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.

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  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).
  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).
  5. V. Twersky, J. Opt. Soc. Am. 60, 1084 (1970).
  6. C. C. Johnson, IEEE Trans. Bio-Med. Eng. 17, 129 (1970).
  7. K. M. Case, Ann. Phys. (N.Y.) 9, 1 (1960); Ref. 1.
  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).
  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).
  14. Ref. 4; V. Twersky, J. Math. Phys. 3, 724 (1962); J. Opt. Soc. Am. 52, 145 (1962).
  15. D. A. Cross and P. Latimer, Appl. Opt. 11, 1225 (1972).
  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).
  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).

Bobyrenko, Y. Y.

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

Case, K. M.

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

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

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.

D. A. Cross and P. Latimer, Appl. Opt. 11, 1225 (1972).

Duntley, S. Q.

S. Q. Duntley, J. Opt. Soc. Am. 53, 214 (1963).

Johnson, C. C.

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

Kattawar, G. W.

G. W. Kattawar and G. N. Plass, Appl. Opt. 11, 2851 (1972); Appl. Opt. 11, 2866 (1972).

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).

Latimer, P.

D. A. Cross and P. Latimer, Appl. Opt. 11, 1225 (1972).

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).

Özisik, M. W.

y. Yener and M. W. Özisik, J. Math. Phys. 13, 2013 (1972).

Plass, G. N.

G. W. Kattawar and G. N. Plass, Appl. Opt. 11, 2851 (1972); Appl. Opt. 11, 2866 (1972).

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).

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).

Twersky, V.

V. Twersky, J. Opt. Soc. Am. 60, 1084 (1970).

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).

Yener, y.

y. Yener and M. W. Özisik, J. Math. Phys. 13, 2013 (1972).

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).

Zweifel, P. F.

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

Other (22)

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

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

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

S. Q. Duntley, J. Opt. Soc. Am. 53, 214 (1963).

V. Twersky, J. Opt. Soc. Am. 60, 1084 (1970).

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

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

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

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).

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).

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

Reference 1, p. 196.

G. W. Kattawar and G. N. Plass, Appl. Opt. 11, 2851 (1972); Appl. Opt. 11, 2866 (1972).

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

D. A. Cross and P. Latimer, Appl. Opt. 11, 1225 (1972).

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).

Reference 1, Ch. 5.

Reference 1, Appendix F.

Reference 1, p. 106.

Reference 1, p. 105.

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

y. Yener and M. W. Özisik, J. Math. Phys. 13, 2013 (1972).

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