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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 10 — Oct. 31, 2007
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Retrieval of particle size distribution in the dependent model using the moment method

Xiaogang Sun, Hong Tang, and Jingmin Dai  »View Author Affiliations


Optics Express, Vol. 15, Issue 18, pp. 11507-11516 (2007)
http://dx.doi.org/10.1364/OE.15.011507


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Abstract

The problem of determining particle size distribution using the moment method in the spectral extinction technique is studied. The feasibility and reliability of the retrieval of spherical particle size distribution using the moment method are investigated. The single spherical particle extinction efficiency, which is derived theoretically using the Mie’s solution to Maxwell’s equation, is approximated with a higher order polynomial in order to apply the moment method. Simulation and experimental results indicate that a fairly reasonable representation of the particle size distribution can be obtained using the moment method in the dependent model algorithm. The method has advantages of simplicity, rapidity, and suitability for in-line particle size measurement.

© 2007 Optical Society of America

1. Introduction

Light scattering particle sizing techniques have been widely used during the recent years since they provide an important tool for the characterization of a large number of industrial production processes. These techniques mostly contain the spectral extinction, angle light scattering, diffraction light scattering and dynamic light scattering [1

1. F. Ferri, A. Bassini, and E. Paganini, “Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing,” Appl. Opt. 34, 5829–5839 (1995). [CrossRef] [PubMed]

4

4. B. N. Khlebtsov, L. A. Kovler, V. A. Bogatyrev, N. G. Khlebtsov, and S. Y. Shchyogolev, “Studies of phosphatidylcholine vesicles by spectroturbidimetric and dynamic light scattering methods,” J. Quant. Spectrosc. Radiat. Transf. 7980, 825–838 (2003). [CrossRef]

]. The measurement range of them ranges form nanometer to millimeter. Among these techniques, the spectral extinction technique is probably the most attractive one because it requires a simple optical layout and can be realized by adaptation of a commercial spectrophotometer. It can in fact be used for in-line monitoring of micron or sub-micron particle systems thus providing real time measurements of both particle size distributions and particle concentration. With the urgent demand for in situ particle sizing, the spectral extinction technique has grown with rapid developments and broad potential applications.

The particle size distribution can be retrieved by some inversion methods based on the measurements of extinction values at multiple wavelengths. The developments of stable methods to retrieve the particle size distribution have long been a subject of research effort. These methods can be divided into three categories [5

5. A. K. Roy and S. K. Sharma, “A simple analysis of the extinction spectrum of a size distritution of Mie particles,” J. Opt. A: Pure Appl. Opt. 7, 675–684 (2005). [CrossRef]

]. The first category is the analytic inversion method in which an approximate scattering kernel is used and the integral equation is retrieved analytically. The second category is the independent model algorithm in which no a prior information about the particle size distribution is available and the particle size distribution is retrieved by the discrete liner equations set. The third category is the dependent model algorithm in which some a prior information about the shape of the size distribution is assumed and the true particle size distribution is retrieved using a certain optimization algorithm [6

6. M. L. Arias and G. L. Frontini, “Particle size distribution retrieval from elastic light scattering measurements by a modified regularization method,” Part. Part. Syst. Charact. 23, 374–380 (2007). [CrossRef]

, 7

7. A. P. Nefedov, O. F. Petrov, and O. S. Vaulina, “Analysis of particle sizes, concentration, and refractive index in measurement of light transmittance in the forward-scattering-angle range,” Appl. Opt. 36, 1357–1366 (1997). [CrossRef] [PubMed]

].

2. Theory

The spectral extinction particle sizing technique is based on the light scattering theory. When a beam of parallel monochromatic radiation light of intensity I0 passes through a suspension of particle system (thickness L) with an refraction index different from that of the dispersant medium, scattering and absorption lead to an attenuation of the transmitted light. According to the Lambert-Beer law, if the suspension of particle system is polydisperse spherical and the multiple scattering and interaction effects can be neglected, the transmitted light intensity I is defined as follows[10

10. E. Marioth, B. Koenig, H. Krause, and S. Loebbecke, “Fast particle size and droplet size measurements in supercritical CO2,” Ind. Eng. Chem. Res. 339, 4853–4857 (2000). [CrossRef]

]:

lnI(λ)I0(λ)=32×L×ND×DminDmaxQext(λ,m,D)D×f(D)dD.
(1)

where I0(λ) is the incident light intensity at wavelength λ(i.e. the intensity of transmitted light through the suspending medium-in the absence of suspended particles), the extinction value I(λ)/I0(λ) is obtained by actual measurements, Qext(λ,m,D) is the Mie extinction efficiency of a single particle which is a complex function of particle diameter D, wavelength λ in the medium and relative refractive index m (the ratio between the particle and medium refractive index), ND is the total number of particles, the lower and upper integration limits are denoted by D min and D max, f(D) is the volume frequency distribution of particle system with a diameter between D and D+dD, which is the particle size distribution function to be determined[11

11. A. Katz, A. Alimova, M. Xu, P. Gottlieb, E. Rudolph, J. C. Steiner, and R. R. Alfano, “In situ determination of refractive index and size of bacillus spores by light transmission,” Opt. Lett. 30,589–591 (2005). [CrossRef] [PubMed]

13

13. W. Liang, Y. Xu, Y. Y. Huang, A. Yariv, J. G. Fleming, and S. W. Yu, “Mie scattering analysis of spherical Bragg “onion” resonators,” Opt. Express 12, 657–669 (2004). [CrossRef] [PubMed]

].

Equation (1) is Fredholm integral equation of the first kind. This is a classic ill conditioned problem, which means that different distributions can fit the data I(λ)/I0(λ). Therefore the retrieval of Eq. (1) is not a trivial task, because the retrieved distribution might be highly unstable. Hence, frequently one has to resort to approximation schemes. To retrieve the particle size distribution from spectral extinction data in the dependent model one need to solve the optimal problem. The optimization method proceeds to fit the spectral extinction data by varying the form of particle size distribution until the best characteristic parameters are achieved.

In the dependent model, different types of particle size distribution functions have been used such as the log-normal function, Rosin-Rammler function, gamma function and so on. Among these available well-behaved distribution functions, we have chosen to work with the log-normal and Rosin-Rammler function.

The monomodal log-normal(L-N) volume frequency distribution function f(D) is described by:

fLN(D)=12πDlnσ×exp((lnDlnu)22(lnσ)2).
(2)

where D is the particle diameter(in µm), u is the geometric number mean diameter(in µm),σ is the geometric standard deviation.

In this work, the particle size distribution from the spectral extinction data is reconstructed in combination with the moments of particle size distribution in the dependent model algorithm. The moment method is a low order method that reduces the initial problem to a simpler one, and the computation time is substantially reduced compared with the numerical integration methods, which is especially suitable for in-line particle size measurement.

Conventional moment models are computationally less demanding, but have been restricted in their application to those systems for which the set of the moment equations is closed [14

14. D. L. Wright, S. C. Yu, P. S. Kasibhatla, R. Mcgraw, S. E. Schwartz, V. K. Saxena, and G. K. Yue, “Retrieval of aerosol properties from moments of the particle size distribution for kernel involving the step function:cloud droplet activation,” J. Aerosol Sci. 33, 319–337 (2002). [CrossRef]

]. For a size distribution f(D) of spherical particles, the lth origin moment Ml is written as:

Ml=0+Dlf(D)dD.
(3)

where l is an arbitrary real number. If l becomes 0, M0=1. When l is 1, M1 is equivalent to the mean value of f(D), and when l is 2, the variance of f(D) is M2-(M1)2.

Applying the monomodal L-N distribution to the moment relation, the relation between each moment is represented by the expression[14

14. D. L. Wright, S. C. Yu, P. S. Kasibhatla, R. Mcgraw, S. E. Schwartz, V. K. Saxena, and G. K. Yue, “Retrieval of aerosol properties from moments of the particle size distribution for kernel involving the step function:cloud droplet activation,” J. Aerosol Sci. 33, 319–337 (2002). [CrossRef]

,15

15. D. L. Wright, “Retrieval of optical properties of atmospheric aerosols from moments of the particle size distribution,” J. Aerosol Sci. 31, 1–18 (2000). [CrossRef]

]:

Ml=ulexp(l22(lnσ)2).
(4)

The characteristic parameters u andσ of L-N distribution can be derived by the first-order and second-order moments:

u=(M1)2M2.
(5)
σ=exp(ln(M2(M1)2)).
(6)

Using the two ordinary equations in regards to the moment relation, the geometric mean diameter u and geometric standard deviation σ can be obtained. With the values of u and σ given by Eqs. (5) and (6), it is possible to construct the L-N size distribution function.

The monomodal Rosin-Rammler(R-R) volume frequency distribution function f(D) is defined as follows[16

16. J. P. Wang, S. Z. Xie, Y. M. Zhang, and W. Li, “Improved projection to invert forward scattered light for particle sizing,” Appl. Opt. 40, 3937–3945 (2001). [CrossRef]

]:

fRR(D)=kD¯×(DD¯)k1×exp((DD¯)k).
(7)

where D is the particle diameter(in µm), D is the is the characteristic diameter of the distribution (in µm), k is narrowness index of the distribution.

The relation between each moment of the R-R distribution can be expressed as:

Ml=D¯lgamma((k+l)k).
(8)

So the characteristic parameters of monomodal particle size distribution may be obtained by the first-order and second-order moments of the particle size distribution.

The original extinction efficiency Q ext(λ,m,D) can be obtained from the complicated Mie’s solution to Maxell’s equation for spherical particle. However, the mathematical expression of Q ext(λ,m,D) requires the summation of an infinite series of complicated terms containing spherical Bessel functions, and the integration equation cannot be evaluated as a moment formula in its present from. For this reason, Q ext(λ,m,D) should be expressed as a function of the power of particle diameter in order to apply the moment method. In this study, the single particle extinction efficiency Q ext(λ,m,D) which is derived theoretically using Mie’s theory, is approximated with a higher order polynomial. Subsequently, the extinction efficiency can be obtained as follows [8

8. C. H. Jung and Y. P. Kim, “Numerical estimation of the effects of condensation and coagulation on visibility using the moment method,” J. Aerosol Sci. 37, 143–161 (2006). [CrossRef]

]:

Qext(λj,m,D)=i=0PAijDPi.
(9)

where i=0…P, j is the jth wavelength, Aij is the approximated coefficient in regard to P order polynomials which is known beforehand and varied with λj, m, and the particle size range. The P value also depends on λj, m, and the particle size range.

By applying this moment operator to Eq. (1), the extinction value I(λ)/I0(λ) can be developed with the general form:

ln(IIO)j=32×L×N×i=0PAijMPi1.
(10)

Combination Eq. (10) with Eq. (4) or Eq. (8), one can derive the characteristic parameters of the L-N or R-R particle size distribution using a certain optimization algorithm.

Figure 1 shows the comparison of the approximated single spherical particle extinction efficiency with the one which is theoretically derived from the Mie theory. The relative refractive index m=1.235. The curve of the extinction efficiency is oscillatory as a function of the particle diameter, and the shape of the extinction efficiency curve is different in different particle size range. So the fitted polynomial is also different in different particle size range. For the case of (a) where the particles are in the range from 0.1~1 um in diameter, the third-order polynomial is used to fit the extinction efficiency. For the case of (b) where the particles are in the range from 1~5 um in diameter, the tenth-order polynomial is used. For the case of (c) and (d), the twentieth-order polynomial is used to fit the extinction efficiency. As shown in Fig. 1, both results show good agreement without a great loss of accuracy in different particle size range. Here, the overall particle size measurement range is limited from 0.1~10 um in diameter, which is the optimal measurement rang in spectral extinction technique. Thus, we can conclude that the approximated polynomial can be alternatively applied to the retrieval of the particle size distribution instead of the Mie solution for spherical particle.

Figure 2 shows the comparison of the approximated single spherical particle extinction efficiency with the one which is theoretically derived from the Mie theory with different relative refractive indices. For the case of (a), the twentieth-order polynomial is used to fit the extinction efficiency. For the case of (b), the curve of the extinction efficiency is smoother because the imaginary part of the relative refractive index is not zero. So the sixteenth-order polynomial is used. As shown in Fig. 2, both results also show good agreement especially for the case of (b), and the P values also vary with the varied relative refractive indices.

Fig. 1. Fitting curve of the extinction efficiency for spherical particle in different particle size range
Fig. 2. Same as Fig.1 but with different relative refractive indices

3. Computer simulations and experiments

Table 1 shows the inversion results of the monomodal R-R distributions in different particle size range using the moment method. The relative refractive index m=1.235. The order of the fitted polynomial is determined according to the different particle size range. Since the fitted polynomial of the extinction efficiency is still an oscillatory function. The oscillatory nature of the extinction efficiency determines that it needs more than two equations to solve the two parameters D and k. Here four incident wavelengths(0.4 um, 0.55 um, 0.65 um, 0.8 um) are selected in the visible spectrum for the monomodal distributions. It is possible to see that the retrieved size parameter D provides almost perfect agreement with the given one, but the distribution parameter k is slightly sensitive to the noise. From the practical point of view the results can be considered satisfactory in all the size range, especially for the case of putting 2% random noise in the extinction values. It is important to note that the accuracy of the inversion results is increased by using a higher order polynomial as the fitted extinction efficiency. However, further increasing the order of the polynomial leads to the fitted extinction efficiency become more and more oscillatory and thus the quality of the results cannot be improved.

Table 1. Inversion Results of the Monomodal R-R Distributions

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To further investigate the feature, we also use the numerical integration method to retrieve the particle size distribution as a comparison with the moment method, and the inversion results in 1~5 um and 0.1~10 um size range for the monomodal R-R distributions are shown in Fig. 3. The set parameters are listed in Table 1. Through comparison, one can see that the inversion curves coincide nearly with the set curves in the figure for both the moment and numerical integration method when the 0% random noise is added to the extinction values, suggesting that it is possible to retrieve the particle size distribution with good results using the moment method. When putting 2% random noise into the extinction values, the inversion results using the moment method are better than that using the numerical integration method, which turns out that the moment method is more insensitivity to the random noise. The comparison of reproducibility for the moment method and numerical integration method in 0.1~10 um range is given in Table 2, and the set parameters are also listed in Table 1. According to the results in Table 2, we can draw a conclusion that it is very feasible to retrieve the particle size distribution in the visible spectrum using the moment method, and the reproducibility for the moment method are superior to the numerical integration method, though the inversion results using the numerical integration method are better than that using the moment method under the ideal condition.

Fig. 3. Comparison of the inversion results for R-R distributions with moment and numerical integration method

Table 2. Comparison of Reproducibility for the Moment Method and Numerical Integration Method with R-R Distributions

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Table 3. Inversion Results of the Monomodal L-N Distributions

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The second test is devised to study the monomodal L-N distributions using the moment method. The inversion results with different monomodal L-N distribution parameters are summarized in Table 3. The relative refractive index of particle m=1.235. The good features of retrieved results mentioned in Table 2 are still presented for L-N distribution function.

Figure 4 depicts the comparison of the inversion results for the L-N distributions with moment and numerical integration method in the 0.1~1 um and 1~10 um size range, respectively. In the range from 0.1~1 um, the retrieved distribution becomes somewhat noisier, while its shape is fairly close to the shape of the given one, which is due to the fact that the parameter σ is more sensitive to the random noise, but the retrieved result is still acceptable. Table 4 lists the monomodal L-N distributions inversion results varied with different random noise using the moment and numerical integration method, respectively. The retrieved results using the moment method are superior to that using the numerical integration method except in the ideal case.

Fig. 4. Comparison of the inversion results for L-N distributions with moment and numerical integration method

Table 4. Comparison of Reproducibility for the Moment and Numerical Integration Method with L-N Distributions

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As a further test for investigating the reliability, we consider the bimodal distributions. The bimodal R-R volume frequency distribution function f(D) is defined as follows:

f(D)=n*(k1D¯1×(DD¯1)k11×exp((DD¯1)k1))+(1n)*(k2D¯2×(DD¯2)k21)×exp((DD¯2)k2))
(11)

where D̄1, D̄2 are the characteristic diameters of the distribution(in µm), k 1, k 2 are the narrowness indices of the distribution, n is the weight coefficient between the two peaks.

Table 5. Inversion Results of the Bimodal R-R Distributions

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The inversion results of bimodal R-R distributions in the range from 0.1~10 um in diameter are shown in Table 5. The relative refractive index of particle m=1.235. There are five parameters (D̄1,k 1,D̄,k 2,n) to be retrieved and six incident wavelengths(0.4 um, 0.5 um, 0.6 um, 0.7 um, 0.75 um, 0.8 um) are selected for the bimodal distributions. In order to present the results clearly, the retrieved size distributions are also depicted in Fig. 5. The same conclusion can be drawn that it is very feasible to retrieve the particle size distribution using the moment method for the bimodal size distributions. The bimodal R-R distributions inversion results with different random noise for moment and numerical integration method are listed in Table 6.

From all the simulations results above mentioned, it can be inferred that it is very feasible and reliable to use the moment method to retrieve the particle size distribution in the dependent model. The reproducibility for the moment method are superior to the numerical integration method, especially for the case of putting 2% or 4% random noise in the extinction measurement values. Therefore, the size parameters can be retrieved with a good accuracy.

Fig. 5. Comparison of the inversion results for bimodal R-R distributions with moment and numerical integration method in the range from 0.1~10 um in diameter

Table 6. Comparison of Reproducibility for the Moment and Numerical Integration Method with Bimodal R-R Distributions

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Table 7. Inversion Results of Standard Particles

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The reliability of the moment method is also verified by the retrieval of reference material. The experimental data of reference material is provided by Prof. X.S. Cai. The standard monodisperse spherical polystyrene particle samples with labeled diameter D=1.98 um and D=3.17 um are produced by Beijing Institute of Nuclear Engineering. The particles are dispersed in water, whose relative refractive indices are m=1.235. The incident wavelengths are selected as λ 1=0.8 um, λ 2=0.6728 um, λ 3=0.5695 um and λ 4=0.4098 um. In this experiment, the extinction intensity ratios at any two wavelengths are used as the input data, and the R-R distribution function is assumed, then the data are processed using the genetic algorithm in combination with the moment relation. The results are presented in Table 7. For a monodisperse or polydisperse particle system, the Sauter mean diameter D 32 [defined in Eq. (12)] is the optimal expression of mean diameter for the particle system. As a comparison, we also use the data to retrieve the particle size distribution with numerical integration method. It is possible to see that the retrieved mean diameter D 32 is nearly closed to the labeled one. The difference between the retrieved result and the labeled diameter is less than 10%, which satisfies the demand of the standard monodisperse spherical polystyrene particles.

D32=DminDmaxD3f(D)dDDminDmaxD2f(D)dD
(12)

4. Conclusion

In spectral extinction particle sizing, the moment method is applied to retrieve the particle size distribution in the dependent model algorithm. The relationships of the origin moments of the monomodal, bimodal R-R and L-N particle size distributions with the corresponding characteristic parameters are investigated to analyze the feasibility of retrieval using the moment method. The higher order polynomial with regard to the particle diameter is used to fit the single spherical particle extinction efficiency which is derived theoretically using the Mie’s solution to Maxwell’s equation in order to apply the moment method. Simulation and experimental results indicate that fairly reasonable results in different size range are obtained even in the case where the bimodal distributions are retrieved. The genetic algorithm is able to yield good results and has high stability in presence of random noise. The computation time is substantially reduced compared with the numerical integration methods, which is especially suitable for in-line particle size measurement.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No.50336010). The authors thank X. S. Cai (Institute of Particle & Two phase Flow Measurement, University of Shanghai for Science & Technology) for the offer of the experimental data.

References and links

1.

F. Ferri, A. Bassini, and E. Paganini, “Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing,” Appl. Opt. 34, 5829–5839 (1995). [CrossRef] [PubMed]

2.

A. V. Kharchenko and D. Gresillon, “Nonparticle laser velocimetry and permanent velocity measurement by enhanced light scattering,” Meas. Sci. Technol. 14, 228–233 (2003). [CrossRef]

3.

F. Pedocchi and M. H. Garcia, “Noise-resolution trade-off in projection algorithms for laser diffraction particle sizing,” Appl. Opt. 45, 3620–3628 (2006). [CrossRef] [PubMed]

4.

B. N. Khlebtsov, L. A. Kovler, V. A. Bogatyrev, N. G. Khlebtsov, and S. Y. Shchyogolev, “Studies of phosphatidylcholine vesicles by spectroturbidimetric and dynamic light scattering methods,” J. Quant. Spectrosc. Radiat. Transf. 7980, 825–838 (2003). [CrossRef]

5.

A. K. Roy and S. K. Sharma, “A simple analysis of the extinction spectrum of a size distritution of Mie particles,” J. Opt. A: Pure Appl. Opt. 7, 675–684 (2005). [CrossRef]

6.

M. L. Arias and G. L. Frontini, “Particle size distribution retrieval from elastic light scattering measurements by a modified regularization method,” Part. Part. Syst. Charact. 23, 374–380 (2007). [CrossRef]

7.

A. P. Nefedov, O. F. Petrov, and O. S. Vaulina, “Analysis of particle sizes, concentration, and refractive index in measurement of light transmittance in the forward-scattering-angle range,” Appl. Opt. 36, 1357–1366 (1997). [CrossRef] [PubMed]

8.

C. H. Jung and Y. P. Kim, “Numerical estimation of the effects of condensation and coagulation on visibility using the moment method,” J. Aerosol Sci. 37, 143–161 (2006). [CrossRef]

9.

S. H. Park, R. Xiang, and K. W. Lee, “Brownian coagulation of fractal agglomerates:analytical solution using the log-normal size distribution assumption,” J. Colloid Interface Sci. 231, 129–135 (2000). [CrossRef] [PubMed]

10.

E. Marioth, B. Koenig, H. Krause, and S. Loebbecke, “Fast particle size and droplet size measurements in supercritical CO2,” Ind. Eng. Chem. Res. 339, 4853–4857 (2000). [CrossRef]

11.

A. Katz, A. Alimova, M. Xu, P. Gottlieb, E. Rudolph, J. C. Steiner, and R. R. Alfano, “In situ determination of refractive index and size of bacillus spores by light transmission,” Opt. Lett. 30,589–591 (2005). [CrossRef] [PubMed]

12.

D. Rosskamp, F. Truffer, S. Bolay, and M. Geiser, “Forward scattering measurement device with a high angular resolution,” Opt. Express 15, 2683–2690 (2007). [CrossRef] [PubMed]

13.

W. Liang, Y. Xu, Y. Y. Huang, A. Yariv, J. G. Fleming, and S. W. Yu, “Mie scattering analysis of spherical Bragg “onion” resonators,” Opt. Express 12, 657–669 (2004). [CrossRef] [PubMed]

14.

D. L. Wright, S. C. Yu, P. S. Kasibhatla, R. Mcgraw, S. E. Schwartz, V. K. Saxena, and G. K. Yue, “Retrieval of aerosol properties from moments of the particle size distribution for kernel involving the step function:cloud droplet activation,” J. Aerosol Sci. 33, 319–337 (2002). [CrossRef]

15.

D. L. Wright, “Retrieval of optical properties of atmospheric aerosols from moments of the particle size distribution,” J. Aerosol Sci. 31, 1–18 (2000). [CrossRef]

16.

J. P. Wang, S. Z. Xie, Y. M. Zhang, and W. Li, “Improved projection to invert forward scattered light for particle sizing,” Appl. Opt. 40, 3937–3945 (2001). [CrossRef]

OCIS Codes
(290.2200) Scattering : Extinction
(290.5820) Scattering : Scattering measurements
(290.5850) Scattering : Scattering, particles
(300.6360) Spectroscopy : Spectroscopy, laser

ToC Category:
Scattering

History
Original Manuscript: July 3, 2007
Revised Manuscript: August 14, 2007
Manuscript Accepted: August 22, 2007
Published: August 24, 2007

Virtual Issues
Vol. 2, Iss. 10 Virtual Journal for Biomedical Optics

Citation
Xiaogang Sun, Hong Tang, and Jingmin Dai, "Retrieval of particle size distribution in the dependent model using the moment method," Opt. Express 15, 11507-11516 (2007)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-18-11507


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References

  1. F. Ferri, A. Bassini, and E. Paganini, "Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing," Appl. Opt. 34, 5829-5839 (1995). [CrossRef] [PubMed]
  2. A. V. Kharchenko, and D. Gresillon, "Nonparticle laser velocimetry and permanent velocity measurement by enhanced light scattering," Meas. Sci. Technol. 14, 228-233 (2003). [CrossRef]
  3. F. Pedocchi, and M. H. Garcia, "Noise-resolution trade-off in projection algorithms for laser diffraction particle sizing," Appl. Opt. 45, 3620-3628 (2006). [CrossRef] [PubMed]
  4. B. N. Khlebtsov, L. A. Kovler, V. A. Bogatyrev, N. G. Khlebtsov, and S. Y. Shchyogolev, "Studies of phosphatidylcholine vesicles by spectroturbidimetric and dynamic light scattering methods," J. Quant. Spectrosc. Radiat. Transf. 79-80, 825-838 (2003). [CrossRef]
  5. A. K. Roy and S. K. Sharma, "A simple analysis of the extinction spectrum of a size distritution of Mie particles," J. Opt. A: Pure Appl. Opt. 7, 675-684 (2005). [CrossRef]
  6. M. L. Arias, and G. L. Frontini, "Particle size distribution retrieval from elastic light scattering measurements by a modified regularization method," Part. Part. Syst. Charact. 23, 374-380 (2007). [CrossRef]
  7. A. P. Nefedov, O. F. Petrov, and O. S. Vaulina, "Analysis of particle sizes, concentration, and refractive index in measurement of light transmittance in the forward-scattering-angle range," Appl. Opt. 36, 1357-1366 (1997). [CrossRef] [PubMed]
  8. C. H. Jung and Y. P. Kim, "Numerical estimation of the effects of condensation and coagulation on visibility using the moment method," J. Aerosol Sci. 37, 143-161 (2006). [CrossRef]
  9. S. H. Park, R. Xiang, and K. W. Lee, "Brownian coagulation of fractal agglomerates:analytical solution using the log-normal size distribution assumption," J. Colloid Interface Sci. 231, 129-135 (2000). [CrossRef] [PubMed]
  10. E. Marioth, B. Koenig, H. Krause, and S. Loebbecke, "Fast particle size and droplet size measurements in supercritical CO2," Ind. Eng. Chem. Res. 339, 4853-4857 (2000). [CrossRef]
  11. A. Katz, A. Alimova, M. Xu, P. Gottlieb, E. Rudolph, J. C. Steiner, and R. R. Alfano, "In situ determination of refractive index and size of bacillus spores by light transmission," Opt. Lett. 30, 589-591 (2005). [CrossRef] [PubMed]
  12. D. Rosskamp, F. Truffer, S. Bolay, and M. Geiser, "Forward scattering measurement device with a high angular resolution," Opt. Express 15, 2683-2690 (2007). [CrossRef] [PubMed]
  13. W. Liang, Y. Xu, Y. Y. Huang, A. Yariv, J. G. Fleming, and S. W. Yu, "Mie scattering analysis of spherical Bragg "onion" resonators," Opt. Express 12, 657-669 (2004). [CrossRef] [PubMed]
  14. D. L. Wright, S. C. Yu, P. S. Kasibhatla, R. Mcgraw, S. E. Schwartz, V. K. Saxena, and G. K. Yue, "Retrieval of aerosol properties from moments of the particle size distribution for kernel involving the step function:cloud droplet activation," J. Aerosol Sci. 33, 319-337 (2002). [CrossRef]
  15. D. L. Wright, "Retrieval of optical properties of atmospheric aerosols from moments of the particle size distribution," J. Aerosol Sci. 31, 1-18 (2000). [CrossRef]
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