Abstract

We investigated the limits of applicability of the interference approximation (ITA), which is often used to calculate light fields in dispersive media with high particle concentrations. We analyzed the conditions under which the use of the ITA leads to an error in calculation of the dispersive-medium extinction coefficient equal to 10%, 15%, and 25%. The analysis was carried out by comparing the values of calculated in the ITA with the results of the calculation of in a much more exact quasi-crystalline approximation (QCA). The validity of the QCA in estimating the error of the ITA is shown by comparing the values of calculated in the ITA and the QCA with experimental data [J. Opt. Soc. Am. 72, 1317 (1982)]. Our investigation shows that the ITA can be used only at relatively small values of the particle size parameter x. When the relative refractive index of the particles is n1.8, the applicability of the ITA is limited to the case in which the particle size parameter xxmax1.5. A decrease in n leads to an increase in xmax, and for nonabsorbing particles at n1.3, xmax is approximately proportional: 1/(n-1). An increase in the imaginary part of the relative refractive index of the particles κ leads to a decrease in xmax. The smaller the n, the greater the dependence of xmax on κ.

© 1999 Optical Society of America

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References

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  1. A. P. Ivanov, Optics of Scattered Media (Nauka I Technica, Minsk, 1969), Chap. 1 (in Russian).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 1, Chap. 6.
  3. A. P. Ivanov, V. A. Loiko, V. P. Dick, Propagation of Light in Densely Packed Dispersive Media (Nauka I Technica, Minsk, 1988), Chap. 2 (in Russian).
  4. 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]
  5. B. L. Drolen, S. Kumar, C. L. Tien, “Experiments on dependent scattering of radiation,” , 1–8 (1987).
  6. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 2, Chap. 14.
  7. J. M. Ziman, Models of Disorder (Cambridge U. Press, Cambridge, UK, 1979), Chap. 4.
  8. 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]
  9. 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]
  10. S. Fraden, G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
    [CrossRef] [PubMed]
  11. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 3.
  12. M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
    [CrossRef]
  13. 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]
  14. 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]
  15. 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]
  16. L. M. Zurk, L. Tsang, K. H. Ding, D. P. Winebrenner, “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]
  17. 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]
  18. 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]
  19. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Chap. 6.
  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. Sommerfeld, Optics, Vol. 4 of Lectures on Theoretical Physics (Academic, New York, 1954), Chap. 5.
  22. F. T. S. Yu, Introduction to Diffraction, Information Processing and Holography (MIT Press, Cambridge, Mass., 1973), Chap. 4.
  23. 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]
  24. V. P. Dick, “Applicability limits of Beer’s law for dispersion media with a high concentration of particles,” Appl. Opt. 37, 4998–5004 (1998).
    [CrossRef]

1998

1995

1994

1990

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]

S. Fraden, G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[CrossRef] [PubMed]

1983

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

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]

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]

1979

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]

1958

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

1952

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

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 3.

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]

Dick, V. P.

V. P. Dick, “Applicability limits of Beer’s law for dispersion media with a high concentration of particles,” Appl. Opt. 37, 4998–5004 (1998).
[CrossRef]

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

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,” , 1–8 (1987).

Fraden, S.

S. Fraden, G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[CrossRef] [PubMed]

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. 3.

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. 6.

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 Dispersive Media (Nauka I Technica, Minsk, 1988), Chap. 2 (in Russian).

A. P. Ivanov, Optics of Scattered Media (Nauka I Technica, Minsk, 1969), Chap. 1 (in Russian).

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), Chap. 6.

Kuga, Y.

Kumar, S.

B. L. Drolen, S. Kumar, C. L. Tien, “Experiments on dependent scattering of radiation,” , 1–8 (1987).

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 Dispersive Media (Nauka I Technica, Minsk, 1988), Chap. 2 (in Russian).

Maret, G.

S. Fraden, G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[CrossRef] [PubMed]

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]

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), Chap. 6.

Sommerfeld, A.

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

Tien, C. L.

B. L. Drolen, S. Kumar, C. L. Tien, “Experiments on dependent scattering of radiation,” , 1–8 (1987).

Tsang, L.

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]

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.

Winebrenner, D. P.

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.

Ziman, J. M.

J. M. Ziman, Models of Disorder (Cambridge U. Press, Cambridge, UK, 1979), 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.

Appl. Opt.

J. Appl. Phys.

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.

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. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Rev.

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

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

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]

Phys. Rev. Lett.

S. Fraden, G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[CrossRef] [PubMed]

Radio Sci.

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

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Chap. 6.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 3.

A. P. Ivanov, Optics of Scattered Media (Nauka I Technica, Minsk, 1969), Chap. 1 (in Russian).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 1, Chap. 6.

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

B. L. Drolen, S. Kumar, C. L. Tien, “Experiments on dependent scattering of radiation,” , 1–8 (1987).

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

J. M. Ziman, Models of Disorder (Cambridge U. Press, Cambridge, UK, 1979), Chap. 4.

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.

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Figures (5)

Fig. 1
Fig. 1

Extinction coefficient normalized to the wave number k for a suspension of latex particles in water versus the particles’ volume concentration cv. Dots, experimental data of Ishimaru and Kuga (Ref. 17); solid curves, results of the calculation in the QCA; dotted curves, results of the calculations in the ITA; dashed curves, results of the calculation in the INA.

Fig. 2
Fig. 2

ITA error δ for particles with the relative refractive index m=1.2+i0 versus the particles’ concentration cv. Numbers by the curves give the value of the particle size parameter x.

Fig. 3
Fig. 3

Volume concentration of particles cv at which, for the dispersive medium consisting of nonabsorbing particles with the relative refractive index n=1.2, calculation by formulas (3) and (4) leads to errors δ equal to 0.1, 0.15, 0.2, and 0.25 (curves 1, 2, 3, and 4, respectively) as a function of the particle size parameter x. Solid curves, results of the calculation in the ITA; dashed curves, results of the calculation in the INA.

Fig. 4
Fig. 4

ITA error as a function of the particle size parameter x for particles with different values of the relative refractive index m=n+iκ. Solid curves, κ=0; squares, κ=10-4; triangles, κ=10-2.

Fig. 5
Fig. 5

Values of the particle size parameter x and the real part of the relative refractive index of particles n at which the maximum error of the interference approximation δmax is equal to 0.1, 0.15, and 0.25. Solid curves, results for the imaginary part of the relative refractive index of particles κ=0; dotted curves, results for κ=10-2.

Equations (8)

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I=I0 exp(-l),
=αnασαext,
=3cv2d{Qext+Qsca0π[1-S3(θ,cv)]p(θ)sin(θ)dθ},
=3cv2dQext,
δ=ITA(cv)-(cv)ITA(cv),
=2 Im K,
B exp(iKr1)=A exp(iKr1)+n0q(r1-r2)×σ(r1-r2)TB×exp[iK(r2-r1)]d3r2,
INA(cv)-(cv)INA(cv)=δ,

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