Abstract

We propose to study the scattering properties of dense distributions of spherical scatterers by resorting to an iterative solution of the Foldy–Twersky equation for the propagation of the coherent field. As a result of the first step of the iterative procedure, the host medium is substituted by an effective medium of complex refractive index to account for the multiple-scattering processes that occur among the particles. Although we truncate the above-mentioned iterative procedure to the second step, the results of our calculations are in excellent agreement with previous experimental results of Zaccanti et al. (“Measurement of optical properties of high-density media,” to be published in Applied Optics) for the scattering coefficient of Intralipid solutions up to a volume density of 15% and show a limited disagreement at a volume density of 22%.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).
  2. M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).
  3. A. Ishimaru, Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. A 72, 1317–1320 (1982).
    [CrossRef]
  4. V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).
    [CrossRef]
  5. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  6. L. Tsang, A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. A 1, 836–839 (1984).
    [CrossRef]
  7. A. Ishimaru, L. Tsang, “Backscattering enhancement of random discrete scatterers of moderate sizes,” J. Opt. Soc. Am. A 5, 228–236 (1988).
    [CrossRef]
  8. M. I. Mishchenko, “Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics,” Appl. Opt. 41, 7114–7134 (2002).
    [CrossRef] [PubMed]
  9. L. Tsang, K. H. Ding, S. E. Shih, J. A. Kong, “Scattering of electromagnetic waves from dense distributions of spheroidal particles based on Monte Carlo simulations,” J. Opt. Soc. Am. A 15, 2660–2669 (1998).
    [CrossRef]
  10. H. G. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 30, 4507–4514 (1991).
    [CrossRef] [PubMed]
  11. J. E. Choukeife, J. P. L’Huilier, “Measurements of scattering effects within tissue-like media at two wavelengths of 632.8 nm and 680 nm,” Lasers Med. Sci. 14, 286–296 (1999).
    [CrossRef]
  12. C. J. M. Moes, M. J. C. van Gemert, W. M. Star, J. P. A. Marijnissen, S. A. Prahl, “Measurements and calculations of the energy fluence rate in the scattering and absorbing phantom at 633 nm,” Appl. Opt. 28, 2292–2296 (1989).
    [CrossRef] [PubMed]
  13. W. M. Star, “Comparing the P3-approximation with diffusion theory and Monte Carlo calculations of light propagation in a slab geometry,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Müller, D. H. Sliney, eds., Vol. ISO5 of the SPIE Institute Series (SPIE, Bellingham, Wash., 1989), pp. 146–154.
  14. W. M. Star, J. P. A. Marijnissen, “Calculating the response of isotropic light dosimetry probes as a function of tissue refractive index,” Appl. Opt. 28, 2288–2291 (1989).
    [CrossRef] [PubMed]
  15. S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Gemert, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
    [CrossRef] [PubMed]
  16. G. Zaccanti, S. Del Bianco, F. Martelli, “Measurements of optical properties of high-density media,” Appl. Opt. (to be published).
  17. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  18. I. W. Sudiarta, P. Chýlek, “Mie-scattering formalism for spherical particles embedded in an absorbing medium,” J. Opt. Soc. Am. A 18, 1275–1278 (2001).
    [CrossRef]
  19. C. F. Bohren, D. P. Gilra, “Extinction by spherical particles in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
    [CrossRef]
  20. G. Videen, R. G. Pinnick, D. Ngo, Q. Fu, P. Chýlek, “Asymmetry parameter and aggregate particles,” Appl. Opt. 37, 1104–1109 (1998).
    [CrossRef]
  21. M. I. Mishchenko, J. W. Hovenier, L. D. Travis, “Concepts, terms, notation,” in Light Scattering by Nonspherical Particles (Academic, New York, 2000), pp. 3–27.
    [CrossRef]

2002 (1)

2001 (1)

1999 (1)

J. E. Choukeife, J. P. L’Huilier, “Measurements of scattering effects within tissue-like media at two wavelengths of 632.8 nm and 680 nm,” Lasers Med. Sci. 14, 286–296 (1999).
[CrossRef]

1998 (2)

1992 (1)

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Gemert, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
[CrossRef] [PubMed]

1991 (1)

1989 (2)

1988 (1)

1984 (1)

1982 (1)

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

1979 (1)

C. F. Bohren, D. P. Gilra, “Extinction by spherical particles in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

1964 (1)

V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. P. Gilra, “Extinction by spherical particles in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

Choukeife, J. E.

J. E. Choukeife, J. P. L’Huilier, “Measurements of scattering effects within tissue-like media at two wavelengths of 632.8 nm and 680 nm,” Lasers Med. Sci. 14, 286–296 (1999).
[CrossRef]

Chýlek, P.

Del Bianco, S.

G. Zaccanti, S. Del Bianco, F. Martelli, “Measurements of optical properties of high-density media,” Appl. Opt. (to be published).

Ding, K. H.

Flock, S. T.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Gemert, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
[CrossRef] [PubMed]

Fu, Q.

Gilra, D. P.

C. F. Bohren, D. P. Gilra, “Extinction by spherical particles in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

Hovenier, J. W.

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, “Concepts, terms, notation,” in Light Scattering by Nonspherical Particles (Academic, New York, 2000), pp. 3–27.
[CrossRef]

Ishimaru, A.

A. Ishimaru, L. Tsang, “Backscattering enhancement of random discrete scatterers of moderate sizes,” J. Opt. Soc. Am. A 5, 228–236 (1988).
[CrossRef]

L. Tsang, A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. A 1, 836–839 (1984).
[CrossRef]

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

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Jacques, S. L.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Gemert, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
[CrossRef] [PubMed]

Kong, J. A.

Kuga, Y.

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

L’Huilier, J. P.

J. E. Choukeife, J. P. L’Huilier, “Measurements of scattering effects within tissue-like media at two wavelengths of 632.8 nm and 680 nm,” Lasers Med. Sci. 14, 286–296 (1999).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

Marijnissen, J. P. A.

Martelli, F.

G. Zaccanti, S. Del Bianco, F. Martelli, “Measurements of optical properties of high-density media,” Appl. Opt. (to be published).

Mishchenko, M. I.

M. I. Mishchenko, “Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics,” Appl. Opt. 41, 7114–7134 (2002).
[CrossRef] [PubMed]

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, “Concepts, terms, notation,” in Light Scattering by Nonspherical Particles (Academic, New York, 2000), pp. 3–27.
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

Moes, C. J. M.

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).

Ngo, D.

Pinnick, R. G.

Prahl, S. A.

Shih, S. E.

Star, W. M.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Gemert, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
[CrossRef] [PubMed]

W. M. Star, J. P. A. Marijnissen, “Calculating the response of isotropic light dosimetry probes as a function of tissue refractive index,” Appl. Opt. 28, 2288–2291 (1989).
[CrossRef] [PubMed]

C. J. M. Moes, M. J. C. van Gemert, W. M. Star, J. P. A. Marijnissen, S. A. Prahl, “Measurements and calculations of the energy fluence rate in the scattering and absorbing phantom at 633 nm,” Appl. Opt. 28, 2292–2296 (1989).
[CrossRef] [PubMed]

W. M. Star, “Comparing the P3-approximation with diffusion theory and Monte Carlo calculations of light propagation in a slab geometry,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Müller, D. H. Sliney, eds., Vol. ISO5 of the SPIE Institute Series (SPIE, Bellingham, Wash., 1989), pp. 146–154.

Sudiarta, I. W.

Travis, L. D.

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, “Concepts, terms, notation,” in Light Scattering by Nonspherical Particles (Academic, New York, 2000), pp. 3–27.
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

Tsang, L.

Twersky, V.

V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).
[CrossRef]

van de Hulst, H. C.

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

van Gemert, M. J. C.

van Marle, J.

van Staveren, H. G.

Videen, G.

Wilson, B. C.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Gemert, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
[CrossRef] [PubMed]

Zaccanti, G.

G. Zaccanti, S. Del Bianco, F. Martelli, “Measurements of optical properties of high-density media,” Appl. Opt. (to be published).

Appl. Opt. (5)

J. Colloid Interface Sci. (1)

C. F. Bohren, D. P. Gilra, “Extinction by spherical particles in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

J. Opt. Soc. Am. A (5)

Lasers Med. Sci. (1)

J. E. Choukeife, J. P. L’Huilier, “Measurements of scattering effects within tissue-like media at two wavelengths of 632.8 nm and 680 nm,” Lasers Med. Sci. 14, 286–296 (1999).
[CrossRef]

Lasers Surg. Med. (1)

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. C. van Gemert, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12, 510–519 (1992).
[CrossRef] [PubMed]

Proc. Symp. Appl. Math. (1)

V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).
[CrossRef]

Other (7)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

G. Zaccanti, S. Del Bianco, F. Martelli, “Measurements of optical properties of high-density media,” Appl. Opt. (to be published).

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

W. M. Star, “Comparing the P3-approximation with diffusion theory and Monte Carlo calculations of light propagation in a slab geometry,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Müller, D. H. Sliney, eds., Vol. ISO5 of the SPIE Institute Series (SPIE, Bellingham, Wash., 1989), pp. 146–154.

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, “Concepts, terms, notation,” in Light Scattering by Nonspherical Particles (Academic, New York, 2000), pp. 3–27.
[CrossRef]

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 (3)

Fig. 1
Fig. 1

Scattering coefficient μSca of Intralipid suspensions as a function of the wavelength. The dotted curve reports the results according to Mie theory (that are independent of the volume density); the results of our calculations (solid curve) are reported for a volume density ρ = 11.37 that is the actual density of the stock Intralipid-10%. The circles are the measurements of van Staveren et al.10 for a suspension of Intralipid diluted to 3%, and the triangles are the measurements of Flock et al.15 for the stock Intralipid-10%. The cross at λ = 632.8 nm reports one of the measurements of Zaccanti et al.16 at ρ = 11.37.

Fig. 2
Fig. 2

Asymmetry factor g of Intralipid suspensions as a function of the wavelength. The crosses and the solid line report the results of the calculations according to Mie theory and to our procedure at ρ = 11.37, respectively. The circles and the triangles report the estimates of van Staveren et al.10 by the diffusion approximation P15 and P3,13 respectively, for a suspension of stock Intralipid-10% diluted to 3%. The diamonds report the estimates of Flock et al.15 for a suspension of stock Intralipid-10% at ρ = 11.37.

Fig. 3
Fig. 3

Scattering coefficient μSca for a suspension of Intralipid as a function of volume density at λ = 632.8 nm. The triangles and circles refer to the results of our calculations and to the measurements of Zaccanti et al.,16 respectively.

Equations (6)

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

ψa=ϕIa+ usaψsρrsdrs,
usaψs=fkˆS, kˆIexpin0k|ra-rs||ra-rs| ψs,
ψz=expiKz,
n1=n01+2πn02k2 ρf0.
2+K2ψ=0.
μSca=ρ0i σSca,iνi,

Metrics