Abstract

This study analyzes the values of volume concentrations of scatterers at which radiation extinction in dispersion media obeys Beer’s law. The dependence of the maximum particle concentration at which Beer’s law holds on the properties of the dispersion medium is investigated. It is shown that the maximum concentration is strongly dependent on the scatterers’ parameters and varies over a wide range, from tenths to tens of percent.

© 1998 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 2, Chap. 14.
  2. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), pp. 425–575.
  3. A. P. Ivanov, V. A. Loiko, V. P. Dick, Propagation of Light in Densely Packed Disperse Media (Nauka I Technica, Minsk, 1988), Chap. 2.
  4. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 1, Chap. 7.
  5. Ref. 2, pp. 128–131.
  6. R. West, D. Gibbs, L. Tsang, A. K. Fund, “Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media,” J. Opt. Soc. Am. A 11, 1854–1858 (1994).
    [CrossRef]
  7. B. L. Drolen, S. Kumar, C. L. Tien, Experiments on dependent scattering of radiation, AIAA Paper 1485 (American Institute of Aeronautics and Astronautics, Reston, Va., 1987), pp. 1–8.
  8. U. Riebel, U. Krauter, “Extinction of radiation in sterically interacted systems of monodisperse spheres. Part 1: theory,” Part. Part. Syst. Charact. 11, 212–221 (1994).
    [CrossRef]
  9. S. W. Churchill, G. C. Clark, “Light-scattering by very dense monodispersions of latex particles,” Discuss. Faraday Soc. 30, 192–199 (1960).
    [CrossRef]
  10. C. R. Berry, “On the need to apply electromagnetic theory to optical behaviour of photographic emulsion,” Photogr. Sci. Eng. 17, 394–399 (1973).
  11. Y. Yamada, J. D. Cartigny, C. L. Tien, “Radiative transfer with dependent scattering by particles: Part 2—experimental investigation,” J. Heat Transfer 108, 614–618 (1986).
    [CrossRef]
  12. C. R. Berry, “Turbidity of monodispersions of AgBr,” J. Opt. Soc. Am. 52, 888–895 (1962).
    [CrossRef]
  13. E. A. Vedernikova, M. V. Kabanov, “Scattering of optical radiation by a system of closely spaced scatters,” Opt. Spectrosc. 37, 130–135 (1974).
  14. M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
    [CrossRef]
  15. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 8.
  16. V. N. Bringi, V. V. Varadan, V. K. Varadan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. 17, 946–952 (1982).
    [CrossRef]
  17. V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
    [CrossRef]
  18. L. M. Zurk, L. Tsang, K. H. Ding, D. P. Winerbrenner, “Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometries,” J. Opt. Soc. Am. A 12, 1772–1781 (1995).
    [CrossRef]
  19. L. Tsang, J. A. Kong, “Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatters,” J. Appl. Phys. 53, 7162–7173 (1982).
    [CrossRef]
  20. J. K. Percus, G. Y. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
    [CrossRef]
  21. A. Ishimaru, Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
    [CrossRef]
  22. P. M. Saulnier, M. P. Zinkin, G. H. Watson, “Scatterer correlation effects on photon transport in dense random media,” Phys. Rev. B 42, 2621–2623 (1990).
    [CrossRef]
  23. P. D. Kaplan, A. D. Dinsmore, A. C. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
    [CrossRef]
  24. V. G. Vereshchagin, A. N. Ponyavina, “Effect of the packing density of scattering layers on their transmission,” J. Appl. Spectrosc. 31, 140–143 (1979).
    [CrossRef]
  25. A. Sommerfeld, Optics, Vol. 4 of Lectures on Theoretical Physics (Academic, New York, 1954), Chap. 5.
  26. F. T. S. Yu, Introduction to Diffraction, Information Processing and Holography (MIT Press, Cambridge, Mass., 1973), Chap. 4.

1995 (1)

1994 (3)

P. D. Kaplan, A. D. Dinsmore, A. C. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
[CrossRef]

R. West, D. Gibbs, L. Tsang, A. K. Fund, “Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media,” J. Opt. Soc. Am. A 11, 1854–1858 (1994).
[CrossRef]

U. Riebel, U. Krauter, “Extinction of radiation in sterically interacted systems of monodisperse spheres. Part 1: theory,” Part. Part. Syst. Charact. 11, 212–221 (1994).
[CrossRef]

1990 (1)

P. M. Saulnier, M. P. Zinkin, G. H. Watson, “Scatterer correlation effects on photon transport in dense random media,” Phys. Rev. B 42, 2621–2623 (1990).
[CrossRef]

1986 (1)

Y. Yamada, J. D. Cartigny, C. L. Tien, “Radiative transfer with dependent scattering by particles: Part 2—experimental investigation,” J. Heat Transfer 108, 614–618 (1986).
[CrossRef]

1983 (1)

V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

1982 (3)

L. Tsang, J. A. Kong, “Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatters,” J. Appl. Phys. 53, 7162–7173 (1982).
[CrossRef]

V. N. Bringi, V. V. Varadan, V. K. Varadan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. 17, 946–952 (1982).
[CrossRef]

A. Ishimaru, Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

1979 (1)

V. G. Vereshchagin, A. N. Ponyavina, “Effect of the packing density of scattering layers on their transmission,” J. Appl. Spectrosc. 31, 140–143 (1979).
[CrossRef]

1974 (1)

E. A. Vedernikova, M. V. Kabanov, “Scattering of optical radiation by a system of closely spaced scatters,” Opt. Spectrosc. 37, 130–135 (1974).

1973 (1)

C. R. Berry, “On the need to apply electromagnetic theory to optical behaviour of photographic emulsion,” Photogr. Sci. Eng. 17, 394–399 (1973).

1962 (1)

1960 (1)

S. W. Churchill, G. C. Clark, “Light-scattering by very dense monodispersions of latex particles,” Discuss. Faraday Soc. 30, 192–199 (1960).
[CrossRef]

1958 (1)

J. K. Percus, G. Y. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
[CrossRef]

1952 (1)

M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[CrossRef]

Berry, C. R.

C. R. Berry, “On the need to apply electromagnetic theory to optical behaviour of photographic emulsion,” Photogr. Sci. Eng. 17, 394–399 (1973).

C. R. Berry, “Turbidity of monodispersions of AgBr,” J. Opt. Soc. Am. 52, 888–895 (1962).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 8.

Bringi, V. N.

V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

V. N. Bringi, V. V. Varadan, V. K. Varadan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. 17, 946–952 (1982).
[CrossRef]

Cartigny, J. D.

Y. Yamada, J. D. Cartigny, C. L. Tien, “Radiative transfer with dependent scattering by particles: Part 2—experimental investigation,” J. Heat Transfer 108, 614–618 (1986).
[CrossRef]

Churchill, S. W.

S. W. Churchill, G. C. Clark, “Light-scattering by very dense monodispersions of latex particles,” Discuss. Faraday Soc. 30, 192–199 (1960).
[CrossRef]

Clark, G. C.

S. W. Churchill, G. C. Clark, “Light-scattering by very dense monodispersions of latex particles,” Discuss. Faraday Soc. 30, 192–199 (1960).
[CrossRef]

Dick, V. P.

A. P. Ivanov, V. A. Loiko, V. P. Dick, Propagation of Light in Densely Packed Disperse Media (Nauka I Technica, Minsk, 1988), Chap. 2.

Ding, K. H.

Dinsmore, A. D.

P. D. Kaplan, A. D. Dinsmore, A. C. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
[CrossRef]

Drolen, B. L.

B. L. Drolen, S. Kumar, C. L. Tien, Experiments on dependent scattering of radiation, AIAA Paper 1485 (American Institute of Aeronautics and Astronautics, Reston, Va., 1987), pp. 1–8.

Fund, A. K.

Gibbs, D.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 8.

Ishimaru, A.

V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

A. Ishimaru, Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 1, Chap. 7.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 2, Chap. 14.

Ivanov, A. P.

A. P. Ivanov, V. A. Loiko, V. P. Dick, Propagation of Light in Densely Packed Disperse Media (Nauka I Technica, Minsk, 1988), Chap. 2.

Kabanov, M. V.

E. A. Vedernikova, M. V. Kabanov, “Scattering of optical radiation by a system of closely spaced scatters,” Opt. Spectrosc. 37, 130–135 (1974).

Kaplan, P. D.

P. D. Kaplan, A. D. Dinsmore, A. C. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, “Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatters,” J. Appl. Phys. 53, 7162–7173 (1982).
[CrossRef]

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), pp. 425–575.

Krauter, U.

U. Riebel, U. Krauter, “Extinction of radiation in sterically interacted systems of monodisperse spheres. Part 1: theory,” Part. Part. Syst. Charact. 11, 212–221 (1994).
[CrossRef]

Kuga, Y.

Kumar, S.

B. L. Drolen, S. Kumar, C. L. Tien, Experiments on dependent scattering of radiation, AIAA Paper 1485 (American Institute of Aeronautics and Astronautics, Reston, Va., 1987), pp. 1–8.

Lax, M.

M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[CrossRef]

Loiko, V. A.

A. P. Ivanov, V. A. Loiko, V. P. Dick, Propagation of Light in Densely Packed Disperse Media (Nauka I Technica, Minsk, 1988), Chap. 2.

Percus, J. K.

J. K. Percus, G. Y. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
[CrossRef]

Pine, D. J.

P. D. Kaplan, A. D. Dinsmore, A. C. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
[CrossRef]

Ponyavina, A. N.

V. G. Vereshchagin, A. N. Ponyavina, “Effect of the packing density of scattering layers on their transmission,” J. Appl. Spectrosc. 31, 140–143 (1979).
[CrossRef]

Riebel, U.

U. Riebel, U. Krauter, “Extinction of radiation in sterically interacted systems of monodisperse spheres. Part 1: theory,” Part. Part. Syst. Charact. 11, 212–221 (1994).
[CrossRef]

Saulnier, P. M.

P. M. Saulnier, M. P. Zinkin, G. H. Watson, “Scatterer correlation effects on photon transport in dense random media,” Phys. Rev. B 42, 2621–2623 (1990).
[CrossRef]

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), pp. 425–575.

Sommerfeld, A.

A. Sommerfeld, Optics, Vol. 4 of Lectures on Theoretical Physics (Academic, New York, 1954), Chap. 5.

Tien, C. L.

Y. Yamada, J. D. Cartigny, C. L. Tien, “Radiative transfer with dependent scattering by particles: Part 2—experimental investigation,” J. Heat Transfer 108, 614–618 (1986).
[CrossRef]

B. L. Drolen, S. Kumar, C. L. Tien, Experiments on dependent scattering of radiation, AIAA Paper 1485 (American Institute of Aeronautics and Astronautics, Reston, Va., 1987), pp. 1–8.

Tsang, L.

L. M. Zurk, L. Tsang, K. H. Ding, D. P. Winerbrenner, “Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometries,” J. Opt. Soc. Am. A 12, 1772–1781 (1995).
[CrossRef]

R. West, D. Gibbs, L. Tsang, A. K. Fund, “Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media,” J. Opt. Soc. Am. A 11, 1854–1858 (1994).
[CrossRef]

L. Tsang, J. A. Kong, “Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatters,” J. Appl. Phys. 53, 7162–7173 (1982).
[CrossRef]

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), pp. 425–575.

Varadan, V. K.

V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

V. N. Bringi, V. V. Varadan, V. K. Varadan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. 17, 946–952 (1982).
[CrossRef]

Varadan, V. V.

V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

V. N. Bringi, V. V. Varadan, V. K. Varadan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. 17, 946–952 (1982).
[CrossRef]

Vedernikova, E. A.

E. A. Vedernikova, M. V. Kabanov, “Scattering of optical radiation by a system of closely spaced scatters,” Opt. Spectrosc. 37, 130–135 (1974).

Vereshchagin, V. G.

V. G. Vereshchagin, A. N. Ponyavina, “Effect of the packing density of scattering layers on their transmission,” J. Appl. Spectrosc. 31, 140–143 (1979).
[CrossRef]

Watson, G. H.

P. M. Saulnier, M. P. Zinkin, G. H. Watson, “Scatterer correlation effects on photon transport in dense random media,” Phys. Rev. B 42, 2621–2623 (1990).
[CrossRef]

West, R.

Winerbrenner, D. P.

Yamada, Y.

Y. Yamada, J. D. Cartigny, C. L. Tien, “Radiative transfer with dependent scattering by particles: Part 2—experimental investigation,” J. Heat Transfer 108, 614–618 (1986).
[CrossRef]

Yevick, G. Y.

J. K. Percus, G. Y. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
[CrossRef]

Yodh, A. C.

P. D. Kaplan, A. D. Dinsmore, A. C. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
[CrossRef]

Yu, F. T. S.

F. T. S. Yu, Introduction to Diffraction, Information Processing and Holography (MIT Press, Cambridge, Mass., 1973), Chap. 4.

Zinkin, M. P.

P. M. Saulnier, M. P. Zinkin, G. H. Watson, “Scatterer correlation effects on photon transport in dense random media,” Phys. Rev. B 42, 2621–2623 (1990).
[CrossRef]

Zurk, L. M.

Discuss. Faraday Soc. (1)

S. W. Churchill, G. C. Clark, “Light-scattering by very dense monodispersions of latex particles,” Discuss. Faraday Soc. 30, 192–199 (1960).
[CrossRef]

J. Appl. Phys. (1)

L. Tsang, J. A. Kong, “Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatters,” J. Appl. Phys. 53, 7162–7173 (1982).
[CrossRef]

J. Appl. Spectrosc. (1)

V. G. Vereshchagin, A. N. Ponyavina, “Effect of the packing density of scattering layers on their transmission,” J. Appl. Spectrosc. 31, 140–143 (1979).
[CrossRef]

J. Heat Transfer (1)

Y. Yamada, J. D. Cartigny, C. L. Tien, “Radiative transfer with dependent scattering by particles: Part 2—experimental investigation,” J. Heat Transfer 108, 614–618 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Spectrosc. (1)

E. A. Vedernikova, M. V. Kabanov, “Scattering of optical radiation by a system of closely spaced scatters,” Opt. Spectrosc. 37, 130–135 (1974).

Part. Part. Syst. Charact. (1)

U. Riebel, U. Krauter, “Extinction of radiation in sterically interacted systems of monodisperse spheres. Part 1: theory,” Part. Part. Syst. Charact. 11, 212–221 (1994).
[CrossRef]

Photogr. Sci. Eng. (1)

C. R. Berry, “On the need to apply electromagnetic theory to optical behaviour of photographic emulsion,” Photogr. Sci. Eng. 17, 394–399 (1973).

Phys. Rev. (2)

M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[CrossRef]

J. K. Percus, G. Y. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
[CrossRef]

Phys. Rev. B (1)

P. M. Saulnier, M. P. Zinkin, G. H. Watson, “Scatterer correlation effects on photon transport in dense random media,” Phys. Rev. B 42, 2621–2623 (1990).
[CrossRef]

Phys. Rev. E (1)

P. D. Kaplan, A. D. Dinsmore, A. C. Yodh, D. J. Pine, “Diffuse-transmission spectroscopy: a structural probe of opaque colloidal mixtures,” Phys. Rev. E 50, 4827–4835 (1994).
[CrossRef]

Radio Sci. (2)

V. N. Bringi, V. V. Varadan, V. K. Varadan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. 17, 946–952 (1982).
[CrossRef]

V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

Other (9)

A. Sommerfeld, Optics, Vol. 4 of Lectures on Theoretical Physics (Academic, New York, 1954), Chap. 5.

F. T. S. Yu, Introduction to Diffraction, Information Processing and Holography (MIT Press, Cambridge, Mass., 1973), Chap. 4.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 8.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 2, Chap. 14.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), pp. 425–575.

A. P. Ivanov, V. A. Loiko, V. P. Dick, Propagation of Light in Densely Packed Disperse Media (Nauka I Technica, Minsk, 1988), Chap. 2.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 1, Chap. 7.

Ref. 2, pp. 128–131.

B. L. Drolen, S. Kumar, C. L. Tien, Experiments on dependent scattering of radiation, AIAA Paper 1485 (American Institute of Aeronautics and Astronautics, Reston, Va., 1987), pp. 1–8.

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Figures (7)

Fig. 1
Fig. 1

Ratio Im(K/ k) as a function of the volume particle concentration c v for media formed of monodispersed spherical particles. Circles, experimental data of Ishimaru and Kuga21; solid curves, results of the calculation in the quasi-crystalline approximation; dotted curves, data obtained by calculation in the independent-scattering approximation.

Fig. 2
Fig. 2

Dependences of the parameter γ on the volume concentration of particles, c v , calculated in the quasi-crystalline approximation for nonabsorbing particles with the relative refractive index m = 1.4 + i0. The numerical labels on the curves represent values of the particle size parameter x.

Fig. 3
Fig. 3

a, b, Maximum allowable concentration of particles, c v end, as a function of the parameter x| m - 1| at various δ1 for particles with the relative refractive index m = 1.2 + i0. Curves 1 and 4 correspond to δ1 = 0.05, curves 2 and 5 to δ1 = 0.02, and curves 3 and 6 to δ1 = 0.007. Solid curves correspond to the calculation in the quasi-crystalline approximation; dashed curves 1–3 and 4–6 correspond to calculations from formulas (8) and (7), respectively. Hatched regions represent ranges of c v values for which Beer’s law holds provided that δ1 = 0.007. c, Factor of extinction efficiency of a single particle, Q ext, as a function of the parameter x| m - 1| for particles with the relative refractive index m = 1.2 + i0.

Fig. 4
Fig. 4

Same as Fig. 3, but for particles with the relative refractive index m = 1.4 + i0.

Fig. 5
Fig. 5

Same as Fig. 3, but for particles with the relative refractive index m = 1.8 + i0.

Fig. 6
Fig. 6

Same as Fig. 3, but for particles with the relative refractive index m = 1.4 + i0.0001.

Fig. 7
Fig. 7

Same as Fig. 3, but for particles with the relative refractive index m = 1.4 + i0.01.

Equations (10)

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

T c = exp - 2   Im   Kl ,
Im   K = 3 Q ext c v 8 a ,
T c B = exp - l ,
T c c v end - T c B c v end T c B c v end = δ ,
| γ c v end - 1 | = δ 1 ,
γ c v = 2   Im   K = 8 3 Im   K c v a Q ext c v ,     δ 1 = δ - ln   T c B .
B   exp i Kr 1 = A   exp i kr 1 + n 0     q r 1 - r 2 × σ r 1 - r 2 TB   exp i K r 2 - r 1 d 3 r 2 ,
γ c v = 1 - Q sca Q ext 0 π 1 - S 3 θ ,   c v p θ sin θ d θ ,
γ c v = 1 1.5 Q ext c v ln 1 + 1.5 Q ext c v exp 1.5 Q ext c v .
Im   K = 1 4 a ln 1 + 1.5 Q ext c v exp 1.5 Q ext c v .

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