Abstract

Dynamic ligh- scattering signals in the current detection mode are normally analyzed in the frequency domain. We propose an optimized inversion procedure for retrieval of the size distributions of the measured particles from such signals, including proper signal preprocessing and application of a Chahine-like relaxation technique, to solve a matrix equation. As for the results of the numerical experiments, the median value of the index of the discrepancy between the output distribution and the true one was mostly under 25% with 2% rms noise on the power spectral density of raw signals.

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References

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  1. B. Chu, Laser Light Scattering, 2nd ed. (Academic, 1991).
  2. G. Popescu and A. Dogariu, Opt. Lett. 26, 551 (2001).
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  3. G. C. Fletcher and J. I. Harnett, Aust. J. Phys. 34, 575 (1981).
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    [CrossRef]
  5. D. E. Chauveau, A. C. M. Vanrooij, and F. H. Ruymgaart, Adv. in Appl. Math. 15, 186 (1994).
    [CrossRef]
  6. E. Trakhovsky and E. P. Shettle, J. Opt. Soc. Am. A 2, 2054 (1985).
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  7. M. Tröbs and G. Heinzel, Measurement 39, 120 (2006).
    [CrossRef]
  8. F. Ferri, G. Righini, and E. Paganini, Appl. Opt. 36, 7539 (1997).
    [CrossRef]
  9. M. T. Chahine, J. Opt. Soc. Am. 58, 1634 (1968).
    [CrossRef]
  10. A. A. Timchenko, N. B. Griko, and I. N. Serdyuk, Biopolymers 29, 303 (1990).
    [CrossRef]

2006 (1)

M. Tröbs and G. Heinzel, Measurement 39, 120 (2006).
[CrossRef]

2001 (1)

1997 (1)

1994 (1)

D. E. Chauveau, A. C. M. Vanrooij, and F. H. Ruymgaart, Adv. in Appl. Math. 15, 186 (1994).
[CrossRef]

1990 (1)

A. A. Timchenko, N. B. Griko, and I. N. Serdyuk, Biopolymers 29, 303 (1990).
[CrossRef]

1985 (1)

1981 (1)

G. C. Fletcher and J. I. Harnett, Aust. J. Phys. 34, 575 (1981).

1979 (1)

S. W. Provencher, Die Makrom. Chemie 180, 201 (1979).
[CrossRef]

1968 (1)

Chahine, M. T.

Chauveau, D. E.

D. E. Chauveau, A. C. M. Vanrooij, and F. H. Ruymgaart, Adv. in Appl. Math. 15, 186 (1994).
[CrossRef]

Chu, B.

B. Chu, Laser Light Scattering, 2nd ed. (Academic, 1991).

Dogariu, A.

Ferri, F.

Fletcher, G. C.

G. C. Fletcher and J. I. Harnett, Aust. J. Phys. 34, 575 (1981).

Griko, N. B.

A. A. Timchenko, N. B. Griko, and I. N. Serdyuk, Biopolymers 29, 303 (1990).
[CrossRef]

Harnett, J. I.

G. C. Fletcher and J. I. Harnett, Aust. J. Phys. 34, 575 (1981).

Heinzel, G.

M. Tröbs and G. Heinzel, Measurement 39, 120 (2006).
[CrossRef]

Paganini, E.

Popescu, G.

Provencher, S. W.

S. W. Provencher, Die Makrom. Chemie 180, 201 (1979).
[CrossRef]

Righini, G.

Ruymgaart, F. H.

D. E. Chauveau, A. C. M. Vanrooij, and F. H. Ruymgaart, Adv. in Appl. Math. 15, 186 (1994).
[CrossRef]

Serdyuk, I. N.

A. A. Timchenko, N. B. Griko, and I. N. Serdyuk, Biopolymers 29, 303 (1990).
[CrossRef]

Shettle, E. P.

Timchenko, A. A.

A. A. Timchenko, N. B. Griko, and I. N. Serdyuk, Biopolymers 29, 303 (1990).
[CrossRef]

Trakhovsky, E.

Tröbs, M.

M. Tröbs and G. Heinzel, Measurement 39, 120 (2006).
[CrossRef]

Vanrooij, A. C. M.

D. E. Chauveau, A. C. M. Vanrooij, and F. H. Ruymgaart, Adv. in Appl. Math. 15, 186 (1994).
[CrossRef]

Adv. in Appl. Math. (1)

D. E. Chauveau, A. C. M. Vanrooij, and F. H. Ruymgaart, Adv. in Appl. Math. 15, 186 (1994).
[CrossRef]

Appl. Opt. (1)

Aust. J. Phys. (1)

G. C. Fletcher and J. I. Harnett, Aust. J. Phys. 34, 575 (1981).

Biopolymers (1)

A. A. Timchenko, N. B. Griko, and I. N. Serdyuk, Biopolymers 29, 303 (1990).
[CrossRef]

Die Makrom. Chemie (1)

S. W. Provencher, Die Makrom. Chemie 180, 201 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Measurement (1)

M. Tröbs and G. Heinzel, Measurement 39, 120 (2006).
[CrossRef]

Opt. Lett. (1)

Other (1)

B. Chu, Laser Light Scattering, 2nd ed. (Academic, 1991).

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Figures (4)

Fig. 1
Fig. 1

Power spectra over the preset channels P i F ¯ i of the signals from monodispersions.

Fig. 2
Fig. 2

Typical track records of goodness of fit, δ P . The input “true” h * ( a ) is a normal distribution whose mean is denoted by a ¯ 0 and whose coefficient of variation is denoted by cv 0 . ▵ Input: a ¯ 0 = 50 nm , cv 0 = 30 % , algorithm C1. ∘ Input: a ¯ 0 = 50 nm , cv 0 = 30 % , algorithm C2. • Input: a ¯ 0 = 200 nm , cv 0 = 30 % , algorithm C1. × Input: a ¯ 0 = 200 nm , cv 0 = 30 % , algorithm C2. δ P 1 2 i | P i * P i ( n ) | . N I : iteration count; C1/C2: previous/present version of the Chahine-like iteration.

Fig. 3
Fig. 3

δ cv 0 with the lognormal distribution model taken for h * ( a ) (median values in the noisy cases): a ¯ 0 , mean; cv 0 , coefficient of variation.

Fig. 4
Fig. 4

Output distributions with the bimodal normal distribution model taken for h * ( a ) (correspond to the median values of δ in the noisy cases). (a) a ¯ 0 · L = 42.9 nm , a ¯ 0 · R = 129 nm ; (b) a ¯ 0 · L = 33.3 nm , a ¯ 0 · R = 133 nm . a ¯ 0 · L , a ¯ 0 · R : the respective means of the two modes. Of each mode, the coefficient of variation is 25%.

Equations (7)

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S ( f ) 0 h ( a ) Γ ( 1 + 4 π 2 f 2 Γ 2 ) d a ,
P i = + ( arctan 2 π F i + 1 Γ arctan 2 π F i Γ ) × a h ( a ) d ln a .
F * ( F i F i + 1 ) 1 2 Γ 2 π .
H ( a ) H ( a i ) + [ H ( a i + 1 ) H ( a i ) ] ln a ln a i ln a i + 1 ln a i , a [ a i , a i + 1 ) .
P = KH ,
P ( n 1 ) = K H ( n 1 ) , H j ( n ) = H j ( n 1 ) i K i · j i K i · j · P i * P i ( n 1 ) , n N ,
δ i | ln a i - 1 ln a i H ( a ) d ln a a i 1 a i h * ( a ) d a | + H ( a ) d ln a + 0 + h * ( a ) d a = 1 2 i | 1 2 [ H i 1 + H i ] ln a i a i 1 a i 1 a i h * ( a ) d a | .

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