Abstract

Equations are derived to determine the diffuse reflectance and transmittance of inhomogeneous materials. The equations are valid for collimated incident radiation for any angle of incidence. The effects of boundary reflectance and anisotropic scattering are included. The equations are derived from the equation of radiative transport, using the Schuster-Schwartzchild approximation. They are sufficiently simple to be used for spectroscopic determination of the absorption and scattering coefficients. Numerical comparison with more exact solutions of the equation of radiative transfer show very good agreement for all cases except for reflectance in the highly anisotropic case, where agreement is only fair.

© 1973 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Kubelka, F. Munk, Z. Phys., 12, 593 (1931).
  2. J. C. Richmond, J. Res. Natl. Bur. Stand. 67C, 217 (1963).
  3. C. Sagan, J. B. Pollack, J. Geophys. Res. 72, 469 (1967).
    [CrossRef]
  4. J. W. Ryde, Proc. Roy. Soc. (London) A131, 451 (1931).
  5. C. M. Chu, S. W. Churchill, J. Phys. Chem. 59, 855 (1955).
    [CrossRef]
  6. S. Chandrasekhar, Radiative Transfer (Dover Publications, Inc., New York, 1960).
  7. L. B. Evans, C. M. Chu, S. W. Churchill, J. Heat Transfer 87C, 381 (1965).
    [CrossRef]
  8. W. M. Irvine, Astrophys. J. 152, 823 (1968).
    [CrossRef]
  9. S. E. Orchard, J. Opt. Soc. Am. 59, 1584 (1969).
    [CrossRef]
  10. K. Schwartzchild, in Selected Papers on the Transfer of Radiation, D. Menzel, Ed. (Dover Publications, Inc., New York, 1966).
  11. C. M. Chu, S. W. Churchill, S. C. Pang, in Electromagnetic Scattering, M. Kerker, Ed. (Macmillan Company, New York, 1963).

1969 (1)

1968 (1)

W. M. Irvine, Astrophys. J. 152, 823 (1968).
[CrossRef]

1967 (1)

C. Sagan, J. B. Pollack, J. Geophys. Res. 72, 469 (1967).
[CrossRef]

1965 (1)

L. B. Evans, C. M. Chu, S. W. Churchill, J. Heat Transfer 87C, 381 (1965).
[CrossRef]

1963 (1)

J. C. Richmond, J. Res. Natl. Bur. Stand. 67C, 217 (1963).

1955 (1)

C. M. Chu, S. W. Churchill, J. Phys. Chem. 59, 855 (1955).
[CrossRef]

1931 (2)

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

J. W. Ryde, Proc. Roy. Soc. (London) A131, 451 (1931).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover Publications, Inc., New York, 1960).

Chu, C. M.

L. B. Evans, C. M. Chu, S. W. Churchill, J. Heat Transfer 87C, 381 (1965).
[CrossRef]

C. M. Chu, S. W. Churchill, J. Phys. Chem. 59, 855 (1955).
[CrossRef]

C. M. Chu, S. W. Churchill, S. C. Pang, in Electromagnetic Scattering, M. Kerker, Ed. (Macmillan Company, New York, 1963).

Churchill, S. W.

L. B. Evans, C. M. Chu, S. W. Churchill, J. Heat Transfer 87C, 381 (1965).
[CrossRef]

C. M. Chu, S. W. Churchill, J. Phys. Chem. 59, 855 (1955).
[CrossRef]

C. M. Chu, S. W. Churchill, S. C. Pang, in Electromagnetic Scattering, M. Kerker, Ed. (Macmillan Company, New York, 1963).

Evans, L. B.

L. B. Evans, C. M. Chu, S. W. Churchill, J. Heat Transfer 87C, 381 (1965).
[CrossRef]

Irvine, W. M.

W. M. Irvine, Astrophys. J. 152, 823 (1968).
[CrossRef]

Kubelka, P.

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

Munk, F.

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

Orchard, S. E.

Pang, S. C.

C. M. Chu, S. W. Churchill, S. C. Pang, in Electromagnetic Scattering, M. Kerker, Ed. (Macmillan Company, New York, 1963).

Pollack, J. B.

C. Sagan, J. B. Pollack, J. Geophys. Res. 72, 469 (1967).
[CrossRef]

Richmond, J. C.

J. C. Richmond, J. Res. Natl. Bur. Stand. 67C, 217 (1963).

Ryde, J. W.

J. W. Ryde, Proc. Roy. Soc. (London) A131, 451 (1931).

Sagan, C.

C. Sagan, J. B. Pollack, J. Geophys. Res. 72, 469 (1967).
[CrossRef]

Schwartzchild, K.

K. Schwartzchild, in Selected Papers on the Transfer of Radiation, D. Menzel, Ed. (Dover Publications, Inc., New York, 1966).

Astrophys. J. (1)

W. M. Irvine, Astrophys. J. 152, 823 (1968).
[CrossRef]

J. Geophys. Res. (1)

C. Sagan, J. B. Pollack, J. Geophys. Res. 72, 469 (1967).
[CrossRef]

J. Heat Transfer (1)

L. B. Evans, C. M. Chu, S. W. Churchill, J. Heat Transfer 87C, 381 (1965).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. Chem. (1)

C. M. Chu, S. W. Churchill, J. Phys. Chem. 59, 855 (1955).
[CrossRef]

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

J. C. Richmond, J. Res. Natl. Bur. Stand. 67C, 217 (1963).

Proc. Roy. Soc. (London) (1)

J. W. Ryde, Proc. Roy. Soc. (London) A131, 451 (1931).

Z. Phys. (1)

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

Other (3)

K. Schwartzchild, in Selected Papers on the Transfer of Radiation, D. Menzel, Ed. (Dover Publications, Inc., New York, 1966).

C. M. Chu, S. W. Churchill, S. C. Pang, in Electromagnetic Scattering, M. Kerker, Ed. (Macmillan Company, New York, 1963).

S. Chandrasekhar, Radiative Transfer (Dover Publications, Inc., New York, 1960).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Comparison of calculated reflectance as a function of angle of incidence (μ0) anisotropic scattering case for optical thickness τ = 1.0, single scattering albedo ω = 0.9, and phase functions a1 = 1, 1.732. Squares refer to calculations of Ref. 7.

Fig. 2
Fig. 2

Comparison of calculated transmittance as a function of angle of incidence (μ0). Anisotropic scattering case for optical thickness τ = 1.0, single scattering albedo ω = 0.9, and phase functions a1 = 1, 1.732. Squares refer to calculations of Ref. 7.

Fig. 3
Fig. 3

Comparison of calculated reflectance as a function of optical thickness (τ). Anisotropic scattering case for single scattering albedo ω = 0.9, phase function a1 = 1.732, and cosine incident angles μ0 = 0.045, 0.5, 0.95. Squares refer to calculations of Ref. 7.

Fig. 4
Fig. 4

Comparison of calculated transmittance as a function of optical thickness (τ). Anisotropic scattering case for single scattering albedo ω = 0.9, phase function a1 = 1.732, and cosine incident angle μ0 = 0.045, 0.5, 0.95. Open circles refer to calculations of Ref. 7.

Fig. 5
Fig. 5

Comparison of calculated reflectance as a function of angle of incidence (μ0). Isotropic scattering case for single scattering albedos ω = 1.0, 0.9, 0.5, and optical thickness τ = 1.0. Open circles refer to calculations of Ref. 10.

Fig. 6
Fig. 6

Comparison of calculated transmittance as a function of angle of incidence (μ0). Isotropic scattering case for single scattering albedos ω = 1.0, 0.9, 0.5, and optical thickness τ = 1.0. Open circles refer to calculations of Ref. 10.

Equations (28)

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

μ d I ( μ ) d τ = - I ( μ ) + ω 4 π exp ( - τ / μ 0 ) × - 1 1 0 2 π p ( μ , φ , μ , φ ) δ ( μ - μ 0 ) d μ d φ + ω 4 π - 1 1 0 2 π I ( μ ) p ( μ , φ , μ , φ ) d μ d φ ,
μ d I ( μ ) d τ = - I ( μ ) + ω I 0 exp ( - τ / μ 0 ) F ( μ , μ 0 ) + ω - 1 1 F ( μ , μ ) I ( μ ) d μ ,
F ( μ , μ ) = 1 4 π 0 2 π p ( μ , 0 , μ , φ ) d φ .
½ ( d I + / d τ ) = - ( 1 - ω F d ) I + + ω ( 1 - F d ) I - + ω I 0 F c ( μ 0 ) exp ( - τ / μ 0 ) ,
- ½ ( d I - / d τ ) = - ( 1 - ω F d ) I - + ω ( 1 - F d ) I + + ω I 0 exp ( - τ / μ 0 ) [ 1 - F c ( μ 0 ) ] ,
F d = 0 1 0 1 F ( μ , μ ) d μ d μ ,
F c ( μ 0 ) = 0 1 F ( μ , μ 0 ) d μ .
F ( μ , μ ) = ½ s = 0 a s P s ( μ ) P s ( μ ) .
F d = ½ ( 1 + s = 1 a s g s 2 ) , F c ( μ 0 ) = ½ [ 1 + s = 1 a s g s P s ( μ 0 ) ] ,
I + = A exp ( - α τ 1 ) + B exp ( α τ 1 ) - G I 0 exp ( - τ 1 / μ 0 ) ,
I - = R A exp ( - α τ 1 ) + ( B / R ) exp ( α τ 1 ) - G H I 0 exp ( - τ 1 / μ 0 ) ,
α = 2 ( 1 + 2 ω 2 F d - ω 2 - 2 ω F d ) 1 / 2 ,
G = 4 ω ( F c + ω - ω F c - ω F d + F c / 2 μ 0 ) / ( 1 / μ 0 2 - α 2 ) ,
H = [ 1 - F c + ω F c - ω F d - ( 1 - F c ) / 2 μ 0 ] ( F c + ω - ω F c - ω F d + F c / 2 μ 0 ) ,
R = ( 2 - 2 ω F d - α ) / 2 ( 1 - F d ) ω .
R = I - ( 0 ) / 2 μ 0 I 0 , T = I + ( τ 1 ) / 2 μ 0 I 0 + exp ( - τ 1 / μ 0 ) ,
R = G { R - H ( 1 - R 2 ) exp [ - ( τ 1 / μ 0 + α τ 1 ) ] - H - R ( 1 - R H ) exp ( - 2 α τ 1 ) } / 2 μ 0 [ 1 - R 2 exp ( - 2 α τ 1 ) ] ,
T = ( G { ( 1 - R 2 ) exp ( - α τ 1 ) - ( 1 - R H ) exp ( - τ 1 / μ 0 ) + R ( R - H ) exp [ - ( τ 1 / μ 0 + 2 α τ 1 ) ] } / 2 μ 0 [ 1 - R 2 exp ( - 2 α τ 1 ) ] ) + exp ( - τ 1 / μ 0 ) .
α = 2 ( 1 - ω ) 1 / 2 , G = 2 ω ( 1 + 1 / μ 0 ) / [ 1 / μ 0 2 - 4 ( 1 - ω ) ] , H = ( 2 μ 0 - 1 ) / ( 2 μ 0 + 1 ) , R = 2 / ω - 1 - 2 ( 1 - ω ) 1 / 2 / ω .
R = ( ω / 2 ) ( { 3 R - ( 1 - R 2 ) exp [ - ( 1 + α ) τ 1 ] - 1 - R ( 3 - R ) exp ( - 2 α τ 1 ) } / ( 4 ω - 3 ) [ 1 - R 2 exp ( - 2 α τ 1 ) ] ) , T = ( ω / 2 ) ( { 3 ( 1 - R 2 ) exp ( - α τ 1 ) - ( 3 - R ) exp ( - τ 1 ) + R ( 3 R - 1 ) exp [ - ( τ 1 + 2 α τ 1 ) ] } / ( 4 ω - 3 ) [ 1 - R 2 exp ( - 2 α τ 1 ) ] ) + e - τ 1 .
T = { 2 μ 0 ( 1 - F d + F c / 2 μ 0 ) [ 1 - exp ( - τ 1 / μ 0 ) ] + exp ( τ 1 / μ 0 ) } / [ 2 τ 1 ( 1 - F d ) + 1 ]
R = 1 - T .
T = 0.5 [ 3 - exp ( - τ 1 ) ] / ( τ 1 + 1 ) .
R = G ( R - H ) / 2 μ 0 ,
R = { ( ω / 2 ) ( 2 μ 0 + 1 ) / [ 1 - 4 μ 0 ( 1 - ω ) ] } { R - [ ( 2 μ 0 - 1 ) / ( 2 μ 0 + 1 ) ] } .
R = [ ( ω / 2 ) / ( 4 ω - 3 ) ] ( 3 R - 1 ) .
R = ( ( 1 - r i ) ( 1 - r e ) { r i T d T + r i r e T d R exp ( - τ 1 / μ 0 ) + [ R + r e T exp ( - τ 1 / μ 0 ) ] ( 1 - r i R d ) } / [ 1 - r e 2 exp ( - 2 τ 1 / μ 0 ) ] [ ( 1 - r i R d ) 2 - r i 2 T d 2 ] ) + { [ r e + r e ( 1 - 2 r e ) exp ( - 2 τ 1 / μ 0 ) ] / [ 1 - r e 2 exp ( - 2 τ 1 / μ 0 ) ] } , J = ( ( 1 - r i ) ( 1 - r e ) { [ T + r e R exp ( - τ 1 / μ 0 ) ] ( 1 - r i R d ) + r i T d [ R + r e T exp ( - τ 1 / μ 0 ) ] } / [ 1 - r e 2 exp ( - 2 τ 1 / μ 0 ) ] [ ( 1 - r i R d ) 2 - r i 2 T d 2 ] ) + { ( 1 - r e ) 2 exp ( - τ 1 / μ 0 ) / [ 1 - r r 2 exp ( - 2 τ 1 / μ 0 ) ] } ,
R d = R [ 1 - exp ( - 2 α τ 1 ) ] / [ 1 - R 2 exp ( - 2 α τ 1 ) ] , T d = ( 1 - R 2 ) exp ( - α τ 1 ) / [ 1 - R 2 exp ( - 2 α τ 1 ) ] ,

Metrics