Abstract

A new solution to the inversion of Fraunhofer diffraction for particle sizing was introduced. Com pared with the well-known Chin–Shifrin inversion, it is an inversion of the form of integral transform and less sensitive to noise. Simulation results with noise-contaminated data were obtained and showed that the new inversion is better than the Chin–Shifrin inversion. Especially when the particle diameter was small, the new inversion still performed well, whereas the Chin–Shifrin inversion did not converge.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. B. Zhao, Z. Yang, M. V. Johnston, H. Wang, A. S. Wexler, M. Balthasar, and M. Kraft, “Measurement and numerical simulation of soot particle size distribution functions in a laminar premixed ethylene-oxygen-argon flame,” Combust. Flame 133, 173-188 (2003).
    [CrossRef]
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    [CrossRef]

2009

2008

A. Garcia-Valenzuela, R. G. Barrera, and E. Gutierrez-Reyes, “Rigorous theoretical framework for particle sizing in turbid colloids using light refraction,” Opt. Express 16, 19741-19756(2008).
[CrossRef] [PubMed]

D. Greaves, J. Boxall, J. Mulligan, A. Montesi, J. Creek, E. Dendy Sloan, and C. A. Koh, “Measuring the particle size of a known distribution using the focused beam reflectance measurement technique,” Chem. Eng. Sci. 63, 5410-5419(2008).
[CrossRef]

N. Riefler and T. Wriedt, “Intercomparison of inversion algorithms for particle-sizing using Mie scattering,” Part. Part. Syst. Charact. 25, 216-230 (2008).
[CrossRef]

2007

D. B. Curtis, M. Aycibin, M. A. Young, V. H. Grassian, and P. D. Kleiber, “Simultaneous measurement of light-scattering properties and particle size distribution for aerosols: application to ammonium sulfate and quartz aerosol particles,” Atmos. Environ. 41, 4748-4758 (2007).
[CrossRef]

J. V. Ubera, J. F. Aguilar, and D. M. Gale, “Reconstruction of particle-size distributions from light-scattering patterns using three inversion methods,” Appl. Opt. 46, 124-132(2007).
[CrossRef]

2005

B. Ge, Z. Luan, and Q. Lu, “Solution of the particle size distribution with improved Newton algorithm,” Opt. Eng. 44, 058003 (2005).
[CrossRef]

2003

B. Zhao, Z. Yang, M. V. Johnston, H. Wang, A. S. Wexler, M. Balthasar, and M. Kraft, “Measurement and numerical simulation of soot particle size distribution functions in a laminar premixed ethylene-oxygen-argon flame,” Combust. Flame 133, 173-188 (2003).
[CrossRef]

2000

A. P. Weber, L. J. Xu, and G. Kasper, “Simultaneous in situ measurement of size, charge and velocity of single aerosol particles,” J. Aerosol Sci. 31, 1015-1016 (2000).
[CrossRef]

1997

1991

1983

1966

1956

J. R. Hatcher, “A method for solving Schlomilch's integral equation,” Am. Math. Mon. 63, 487-488 (1956).
[CrossRef]

Aguilar, J. F.

Aycibin, M.

D. B. Curtis, M. Aycibin, M. A. Young, V. H. Grassian, and P. D. Kleiber, “Simultaneous measurement of light-scattering properties and particle size distribution for aerosols: application to ammonium sulfate and quartz aerosol particles,” Atmos. Environ. 41, 4748-4758 (2007).
[CrossRef]

Balthasar, M.

B. Zhao, Z. Yang, M. V. Johnston, H. Wang, A. S. Wexler, M. Balthasar, and M. Kraft, “Measurement and numerical simulation of soot particle size distribution functions in a laminar premixed ethylene-oxygen-argon flame,” Combust. Flame 133, 173-188 (2003).
[CrossRef]

Barrera, R. G.

Boxall, J.

D. Greaves, J. Boxall, J. Mulligan, A. Montesi, J. Creek, E. Dendy Sloan, and C. A. Koh, “Measuring the particle size of a known distribution using the focused beam reflectance measurement technique,” Chem. Eng. Sci. 63, 5410-5419(2008).
[CrossRef]

Coston, S. D.

Creek, J.

D. Greaves, J. Boxall, J. Mulligan, A. Montesi, J. Creek, E. Dendy Sloan, and C. A. Koh, “Measuring the particle size of a known distribution using the focused beam reflectance measurement technique,” Chem. Eng. Sci. 63, 5410-5419(2008).
[CrossRef]

Curtis, D. B.

D. B. Curtis, M. Aycibin, M. A. Young, V. H. Grassian, and P. D. Kleiber, “Simultaneous measurement of light-scattering properties and particle size distribution for aerosols: application to ammonium sulfate and quartz aerosol particles,” Atmos. Environ. 41, 4748-4758 (2007).
[CrossRef]

Gale, D. M.

Garcia-Valenzuela, A.

Ge, B.

B. Ge, Z. Luan, and Q. Lu, “Solution of the particle size distribution with improved Newton algorithm,” Opt. Eng. 44, 058003 (2005).
[CrossRef]

George, N.

Grassian, V. H.

D. B. Curtis, M. Aycibin, M. A. Young, V. H. Grassian, and P. D. Kleiber, “Simultaneous measurement of light-scattering properties and particle size distribution for aerosols: application to ammonium sulfate and quartz aerosol particles,” Atmos. Environ. 41, 4748-4758 (2007).
[CrossRef]

Greaves, D.

D. Greaves, J. Boxall, J. Mulligan, A. Montesi, J. Creek, E. Dendy Sloan, and C. A. Koh, “Measuring the particle size of a known distribution using the focused beam reflectance measurement technique,” Chem. Eng. Sci. 63, 5410-5419(2008).
[CrossRef]

Gutierrez-Reyes, E.

Hatcher, J. R.

J. R. Hatcher, “A method for solving Schlomilch's integral equation,” Am. Math. Mon. 63, 487-488 (1956).
[CrossRef]

Hodkinson, J. R.

Jagodnicka, A. K.

Johnston, M. V.

B. Zhao, Z. Yang, M. V. Johnston, H. Wang, A. S. Wexler, M. Balthasar, and M. Kraft, “Measurement and numerical simulation of soot particle size distribution functions in a laminar premixed ethylene-oxygen-argon flame,” Combust. Flame 133, 173-188 (2003).
[CrossRef]

Karasinski, G.

Kasper, G.

A. P. Weber, L. J. Xu, and G. Kasper, “Simultaneous in situ measurement of size, charge and velocity of single aerosol particles,” J. Aerosol Sci. 31, 1015-1016 (2000).
[CrossRef]

Kleiber, P. D.

D. B. Curtis, M. Aycibin, M. A. Young, V. H. Grassian, and P. D. Kleiber, “Simultaneous measurement of light-scattering properties and particle size distribution for aerosols: application to ammonium sulfate and quartz aerosol particles,” Atmos. Environ. 41, 4748-4758 (2007).
[CrossRef]

Koh, C. A.

D. Greaves, J. Boxall, J. Mulligan, A. Montesi, J. Creek, E. Dendy Sloan, and C. A. Koh, “Measuring the particle size of a known distribution using the focused beam reflectance measurement technique,” Chem. Eng. Sci. 63, 5410-5419(2008).
[CrossRef]

Kraft, M.

B. Zhao, Z. Yang, M. V. Johnston, H. Wang, A. S. Wexler, M. Balthasar, and M. Kraft, “Measurement and numerical simulation of soot particle size distribution functions in a laminar premixed ethylene-oxygen-argon flame,” Combust. Flame 133, 173-188 (2003).
[CrossRef]

Liu, J. J.

Lu, Q.

B. Ge, Z. Luan, and Q. Lu, “Solution of the particle size distribution with improved Newton algorithm,” Opt. Eng. 44, 058003 (2005).
[CrossRef]

Luan, Z.

B. Ge, Z. Luan, and Q. Lu, “Solution of the particle size distribution with improved Newton algorithm,” Opt. Eng. 44, 058003 (2005).
[CrossRef]

Malinowski, S. P.

Montesi, A.

D. Greaves, J. Boxall, J. Mulligan, A. Montesi, J. Creek, E. Dendy Sloan, and C. A. Koh, “Measuring the particle size of a known distribution using the focused beam reflectance measurement technique,” Chem. Eng. Sci. 63, 5410-5419(2008).
[CrossRef]

Mulligan, J.

D. Greaves, J. Boxall, J. Mulligan, A. Montesi, J. Creek, E. Dendy Sloan, and C. A. Koh, “Measuring the particle size of a known distribution using the focused beam reflectance measurement technique,” Chem. Eng. Sci. 63, 5410-5419(2008).
[CrossRef]

Nakadate, S.

Posyniak, M. L.

Riefler, N.

N. Riefler and T. Wriedt, “Intercomparison of inversion algorithms for particle-sizing using Mie scattering,” Part. Part. Syst. Charact. 25, 216-230 (2008).
[CrossRef]

Saito, H.

Shifrin, K. S.

Sloan, E. Dendy

D. Greaves, J. Boxall, J. Mulligan, A. Montesi, J. Creek, E. Dendy Sloan, and C. A. Koh, “Measuring the particle size of a known distribution using the focused beam reflectance measurement technique,” Chem. Eng. Sci. 63, 5410-5419(2008).
[CrossRef]

Stacewicz, T.

Ubera, J. V.

Wang, H.

B. Zhao, Z. Yang, M. V. Johnston, H. Wang, A. S. Wexler, M. Balthasar, and M. Kraft, “Measurement and numerical simulation of soot particle size distribution functions in a laminar premixed ethylene-oxygen-argon flame,” Combust. Flame 133, 173-188 (2003).
[CrossRef]

Weber, A. P.

A. P. Weber, L. J. Xu, and G. Kasper, “Simultaneous in situ measurement of size, charge and velocity of single aerosol particles,” J. Aerosol Sci. 31, 1015-1016 (2000).
[CrossRef]

Wexler, A. S.

B. Zhao, Z. Yang, M. V. Johnston, H. Wang, A. S. Wexler, M. Balthasar, and M. Kraft, “Measurement and numerical simulation of soot particle size distribution functions in a laminar premixed ethylene-oxygen-argon flame,” Combust. Flame 133, 173-188 (2003).
[CrossRef]

Wriedt, T.

N. Riefler and T. Wriedt, “Intercomparison of inversion algorithms for particle-sizing using Mie scattering,” Part. Part. Syst. Charact. 25, 216-230 (2008).
[CrossRef]

Xu, L. J.

A. P. Weber, L. J. Xu, and G. Kasper, “Simultaneous in situ measurement of size, charge and velocity of single aerosol particles,” J. Aerosol Sci. 31, 1015-1016 (2000).
[CrossRef]

Yang, Z.

B. Zhao, Z. Yang, M. V. Johnston, H. Wang, A. S. Wexler, M. Balthasar, and M. Kraft, “Measurement and numerical simulation of soot particle size distribution functions in a laminar premixed ethylene-oxygen-argon flame,” Combust. Flame 133, 173-188 (2003).
[CrossRef]

Young, M. A.

D. B. Curtis, M. Aycibin, M. A. Young, V. H. Grassian, and P. D. Kleiber, “Simultaneous measurement of light-scattering properties and particle size distribution for aerosols: application to ammonium sulfate and quartz aerosol particles,” Atmos. Environ. 41, 4748-4758 (2007).
[CrossRef]

Zhao, B.

B. Zhao, Z. Yang, M. V. Johnston, H. Wang, A. S. Wexler, M. Balthasar, and M. Kraft, “Measurement and numerical simulation of soot particle size distribution functions in a laminar premixed ethylene-oxygen-argon flame,” Combust. Flame 133, 173-188 (2003).
[CrossRef]

Zolotov, I. G.

Am. Math. Mon.

J. R. Hatcher, “A method for solving Schlomilch's integral equation,” Am. Math. Mon. 63, 487-488 (1956).
[CrossRef]

Appl. Opt.

Atmos. Environ.

D. B. Curtis, M. Aycibin, M. A. Young, V. H. Grassian, and P. D. Kleiber, “Simultaneous measurement of light-scattering properties and particle size distribution for aerosols: application to ammonium sulfate and quartz aerosol particles,” Atmos. Environ. 41, 4748-4758 (2007).
[CrossRef]

Chem. Eng. Sci.

D. Greaves, J. Boxall, J. Mulligan, A. Montesi, J. Creek, E. Dendy Sloan, and C. A. Koh, “Measuring the particle size of a known distribution using the focused beam reflectance measurement technique,” Chem. Eng. Sci. 63, 5410-5419(2008).
[CrossRef]

Combust. Flame

B. Zhao, Z. Yang, M. V. Johnston, H. Wang, A. S. Wexler, M. Balthasar, and M. Kraft, “Measurement and numerical simulation of soot particle size distribution functions in a laminar premixed ethylene-oxygen-argon flame,” Combust. Flame 133, 173-188 (2003).
[CrossRef]

J. Aerosol Sci.

A. P. Weber, L. J. Xu, and G. Kasper, “Simultaneous in situ measurement of size, charge and velocity of single aerosol particles,” J. Aerosol Sci. 31, 1015-1016 (2000).
[CrossRef]

Opt. Eng.

B. Ge, Z. Luan, and Q. Lu, “Solution of the particle size distribution with improved Newton algorithm,” Opt. Eng. 44, 058003 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Part. Part. Syst. Charact.

N. Riefler and T. Wriedt, “Intercomparison of inversion algorithms for particle-sizing using Mie scattering,” Part. Part. Syst. Charact. 25, 216-230 (2008).
[CrossRef]

Other

H.G.Barth, Modern Methods of Particle Size Analysis (Wiley-Interscience, 1984).

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Figures (6)

Fig. 1
Fig. 1

Responses of the proposed new inversion and the Chin–Shifrin inversion to the Dirac delta function at angle θ d = 5 ° .

Fig. 2
Fig. 2

Comparison of the new inversion with the Chin–Shifrin inversion by using noise-free data. The original particle-size distributions are (a) gamma, (b) lognormal, and (c) bimodal exponential distributions.

Fig. 3
Fig. 3

Comparison of the error bar plots of the s values of the new inversion with those of the Chin–Shifrin inversion with data contaminated by multiplicative noise. One hundred repetitive measurements at each SNR m were used to generate the plots. Three cases of particle-size distributions, i.e., (a) gamma, (b) lognormal, (c) bimodal exponential distributions, are depicted.

Fig. 4
Fig. 4

Comparison of the error bar plots of the s values of new inversion with those of the Chin–Shifrin inversion by using repetitive measurements. One hundred repetitive measurements contaminated by multiplicative noise at each particle size were used to generate the plots. The values of SNR m are 24, 26, and 36 dB for (a) gamma, (b) lognormal, (c) bimodal exponential distributions, respectively.

Fig. 5
Fig. 5

Comparison of the error bar plots of the s values of new inversion with those of the Chin–Shifrin inversion with data contaminated by additive noise. One hundred repetitive measurements at each SNR a were used to generate the plots. Three particle size distributions, i.e., (a) gamma, (b) lognormal, and (c) bimodal exponential distributions, are depicted. The data are plotted in linear scales for the x axis and in logarithmic scales for the y axis.

Fig. 6
Fig. 6

Comparison of the error bar plots of the s values of new inversion with those of the Chin–Shifrin inversion by using repetitive measurements. One hundred repetitive measurements contaminated by additive noise at each particle size were used to generate the plots. Three particle-size distributions, i.e., (a) gamma, (b) lognormal, and (c) bimodal exponential distributions, obtained by using both the new inversion and the Chin–Shifrin inversion are depicted.

Equations (18)

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I ( θ ) = I 0 [ a J 1 ( x θ ) θ ] 2 .
I ( θ ) = λ 2 4 π 2 θ 2 I 0 0 x 2 J 1 2 ( x θ ) f ( x ) d x .
f ( x ) = 8 π 3 λ 2 I 0 x 0 θ J 1 ( θ x ) Y 1 ( θ x ) d d θ [ θ 3 I ( θ ) ] d θ ,
4 π 2 F 2 θ 2 λ 2 I 0 I ( θ ) = 2 π 0 π 2 0 J 2 ( 2 x θ sin φ ) x 2 f ( x ) d x d φ .
d 0 π 2 π 2 ( θ sin φ ) 3 λ 2 I 0 I ( θ sin φ 2 ) d φ d θ = 0 x J 2 ( θ x ) x f ( x ) d x .
f ( x ) = π 2 λ 2 I 0 x 0 θ J 2 ( θ x ) d [ 0 π 2 ( θ sin φ ) 3 I ( θ sin φ 2 ) d φ ] d θ d θ .
f c s ( x ) = 8 π 3 λ 2 I 0 x 0 J 1 ( θ x ) Y 1 ( θ x ) θ d d θ [ θ 3 δ ( θ θ d ) ] d θ = 8 π 3 θ d 3 λ 2 I 0 x { J 1 ( θ d x ) Y 1 ( θ d x ) + θ d 2 x [ J 0 ( θ d x ) J 2 ( θ d x ) ] Y 1 ( θ d x ) + θ d 2 x J 1 ( θ d x ) [ Y 0 ( θ d x ) Y 2 ( θ d x ) ] } ,
f new ( x ) = π 2 λ 2 I 0 x 0 θ J 2 ( θ x ) d 0 π 2 ( θ sin φ ) 3 δ [ sin φ 2 ( θ θ d ) ] d φ d θ d θ = π 3 θ d 3 2 λ 2 I 0 x [ J 2 ( θ d x ) + θ d 2 x J 1 ( θ d x ) θ d 2 x J 3 ( θ d x ) ] .
f ( x ) = 8 π 3 λ 2 I 0 x 0 J 1 ( θ x ) Y 1 ( θ x ) θ d d θ [ θ 3 I ( θ ) ] d θ = 8 π 3 λ 2 I 0 x n = 0 N θ n J 1 ( θ n x ) Y 1 ( θ n x ) Δ [ θ 3 I ( θ ) ] Δ θ Δ θ = 8 π 3 λ 2 I 0 x n = 0 N θ n J 1 ( θ n x ) Y 1 ( θ n x ) [ θ n + 1 3 I ( θ n + 1 ) θ n 3 I ( θ n ) ] ,
f ( x ) = 1 x n = 0 N θ n J 2 ( θ n x ) 0 π 2 [ I ( θ n + 1 sin φ 2 ) θ n + 1 3 I ( θ n sin φ 2 ) θ n 3 ] sin 3 φ d φ = π 4 x n = 0 N θ n J 2 ( θ n x ) m = 1 M l m [ α ( θ n + 1 , t m ) α ( θ n , t m ) ] ,
f G ( a ) = 1 Γ ( n ) a 0 n a n 1 exp [ ( a a 0 ) n ] ,
f L ( a ) = N 2 π σ ( a R 0 a 0 ) exp { [ ln ( a a 0 R 0 ) 2 σ ] 2 } ,
f B ( a ) = 1 N ( a a 0 ) n 1 exp [ ( a a 0 R 1 ) n ] + 2 5 N ( a a 0 ) n 1 exp [ ( a a 0 R 2 ) n ] .
s = 1 N s i = 1 N s ( f i f ref i ) 2 1 N s i = 1 N s f ref i 2 = i = 1 N s ( f i f ref i ) 2 i = 1 N s f ref i 2 ,
P i = P 0 , i + n i ,
SNR m = 20 log ( P N ) .
C j = C 0 , j + n ,
SNR a = 20 log ( C N ) ,

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