Abstract

By means of a numerical study we show particle-size distributions retrieved with the Chin–Shifrin, Phillips–Twomey, and singular value decomposition methods. Synthesized intensity data are generated using Mie theory, corresponding to unimodal normal, gamma, and lognormal distributions of spherical particles, covering the size parameter range from 1 to 250. Our results show the advantages and disadvantages of each method, as well as the range of applicability for the Fraunhofer approximation as compared to rigorous Mie theory.

© 2007 Optical Society of America

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  1. H. C. van de Hulst, Light Scattering by Small Particles, 2nd ed. (Dover 1981).
  2. J. R. Hodkinson, "Particle sizing by means of the forward scattering lobe," Appl. Opt. 5, 839-844 (1966).
    [CrossRef] [PubMed]
  3. A. R. Jones, "Error contour charts relevant to particle sizing by forward-scattered lobe methods," J. Phys. D 10, L163-L165 (1977).
    [CrossRef]
  4. 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]
  5. J. Liu, "Essential parameters in particle sizing by integral transform inversions," Appl. Opt. 36, 5535-5545 (1997).
    [CrossRef] [PubMed]
  6. 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]
  7. S. D. Coston and N. George, "Particle sizing by inversion of the optical transform pattern," Appl. Opt. 30, 4785-4794 (1991).
    [CrossRef] [PubMed]
  8. 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]
  9. J. B. Riley and Y. C. Agrawal, "Sampling and inversion of data in diffraction particle sizing," Appl. Opt. 30, 4800-4817 (1991).
    [CrossRef] [PubMed]
  10. 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]
  11. E. D. Hirleman, "Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature," Part. Part. Syst. Charact. 4, 128-133 (1987).
    [CrossRef]
  12. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).
  13. S. Twomey, "On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by quadrature," J. Assoc. Comput. Mach. 10, 97-101 (1963).
  14. D. A. Ligon, T. W. Chen, and J. B. Gillespie, "Determination of aerosol parameters from light-scattering data using an inverse Monte Carlo technique," Appl. Opt. 35, 4297-4303 (1996).
    [CrossRef] [PubMed]
  15. N. S. Mera, L. Elliott, and D. B. Ingham, "On the use of genetic algorithms for solving ill-posed problems," Inverse Probl. Eng. 11, 105-121 (2003).
    [CrossRef]
  16. M. Ye, S. Wang, Y. Lu, T. Hu, Z. Zhu, and Y. Xu, "Inversion of particle-size distribution from angular light-scattering data with genetic algorithms," Appl. Opt. 38, 2677-2685 (1999).
    [CrossRef]
  17. S. D. Coston and N. George, "Recovery of particle-size distributions by inversion of the optical transform intensity," Opt. Lett. 16, 1918-1920 (1991).
    [CrossRef] [PubMed]

2003

N. S. Mera, L. Elliott, and D. B. Ingham, "On the use of genetic algorithms for solving ill-posed problems," Inverse Probl. Eng. 11, 105-121 (2003).
[CrossRef]

1999

1997

1996

1992

1991

1987

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

1978

1977

A. R. Jones, "Error contour charts relevant to particle sizing by forward-scattered lobe methods," J. Phys. D 10, L163-L165 (1977).
[CrossRef]

1976

1966

1963

S. Twomey, "On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by quadrature," J. Assoc. Comput. Mach. 10, 97-101 (1963).

Agrawal, Y. C.

Ball, D.

Chen, T. W.

Chow, L. C.

Coston, S. D.

Elliott, L.

N. S. Mera, L. Elliott, and D. B. Ingham, "On the use of genetic algorithms for solving ill-posed problems," Inverse Probl. Eng. 11, 105-121 (2003).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Fymat, A. L.

George, N.

Gillespie, J. B.

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. Part. Syst. Charact. 4, 128-133 (1987).
[CrossRef]

Hodkinson, J. R.

Hu, T.

Ingham, D. B.

N. S. Mera, L. Elliott, and D. B. Ingham, "On the use of genetic algorithms for solving ill-posed problems," Inverse Probl. Eng. 11, 105-121 (2003).
[CrossRef]

Jones, A. R.

A. R. Jones, "Error contour charts relevant to particle sizing by forward-scattered lobe methods," J. Phys. D 10, L163-L165 (1977).
[CrossRef]

Knight, J. C.

Koo, J. H.

Ligon, D. A.

Liu, J.

Lu, Y.

Mera, N. S.

N. S. Mera, L. Elliott, and D. B. Ingham, "On the use of genetic algorithms for solving ill-posed problems," Inverse Probl. Eng. 11, 105-121 (2003).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Riley, J. B.

Robertson, G. N.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Tien, C. L.

Twomey, S.

S. Twomey, "On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by quadrature," J. Assoc. Comput. Mach. 10, 97-101 (1963).

van de Hulst, H. C.

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

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Wang, S.

Xu, Y.

Ye, M.

Zhu, Z.

Appl. Opt.

J. R. Hodkinson, "Particle sizing by means of the forward scattering lobe," Appl. Opt. 5, 839-844 (1966).
[CrossRef] [PubMed]

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. 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]

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]

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

M. Ye, S. Wang, Y. Lu, T. Hu, Z. Zhu, and Y. Xu, "Inversion of particle-size distribution from angular light-scattering data with genetic algorithms," Appl. Opt. 38, 2677-2685 (1999).
[CrossRef]

D. A. Ligon, T. W. Chen, and J. B. Gillespie, "Determination of aerosol parameters from light-scattering data using an inverse Monte Carlo technique," Appl. Opt. 35, 4297-4303 (1996).
[CrossRef] [PubMed]

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]

Inverse Probl. Eng.

N. S. Mera, L. Elliott, and D. B. Ingham, "On the use of genetic algorithms for solving ill-posed problems," Inverse Probl. Eng. 11, 105-121 (2003).
[CrossRef]

J. Assoc. Comput. Mach.

S. Twomey, "On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by quadrature," J. Assoc. Comput. Mach. 10, 97-101 (1963).

J. Phys. D

A. R. Jones, "Error contour charts relevant to particle sizing by forward-scattered lobe methods," J. Phys. D 10, L163-L165 (1977).
[CrossRef]

Opt. Lett.

Part. Part. Syst. Charact.

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

Other

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

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

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Figures (6)

Fig. 1
Fig. 1

(a), (c) Normal, gamma, and lognormal distributions centered at α 50 and α 30 , respectively. (b), (d) Intensity patterns for the normal distribution of (a) and (c), respectively.

Fig. 2
Fig. 2

Recovered distributions using the Fraunhofer approximation with the SVD and CS inversion methods. Graphs (a), (b), and (c) correspond to normal, gamma, and lognormal distributions, respectively, with modal peaks at α 50 . Graphs (d), (e), and (f) as above, for α 30 .

Fig. 3
Fig. 3

Normal distributions recovered using the Fraunhofer approximation with the SVD inversion method in the critical size interval.

Fig. 4
Fig. 4

Distributions recovered using Mie theory with the PT and SVD inversion methods. The regularization parameter γ = 10 - 18 is used throughout. Graphs (a), (d), (b), (e), and (c), (f) correspond to the normal, gamma and lognormal distributions, respectively.

Fig. 5
Fig. 5

Evolution of the regularization parameter. Plots (a) and (c) are the distributions retrieved with values of γ 0 and γ opt , respectively. In (b) the values of γ 0 = 5.9938 × 10 - 11 (∅) and γ opt = 5.9938 × 10 - 15 ( ) generate the distributions of the large particles shown in (a). Similarly in (d), the values γ 0 = 9.0564 × 10 - 21 (∅) and γ opt = 2.0 × 10 - 22 ( ) generate the distributions for the small particles shown in (c).

Fig. 6
Fig. 6

Comparison between Mie theory and the Fraunhofer approximation in the region of large particle sizes.

Tables (2)

Tables Icon

Table 1 Parameters of the Proposed Distribution Functions a

Tables Icon

Table 2 Error in the Retrieval of Fig. 4 using PT and SVD Methods with Mie Theory

Equations (137)

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

2.5 μ m
λ = 0.5 μ m
20 %
f G ( α ) = { N ( α α 0 ) ( 1 / b ) 3 ( b t ) ( 1 / b ) 2 Γ [ ( 1 / b ) 2 ]   exp [ α 0 α b t ] for   α > α 0 0 for   α α 0 ,
α = k r
k = 2 π / λ
α 0
f L ( α ) = N σ ( α α 0 ) 2 π   exp [ - ( ln ( α - α 0 A 0 ) σ 2 ) 2 ] ,
α 0
A 0
( μ )
μ = exp [ ln ( A 0 ) + σ 2 2 ] .
f N ( α ) = N σ 2 π   exp [ 1 2 ( α μ σ ) 2 ] ,
m = n p / n m = 1.5
n p
n m
I ( θ )
f ( α )
I ( θ ) = 0 I ( θ , α ) f ( α ) d α ,
I ( θ , α )
f ( α )
f ( α ) d α
α + d α
f ( α )
y = A x ,
a i j
I ( θ , α )
I ( θ ) = I 0 k 2 F 2 0 α 2 J 1 2 ( α θ ) θ 2 f ( α ) d α ,
I 0
J 1
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 ,
Y 1
θ max
Δ θ
θ min
A ( m × n )
U ( m × p )
V ( n × p )
P ( p × p )
A = U P V T ,
Q = A - 1 = V P - 1 U T .
x = Q y .
I ( θ ) = 0 I 0 ( i 1 + i 2 ) 2 k 2 R 2 f ( α ) d α ,
i 1
i 2
ϵ 2 + γ 2 B x 2 min .
ϵ A x y
·
x = ( A T A + γ H ) - 1 A T y ,
H = B T B
( A T A + γ H ) - 1
γ 0 = T r ( A T A ) T r ( H ) ,
α = 50
σ = 15
1 α 100
Δ α = 1
α = 50
σ = 15
α = 50
α = 30
α = 50
θ = 4 °
α = 30
θ = 5 °
sin   θ θ
1 α 60
10 α 30
68 %
100 %
α 45
0 ° θ 4 °
30 α 170
50 α 250
0 ° θ 2.5 °
1 α 100
30 α 50
θ min θ θ max
θ min = 0.057 °
θ max = 10 °
Δ θ = 0.057 °
θ max
θ max
θ max
θ max
θ max
θ max
θ max
Δ θ
α = 50
α = 30
α = 48
s = 1 N s i = 1 N s ( f i x i ) 2 f i 2 ,
f i
x i
N s
42 α 48
α = 46
( α = 46 )
γ = 10 - 18
1.56 × 10 21
( 5.85 × 10 19 )
α 30
γ 0
γ opt
γ opt
γ opt
γ 0 = 5.9938 × 10 - 11
γ opt = 5.9938 × 10 - 15
γ 0 = 9.0564 × 10 - 21
γ opt = 2.0 × 10 - 22
γ opt
50 α 250
α = 150
σ = 30
m = 1.5
λ = 0.6328 μ m
( Δ θ )
γ = 1.0 × 10 - 12
θ max
m = 1.5
42 α 48
α = 46
1 α 100
A 0
α 50
α 30
α 50
α 30
γ = 10 - 18
γ 0
γ opt
γ 0 = 5.9938 × 10 - 11
γ opt = 5.9938 × 10 - 15
( )
γ 0 = 9.0564 × 10 - 21
γ opt = 2.0 × 10 - 22
( )

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