Abstract

A set of algorithms is proposed to retrieve the size of spherically symmetric particles from the measured intensity of angular scatter data. Of special interest are low-contrast particles whose real part of the index of refraction is between 1.03 and 1.09 and whose size ka is constrained so that πka16π, where k=2πλ and a is particle radius. Several algorithms are evaluated and compared that are based on either simple matching to the Mie theory predictions or inverse tomography methods. In the tomography methods, a previously proposed algorithm [Opt. Express. 15, 12217 (2007) ] was used after estimating the phase of the scattered data or adapted to use intensity-only data. In order to ensure stability, all algorithms’ performance was evaluated in the presence of moderate noise. The performance varied as a function of particle size, refractive index, and algorithm. Results suggest that a scattering device that collects only the angular scatter that is perpendicular to the polarization of incident light, usually denoted as S1, can be used to accurately estimate the size of homogeneous, low-contrast, spherical particles whose diameters are close to the wavelength of the incident light.

© 2010 Optical Society of America

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References

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2007 (1)

2006 (2)

B. Shao, J. S. Jaffe, M. Chachisvilis, and S. C. Esener, “Angular resolved light scattering for discriminating among marine picoplankton: modeling and experimental measurements,” Opt. Express 14, 12473-12484 (2006).
[CrossRef] [PubMed]

V. V. Berdnik and V. A. Loiko, “Particle sizing by multiangle light-scattering data using the high-order neural networks,” J. Quant. Spectrosoc. Ra diat. Transfer 100, 55-63 (2006).
[CrossRef]

2005 (1)

K. Arrigo, “Marine microorganisms and global nutrient cycles,” Nature 437, 349-355 (2005).
[CrossRef] [PubMed]

2004 (2)

2003 (2)

1999 (2)

1998 (2)

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, and W. Vassen, “Laboratory measurements of angular distributions of light scattered by phytoplankton and silt,” Limnol. Oceanogr. 43, 1180 (1998).
[CrossRef]

S. Oshchepkov and A. Sinyuk, “Optical sizing of ultrafine metallic particles. Retrieval of particle size distribution from spectral extinction measurements,” J. Colloid Interface Sci. 208, 137-146 (1998).
[CrossRef] [PubMed]

1991 (1)

J. Everitt and I. Ludlow, “Particle sizing using methods of discrete Legendre analysis,” Biochem. Soc. Trans. 19, 504-505 (1991).
[PubMed]

1988 (1)

W. S. Cleveland and S. J. Devlin, “Locally weighted regression: an approach to regression analysis by local fitting,” J. Am. Stat. Assoc. 83, 596-610 (1988).
[CrossRef]

1982 (1)

Alexander, M.

Aptowicz, K.

Arrigo, K.

K. Arrigo, “Marine microorganisms and global nutrient cycles,” Nature 437, 349-355 (2005).
[CrossRef] [PubMed]

Berdnik, V. V.

V. V. Berdnik and V. A. Loiko, “Particle sizing by multiangle light-scattering data using the high-order neural networks,” J. Quant. Spectrosoc. Ra diat. Transfer 100, 55-63 (2006).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (WILEY-VCH Verlag GmbH&Co., KGaA, Weinheim, 2004).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge Univ. Press, 1999).
[PubMed]

Chachisvilis, M.

Chang, R.

Chernyshev, A.

Clare, R. M.

Cleveland, W. S.

W. S. Cleveland and S. J. Devlin, “Locally weighted regression: an approach to regression analysis by local fitting,” J. Am. Stat. Assoc. 83, 596-610 (1988).
[CrossRef]

de Haan, J. F.

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, and W. Vassen, “Laboratory measurements of angular distributions of light scattered by phytoplankton and silt,” Limnol. Oceanogr. 43, 1180 (1998).
[CrossRef]

Devlin, S. J.

W. S. Cleveland and S. J. Devlin, “Locally weighted regression: an approach to regression analysis by local fitting,” J. Am. Stat. Assoc. 83, 596-610 (1988).
[CrossRef]

Esener, S. C.

Everitt, J.

J. Everitt and I. Ludlow, “Particle sizing using methods of discrete Legendre analysis,” Biochem. Soc. Trans. 19, 504-505 (1991).
[PubMed]

Eversole, J.

Fienup, J. R.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Hallett, F.

Hart, M.

Hoekstra, A.

Hovenier, J. W.

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, and W. Vassen, “Laboratory measurements of angular distributions of light scattered by phytoplankton and silt,” Limnol. Oceanogr. 43, 1180 (1998).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (WILEY-VCH Verlag GmbH&Co., KGaA, Weinheim, 2004).

Jaffe, J.

Jaffe, J. S.

Jones, A.

A. Jones, “Light scattering for particle characterization,” Prog. Energ. Combust. 25, 1-53 (1999).
[CrossRef]

Lane, R. G.

Loiko, V. A.

V. V. Berdnik and V. A. Loiko, “Particle sizing by multiangle light-scattering data using the high-order neural networks,” J. Quant. Spectrosoc. Ra diat. Transfer 100, 55-63 (2006).
[CrossRef]

Ludlow, I.

J. Everitt and I. Ludlow, “Particle sizing using methods of discrete Legendre analysis,” Biochem. Soc. Trans. 19, 504-505 (1991).
[PubMed]

Maltsev, V.

Matzler, C.

C. Matzler, “Matlab codes for Mie scattering and absorption,” (2004), http://diogenes.iwt.uni-bremen.de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.html.

Oshchepkov, S.

S. Oshchepkov and A. Sinyuk, “Optical sizing of ultrafine metallic particles. Retrieval of particle size distribution from spectral extinction measurements,” J. Colloid Interface Sci. 208, 137-146 (1998).
[CrossRef] [PubMed]

Pan, Y.

Schreurs, R.

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, and W. Vassen, “Laboratory measurements of angular distributions of light scattered by phytoplankton and silt,” Limnol. Oceanogr. 43, 1180 (1998).
[CrossRef]

Semyanov, K.

Shao, B.

Sinyuk, A.

S. Oshchepkov and A. Sinyuk, “Optical sizing of ultrafine metallic particles. Retrieval of particle size distribution from spectral extinction measurements,” J. Colloid Interface Sci. 208, 137-146 (1998).
[CrossRef] [PubMed]

Tarasov, P.

Vassen, W.

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, and W. Vassen, “Laboratory measurements of angular distributions of light scattered by phytoplankton and silt,” Limnol. Oceanogr. 43, 1180 (1998).
[CrossRef]

Volten, H.

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, and W. Vassen, “Laboratory measurements of angular distributions of light scattered by phytoplankton and silt,” Limnol. Oceanogr. 43, 1180 (1998).
[CrossRef]

Wilson, W.

W. Wilson, “Light scattering as a diagnostic for protein crystal growth--A practical approach,” J. Struct. Biol. 142, 56-65 (2003).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge Univ. Press, 1999).
[PubMed]

Zharinov, A.

Appl. Opt. (4)

Biochem. Soc. Trans. (1)

J. Everitt and I. Ludlow, “Particle sizing using methods of discrete Legendre analysis,” Biochem. Soc. Trans. 19, 504-505 (1991).
[PubMed]

J. Am. Stat. Assoc. (1)

W. S. Cleveland and S. J. Devlin, “Locally weighted regression: an approach to regression analysis by local fitting,” J. Am. Stat. Assoc. 83, 596-610 (1988).
[CrossRef]

J. Colloid Interface Sci. (1)

S. Oshchepkov and A. Sinyuk, “Optical sizing of ultrafine metallic particles. Retrieval of particle size distribution from spectral extinction measurements,” J. Colloid Interface Sci. 208, 137-146 (1998).
[CrossRef] [PubMed]

J. Quant. Spectrosoc. Ra (1)

V. V. Berdnik and V. A. Loiko, “Particle sizing by multiangle light-scattering data using the high-order neural networks,” J. Quant. Spectrosoc. Ra diat. Transfer 100, 55-63 (2006).
[CrossRef]

J. Struct. Biol. (1)

W. Wilson, “Light scattering as a diagnostic for protein crystal growth--A practical approach,” J. Struct. Biol. 142, 56-65 (2003).
[CrossRef] [PubMed]

Limnol. Oceanogr. (1)

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, and W. Vassen, “Laboratory measurements of angular distributions of light scattered by phytoplankton and silt,” Limnol. Oceanogr. 43, 1180 (1998).
[CrossRef]

Nature (1)

K. Arrigo, “Marine microorganisms and global nutrient cycles,” Nature 437, 349-355 (2005).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Prog. Energ. Combust. (1)

A. Jones, “Light scattering for particle characterization,” Prog. Energ. Combust. 25, 1-53 (1999).
[CrossRef]

Other (5)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (WILEY-VCH Verlag GmbH&Co., KGaA, Weinheim, 2004).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge Univ. Press, 1999).
[PubMed]

The MathWorks Inc., “Local regression smoothing,” (1984-2009), http://www.mathworks.com/access/helpdesk/help/toolbox/curvefit/index.html?/access/helpdesk/help/toolbox/curvefit/bq_6yqb.html.

C. Matzler, “Matlab codes for Mie scattering and absorption,” (2004), http://diogenes.iwt.uni-bremen.de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.html.

J. W. Goodman, Statistical Optics (Wiley, 2000).

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Figures (9)

Fig. 1
Fig. 1

Reference frame for scattering and incident light.

Fig. 2
Fig. 2

Noiseless, noisy, and smoothed | S 1 - obs ( θ ) | for particle x = 4 π .

Fig. 3
Fig. 3

Real and imaginary components of the S 1 - new ( θ ) function after reflection about the horizontal axes x = 4 π .

Fig. 4
Fig. 4

Real and imaginary components of the S 1 - new ( θ ) function after reflection about the zigzag axes, dotted–dashed curve (blue online) for x = 4 π .

Fig. 5
Fig. 5

Error and standard deviations of retrieved values for size index x = { π , 2 π , 4 π , 8 π , 16 π } and m = 1.035 + 0.01 i .

Fig. 6
Fig. 6

Results of retrieving random particle sizes in the range of k a [ 0.5 π , 1.5 π ] .

Fig. 7
Fig. 7

Size estimation error and standard at different refractive index and theoretical size.

Fig. 8
Fig. 8

Average error for each algorithm.

Fig. 9
Fig. 9

Refractive index estimates obtained using the S 1 matching algorithm.

Equations (14)

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( E s E s ) = exp ( i k ( r z ) ) i k r ( S 2 S 1 ) ( E i E i ) .
S 1 - new ( θ ) = ( re ( S 1 - new ( θ ) ) , i i m a g ( S 1 - new ( θ ) ) ) ,
| S 1 - new ( θ ) | = | S 1 - obs ( θ ) | ,
re ( S 1 - new ( θ ) ) = { 2 | S 1 - obs ( θ n 1 ) | | S 1 - obs ( θ ) | { θ { ( θ n 1 , θ n ) | n = 2 , 4 , N 1 ( N is odd ) or n = 2 , 4 , N ( N is even ) } } | S 1 - obs ( θ ) | otherwise } .
i m a g ( S 1 - new ( θ ) ) = + ( | S 1 - obs ( θ ) | 2 re ( S 1 - new ( θ ) ) 2 ) 1 2 ,
w i = ( 1 | x x i d ( x ) | 3 ) 3 ,
S 1 - refl = k n θ + b n ,
k n = | S 1 - obs ( θ n ) | | S 1 - obs ( θ n 1 ) | θ n θ n 1 ,
b n = | S 1 - obs ( θ n ) | k n θ n .
r e ( S 1 - new ( θ ) ) = { 2 ( k n θ + b n ) | S 1 - obs ( θ ) | { θ { ( θ n 1 , θ n ) | n = 2 , 4 , N 1 ( N is o d d ) or n = 2 , 4 , N ( N is e v e n ) } } | S 1 - obs ( θ ) | o t h e r w i s e } .
E x s ( k o b s ) = k 2 exp ( i k d ) d α j = 1 N E x ( r j ) exp ( i ( k o b s k inc ) int r j ) .
E x s ( k obs ) = C F { E x ( r ) } w h e r e C = [ k 2 exp ( ikd ) d α ] ,
F 1 { I ( k obs ) } = C 2 E x int ( i ) E x int ( r r ) d r ,
I ( k obs ) = | S 1 - obs | 2 .

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