Abstract

The diffuse reflection and transmission of light by a plane-parallel suspension of randomly oriented particles, scattering independently, are treated using radiative-transfer theory. Exact allowance is made for external and internal reflection at the plane surfaces of the matrix. General equations are derived in a form suitable for automatic computation. Some comparisons made between theoretical predictions and experimental measurements of the total transmittance of an optically thick diffusing suspension show satisfactory agreement with radiative-transfer theory and also with the Eddington approximation.

© 1969 Optical Society of America

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References

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  1. P. Debye, in Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell and R. S. Stein, Eds.(Gordon and Breach, New York, 1967).
  2. S. Chandrasekhar, Radiative Transfer (Dover Publications, Inc., New York, 1960).
  3. V. V. Sobolev, A Treatise on Radiative Transfer (D. van Nostrand Co., Inc., Princeton, N. J., 1963).
  4. G. W. Kattawar and G. N. Plass, Appl. Opt. 7, 1519 (1968).
  5. S. E. Orchard, J. Oil Col. Chem. Assoc. 51, 44 (1968).
  6. P. Kubelka and F. Munk, Z. Phys. 12, 593 (1931).
  7. E. Pitts, Proc. Phys. Soc. B67, 105 (1954).
  8. H. C. van de Hulst, Astrophys. J. 107, 220 (1948).
  9. C. M. Chu and S. W. Churchill, IRE Trans. AP-4, 142 (1956).
  10. G. N. Plass and G. W. Kattawar, Appl. Opt. 7, 361 (1968).
  11. M. de Belder, J. de Kerf, J. Jespers, and R. Verbrugghe, J. Opt. Soc. Am. 55, 1261 (1965).
  12. E. M. Feigel’son, Light and Heat Conduction in Stratus Clouds (Oldbourne Press, London, 1966).
  13. Another possibility is to rewrite the equation of transfer as an integral equation, and express the solution as a Neumann series [W. M. Irvine, Astrophys. J. 142, 1563 (1965)].For thick diffusing layers the convergence is very slow, but can be speeded up by a thickness-doubling method due to Van de Hulst [H. C. van de Hulst, in The Atmospheres of Venus and Mars, J. C. Brandt and M. B. McElroy, Eds. (Gordon and Breach, New York, 1968)]; this method has been applied by Irvine31 since the present work was done.
  14. I. W. Busbridge, The Mathematics of Radiative Transfer (Cambridge University Press, New York, 1960).
  15. T. W. Mullikin, Trans. Am. Math. Soc. 113, 316 (1964).
  16. I. W. Busbridge and S. E. Orchard, Astrophys. J. 149, 655 (1967).
  17. D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry (John Wiley & Sons, Inc., New York, 1963).
  18. R. E. Bellman, R. E. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (American Elsevier Publishing Co., New York, 1963).
  19. T. W. Mullikin, in Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, Ed. (Pergamon Press, Ltd., Oxford, 1963).
  20. L. B. Evans, C. M. Chu, and S. W. Churchill, J. Heat Transfer 87C, 381 (1965).
  21. J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 464 (1931).
  22. A. Brockes, Farbe 9, 53 (1960).
  23. R. G. Giovanelli, Opt. Acta 2, 153 (1955).
  24. L. Fox, Numerical Solution of Ordinary and Partial Differential Equations (Pergamon Press, Inc., New York, 1962).
  25. J. L. Carlstedt and T. W. Mullikin, Astrophys J. Suppl. 12, 449 (1966).
  26. R. Bellman, H. Kagiwada, R. Kalaba, and S. Ueno, J. Quant. Spectry. Radiative Transfer 6, 479 (1966).
  27. S. E. Orchard, Astrophys. J. 149, 665 (1967).
  28. I. W. Busbridge and S. E. Orchard, Astrophys. J. 154, 729 (1968).
  29. H. H. Kagiwada and R. E. Kalaba, J. Quant. Spectry. Radiative Transfer 7, 295 (1967).
  30. J. W. Ryde, Proc. Roy. Soc. (London) A131, 451 (1931).
  31. W. M. Irvine, Astrophys. J. 152, 823 (1968).

1968 (5)

I. W. Busbridge and S. E. Orchard, Astrophys. J. 154, 729 (1968).

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

S. E. Orchard, J. Oil Col. Chem. Assoc. 51, 44 (1968).

G. N. Plass and G. W. Kattawar, Appl. Opt. 7, 361 (1968).

G. W. Kattawar and G. N. Plass, Appl. Opt. 7, 1519 (1968).

1967 (3)

H. H. Kagiwada and R. E. Kalaba, J. Quant. Spectry. Radiative Transfer 7, 295 (1967).

S. E. Orchard, Astrophys. J. 149, 665 (1967).

I. W. Busbridge and S. E. Orchard, Astrophys. J. 149, 655 (1967).

1966 (2)

J. L. Carlstedt and T. W. Mullikin, Astrophys J. Suppl. 12, 449 (1966).

R. Bellman, H. Kagiwada, R. Kalaba, and S. Ueno, J. Quant. Spectry. Radiative Transfer 6, 479 (1966).

1965 (3)

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

Another possibility is to rewrite the equation of transfer as an integral equation, and express the solution as a Neumann series [W. M. Irvine, Astrophys. J. 142, 1563 (1965)].For thick diffusing layers the convergence is very slow, but can be speeded up by a thickness-doubling method due to Van de Hulst [H. C. van de Hulst, in The Atmospheres of Venus and Mars, J. C. Brandt and M. B. McElroy, Eds. (Gordon and Breach, New York, 1968)]; this method has been applied by Irvine31 since the present work was done.

M. de Belder, J. de Kerf, J. Jespers, and R. Verbrugghe, J. Opt. Soc. Am. 55, 1261 (1965).

1964 (1)

T. W. Mullikin, Trans. Am. Math. Soc. 113, 316 (1964).

1960 (1)

A. Brockes, Farbe 9, 53 (1960).

1956 (1)

C. M. Chu and S. W. Churchill, IRE Trans. AP-4, 142 (1956).

1955 (1)

R. G. Giovanelli, Opt. Acta 2, 153 (1955).

1954 (1)

E. Pitts, Proc. Phys. Soc. B67, 105 (1954).

1948 (1)

H. C. van de Hulst, Astrophys. J. 107, 220 (1948).

1931 (3)

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

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

J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 464 (1931).

Bellman, R.

R. Bellman, H. Kagiwada, R. Kalaba, and S. Ueno, J. Quant. Spectry. Radiative Transfer 6, 479 (1966).

Bellman, R. E.

R. E. Bellman, R. E. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (American Elsevier Publishing Co., New York, 1963).

Brockes, A.

A. Brockes, Farbe 9, 53 (1960).

Busbridge, I. W.

I. W. Busbridge and S. E. Orchard, Astrophys. J. 154, 729 (1968).

I. W. Busbridge and S. E. Orchard, Astrophys. J. 149, 655 (1967).

I. W. Busbridge, The Mathematics of Radiative Transfer (Cambridge University Press, New York, 1960).

Carlstedt, J. L.

J. L. Carlstedt and T. W. Mullikin, Astrophys J. Suppl. 12, 449 (1966).

Chandrasekhar, S.

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

Chu, C. M.

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

C. M. Chu and S. W. Churchill, IRE Trans. AP-4, 142 (1956).

Churchill, S. W.

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

C. M. Chu and S. W. Churchill, IRE Trans. AP-4, 142 (1956).

Cooper, B. S.

J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 464 (1931).

de Belder, M.

de Kerf, J.

Debye, P.

P. Debye, in Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell and R. S. Stein, Eds.(Gordon and Breach, New York, 1967).

Evans, L. B.

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

Feigel’son, E. M.

E. M. Feigel’son, Light and Heat Conduction in Stratus Clouds (Oldbourne Press, London, 1966).

Fox, L.

L. Fox, Numerical Solution of Ordinary and Partial Differential Equations (Pergamon Press, Inc., New York, 1962).

Giovanelli, R. G.

R. G. Giovanelli, Opt. Acta 2, 153 (1955).

Irvine, W. M.

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

Another possibility is to rewrite the equation of transfer as an integral equation, and express the solution as a Neumann series [W. M. Irvine, Astrophys. J. 142, 1563 (1965)].For thick diffusing layers the convergence is very slow, but can be speeded up by a thickness-doubling method due to Van de Hulst [H. C. van de Hulst, in The Atmospheres of Venus and Mars, J. C. Brandt and M. B. McElroy, Eds. (Gordon and Breach, New York, 1968)]; this method has been applied by Irvine31 since the present work was done.

Jespers, J.

Judd, D. B.

D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry (John Wiley & Sons, Inc., New York, 1963).

Kagiwada, H.

R. Bellman, H. Kagiwada, R. Kalaba, and S. Ueno, J. Quant. Spectry. Radiative Transfer 6, 479 (1966).

Kagiwada, H. H.

H. H. Kagiwada and R. E. Kalaba, J. Quant. Spectry. Radiative Transfer 7, 295 (1967).

Kalaba, R.

R. Bellman, H. Kagiwada, R. Kalaba, and S. Ueno, J. Quant. Spectry. Radiative Transfer 6, 479 (1966).

Kalaba, R. E.

H. H. Kagiwada and R. E. Kalaba, J. Quant. Spectry. Radiative Transfer 7, 295 (1967).

R. E. Bellman, R. E. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (American Elsevier Publishing Co., New York, 1963).

Kattawar, G. W.

Kubelka, P.

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

Mullikin, T. W.

J. L. Carlstedt and T. W. Mullikin, Astrophys J. Suppl. 12, 449 (1966).

T. W. Mullikin, Trans. Am. Math. Soc. 113, 316 (1964).

T. W. Mullikin, in Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, Ed. (Pergamon Press, Ltd., Oxford, 1963).

Munk, F.

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

Orchard, S. E.

S. E. Orchard, J. Oil Col. Chem. Assoc. 51, 44 (1968).

I. W. Busbridge and S. E. Orchard, Astrophys. J. 154, 729 (1968).

S. E. Orchard, Astrophys. J. 149, 665 (1967).

I. W. Busbridge and S. E. Orchard, Astrophys. J. 149, 655 (1967).

Pitts, E.

E. Pitts, Proc. Phys. Soc. B67, 105 (1954).

Plass, G. N.

Prestrud, M. C.

R. E. Bellman, R. E. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (American Elsevier Publishing Co., New York, 1963).

Ryde, J. W.

J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 464 (1931).

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

Sobolev, V. V.

V. V. Sobolev, A Treatise on Radiative Transfer (D. van Nostrand Co., Inc., Princeton, N. J., 1963).

Ueno, S.

R. Bellman, H. Kagiwada, R. Kalaba, and S. Ueno, J. Quant. Spectry. Radiative Transfer 6, 479 (1966).

van de Hulst, H. C.

H. C. van de Hulst, Astrophys. J. 107, 220 (1948).

Verbrugghe, R.

Wyszecki, G.

D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry (John Wiley & Sons, Inc., New York, 1963).

Appl. Opt. (2)

Astrophys J. Suppl. (1)

J. L. Carlstedt and T. W. Mullikin, Astrophys J. Suppl. 12, 449 (1966).

Astrophys. J. (6)

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

S. E. Orchard, Astrophys. J. 149, 665 (1967).

I. W. Busbridge and S. E. Orchard, Astrophys. J. 154, 729 (1968).

H. C. van de Hulst, Astrophys. J. 107, 220 (1948).

Another possibility is to rewrite the equation of transfer as an integral equation, and express the solution as a Neumann series [W. M. Irvine, Astrophys. J. 142, 1563 (1965)].For thick diffusing layers the convergence is very slow, but can be speeded up by a thickness-doubling method due to Van de Hulst [H. C. van de Hulst, in The Atmospheres of Venus and Mars, J. C. Brandt and M. B. McElroy, Eds. (Gordon and Breach, New York, 1968)]; this method has been applied by Irvine31 since the present work was done.

I. W. Busbridge and S. E. Orchard, Astrophys. J. 149, 655 (1967).

Farbe (1)

A. Brockes, Farbe 9, 53 (1960).

IRE Trans. (1)

C. M. Chu and S. W. Churchill, IRE Trans. AP-4, 142 (1956).

J. Heat Transfer (1)

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

J. Oil Col. Chem. Assoc. (1)

S. E. Orchard, J. Oil Col. Chem. Assoc. 51, 44 (1968).

J. Opt. Soc. Am. (1)

J. Quant. Spectry. Radiative Transfer (2)

H. H. Kagiwada and R. E. Kalaba, J. Quant. Spectry. Radiative Transfer 7, 295 (1967).

R. Bellman, H. Kagiwada, R. Kalaba, and S. Ueno, J. Quant. Spectry. Radiative Transfer 6, 479 (1966).

Opt. Acta (1)

R. G. Giovanelli, Opt. Acta 2, 153 (1955).

Proc. Phys. Soc. (1)

E. Pitts, Proc. Phys. Soc. B67, 105 (1954).

Proc. Roy. Soc. (London) (2)

J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 464 (1931).

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

Trans. Am. Math. Soc. (1)

T. W. Mullikin, Trans. Am. Math. Soc. 113, 316 (1964).

Z. Phys. (1)

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

Other (9)

P. Debye, in Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell and R. S. Stein, Eds.(Gordon and Breach, New York, 1967).

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

V. V. Sobolev, A Treatise on Radiative Transfer (D. van Nostrand Co., Inc., Princeton, N. J., 1963).

E. M. Feigel’son, Light and Heat Conduction in Stratus Clouds (Oldbourne Press, London, 1966).

I. W. Busbridge, The Mathematics of Radiative Transfer (Cambridge University Press, New York, 1960).

D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry (John Wiley & Sons, Inc., New York, 1963).

R. E. Bellman, R. E. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (American Elsevier Publishing Co., New York, 1963).

T. W. Mullikin, in Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, Ed. (Pergamon Press, Ltd., Oxford, 1963).

L. Fox, Numerical Solution of Ordinary and Partial Differential Equations (Pergamon Press, Inc., New York, 1962).

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Figures (12)

F. 1
F. 1

Reflectance of one-dimensional diffuser; addition of thin layer from above. (No surface reflection.)

F. 2
F. 2

Reflectance of one-dimensional diffuser; addition of thin layer from below. (No surface reflection.)

F. 3
F. 3

Transmittance of one-dimensional diffuser; addition of thin layer from above. (No surface reflection.)

F. 4
F. 4

Transmittance of one-dimensional diffuser; addition of thin layer from below. (No surface reflection.)

F. 5
F. 5

Geometry of illumination of two-dimensional diffuser. (No surface reflection.)

F. 6
F. 6

Reflection from front surface of one-dimensional diffuser.

F. 7
F. 7

Reflection from front and rear surfaces of one-dimensional diffuser.

F. 8
F. 8

Reflection from front surface of two-dimensional diffuser.

F. 9
F. 9

Experimental arrangement for measuring the total transmittance of a plane-parallel slab of polystyrene latex with normal monochromatic illumination. (S, source; IF, interference filter; C, cell; L, latex; PM, photomultiplier.)

F. 10
F. 10

Computed total transmittances of a polystyrene latex with normal illumination, using various truncations of the phase function. (Circles, τ = 20; triangles, τ = 30; crosses, τ = 40; dashes, Eddington approximation.)

F. 11
F. 11

Measurements (crosses) and theoretical predictions (solid line) of total transmittance of a polystyrene latex with normal illumination.

F. 12
F. 12

Measurements (crosses) and theoretical predictions (solid line) of reciprocal total transmittance of a polystyrene latex with normal illumination.

Tables (4)

Tables Icon

Table I Conversion of notation between Chandrasekhar2 and Sobolev.3

Tables Icon

Table II Surface reflectance coefficients for plane air–matrix boundary.

Tables Icon

Table III Experimental results for polystyrene latex.

Tables Icon

Table IV Theoretical results for polystyrene latex.

Equations (98)

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R ( τ + Δ τ ) = ( 1 2 Δ τ ) R ( τ ) + 1 2 λ Δ τ + λ Δ τ · R ( τ ) + 1 2 λ Δ τ · R 2 ( τ ) ,
d R / d τ = 2 R + 1 2 λ ( 1 + R ) 2 .
λ = 4 R / ( 1 + R ) 2 ,
α / ( σ / 2 ) = ( 1 R ) 2 / 2 R ,
R = R ( 1 e k τ ) 1 R 2 e k τ ,
R = τ / ( τ + 2 ) .
R ( τ + Δ τ ) = R ( τ ) + 1 2 λ Δ τ · T 2 ( τ ) ,
d R / d τ = 1 2 λ T 2 .
T ( τ + Δ τ ) = ( 1 Δ τ ) T ( τ ) + 1 2 λ Δ τ · T ( τ ) + 1 2 λ Δ τ · R ( τ ) · T ( τ ) ,
d T d τ = 1 2 λ T ( 1 + R ) T .
T ( τ + Δ τ ) = ( 1 Δ τ ) T ( τ ) + 1 2 λ Δ τ · T ( τ ) + 1 2 λ Δ τ · R ( τ ) · T ( τ ) ,
d T / d τ = 1 2 λ T ( 1 + R ) T .
R = λ 4 [ ( 1 + R ) 2 T 2 ] .
( 1 + R ) 2 R = ( 1 + R ) 2 T 2 R ,
and radiance component = π F ζ · ( λ / 4 π ) ( Δ τ / ζ ) I ( η , η ) η π F η · 1 η d ω = λ 2 Δ τ 0 1 I ( η , η ) d η / η .
radiance component = π F ζ I ( η , ζ ) η π F ζ · λ 4 π · Δ τ η · 1 η d ω , = λ 2 Δ τ η 0 1 I ( η , ζ ) d η .
radiance component = π F ζ I ( η , ζ ) η π F ζ · λ 4 π · Δ τ η d ω I ( η , η ) η π F η · 1 η d ω = λ Δ τ F 0 1 I ( η , ζ ) d η 0 1 I ( η , η ) η d η .
d I ( η , ζ ) d τ = I ( η , ζ ) [ 1 η + 1 ζ ] + λ 4 F η + λ 2 0 1 I ( η , η ) d η η + λ 2 n 0 1 I ( η , ζ ) d η + λ F 0 1 I ( η , ζ ) d η 0 1 I ( η , η ) η d η .
ρ ( η , ζ ) = I ( η , ζ ) F ζ .
ζ d ρ ( η , ζ ) d τ = ζ ρ ( η , ζ ) [ 1 η + 1 ζ ] + λ 4 η + λ 2 0 1 ρ ( η , η ) d η + ζ λ 2 η 0 1 ρ ( η , ζ ) d η + ζ λ 0 1 ρ ( η , ζ ) d η 0 1 ρ ( η , η ) d η .
η ζ d ρ ( η , ζ ) d τ + [ η + ζ ] ρ ( η , ζ ) = λ 4 [ 1 + 2 η 0 1 ρ ( η , η ) d η ] [ 1 + 2 ζ 0 1 ρ ( η , ζ ) d η ] .
ϕ ( η ) = 1 + 2 η 0 1 ρ ( η , η ) d η .
η ζ d d τ ρ ( η , ζ ) + [ η + ζ ] ρ ( η , ζ ) = λ 4 ϕ ( η ) ϕ ( ζ ) .
ρ ( η , ζ ) = λ 4 ϕ ( η ) ϕ ( ζ ) η + ζ .
ϕ ( η ) = 1 + λ 2 η ϕ ( η ) 0 1 ϕ ( η ) η + η d η ,
1 4 π γ ( Θ ) d ω = 1 .
γ ( Θ ) = i = 0 N x i P i ( cos Θ ) , ( x 0 = 1 ) .
P i ( cos Θ ) P i ( η ) P i ( η ) ,
λ 4 π i = 0 N x i P i ( η ) P i ( η ) ,
ϕ i ( η ) = P i ( η ) + 2 ( 1 ) i η 0 1 ρ ( η , η ) P i ( η ) d η .
η ζ d d τ ρ ( η , ζ ) + [ η + ζ ] ρ ( η , ζ ) = λ 4 i = 0 N ( 1 ) i x i · ϕ i ( η ) ϕ i ( ζ ) .
ρ ( η , ζ ) = λ 4 i = 0 N ( 1 ) i x i ϕ i ( η ) ϕ i ( ζ ) η + ζ .
ϕ k ( η ) = P k ( η ) + λ 2 η i = 0 N ( 1 ) i + k x i ϕ i ( η ) 0 1 ϕ i ( η ) η + η P k ( η ) d η , ( k = 0 , 1 , N ) .
( 1 μ + 1 μ 0 ) S ( μ , μ 0 ) + S ( μ , μ 0 ) τ 1 = i = 0 N ( 1 ) i ω ¯ i ψ i ( μ ) ψ i ( μ 0 ) ,
S ( μ , μ 0 ) τ 1 = i = 0 N ( 1 ) i ω ¯ i ϕ i ( μ ) ϕ i ( μ 0 ) ,
1 μ T ( μ , μ 0 ) + T ( μ , μ 0 ) τ 1 = i = 0 N ω ¯ i ψ i ( μ ) ϕ i ( μ 0 ) ,
1 μ 0 T ( μ , μ 0 ) + T ( μ , μ 0 ) τ 1 = i = 0 N ω ¯ i ϕ i ( μ ) ψ i ( μ 0 ) ,
ψ k ( μ ) = P k ( μ ) + 1 2 ( 1 ) k 0 1 S ( μ , μ ) P k ( μ ) d μ / μ ,
ϕ k ( μ ) = e τ 1 / μ P k ( μ ) + 1 2 0 1 T ( μ , μ ) P k ( μ ) d μ / μ ,
ψ k ( μ ) = P k ( μ ) + 1 2 μ i = 0 N ( 1 ) i + k ω ¯ i 0 1 d μ P k ( μ ) μ + μ × [ ψ i ( μ ) ψ i ( μ ) ϕ i ( μ ) ϕ i ( μ ) ] ,
ϕ k ( μ ) = e τ 1 / μ P k ( μ ) + 1 2 μ i = 0 N ω ¯ i 0 1 d μ P k ( μ ) μ μ × [ ϕ i ( μ ) ψ i ( μ ) ψ i ( μ ) ϕ i ( μ ) ] .
ψ k ( μ ) τ 1 = 1 2 i = 0 N ( 1 ) i + k ω ¯ i ϕ i ( μ ) 0 1 ϕ i ( μ ) P k ( μ ) d μ μ ,
ϕ k ( μ ) τ 1 = ϕ k ( μ ) μ + 1 2 i = 0 N ω ¯ i ψ i ( μ ) 0 1 ϕ i ( μ ) P k ( μ ) d μ μ .
ψ k ( 0 , μ ) = P k ( μ ) ; ϕ k ( 0 , μ ) = P k ( μ ) , ( μ 0 ) = 0 , ( μ = 0 ) .
ϕ k ( τ 0 , ζ ) τ 0 = λ 2 i = 0 N ( 1 ) i + k x i ψ i ( τ 0 , ζ ) × 0 1 ψ i ( τ 0 , η ) P k ( η ) d η η ,
ψ k ( τ 0 , ζ ) τ 0 = ψ k ( τ 0 , ζ ) ζ + λ 2 i = 0 N x i ϕ i ( τ 0 , ζ ) × 0 1 ψ i ( τ 0 , η ) P k ( η ) d η η ,
ϕ k ( 0 , ζ ) = P k ( ζ ) ; ψ k ( 0 , ζ ) = P k ( ζ ) , ( ζ 0 ) = 0 , ( ζ = 0 ) .
ρ ( η , ζ ) = λ 4 i = 0 N ( 1 ) i x i ϕ i ( η ) ϕ i ( ζ ) ψ i ( η ) ψ i ( ζ ) n + ζ ,
σ ( η , ζ ) = λ 4 i = 0 N x i ψ i ( η ) ϕ i ( ζ ) ϕ i ( η ) ψ i ( ζ ) n ζ .
1 π ρ ( η , ζ ) η d ω = 2 0 1 ρ ( η , ζ ) η d η ,
1 π σ ( η , ζ ) η d ω = 2 0 1 σ ( η , ζ ) η d η .
R ( ζ ) = 1 ϕ 1 ( ζ ) / ζ ,
T ( ζ ) = ψ 1 ( ζ ) / ζ ,
R ( ζ ) = 1 [ 1 λ α 0 / 2 ] ϕ 0 ( ζ ) [ λ / 2 ] β 0 ψ 0 ( ζ ) ,
T ( ζ ) = [ 1 λ α 0 / 2 ] ψ 0 ( ζ ) + [ λ / 2 ] β 0 ϕ 0 ( ζ ) ,
α 0 = 0 1 ϕ 0 ( η ) d η , β 0 = 0 1 ψ 0 ( η ) d η .
R d = 2 0 1 R ( ζ ) ζ d ζ ,
T d = 2 0 1 T ( ζ ) ζ d ζ .
J = I R + J r i R .
R * = R + R * r i R .
R * = J / I = R / ( 1 r i R ) .
I = I 0 t e .
J t i = I 0 t e t i R * .
R 1 = r e + t e t i R * ,
R 1 = r e + t i t e R 1 r i R .
R 1 = r e + t i t e R c / ( 1 r i R d ) ,
J = I R + J r i R + K r i T ,
K = I T + J r i T + K r i R .
R * = R + R * r i R + T * r i T ,
T * = T + R * r i T + T * r i R .
R * = R ( 1 r i R ) + r i T 2 ( 1 r i R ) ( 1 r i R ) r i r i T 2 .
t i ( diffuse ) = t e / n 2 ( diffuse ) ,
1 m 2 = n 2 ( 1 μ 0 ) 2 ,
π J ( μ ) = I ( μ 0 ) μ 0 ρ ( μ , μ 0 ) + 2 π 0 1 J ( μ ) μ r ( μ ) ρ ( μ , μ ) d μ .
ρ * ( μ , μ 0 ) = π J ( μ ) I ( μ 0 ) μ 0 .
ρ * ( μ , μ 0 ) = ρ ( μ , μ 0 ) + 2 0 1 ρ * ( μ , μ 0 ) r ( μ ) ρ ( μ , μ ) μ d μ .
π I ( μ 0 ) μ 0 = π F ( m ) m · t ( m ) .
J ( μ ) · t ( μ ) μ d ω / F ( m ) m .
π J ( μ ) = F ( m ) m · t ( m ) · ρ * ( μ , μ 0 ) .
2 t ( m ) 0 1 ρ * ( μ , μ 0 ) t ( μ ) μ d μ .
R 1 ( m ) = r ( m ) + 2 t ( m ) 0 1 ρ * ( μ , μ 0 ) t ( μ ) μ d μ .
ρ * ( μ , μ 0 ) = ρ ( μ , μ 0 ) + 2 0 1 ρ * ( μ , μ 0 ) r ( μ ) ρ ( μ , μ ) μ d μ + 2 0 1 σ * ( μ , μ 0 ) r ( μ ) σ ( μ , μ ) μ d μ ,
σ * ( μ , μ 0 ) = σ ( μ , μ 0 ) + 2 0 1 ρ * ( μ , μ 0 ) r ( μ ) σ ( μ , μ ) μ d μ + 2 0 1 σ * ( μ , μ 0 ) r ( μ ) ρ ( μ , μ ) μ d μ ,
T 1 ( m ) = 2 t ( m ) 0 1 σ * ( μ , μ 0 ) t ( μ ) μ d μ .
T * = T / [ ( 1 r i R ) 2 r i 2 T 2 ] ,
T * = T / ( 1 r i R ) 2 .
T 1 = T ( 1 r i ) / ( 1 r i R ) 2 .
τ = N σ t ,
efficiency factor Q = σ / π r 2 size parameter x = 2 π r / λ } ,
= 1.000 + 2.772 P 1 + 4.235 P 2 + 5.340 P 3 + 6.121 P 4 + 6.658 P 5 + 6.968 P 6 + 7.104 P 7 + 7.093 P 8 + 6.937 P 9 + 6.677 P 10 + 6.25 P 11 + 5.76 P 12 + 5.20 P 13 + 4.48 P 14 + 3.72 P 15 + 2.85 P 16 + 1.95 P 17 + 1.13 P 18 + 0.50 P 19 + 0.21 P 20 + 0.07 P 21 .
T = k 1 ( 3 x 1 ) τ + k 2 ,
Eddington T = 5 exp ( τ ) ( 3 x 1 ) τ + 4 ,
Ryde T = 1 + [ D C ] [ 1 exp ( τ ) ] D τ + 1 ,
C = 1 2 ( 1 n = 1 x n g n ) ,
D = 1 2 ( 1 n = 1 x n g n 2 ) ,
g 0 = 1 ; g 1 = 1 2 ; g n = ( 2 n 1 + n ) g n 2 .
ω ¯ 0
ω ¯ i / ω ¯ 0