Abstract

A modified form of the linear-transport equation is derived and applied to situations in which the scattering function is strongly peaked in the forward direction. This important case, which includes multiple scattering of light by biological suspensions, is very difficult to handle by use of ordinary linear-transport theory, but quite tractable with the modified equation. The modified equation is a very good approximation to the usual transport equation throughout the scattering medium except in the close vicinity of a δ function (i.e., a unidirectional) source. In the case of no absorption, the modified equation describes the statistics of stiff polymer chains.

© 1974 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison–Wesley, Reading, Mass., 1967).
  2. K. M. Watson, J. Math. Phys. 10, 688 (1969).
    [Crossref]
  3. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 4.
  4. S. Q. Duntley, J. Opt. Soc. Am. 53, 214 (1963).
    [Crossref]
  5. V. Twersky, J. Opt. Soc. Am. 60, 1084 (1970).
    [Crossref] [PubMed]
  6. C. C. Johnson, IEEE Trans. Bio-Med. Eng. 17, 129 (1970).
    [Crossref]
  7. K. M. Case, Ann. Phys. (N.Y.) 9, 1 (1960); Ref. 1.
    [Crossref]
  8. J. R. Mika, Nucl. Sci. Eng. 11, 415 (1961); Ref. 1, p. 87.
  9. S. Chandrasekhar, Radiative Transfer (Oxford, London, 1950); M. Wing, An Introduction to Transport Theory (Wiley, New York, 1962); J. O. Mingle, Nuc. Sci. Eng. 28, 177 (1967).
  10. P. Kubelka and F. Munk, Z. Tech. Phys. 12, 593 (1931); P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948); J. Opt. Soc. Am. 44, 330 (1954); P. S. Mudgett and L. W. Richards, Appl. Opt. 10, 1485 (1971).
    [Crossref] [PubMed]
  11. Y. Y. Bobyrenko, Opt. Spektrosk. 24, 680 (1968) [Opt. Spectrosc. 24, 365 (1968)].
  12. Reference 1, p. 196.
  13. G. W. Kattawar and G. N. Plass, Appl. Opt. 11, 2851 (1972); Appl. Opt. 11, 2866 (1972).
    [Crossref] [PubMed]
  14. Ref. 4; V. Twersky, J. Math. Phys. 3, 724 (1962); J. Opt. Soc. Am. 52, 145 (1962).
    [Crossref]
  15. D. A. Cross and P. Latimer, Appl. Opt. 11, 1225 (1972).
    [Crossref] [PubMed]
  16. N. Saito, K. Takahashi, and Y. Yunoki, J. Phys. Soc. Jap. 22, 219 (1967); Y. Yamakawa, Modern Theory of Polymer Solutions (Harper and Row, New York, 1971).
    [Crossref]
  17. Reference 1, Ch. 5.
  18. Reference 1, Appendix F.
  19. Reference 1, p. 106.
  20. Reference 1, p. 105.
  21. N. J. McCormick and M. R. Mendelson, Nucl. Sci. Eng. 20, 462 (1964).
  22. Y. Yener and M. W. Özisik, J. Math. Phys. 13, 2013 (1972).
    [Crossref]

1972 (3)

1970 (2)

V. Twersky, J. Opt. Soc. Am. 60, 1084 (1970).
[Crossref] [PubMed]

C. C. Johnson, IEEE Trans. Bio-Med. Eng. 17, 129 (1970).
[Crossref]

1969 (1)

K. M. Watson, J. Math. Phys. 10, 688 (1969).
[Crossref]

1968 (1)

Y. Y. Bobyrenko, Opt. Spektrosk. 24, 680 (1968) [Opt. Spectrosc. 24, 365 (1968)].

1967 (1)

N. Saito, K. Takahashi, and Y. Yunoki, J. Phys. Soc. Jap. 22, 219 (1967); Y. Yamakawa, Modern Theory of Polymer Solutions (Harper and Row, New York, 1971).
[Crossref]

1964 (1)

N. J. McCormick and M. R. Mendelson, Nucl. Sci. Eng. 20, 462 (1964).

1963 (1)

1962 (1)

Ref. 4; V. Twersky, J. Math. Phys. 3, 724 (1962); J. Opt. Soc. Am. 52, 145 (1962).
[Crossref]

1961 (1)

J. R. Mika, Nucl. Sci. Eng. 11, 415 (1961); Ref. 1, p. 87.

1960 (1)

K. M. Case, Ann. Phys. (N.Y.) 9, 1 (1960); Ref. 1.
[Crossref]

1931 (1)

P. Kubelka and F. Munk, Z. Tech. Phys. 12, 593 (1931); P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948); J. Opt. Soc. Am. 44, 330 (1954); P. S. Mudgett and L. W. Richards, Appl. Opt. 10, 1485 (1971).
[Crossref] [PubMed]

Bobyrenko, Y. Y.

Y. Y. Bobyrenko, Opt. Spektrosk. 24, 680 (1968) [Opt. Spectrosc. 24, 365 (1968)].

Case, K. M.

K. M. Case, Ann. Phys. (N.Y.) 9, 1 (1960); Ref. 1.
[Crossref]

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison–Wesley, Reading, Mass., 1967).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford, London, 1950); M. Wing, An Introduction to Transport Theory (Wiley, New York, 1962); J. O. Mingle, Nuc. Sci. Eng. 28, 177 (1967).

Cross, D. A.

Duntley, S. Q.

Johnson, C. C.

C. C. Johnson, IEEE Trans. Bio-Med. Eng. 17, 129 (1970).
[Crossref]

Kattawar, G. W.

Kubelka, P.

P. Kubelka and F. Munk, Z. Tech. Phys. 12, 593 (1931); P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948); J. Opt. Soc. Am. 44, 330 (1954); P. S. Mudgett and L. W. Richards, Appl. Opt. 10, 1485 (1971).
[Crossref] [PubMed]

Latimer, P.

McCormick, N. J.

N. J. McCormick and M. R. Mendelson, Nucl. Sci. Eng. 20, 462 (1964).

Mendelson, M. R.

N. J. McCormick and M. R. Mendelson, Nucl. Sci. Eng. 20, 462 (1964).

Mika, J. R.

J. R. Mika, Nucl. Sci. Eng. 11, 415 (1961); Ref. 1, p. 87.

Munk, F.

P. Kubelka and F. Munk, Z. Tech. Phys. 12, 593 (1931); P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948); J. Opt. Soc. Am. 44, 330 (1954); P. S. Mudgett and L. W. Richards, Appl. Opt. 10, 1485 (1971).
[Crossref] [PubMed]

Özisik, M. W.

Y. Yener and M. W. Özisik, J. Math. Phys. 13, 2013 (1972).
[Crossref]

Plass, G. N.

Saito, N.

N. Saito, K. Takahashi, and Y. Yunoki, J. Phys. Soc. Jap. 22, 219 (1967); Y. Yamakawa, Modern Theory of Polymer Solutions (Harper and Row, New York, 1971).
[Crossref]

Takahashi, K.

N. Saito, K. Takahashi, and Y. Yunoki, J. Phys. Soc. Jap. 22, 219 (1967); Y. Yamakawa, Modern Theory of Polymer Solutions (Harper and Row, New York, 1971).
[Crossref]

Twersky, V.

V. Twersky, J. Opt. Soc. Am. 60, 1084 (1970).
[Crossref] [PubMed]

Ref. 4; V. Twersky, J. Math. Phys. 3, 724 (1962); J. Opt. Soc. Am. 52, 145 (1962).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 4.

Watson, K. M.

K. M. Watson, J. Math. Phys. 10, 688 (1969).
[Crossref]

Yener, Y.

Y. Yener and M. W. Özisik, J. Math. Phys. 13, 2013 (1972).
[Crossref]

Yunoki, Y.

N. Saito, K. Takahashi, and Y. Yunoki, J. Phys. Soc. Jap. 22, 219 (1967); Y. Yamakawa, Modern Theory of Polymer Solutions (Harper and Row, New York, 1971).
[Crossref]

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison–Wesley, Reading, Mass., 1967).

Ann. Phys. (N.Y.) (1)

K. M. Case, Ann. Phys. (N.Y.) 9, 1 (1960); Ref. 1.
[Crossref]

Appl. Opt. (2)

IEEE Trans. Bio-Med. Eng. (1)

C. C. Johnson, IEEE Trans. Bio-Med. Eng. 17, 129 (1970).
[Crossref]

J. Math. Phys. (3)

K. M. Watson, J. Math. Phys. 10, 688 (1969).
[Crossref]

Ref. 4; V. Twersky, J. Math. Phys. 3, 724 (1962); J. Opt. Soc. Am. 52, 145 (1962).
[Crossref]

Y. Yener and M. W. Özisik, J. Math. Phys. 13, 2013 (1972).
[Crossref]

J. Opt. Soc. Am. (2)

J. Phys. Soc. Jap. (1)

N. Saito, K. Takahashi, and Y. Yunoki, J. Phys. Soc. Jap. 22, 219 (1967); Y. Yamakawa, Modern Theory of Polymer Solutions (Harper and Row, New York, 1971).
[Crossref]

Nucl. Sci. Eng. (2)

J. R. Mika, Nucl. Sci. Eng. 11, 415 (1961); Ref. 1, p. 87.

N. J. McCormick and M. R. Mendelson, Nucl. Sci. Eng. 20, 462 (1964).

Opt. Spektrosk. (1)

Y. Y. Bobyrenko, Opt. Spektrosk. 24, 680 (1968) [Opt. Spectrosc. 24, 365 (1968)].

Z. Tech. Phys. (1)

P. Kubelka and F. Munk, Z. Tech. Phys. 12, 593 (1931); P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948); J. Opt. Soc. Am. 44, 330 (1954); P. S. Mudgett and L. W. Richards, Appl. Opt. 10, 1485 (1971).
[Crossref] [PubMed]

Other (8)

Reference 1, p. 196.

Reference 1, Ch. 5.

Reference 1, Appendix F.

Reference 1, p. 106.

Reference 1, p. 105.

S. Chandrasekhar, Radiative Transfer (Oxford, London, 1950); M. Wing, An Introduction to Transport Theory (Wiley, New York, 1962); J. O. Mingle, Nuc. Sci. Eng. 28, 177 (1967).

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison–Wesley, Reading, Mass., 1967).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 4.

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 (1)

Fig. 1
Fig. 1

Photon density, ρpt(r), due to an isotropic point source in an infinite scattering medium, plotted against the distance from the source, r, where r is in dimensionless units (see text).

Tables (1)

Tables Icon

Table I Several eigenvalues and normalization factors for Eq. (12); α = 0.

Equations (65)

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

Ω ˆ · ψ ( r , Ω ˆ ) + σ ψ ( r , Ω ˆ ) = σ c ψ ( r , Ω ˆ ) f ( Ω ˆ · Ω ˆ ) d Ω ˆ + q ( r , Ω ˆ ) v .
d Ω ˆ f ( Ω ˆ · Ω ˆ ) = 1.
Ω ˆ · Ω ˆ = cos β ,             d Ω ˆ = sin β d β d γ .
ξ = sin β cos γ ,             η = sin β sin γ .
h ( ξ , η ) = ψ [ r , Ω ˆ ( ξ , η ) ] .
h ( ξ , η ) = h ( 0 , 0 ) + ξ h ξ | ξ , η = 0 + η h η | ξ , η = 0 + ξ 2 2 2 h ξ 2 | ξ , η = 0 + ξ η 2 h ξ η | ξ , η = 0 + η 2 2 2 h η 2 | ξ , η = 0 + .
ψ ( r , Ω ˆ ) f ( Ω ˆ · Ω ˆ ) d Ω ˆ = h ( 0 , 0 ) + κ [ ( 2 ξ 2 + 2 η 2 ) h ] ξ , η = 0 + κ [ ( 2 ξ 2 + 2 η 2 ) 2 h ] ξ , η = 0 + = ψ ( r , Ω ˆ ) + κ Ω ˆ 2 ψ ( r , Ω ˆ ) + κ Ω ˆ 4 ψ ( r , Ω ˆ ) + .
κ = π 2 - 1 1 d ( cos β ) sin 2 β f ( cos β ) ,
κ = π 32 - 1 1 d ( cos β ) sin 4 β f ( cos β ) ,
Ω ˆ = ( sin θ 0 cos φ 0 , sin θ 0 sin θ 0 , cos θ 0 )
Ω ˆ 2 = μ 0 ( 1 - μ 0 2 ) μ 0 + 1 1 - μ 0 2 2 φ 0 2 ,
κ = O ( κ 2 ) .
κ 0.013 ,             κ 0.0002.
Ω ˆ · ψ ( r , Ω ˆ ) + α Ψ ( r , Ω ˆ ) = Ω ˆ 2 ψ ( r , Ω ˆ ) + q ( r , Ω ˆ ) v ,
ψ ~ g ( 1 - ν ) 1 exp [ - ( 1 - ν ) ] ,             ν = Ω ˆ · Ω ˆ = cos β .
lim 0 0 d x φ ( x ) g ( x ) = φ ( 0 )
Ω ˆ 2 = d d ν ( 1 - ν 2 ) d d ν = d d x x ( 2 - x ) d d x ,
ψ ( r . Ω ˆ ) [ 1 - 2 κ - 4 κ 2 ( 1 + 2 ) + ] .
μ ψ ( x , μ ) x + α ψ ( x , μ ) = μ ( 1 - μ 2 ) ψ ( x , μ ) μ ,
ψ ( x , μ ) = e - x φ ( μ ) ,
{ d d μ ( 1 - μ 2 ) d d μ + μ - α } φ ( μ ) = 0.
φ ( μ ) = n = 0 c n ( α ) P n ( μ ) .
[ n ( n + 1 ) + α ] c n = [ n + 1 2 n + 3 c n + 1 + n 2 n - 1 c n - 1 ] .
g n α = [ n ( n + 1 ) + α ] ( 2 n + 1 ) [ ( n - 1 ) ! ! n ! ! ] 2
a n ( α ) = n ! ! ( n - 1 ) ! ! 1 2 n + 1 c n ( α ) ,
g n α a n = [ a n + 1 + a n - 1 ] .
R n = 1 g n + 1 , α / - R n + 1 .
lim n R n 1.
R n = O ( 1 / n 2 ) ,
R n = 1 g n + 1 , α / - 1 g n + 2 , α / - 1 g n + 3 , α / - .
α = 1 g 1 , α / - 1 g 2 , α / - 1 g 3 , α / - .
φ l ( μ ) = n = 0 c n l ( α ) P n ( μ ) .
a 0 = c 0 = 1 ,
1 2 - 1 1 d μ φ l ( μ ) = 1 ,             all l .
φ l ( μ ) = φ - l ( - μ )             and             c n l = ( - 1 ) c n n , - l .
- 1 1 d μ μ φ l ( μ ) φ l ( μ ) = N l ( α ) δ l l ,
N l = - N - l .
q ( x , μ ) v = δ ( x ) δ ( μ - μ 0 ) 2 π .
ψ ( 0 + , μ ; μ 0 ) - ψ ( 0 - , μ ; μ 0 ) = δ ( μ - μ 0 ) 2 π μ .
ψ ( x , μ ; μ 0 ) = ± l = ± 1 ± b l φ l ( μ ) e - l x ,             x 0.
l = - b l φ l ( μ ) = δ ( μ - μ 0 ) 2 π μ
b l = φ l ( μ 0 ) 2 π N l .
ψ ( x , μ ) = ± 1 2 π l = ± 1 ± φ l ( μ ) N l e - l x ,             x 0.
ρ ( x ) = 2 π - 1 1 d μ ψ ( x , μ ) .
ρ pl ( x ) = 2 l = 1 e - l x N l .
ρ pt ( r ) = - 1 2 π r d d r ρ pl ( r ) = 1 π r l = 1 e - l r N l · l .
ρ pt ( r ) = κ σ P π v r k = 1 k e - k κ σ r N k .
ψ ( x , μ ) = l = 1 b l φ l ( μ ) e - l x ,             x 0.
ψ ( 0 , μ ) = l = 1 b l φ l ( μ ) = δ ( μ - μ 0 ) ;             μ , μ 0 > 0.
B l l = 0 1 d μ μ φ l φ l ,
φ l ( μ 0 ) = l = 1 b l B l l .
b l = l = 1 ( B - 1 ) l l φ l ( μ 0 ) ,
ψ ( x , μ ) = l , l = 1 ( B - 1 ) l l φ l ( μ 0 ) φ l ( μ ) e - l x ,             x 0.
ψ ( 0 , μ ) = l , l = 1 ( B - 1 ) l l φ l ( μ 0 ) φ l ( μ ) ,             μ 0.
ρ ( x ) = 2 π l , l = 1 ( B - 1 ) l l φ l ( μ 0 ) e - l x .
B l l = φ l ( 0 ) φ l ( 0 ) - φ l ( 0 ) φ 1 ( 0 ) l - l ,             l l .
- 1 = 1 = 0.
φ 0 = 1.
α / 1 = 1 / g 1 , α · 1 1 - 1 g 1 , α g 2 , α / 1 ϕ - = 1 / g 1 , α { 1 + 1 2 g 1 , α g 2 , α + O ( 1 4 ) } .
g 1 , α = 3 ( 2 + α ) , g 2 , α = 5 ( 6 + α ) / 4.
± 1 = ± ( 6 α ) 1 2 + O ( α 3 2 ) .
φ 1 ( μ ) e - 1 x = [ 1 + 3 α μ / 1 + O ( 1 2 ) ] [ 1 - 1 x + O ( 1 2 ) ] = 1 + 1 ( μ / 2 - x ) + O ( 1 2 ) .
N 1 - 1 1 d μ μ φ 1 2 ( μ ) = 1 3 1 + O ( 1 3 ) .
N l = 2 l n = 0 n ( n + 1 ) + α 2 n + 1 c n l 2 .
r d 0.25 l / κ = 0.25 / κ σ ,