line) is extraneous.

Fig. 3
Fig. 3

Intersection of a ray with a circular interface between two dielectrics. The ray and the circle are both specified relative to a fixed origin.

Fig. 4
Fig. 4

Indicatrices for reflectance and transmittance obtained from model illustrated in Fig. 1(b). (a) Specular interfaces. (b) Diffuse interfaces.

Fig. 5
Fig. 5

(a) Cross section of a 100-μm thick maple leaf taken from a sunlit portion of the tree. (b) Cross section of a 75-μm thick maple leaf taken from a shady portion of the tree. (These leaf cross sections are by courtesy of Turtox, Chicago.)

Equations (30)

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( x - a ) 2 + ( y - b ) 2 = r 2 .
x 1 2 + y 1 2 + a 2 + b 2 - 2 ( a x 1 + b y 1 ) = r 2 , x 2 2 + y 2 2 + a 2 + b 2 - 2 ( a x 2 + b y 2 ) = r 2 , x 3 2 + y 3 2 + a 2 + b 2 - 2 ( a x 3 + b y 3 ) = r 2 .
2 a y 1 - y 2 + 2 b x 1 - x 2 = x 1 + x 2 y 1 - y 2 + y 1 + y 2 x 1 - x 2 , 2 a y 2 - y 3 + 2 b x 2 - x 3 = x 2 + x 3 y 2 - y 3 + y 2 + y 3 x 2 - x 3 ,
( x 3 - x 1 ) 2 + ( y 3 - y 1 ) 2 > ( x - x 1 ) 2 + ( y - y 1 ) 2 + ( x - x 3 ) 2 + ( y - y 3 ) 2 .
c 2 = a 2 + b 2 - 2 a b cos C .
c 2 > a 2 + b 2 .
sin 2 i = ( n × q ) 2 ,
sin 2 i = ( n × q ) 2 ,
sin 2 i = ( n × q ) 2 .
n × q = n × q .
q = q - 2 ( n · q ) n .
n × s = n × s ,
s = s + ( n · s - n · s ) n .
sin i = [ 1 - ( n · q ) 2 ] 1 / 2 ,
sin i = ( μ / μ ) [ 1 - ( n · q ) 2 ] 1 / 2 ,
cos i = { 1 - ( μ / μ ) 2 [ 1 - ( n · q ) 2 ] } 1 / 2 ,
n · s - μ cos i = [ μ 2 - μ 2 + μ 2 ( n · q ) 2 ] 1 / 2 .
s = s - n · s { [ 1 + ( μ 2 - μ 2 ) / ( n · s ) 2 ] 1 / 2 + 1 } n .
n = r - 1 ( T + l q - U ) ,
( R - U ) 2 = r 2 ,
R = T + l q .
[ l q + ( T - U ) ] 2 = r 2 .
l 2 + 2 q · ( T - U ) l + ( T - U ) 2 = r 2 .
l 2 + 2 B l + C = 0 ,
B = q x ( T x - a ) + q y ( T y - b ) ,
C = ( T x - a ) 2 + ( T y - b ) 2 - r 2 .
l = - B ± ( B 2 - C ) 1 / 2 .
T i + 1 = T i + l q i .
R 1 = sin 2 ( i - i ) sin - 2 ( i + i ) , R 2 = tan 2 ( i - i ) tan - 2 ( i + i ) .
I = I 0 exp ( - k l ) ,

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