Abstract

We present a methodology to compare results of classical radiative transfer theory against exact solutions of Maxwell theory for a high number of spheres. We calculated light propagation in a cubic scattering region (20×20×20μm3) consisting of different concentrations of polystyrene spheres in water (diameter 2μm) by an analytical solution of Maxwell theory and by a numerical solution of radiative transfer theory. The relative deviation of differential as well as total scattering cross sections obtained by both approaches was evaluated for each sphere concentration. For the considered case, we found that deviations due to radiative transfer theory remain small, even for concentrations up to ca. 20 vol. %.

© 2009 Optical Society of America

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  1. Y. Liu, R. E. Brand, V. Turzhitsky, Y. L. Kim, H. K. Roy, N. Hasabou, C. Sturgis, D. Shah, C. Hall, and V. Backman, Clin. Cancer Res. 13, 4392 (2007).
    [CrossRef] [PubMed]
  2. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).
  3. S. H. Tseng and B. Huang, Appl. Phys. Lett. 91, 051114 (2007).
    [CrossRef]
  4. J. Schäfer and A. Kienle, Opt. Lett. 33, 2413 (2008).
    [CrossRef] [PubMed]
  5. L. Roux, P. Mareschal, N. Vukadinovic, J.-B. Thibaud, and J.-J. Greffet, J. Opt. Soc. Am. A 18, 374 (2001).
    [CrossRef]
  6. Y. Xu, Appl. Opt. 34, 4573 (1995).
    [CrossRef] [PubMed]
  7. Y. Xu and B. Gustafson, code description: http://www.astro.ufl.edu/xu/codes.htm (obsolete), http://diogenes.iwt.uni-bremen.de/vt/laser/codes/Yu-linXu/description.ps (archived).
  8. Y. Xu and B. Gustafson, Astrophys. J. 513, 894 (1999).
    [CrossRef]
  9. A. Kienle, F. K. Forster, R. Diebolder, and H. Hibst, Phys. Med. Biol. 48, N7 (2003).
    [CrossRef] [PubMed]
  10. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998).
    [CrossRef]
  11. G. Mie, Ann. Phys. 330, 377 (1908).
    [CrossRef]
  12. L. Tsang, J. A. Kong, K.-H. Ding and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley-VCH, 2001).
    [CrossRef]

2008 (1)

2007 (2)

Y. Liu, R. E. Brand, V. Turzhitsky, Y. L. Kim, H. K. Roy, N. Hasabou, C. Sturgis, D. Shah, C. Hall, and V. Backman, Clin. Cancer Res. 13, 4392 (2007).
[CrossRef] [PubMed]

S. H. Tseng and B. Huang, Appl. Phys. Lett. 91, 051114 (2007).
[CrossRef]

2003 (1)

A. Kienle, F. K. Forster, R. Diebolder, and H. Hibst, Phys. Med. Biol. 48, N7 (2003).
[CrossRef] [PubMed]

2001 (1)

1999 (1)

Y. Xu and B. Gustafson, Astrophys. J. 513, 894 (1999).
[CrossRef]

1995 (1)

1908 (1)

G. Mie, Ann. Phys. 330, 377 (1908).
[CrossRef]

Ao, C. O.

L. Tsang, J. A. Kong, K.-H. Ding and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley-VCH, 2001).
[CrossRef]

Backman, V.

Y. Liu, R. E. Brand, V. Turzhitsky, Y. L. Kim, H. K. Roy, N. Hasabou, C. Sturgis, D. Shah, C. Hall, and V. Backman, Clin. Cancer Res. 13, 4392 (2007).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998).
[CrossRef]

Brand, R. E.

Y. Liu, R. E. Brand, V. Turzhitsky, Y. L. Kim, H. K. Roy, N. Hasabou, C. Sturgis, D. Shah, C. Hall, and V. Backman, Clin. Cancer Res. 13, 4392 (2007).
[CrossRef] [PubMed]

Diebolder, R.

A. Kienle, F. K. Forster, R. Diebolder, and H. Hibst, Phys. Med. Biol. 48, N7 (2003).
[CrossRef] [PubMed]

Ding, K.-H.

L. Tsang, J. A. Kong, K.-H. Ding and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley-VCH, 2001).
[CrossRef]

Forster, F. K.

A. Kienle, F. K. Forster, R. Diebolder, and H. Hibst, Phys. Med. Biol. 48, N7 (2003).
[CrossRef] [PubMed]

Greffet, J.-J.

Gustafson, B.

Y. Xu and B. Gustafson, Astrophys. J. 513, 894 (1999).
[CrossRef]

Y. Xu and B. Gustafson, code description: http://www.astro.ufl.edu/xu/codes.htm (obsolete), http://diogenes.iwt.uni-bremen.de/vt/laser/codes/Yu-linXu/description.ps (archived).

Hall, C.

Y. Liu, R. E. Brand, V. Turzhitsky, Y. L. Kim, H. K. Roy, N. Hasabou, C. Sturgis, D. Shah, C. Hall, and V. Backman, Clin. Cancer Res. 13, 4392 (2007).
[CrossRef] [PubMed]

Hasabou, N.

Y. Liu, R. E. Brand, V. Turzhitsky, Y. L. Kim, H. K. Roy, N. Hasabou, C. Sturgis, D. Shah, C. Hall, and V. Backman, Clin. Cancer Res. 13, 4392 (2007).
[CrossRef] [PubMed]

Hibst, H.

A. Kienle, F. K. Forster, R. Diebolder, and H. Hibst, Phys. Med. Biol. 48, N7 (2003).
[CrossRef] [PubMed]

Huang, B.

S. H. Tseng and B. Huang, Appl. Phys. Lett. 91, 051114 (2007).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998).
[CrossRef]

Kienle, A.

J. Schäfer and A. Kienle, Opt. Lett. 33, 2413 (2008).
[CrossRef] [PubMed]

A. Kienle, F. K. Forster, R. Diebolder, and H. Hibst, Phys. Med. Biol. 48, N7 (2003).
[CrossRef] [PubMed]

Kim, Y. L.

Y. Liu, R. E. Brand, V. Turzhitsky, Y. L. Kim, H. K. Roy, N. Hasabou, C. Sturgis, D. Shah, C. Hall, and V. Backman, Clin. Cancer Res. 13, 4392 (2007).
[CrossRef] [PubMed]

Kong, J. A.

L. Tsang, J. A. Kong, K.-H. Ding and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley-VCH, 2001).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

Liu, Y.

Y. Liu, R. E. Brand, V. Turzhitsky, Y. L. Kim, H. K. Roy, N. Hasabou, C. Sturgis, D. Shah, C. Hall, and V. Backman, Clin. Cancer Res. 13, 4392 (2007).
[CrossRef] [PubMed]

Mareschal, P.

Mie, G.

G. Mie, Ann. Phys. 330, 377 (1908).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

Roux, L.

Roy, H. K.

Y. Liu, R. E. Brand, V. Turzhitsky, Y. L. Kim, H. K. Roy, N. Hasabou, C. Sturgis, D. Shah, C. Hall, and V. Backman, Clin. Cancer Res. 13, 4392 (2007).
[CrossRef] [PubMed]

Schäfer, J.

Shah, D.

Y. Liu, R. E. Brand, V. Turzhitsky, Y. L. Kim, H. K. Roy, N. Hasabou, C. Sturgis, D. Shah, C. Hall, and V. Backman, Clin. Cancer Res. 13, 4392 (2007).
[CrossRef] [PubMed]

Sturgis, C.

Y. Liu, R. E. Brand, V. Turzhitsky, Y. L. Kim, H. K. Roy, N. Hasabou, C. Sturgis, D. Shah, C. Hall, and V. Backman, Clin. Cancer Res. 13, 4392 (2007).
[CrossRef] [PubMed]

Thibaud, J.-B.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

Tsang, L.

L. Tsang, J. A. Kong, K.-H. Ding and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley-VCH, 2001).
[CrossRef]

Tseng, S. H.

S. H. Tseng and B. Huang, Appl. Phys. Lett. 91, 051114 (2007).
[CrossRef]

Turzhitsky, V.

Y. Liu, R. E. Brand, V. Turzhitsky, Y. L. Kim, H. K. Roy, N. Hasabou, C. Sturgis, D. Shah, C. Hall, and V. Backman, Clin. Cancer Res. 13, 4392 (2007).
[CrossRef] [PubMed]

Vukadinovic, N.

Xu, Y.

Y. Xu and B. Gustafson, Astrophys. J. 513, 894 (1999).
[CrossRef]

Y. Xu, Appl. Opt. 34, 4573 (1995).
[CrossRef] [PubMed]

Y. Xu and B. Gustafson, code description: http://www.astro.ufl.edu/xu/codes.htm (obsolete), http://diogenes.iwt.uni-bremen.de/vt/laser/codes/Yu-linXu/description.ps (archived).

Ann. Phys. (1)

G. Mie, Ann. Phys. 330, 377 (1908).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. H. Tseng and B. Huang, Appl. Phys. Lett. 91, 051114 (2007).
[CrossRef]

Astrophys. J. (1)

Y. Xu and B. Gustafson, Astrophys. J. 513, 894 (1999).
[CrossRef]

Clin. Cancer Res. (1)

Y. Liu, R. E. Brand, V. Turzhitsky, Y. L. Kim, H. K. Roy, N. Hasabou, C. Sturgis, D. Shah, C. Hall, and V. Backman, Clin. Cancer Res. 13, 4392 (2007).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Phys. Med. Biol. (1)

A. Kienle, F. K. Forster, R. Diebolder, and H. Hibst, Phys. Med. Biol. 48, N7 (2003).
[CrossRef] [PubMed]

Other (4)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998).
[CrossRef]

Y. Xu and B. Gustafson, code description: http://www.astro.ufl.edu/xu/codes.htm (obsolete), http://diogenes.iwt.uni-bremen.de/vt/laser/codes/Yu-linXu/description.ps (archived).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

L. Tsang, J. A. Kong, K.-H. Ding and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley-VCH, 2001).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Differential scattering cross sections d σ d θ due to an incident wave at λ = 633 nm for different sphere concentrations. The results of GMM simulations (solid curves) and Monte Carlo simulations (dots) are compared. In (a) the sphere numbers are N = 50 , 100, and 200, in (b) N = 400 , 600, and 800.

Fig. 2
Fig. 2

(a) Relative error δ ( d σ d θ ) of GMM versus Monte Carlo results for N = 400 , 600, and 800 plotted against the scattering angle that is cropped to a range of θ = 20 ° to θ = 170 ° to exclude the sharp forward- and backscattering peaks. (b) Averaged error δ ( d σ d θ ) , plotted against number of spheres.

Fig. 3
Fig. 3

Simulated total scattering cross section against number of spheres. While uncorrected cross sections (dashed and dotted curves) differ by a factor of approximately 2, the corrected ones are in good agreement. For N 15 the curves deviate from linear increase because of multiple scattering.

Equations (5)

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E sca = e i k ( r z ) i k r ( S 2 S 3 S 4 S 1 ) E inc ,
d L ( r , s ) d s = μ t L ( r , s ) + μ s 4 π d Ω p ( s , s ) L ( r , s ) ,
δ ( d σ d θ ) = ( d σ d θ ) GMM ( d σ d θ ) MonteCarlo ( d σ d θ ) GMM × 100 % .
δ ( d σ d θ ) = 1 n st i = 1 n st | δ i ( d σ d θ ) | ,
C sca = 0 π d θ sin θ d σ d θ .

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