Abstract

An algorithm is presented based on an evolution strategy to retrieve a particle size distribution from angular light-scattering data. The analyzed intensity patterns are generated using the Mie theory, and the algorithm retrieves a series of known normal, gamma, and lognormal distributions by using the Fraunhofer approximation. The distributions scan the interval of modal size parameters 100α¯150. The numerical results show that the evolution strategy can be successfully applied to solve this kind of inverse problem, obtaining a more accurate solution than, for example, the Chin–Shifrin inversion method, and avoiding the use of a priori information concerning the domain of the distribution, commonly necessary for reconstructing the particle size distribution when this analytical inversion method is used.

© 2007 Optical Society of America

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References

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  1. H. C. Van de Hulst, Light Scattering by Small Particles, 2nd ed. (Dover, 1981).
  2. A. L. Fymat, "Analytical inversions in remote sensing of particle-size distributions. 2: Angular and spectral scattering in diffraction approximations," Appl. Opt. 17, 1677-1678 (1978).
    [CrossRef] [PubMed]
  3. J. Liu, "Essential parameters in particle sizing by integral transform inversions," Appl. Opt. 36, 5535-5545 (1997).
    [CrossRef] [PubMed]
  4. J. H. Koo and E. D. Hirleman, "Synthesis of integral transform solutions for the reconstruction of particle-size distributions from forward-scattered light," Appl. Opt. 31, 2130-2140 (1992).
    [CrossRef] [PubMed]
  5. S. D. Coston and N. George, "Particle sizing by inversion of the optical transform pattern," Appl. Opt. 30, 4785-4794 (1991).
    [CrossRef] [PubMed]
  6. J. C. Knight, D. Ball, and G. N. Robertson, "Analytical inversion for laser diffraction spectrometry giving improved resolution and accuracy in size distribution," Appl. Opt. 30, 4795-4799 (1991).
    [CrossRef] [PubMed]
  7. J. B. Riley and Y. C. Agrawal, "Sampling and inversion of data in diffraction particle sizing," Appl. Opt. 30, 4800-4817 (1991).
    [CrossRef] [PubMed]
  8. Z. Ma, H. G. Merkus, and B. Scarlett, "Particle-size analysis by laser diffraction with a complementary metal-oxide semiconductor pixel array," Appl. Opt. 39, 4547-4555 (2000).
    [CrossRef]
  9. L. C. Chow and C. L. Tien, "Inversion techniques for determining the droplet size distribution in clouds: numerical examination," Appl. Opt. 15, 378-383 (1976).
    [CrossRef] [PubMed]
  10. E. D. Hirleman, "Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature," Part. Charact. 4, 128-133 (1987).
    [CrossRef]
  11. I. Rechenberg, Cybernetic Solution Path of an Experimental Problem (Royal Aircraft Establishment, Library Translation No. 1122, August 1965).
  12. H. J. Bremermann, "Optimization through evolution and recombination," in Self-Organizing Systems, M.C.Yovits, G.T.Jacobi, and G.D.Goldstine, eds. (Spartan Books, 1962), pp. 93-106.
  13. T. Back, Evolutionary Algorithms in Theory and Practice, 1st ed. (Oxford U. Press, 1996).
  14. S. Vázquez-Montiel, J. J. Sánchez-Escobar, and O. Fuentes, "Obtaining the phase of an interferogram by use of an evolution strategy: Part I," Appl. Opt. 41, 3448-3452 (2002).
    [CrossRef] [PubMed]
  15. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, 2nd ed. (Wiley, 1983).
  16. J. Vargas-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]

2007 (1)

2002 (1)

2000 (1)

1997 (1)

1992 (1)

1991 (3)

1987 (1)

E. D. Hirleman, "Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature," Part. Charact. 4, 128-133 (1987).
[CrossRef]

1978 (1)

1976 (1)

Agrawal, Y. C.

Aguilar, J. F.

Back, T.

T. Back, Evolutionary Algorithms in Theory and Practice, 1st ed. (Oxford U. Press, 1996).

Ball, D.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, 2nd ed. (Wiley, 1983).

Bremermann, H. J.

H. J. Bremermann, "Optimization through evolution and recombination," in Self-Organizing Systems, M.C.Yovits, G.T.Jacobi, and G.D.Goldstine, eds. (Spartan Books, 1962), pp. 93-106.

Chow, L. C.

Coston, S. D.

Fuentes, O.

Fymat, A. L.

Gale, D. M.

George, N.

Hirleman, E. D.

J. H. Koo and E. D. Hirleman, "Synthesis of integral transform solutions for the reconstruction of particle-size distributions from forward-scattered light," Appl. Opt. 31, 2130-2140 (1992).
[CrossRef] [PubMed]

E. D. Hirleman, "Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature," Part. Charact. 4, 128-133 (1987).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, 2nd ed. (Wiley, 1983).

Knight, J. C.

Koo, J. H.

Liu, J.

Ma, Z.

Merkus, H. G.

Rechenberg, I.

I. Rechenberg, Cybernetic Solution Path of an Experimental Problem (Royal Aircraft Establishment, Library Translation No. 1122, August 1965).

Riley, J. B.

Robertson, G. N.

Sánchez-Escobar, J. J.

Scarlett, B.

Tien, C. L.

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles, 2nd ed. (Dover, 1981).

Vargas-Ubera, J.

Vázquez-Montiel, S.

Appl. Opt. (10)

A. L. Fymat, "Analytical inversions in remote sensing of particle-size distributions. 2: Angular and spectral scattering in diffraction approximations," Appl. Opt. 17, 1677-1678 (1978).
[CrossRef] [PubMed]

J. Liu, "Essential parameters in particle sizing by integral transform inversions," Appl. Opt. 36, 5535-5545 (1997).
[CrossRef] [PubMed]

J. H. Koo and E. D. Hirleman, "Synthesis of integral transform solutions for the reconstruction of particle-size distributions from forward-scattered light," Appl. Opt. 31, 2130-2140 (1992).
[CrossRef] [PubMed]

S. D. Coston and N. George, "Particle sizing by inversion of the optical transform pattern," Appl. Opt. 30, 4785-4794 (1991).
[CrossRef] [PubMed]

J. C. Knight, D. Ball, and G. N. Robertson, "Analytical inversion for laser diffraction spectrometry giving improved resolution and accuracy in size distribution," Appl. Opt. 30, 4795-4799 (1991).
[CrossRef] [PubMed]

J. B. Riley and Y. C. Agrawal, "Sampling and inversion of data in diffraction particle sizing," Appl. Opt. 30, 4800-4817 (1991).
[CrossRef] [PubMed]

Z. Ma, H. G. Merkus, and B. Scarlett, "Particle-size analysis by laser diffraction with a complementary metal-oxide semiconductor pixel array," Appl. Opt. 39, 4547-4555 (2000).
[CrossRef]

L. C. Chow and C. L. Tien, "Inversion techniques for determining the droplet size distribution in clouds: numerical examination," Appl. Opt. 15, 378-383 (1976).
[CrossRef] [PubMed]

S. Vázquez-Montiel, J. J. Sánchez-Escobar, and O. Fuentes, "Obtaining the phase of an interferogram by use of an evolution strategy: Part I," Appl. Opt. 41, 3448-3452 (2002).
[CrossRef] [PubMed]

J. Vargas-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]

Part. Charact. (1)

E. D. Hirleman, "Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature," Part. Charact. 4, 128-133 (1987).
[CrossRef]

Other (5)

I. Rechenberg, Cybernetic Solution Path of an Experimental Problem (Royal Aircraft Establishment, Library Translation No. 1122, August 1965).

H. J. Bremermann, "Optimization through evolution and recombination," in Self-Organizing Systems, M.C.Yovits, G.T.Jacobi, and G.D.Goldstine, eds. (Spartan Books, 1962), pp. 93-106.

T. Back, Evolutionary Algorithms in Theory and Practice, 1st ed. (Oxford U. Press, 1996).

H. C. Van de Hulst, Light Scattering by Small Particles, 2nd ed. (Dover, 1981).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, 2nd ed. (Wiley, 1983).

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Figures (4)

Fig. 1
Fig. 1

Typical behavior in the minimization of the objective function for normal, gamma, and lognormal retrieved distributions in the medium size range.

Fig. 2
Fig. 2

Comparison between the retrieved scattering pattern produced by the ES and the simulated intensity patterns calculated with Mie theory. The plots correspond to the three distributions presented in Fig. 1: (a) normal, (b) lognormal, and (c) gamma.

Fig. 3
Fig. 3

Comparison between the ES and the CS methods applied to the recuperation of a normal distribution. (a) Medium size region and (b) large size regions.

Fig. 4
Fig. 4

Recuperation with the ES from the intensity pattern generated by gamma and lognormal distributions in medium and large size ranges. (a) and (b) correspond to normal, gamma, and lognormal distributions retrieved by the ES from the intensity pattern generated by Mie theory with a gamma distribution at the medium and large sizes. (c) and (d) correspond to normal, gamma, and lognormal distributions retrieved by the ES from the intensity pattern generated by Mie theory with a lognormal distribution at the medium and large sizes.

Tables (3)

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Table 1 Shape Parameters of the Proposed Distribution Functions a

Tables Icon

Table 2 Errors for Proposed Gamma Distributions a

Tables Icon

Table 3 Errors for Proposed Lognormal Distributions a

Equations (11)

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I ( θ ) = 0 I ( θ , α , m ) f ( α ) d α .
I ( θ ) = I 0 k 2 F 2 0 α 2 J 1 2 ( α θ ) θ 2 f ( α ) d α ,
f ( α ) = 2 π k 3 F 2 α 2 0 ( α θ ) J 1 ( α θ ) Y 1 ( α θ ) d d θ [ θ 3 I ( θ ) I 0 ] d θ ,
x = [ x , p , ξ , p ] ,   where ,   p = 1 , , 5.
x , 1 mean   size   parameter ,
x , 2 standard   deviation,
x , 3 first   size   parameter ,
x , 4 second   size   parameter,
x , 5 smallest   size   parameter ,
fitness = i = 1 j [ I i I r e f i ] 2 ,
s = 1 N s i = 1 N s ( f r e f i f i ) 2 f r e f i 2 ,

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