Abstract

The Rayleigh–Debye–Gans theory, though being an approximation, plays an important role in the study of light scattering by aggregates. Therefore, considerable research efforts have been invested in examining its applicable range. Past examinations were predominately performed in terms of the integrative properties (e.g., the scattering and absorption cross sections), with little work done in terms of the angular scattering properties (e.g., the Mueller scattering matrix). However, in practice, many techniques directly measure these angular properties, calling for a close examination of the theory’s accuracy in predicting the angular properties. We describe such an investigation, conducted under the context of soot aggregates. The results are expected to provide useful insights into the optimal design of experiments and instruments that use light scattering for particle characterization.

© 2009 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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2008 (1)

2007 (4)

L. Liu and M. I. Mishchenko, “Scattering and radiative properties of complex soot and soot-containing aggregate particles,” J. Quantum Spectrosc. Radiat. Transfer 106, 262-273(2007).
[CrossRef]

C. J. Huang, Y. F. Liu, and Z. S. Wu, “Numerical calculation of optical cross section and scattering matrix for soot aggregation particles,” Acta Phys. Sin. 56, 4068-4074 (2007).

H. Ding, J. Q. Lu, R. S. Brock, T. J. McConnell, J. F. Ojeda, K. M. Jacobs, and X. Hu, “Angle-resolved Mueller matrix study of light scattering by B-cells at three wavelengths of 442, 633, and 850 nm,” J. Biomed. Opt. 12, 034032 (2007).
[CrossRef] [PubMed]

S. S. Iyer, T. A. Litzinger, S. Lee, and R. T. Santoro, “Determination of soot scattering coefficient from extinction and three-angle scattering in a laminar diffusion flame,” Combust. Flame 149, 206-216 (2007).
[CrossRef]

2006 (2)

D. W. Mackowski, “A simplified model to predict the effects of aggregation on the absorption properties of soot particles,” J. Quantum Spectrosc. Radiat. Transfer 100, 237-249 (2006).
[CrossRef]

A. C. Garcia-Lopez, A. D. Snider, and L. H. Garcia-Rubio, “Rayleigh-Debye-Gans as a model for continuous monitoring of biological particles: Part I, assessment of theoretical limits and approximations,” Opt. Express 14, 8849-8865 (2006).
[CrossRef] [PubMed]

2005 (1)

H. Burtscher, “Physical characterization of particulate emissions from diesel engines: a review,” J. Aerosol Sci. 36, 896-932 (2005).
[CrossRef]

2004 (1)

2001 (2)

C. M. Sorensen, “Light scattering by fractal aggregates: a review,” Aerosol Sci. Technol. 35, 648-687 (2001).

B. R. Stanmore, J. F. Brilhac, and P. Gilot, “The oxidation of soot: a review of experiments, mechanisms and models,” Carbon 39, 2247-2268 (2001).
[CrossRef]

1996 (3)

1991 (1)

1978 (1)

1973 (1)

A. J. Hunt and D. R. Huffman, “New polarization-modulated light-scattering instrument,” Rev. Sci. Instrum. 44, 1753-1762(1973).
[CrossRef]

1970 (1)

Barber, P. W.

Bohren, C. F.

C. F. Bohren and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Brilhac, J. F.

B. R. Stanmore, J. F. Brilhac, and P. Gilot, “The oxidation of soot: a review of experiments, mechanisms and models,” Carbon 39, 2247-2268 (2001).
[CrossRef]

Brock, R. S.

H. Ding, J. Q. Lu, R. S. Brock, T. J. McConnell, J. F. Ojeda, K. M. Jacobs, and X. Hu, “Angle-resolved Mueller matrix study of light scattering by B-cells at three wavelengths of 442, 633, and 850 nm,” J. Biomed. Opt. 12, 034032 (2007).
[CrossRef] [PubMed]

Burtscher, H.

H. Burtscher, “Physical characterization of particulate emissions from diesel engines: a review,” J. Aerosol Sci. 36, 896-932 (2005).
[CrossRef]

Carvalho, M. G.

Ding, H.

H. Ding, J. Q. Lu, R. S. Brock, T. J. McConnell, J. F. Ojeda, K. M. Jacobs, and X. Hu, “Angle-resolved Mueller matrix study of light scattering by B-cells at three wavelengths of 442, 633, and 850 nm,” J. Biomed. Opt. 12, 034032 (2007).
[CrossRef] [PubMed]

Dobbins, R. A.

Farias, T. L.

Garcia-Lopez, A. C.

Garcia-Rubio, L. H.

Gilot, P.

B. R. Stanmore, J. F. Brilhac, and P. Gilot, “The oxidation of soot: a review of experiments, mechanisms and models,” Carbon 39, 2247-2268 (2001).
[CrossRef]

Hu, X.

H. Ding, J. Q. Lu, R. S. Brock, T. J. McConnell, J. F. Ojeda, K. M. Jacobs, and X. Hu, “Angle-resolved Mueller matrix study of light scattering by B-cells at three wavelengths of 442, 633, and 850 nm,” J. Biomed. Opt. 12, 034032 (2007).
[CrossRef] [PubMed]

Huang, C. J.

C. J. Huang, Y. F. Liu, and Z. S. Wu, “Numerical calculation of optical cross section and scattering matrix for soot aggregation particles,” Acta Phys. Sin. 56, 4068-4074 (2007).

Huffman, D. R.

A. J. Hunt and D. R. Huffman, “New polarization-modulated light-scattering instrument,” Rev. Sci. Instrum. 44, 1753-1762(1973).
[CrossRef]

Huffmann, D. R.

C. F. Bohren and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Hull, P.

Hunt, A.

Hunt, A. J.

A. J. Hunt and D. R. Huffman, “New polarization-modulated light-scattering instrument,” Rev. Sci. Instrum. 44, 1753-1762(1973).
[CrossRef]

A. J. Hunt, M. S. Quinby-Hunt, and I. G. Shepard, “Diesel exhaust particle characterization by polarized light scattering,” SAE Technical Paper 982629 (Society of Automotive Engineers, 1998).

Iyer, S. S.

S. S. Iyer, T. A. Litzinger, S. Lee, and R. T. Santoro, “Determination of soot scattering coefficient from extinction and three-angle scattering in a laminar diffusion flame,” Combust. Flame 149, 206-216 (2007).
[CrossRef]

Jacobs, K. M.

H. Ding, J. Q. Lu, R. S. Brock, T. J. McConnell, J. F. Ojeda, K. M. Jacobs, and X. Hu, “Angle-resolved Mueller matrix study of light scattering by B-cells at three wavelengths of 442, 633, and 850 nm,” J. Biomed. Opt. 12, 034032 (2007).
[CrossRef] [PubMed]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

Koylu, U. O.

Lacis, A. A.

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

Lee, S.

S. S. Iyer, T. A. Litzinger, S. Lee, and R. T. Santoro, “Determination of soot scattering coefficient from extinction and three-angle scattering in a laminar diffusion flame,” Combust. Flame 149, 206-216 (2007).
[CrossRef]

Litzinger, T. A.

S. S. Iyer, T. A. Litzinger, S. Lee, and R. T. Santoro, “Determination of soot scattering coefficient from extinction and three-angle scattering in a laminar diffusion flame,” Combust. Flame 149, 206-216 (2007).
[CrossRef]

Liu, L.

L. Liu and M. I. Mishchenko, “Scattering and radiative properties of complex soot and soot-containing aggregate particles,” J. Quantum Spectrosc. Radiat. Transfer 106, 262-273(2007).
[CrossRef]

Liu, Y. F.

C. J. Huang, Y. F. Liu, and Z. S. Wu, “Numerical calculation of optical cross section and scattering matrix for soot aggregation particles,” Acta Phys. Sin. 56, 4068-4074 (2007).

Lu, J. Q.

H. Ding, J. Q. Lu, R. S. Brock, T. J. McConnell, J. F. Ojeda, K. M. Jacobs, and X. Hu, “Angle-resolved Mueller matrix study of light scattering by B-cells at three wavelengths of 442, 633, and 850 nm,” J. Biomed. Opt. 12, 034032 (2007).
[CrossRef] [PubMed]

Ma, L.

L. Ma, “Some practical aspects of soot characterization by techniques based on light scattering,” J. Aerosol Sci. , Ref. JAEROSCI-D-08-00198.
[PubMed]

Mackowski, D. W.

D. W. Mackowski, “A simplified model to predict the effects of aggregation on the absorption properties of soot particles,” J. Quantum Spectrosc. Radiat. Transfer 100, 237-249 (2006).
[CrossRef]

D. W. Mackowski and M. I. Mishchenko, “Calculation of the T-matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266-2278 (1996).
[CrossRef]

McConnell, T. J.

H. Ding, J. Q. Lu, R. S. Brock, T. J. McConnell, J. F. Ojeda, K. M. Jacobs, and X. Hu, “Angle-resolved Mueller matrix study of light scattering by B-cells at three wavelengths of 442, 633, and 850 nm,” J. Biomed. Opt. 12, 034032 (2007).
[CrossRef] [PubMed]

Megaridis, C. M.

Mishchenko, M. I.

L. Liu and M. I. Mishchenko, “Scattering and radiative properties of complex soot and soot-containing aggregate particles,” J. Quantum Spectrosc. Radiat. Transfer 106, 262-273(2007).
[CrossRef]

D. W. Mackowski and M. I. Mishchenko, “Calculation of the T-matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266-2278 (1996).
[CrossRef]

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

Ojeda, J. F.

H. Ding, J. Q. Lu, R. S. Brock, T. J. McConnell, J. F. Ojeda, K. M. Jacobs, and X. Hu, “Angle-resolved Mueller matrix study of light scattering by B-cells at three wavelengths of 442, 633, and 850 nm,” J. Biomed. Opt. 12, 034032 (2007).
[CrossRef] [PubMed]

Olaofe, G. O.

Quinby-Hunt, M. S.

A. J. Hunt, M. S. Quinby-Hunt, and I. G. Shepard, “Diesel exhaust particle characterization by polarized light scattering,” SAE Technical Paper 982629 (Society of Automotive Engineers, 1998).

Santoro, R. T.

S. S. Iyer, T. A. Litzinger, S. Lee, and R. T. Santoro, “Determination of soot scattering coefficient from extinction and three-angle scattering in a laminar diffusion flame,” Combust. Flame 149, 206-216 (2007).
[CrossRef]

Shaddix, C. R.

K. C. Smyth and C. R. Shaddix, “The elusive history of m˜=1.57−0.56i for the refractive index of soot,” Combust. Flame 107, 314-320 (1996).
[CrossRef]

Shepard, I. G.

A. J. Hunt, M. S. Quinby-Hunt, and I. G. Shepard, “Diesel exhaust particle characterization by polarized light scattering,” SAE Technical Paper 982629 (Society of Automotive Engineers, 1998).

Shepherd, I.

Smyth, K. C.

K. C. Smyth and C. R. Shaddix, “The elusive history of m˜=1.57−0.56i for the refractive index of soot,” Combust. Flame 107, 314-320 (1996).
[CrossRef]

Snider, A. D.

Sorensen, C. M.

C. M. Sorensen, “Light scattering by fractal aggregates: a review,” Aerosol Sci. Technol. 35, 648-687 (2001).

Stanmore, B. R.

B. R. Stanmore, J. F. Brilhac, and P. Gilot, “The oxidation of soot: a review of experiments, mechanisms and models,” Carbon 39, 2247-2268 (2001).
[CrossRef]

Travis, L. D.

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

Wang, D. S.

Wu, Z. S.

C. J. Huang, Y. F. Liu, and Z. S. Wu, “Numerical calculation of optical cross section and scattering matrix for soot aggregation particles,” Acta Phys. Sin. 56, 4068-4074 (2007).

Acta Phys. Sin. (1)

C. J. Huang, Y. F. Liu, and Z. S. Wu, “Numerical calculation of optical cross section and scattering matrix for soot aggregation particles,” Acta Phys. Sin. 56, 4068-4074 (2007).

Aerosol Sci. Technol. (1)

C. M. Sorensen, “Light scattering by fractal aggregates: a review,” Aerosol Sci. Technol. 35, 648-687 (2001).

Appl. Opt. (5)

Carbon (1)

B. R. Stanmore, J. F. Brilhac, and P. Gilot, “The oxidation of soot: a review of experiments, mechanisms and models,” Carbon 39, 2247-2268 (2001).
[CrossRef]

Combust. Flame (2)

K. C. Smyth and C. R. Shaddix, “The elusive history of m˜=1.57−0.56i for the refractive index of soot,” Combust. Flame 107, 314-320 (1996).
[CrossRef]

S. S. Iyer, T. A. Litzinger, S. Lee, and R. T. Santoro, “Determination of soot scattering coefficient from extinction and three-angle scattering in a laminar diffusion flame,” Combust. Flame 149, 206-216 (2007).
[CrossRef]

J. Aerosol Sci. (2)

H. Burtscher, “Physical characterization of particulate emissions from diesel engines: a review,” J. Aerosol Sci. 36, 896-932 (2005).
[CrossRef]

L. Ma, “Some practical aspects of soot characterization by techniques based on light scattering,” J. Aerosol Sci. , Ref. JAEROSCI-D-08-00198.
[PubMed]

J. Biomed. Opt. (1)

H. Ding, J. Q. Lu, R. S. Brock, T. J. McConnell, J. F. Ojeda, K. M. Jacobs, and X. Hu, “Angle-resolved Mueller matrix study of light scattering by B-cells at three wavelengths of 442, 633, and 850 nm,” J. Biomed. Opt. 12, 034032 (2007).
[CrossRef] [PubMed]

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

J. Quantum Spectrosc. Radiat. Transfer (2)

L. Liu and M. I. Mishchenko, “Scattering and radiative properties of complex soot and soot-containing aggregate particles,” J. Quantum Spectrosc. Radiat. Transfer 106, 262-273(2007).
[CrossRef]

D. W. Mackowski, “A simplified model to predict the effects of aggregation on the absorption properties of soot particles,” J. Quantum Spectrosc. Radiat. Transfer 100, 237-249 (2006).
[CrossRef]

Opt. Express (2)

Rev. Sci. Instrum. (1)

A. J. Hunt and D. R. Huffman, “New polarization-modulated light-scattering instrument,” Rev. Sci. Instrum. 44, 1753-1762(1973).
[CrossRef]

Other (5)

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

C. F. Bohren and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds., Light Scattering by Nonspherical Particles--Theory, Measurements, and Applications (Academic, 2000).

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

A. J. Hunt, M. S. Quinby-Hunt, and I. G. Shepard, “Diesel exhaust particle characterization by polarized light scattering,” SAE Technical Paper 982629 (Society of Automotive Engineers, 1998).

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Figures (10)

Fig. 1
Fig. 1

Comparison of the scattering cross section of a homogeneous sphere calculated from the RDGA and the exact Mie theory. Region A represents an error of less than 10%; Region B, between 10% and 100%; and Region C, larger than 100%.

Fig. 2
Fig. 2

Comparison of the modulus of (a)  S 1 and (b)  S 2 calculated from the RDGA and the exact Mie theory, at an x = 3.0 and m = 1.6 .

Fig. 3
Fig. 3

Comparison of the two elements the Mueller matrix, (a)  S 11 and (b)  S 12 , calculated from the RDGA and the exact Mie theory, at an x = 3.0 and m = 1.6 .

Fig. 4
Fig. 4

Comparison of C sca of soot aggregates calculated using the RDGA and the T-matrix method ( D f = 1.82 and k f = 1.19 for all three cases). The symbols denote the values of x p at which the calculations were performed; the curves are used only as a visual aid.

Fig. 5
Fig. 5

Contour plots of the error in S 11 calculated by the RDGA relative to that by the T-matrix method. Conditions: N S = 128 , D f = 1.82 , k f = 1.19 , (a)  m = 1.57 + 0.56 i , and (b)  m = 1.95 + 0.79 i .

Fig. 6
Fig. 6

Contour plots of the error in S 11 normalized by its value θ = 0 ° calculated by the RDGA relative to that by the T-matrix method. Conditions are the same as those specified in the Fig. 5 caption.

Fig. 7
Fig. 7

Contour plots of the error in S 12 calculated by the RDGA relative to that by the T-matrix method. Conditions are the same as those specified in the Fig. 5 caption.

Fig. 8
Fig. 8

Contour plots of the error in S 12 normalized by S 11 calculated by the RDGA relative to that by the T-matrix method. Conditions are the same as those specified in the Fig. 5 caption.

Fig. 9
Fig. 9

Contour plots of the error in S 33 calculated by the RDGA relative to that by the T-matrix method. Conditions are the same as those specified in the Fig. 5 caption.

Fig. 10
Fig. 10

Contour plot of the error in S 33 normalized by S 11 calculated by the RDGA relative to that by the T-matrix method. Conditions are the same as those specified in the Fig. 5 caption.

Equations (14)

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

( E S E S ) = e i k ( r z ) i k r ( S 2 S 3 S 4 S 1 ) ( E i E i ) ,
| m 1 | 1 , 2 π λ d | m 1 | 1 ,
S 1 = i k 3 2 π ( m 2 1 ) m 2 + 2 v f ( θ , ϕ ) ,
S 2 = i k 3 2 π ( m 2 1 ) m 2 + 2 v f ( θ , ϕ ) cos θ ,
C sca = 0 2 π 0 π | X | 2 k 2 sin θ d θ d ϕ ,
X = S 2 cos ϕ e S + S 1 sin ϕ e S .
( S 11 S 12 0 0 S 12 S 11 0 0 0 0 S 33 0 0 0 0 S 33 ) ,
S 11 = 1 2 ( | S 1 | 2 + | S 2 | 2 ) ,
S 12 = 1 2 ( | S 2 | 2 | S 1 | 2 ) ,
S 33 = Re ( S 2 S 1 * ) .
f ( θ ) = g ( u ) = 3 u 3 ( sin u u cos u ) , u = 2 x sin θ 2 ,
N S = k f ( R g a ) D f ,
f ( u ) = g ( u ) i = 1 N S j = 1 N S sinc ( r i j u ) ,
error = | Y RDG Y TM | 1 2 ( | Y RDG | + | Y TM | ) × 100 % ,

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