Abstract

A new difference equation of radiative transfer is derived, including the angular dependence of the radiation and the particle scattering properties. The scattering by the particles in close-packed media is treated in terms of suitably defined near-field differential cross sections. The new equation agrees with the standard differential equation in application to typical atmospheric aerosols, but disagrees in application to all close-packed media. Examples are given which imply that the new equation should be significantly more accurate than the standard one in the treatment of strongly reflecting or absorbing paints, powders, and coatings.

© 1977 Optical Society of America

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  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  2. Z. Bodó, "Some Optical Properties of Luminescent Powders," Acta Phys. Hungar. 1, 135–150 (1951).
  3. P. D. Johnson, "Absolute Optical Absorption from Diffuse Reflectance," J. Opt. Soc. Am. 42, 978–981 (1952).
  4. G. T. Bauer, "Some Optical Properties of Luminescent Films Consisting of Microcrystalline Grains," Acta Phys. Hungar. 16, 333–43 (1964) (in German).
  5. J. W. Ter Vrugt, "Optical Properties of Thin Powder Layers," Philips Res. Rep. 20, 23–33 (1965).
  6. N. T. Melamed, "Optical Properties of Powders. Part I. Optical Absorption Coefficients and the Absolute Value of the Diffuse Reflectance. Part II. Properties of Luminescent Powders, " J. Appl. Phys. 34, 560–570 (1963).
  7. E. L. Simmons, "Diffuse Reflectance Spectroscopy: A Comparison of the Theories, " Appl. Opt. 14, 1380–1386 (1975).
  8. P. Kubelka and F. Munk, "Reflection Characteristics of Paints, " Z. Tech. Phys. 12, 593 (1931).
  9. A. Schuster, "Radiation Through a Foggy Atmosphere, " Astrophys. J. 21, 1–22 (1905).
  10. It is worth noting that the general solution of Eqs. (15) may contain two arbitrary but periodic functions, with period l, in place of the constants B+1, B+2. The possible oscillations represented by such nonconstant periodic functions result from the extra (infinitely many) degrees of freedom introduced in going from the correct discrete equations (12) to the continuous difference equations (15); these extra degrees of freedom are clearly spurious.
  11. S. Chandrasekhar, in Ref. 1, p. 11.
  12. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 9.
  13. This has been verified in detail by the author, but the calculation was not included in this paper, inasmuch as it is purely manipulational. The crucial point is that the Wronskian jl(ρ) nl(ρ)−nl(ρ) jl(ρ) =ρ-2, independent of l.
  14. J. E. Hansen, "Radiative Transfer by Doubling Very Thin Layers," Astrophys. J. 155, 565–573 (1969).
  15. P. S. Mudgett and L. W. Richards, "Multiple Scattering Calculations for Technology," Appl. Opt. 10, 1485–1502 (1971).
  16. F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956), Chap. 10.
  17. B. M. Herman and S. R. Brownling, "A numerical Solution to the Equation of Radiative Transfer, " J. Atmos. Sci. 22, 559–566 (1965).
  18. J. D. Lindberg and L. S. Laude, "Measurement of the Absorption Coefficient of Atmospheric Dust," Appl. Opt. 13, 1923–1927 (1974).
  19. G. H. Goedecke and M. Rahman, Final Report, Contract No. DAAG-29-76-G-0077 (unpublished), U. S. Army Research Office, P. O. Box 12211, Research Triangle Park, N. C. 27709.
  20. J. Reichman, "Determination of Absorption and Scattering Coefficients for Nonhomogeneous Media," Appl. Opt. 12, 1811–1815 (1973).

1975

1974

1973

1971

1969

J. E. Hansen, "Radiative Transfer by Doubling Very Thin Layers," Astrophys. J. 155, 565–573 (1969).

1965

B. M. Herman and S. R. Brownling, "A numerical Solution to the Equation of Radiative Transfer, " J. Atmos. Sci. 22, 559–566 (1965).

J. W. Ter Vrugt, "Optical Properties of Thin Powder Layers," Philips Res. Rep. 20, 23–33 (1965).

1964

G. T. Bauer, "Some Optical Properties of Luminescent Films Consisting of Microcrystalline Grains," Acta Phys. Hungar. 16, 333–43 (1964) (in German).

1963

N. T. Melamed, "Optical Properties of Powders. Part I. Optical Absorption Coefficients and the Absolute Value of the Diffuse Reflectance. Part II. Properties of Luminescent Powders, " J. Appl. Phys. 34, 560–570 (1963).

1952

1951

Z. Bodó, "Some Optical Properties of Luminescent Powders," Acta Phys. Hungar. 1, 135–150 (1951).

1931

P. Kubelka and F. Munk, "Reflection Characteristics of Paints, " Z. Tech. Phys. 12, 593 (1931).

1905

A. Schuster, "Radiation Through a Foggy Atmosphere, " Astrophys. J. 21, 1–22 (1905).

Bauer, G. T.

G. T. Bauer, "Some Optical Properties of Luminescent Films Consisting of Microcrystalline Grains," Acta Phys. Hungar. 16, 333–43 (1964) (in German).

Bodó, Z.

Z. Bodó, "Some Optical Properties of Luminescent Powders," Acta Phys. Hungar. 1, 135–150 (1951).

Brownling, S. R.

B. M. Herman and S. R. Brownling, "A numerical Solution to the Equation of Radiative Transfer, " J. Atmos. Sci. 22, 559–566 (1965).

Chandrasekhar, S.

S. Chandrasekhar, in Ref. 1, p. 11.

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

Goedecke, G. H.

G. H. Goedecke and M. Rahman, Final Report, Contract No. DAAG-29-76-G-0077 (unpublished), U. S. Army Research Office, P. O. Box 12211, Research Triangle Park, N. C. 27709.

Hansen, J. E.

J. E. Hansen, "Radiative Transfer by Doubling Very Thin Layers," Astrophys. J. 155, 565–573 (1969).

Herman, B. M.

B. M. Herman and S. R. Brownling, "A numerical Solution to the Equation of Radiative Transfer, " J. Atmos. Sci. 22, 559–566 (1965).

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956), Chap. 10.

Johnson, P. D.

Kubelka, P.

P. Kubelka and F. Munk, "Reflection Characteristics of Paints, " Z. Tech. Phys. 12, 593 (1931).

Laude, L. S.

Lindberg, J. D.

Melamed, N. T.

N. T. Melamed, "Optical Properties of Powders. Part I. Optical Absorption Coefficients and the Absolute Value of the Diffuse Reflectance. Part II. Properties of Luminescent Powders, " J. Appl. Phys. 34, 560–570 (1963).

Mudgett, P. S.

Munk, F.

P. Kubelka and F. Munk, "Reflection Characteristics of Paints, " Z. Tech. Phys. 12, 593 (1931).

Rahman, M.

G. H. Goedecke and M. Rahman, Final Report, Contract No. DAAG-29-76-G-0077 (unpublished), U. S. Army Research Office, P. O. Box 12211, Research Triangle Park, N. C. 27709.

Reichman, J.

Richards, L. W.

Schuster, A.

A. Schuster, "Radiation Through a Foggy Atmosphere, " Astrophys. J. 21, 1–22 (1905).

Simmons, E. L.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 9.

Vrugt, J. W. Ter

J. W. Ter Vrugt, "Optical Properties of Thin Powder Layers," Philips Res. Rep. 20, 23–33 (1965).

Acta Phys. Hungar.

G. T. Bauer, "Some Optical Properties of Luminescent Films Consisting of Microcrystalline Grains," Acta Phys. Hungar. 16, 333–43 (1964) (in German).

Z. Bodó, "Some Optical Properties of Luminescent Powders," Acta Phys. Hungar. 1, 135–150 (1951).

Appl. Opt.

Astrophys. J.

J. E. Hansen, "Radiative Transfer by Doubling Very Thin Layers," Astrophys. J. 155, 565–573 (1969).

A. Schuster, "Radiation Through a Foggy Atmosphere, " Astrophys. J. 21, 1–22 (1905).

J. Appl. Phys.

N. T. Melamed, "Optical Properties of Powders. Part I. Optical Absorption Coefficients and the Absolute Value of the Diffuse Reflectance. Part II. Properties of Luminescent Powders, " J. Appl. Phys. 34, 560–570 (1963).

J. Atmos. Sci.

B. M. Herman and S. R. Brownling, "A numerical Solution to the Equation of Radiative Transfer, " J. Atmos. Sci. 22, 559–566 (1965).

J. Opt. Soc. Am.

Philips Res. Rep.

J. W. Ter Vrugt, "Optical Properties of Thin Powder Layers," Philips Res. Rep. 20, 23–33 (1965).

Z. Tech. Phys.

P. Kubelka and F. Munk, "Reflection Characteristics of Paints, " Z. Tech. Phys. 12, 593 (1931).

Other

It is worth noting that the general solution of Eqs. (15) may contain two arbitrary but periodic functions, with period l, in place of the constants B+1, B+2. The possible oscillations represented by such nonconstant periodic functions result from the extra (infinitely many) degrees of freedom introduced in going from the correct discrete equations (12) to the continuous difference equations (15); these extra degrees of freedom are clearly spurious.

S. Chandrasekhar, in Ref. 1, p. 11.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 9.

This has been verified in detail by the author, but the calculation was not included in this paper, inasmuch as it is purely manipulational. The crucial point is that the Wronskian jl(ρ) nl(ρ)−nl(ρ) jl(ρ) =ρ-2, independent of l.

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

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956), Chap. 10.

G. H. Goedecke and M. Rahman, Final Report, Contract No. DAAG-29-76-G-0077 (unpublished), U. S. Army Research Office, P. O. Box 12211, Research Triangle Park, N. C. 27709.

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