Simultaneous retrieval of the complex refractive index and particle size distribution

Yatao Ren; Hong Qi; Qin Chen; Liming Ruan; Heping Tan

doi:10.1364/OE.23.019328

1. Introduction

Research into microphysical properties of particle systems, such as the optical properties and particle size distributions (PSDs), will elucidate the radiative transfer processes of environmental monitoring [1], remote sensing [2], biofuel production [3], chemical industry [4], and ocean optics [5]. Considerable research has been conducted on estimating the PSDs or optical properties of particle systems [6–14 ]. However, most of the studies concentrate on the measurement of the optical properties or PSDs with the condition that a priori information about the other one is known beforehand. The very few methods that try to simultaneously determine the complex refractive index and PSDs are based on the library method [15], which is very time-consuming. Furthermore, the experimental equipment is expensive and the measurement process is very complex. Hence, the focus of this work is to develop a simple and accurate method which can simultaneously retrieve the complex refractive index and PSDs simultaneously. In the proposed method, only the diffuse reflectance (R), diffuse transmittance (T) and collimated transmittance (T _c) signals need to be measured, which can be easily obtained by the double integrating sphere system [16].

Apart from the measurement of the radiative signals, inversion techniques also need to be investigated when estimating the properties of the particle systems. Inversion techniques have been studied thoroughly in the past decades, and can be roughly classified into two categories [17]: 1) The gradient-based techniques, such as the Gauss-Newton [18,19 ], Levenberg-Marquardt [20], and Conjugate Gradient methods [21]; 2) The stochastic heuristic intelligent optimization techniques (IPTs), such as the Generic Algorithm [22–24 ], Particle Swarm Optimization (PSO) [25–27 ], and Ant Colony Optimization [17,28 ]. The IPTs usually prove to be powerful techniques for the inverse problems in many areas [29,30 ]. Furthermore, the conventional gradient-based techniques are always considered to be powerless for solving the multi-solution and ill-posed problems. Moreover, for the complex refractive index reconstruction problems, the derivatives of the optical characteristics with respect to the real and imaginary parts of the refractive index are extremely difficult to obtain. Hence, in this work, we studied the possibility of retrieving the complex refractive index and PSDs simultaneously using the Quantum Particle Swarm Optimization (QPSO) algorithm which has proved to be effective and robust [31].

To test the performance of the proposed method, a 1-D radiative transfer model in a spherical particle system was applied. In order to retrieve the complex refractive index and PSDs, two experimental samples with different thicknesses are needed. Then the two samples (Samples 1 and 2) were separately exposed to two continuous wave laser (Lasers 1 and 2) of different wavelengths. The diffuse transmittance, diffuse reflectance, and collimated transmittance signals then needed to be calculated by the direct model under all the conditions which are (1) Sample 1 exposed to Laser 1; (2) Sample 1 exposed to Laser 2; (3) Sample 2 exposed to Laser 1; (4) Sample 2 exposed to Laser 2. Finally, the complex refractive index and PSDs were retrieved simultaneously using QPSO. To be precise, the unknown complex refractive index and PSDs were varied in the direct model calculations until best agreement with the measurements was achieved. Additionally, the sensitivity of the radiative signals to the complex refractive index and PSDs were calculated and analyzed, based on which the optimal thicknesses for the samples were chosen. The remainder of this paper is organized as follows. First, the direct model of the radiative transfer in a spherical particle system is introduced in Section 2.1. The inverse method is then presented. In Section 3, the inverse procedure is explained in detail and the corresponding analysis is carried out. Finally, the main conclusions are listed in Section 4.

2. Theory

2.1 The direct model

A 1-D absorbing, scattering, non-emitting particle system was under consideration in the present work (see Fig. 1 ). The left side of the system was exposed to a continuous wave laser beam of different wavelengths. For the short interaction time and low laser intensity, the media was assumed to be cold without considering the thermal effect caused by the interaction of the laser and media. The radiative transfer equation in a 1-D particle system can be expressed as follows [32]:

\frac{\partial I (x, θ)}{\partial x} = - β_{λ} I (x, θ) + \frac{σ_{s λ}}{2} \int_{0}^{π} I (x, θ^{'}) Φ_{λ} (θ^{'}, θ) \sin θ^{'} d θ^{'},

where I is the intensity in direction

θ

at location x. The extinction coefficient is denoted by β_λ which is equal to κ_λ + σ _s _λ, where κ_λ and σ _s _λ are absorption and scattering coefficients, respectively. The subscript λ stands for the wavelength of the incident laser. The scattering phase function

Φ_{λ} (θ^{'}, θ)

represents the probability that radiation which propagates from the incoming direction

θ^{'}

will be scattered into the direction

θ

which is calculated by the Mie theory [33] in the present work.

Fig. 1 The schematic of a 1-D slab absorbing, scattering, but non-emitting particle system exposed to collimated continuous wave laser. The signals that need to be measured include: the diffuse reflectance (R), diffuse transmittance (T), and collimated transmittance (T _c).

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The diffuse reflectance R _mea, diffuse transmittance signals T _mea, and collimated transmittance T _c can be expressed as follows:

(2)

(3)

(4)

where I ₀ is the intensity of the incident laser and I _c is the intensity of the collimated light which can be calculated by the Beer's law.

The absorption coefficient $κ_{λ}$ and scattering coefficient $σ_{s λ}$ for a particle system can be calculated using the following equations:

κ_{λ} = \int_{0}^{\infty} C_{a λ} (D) N (D) d D,

σ_{s λ} = \int_{0}^{\infty} C_{s λ} (D) N (D) d D,

where

C_{a λ}

and

C_{s λ}

denote the absorption and scattering cross-section, respectively. For sphere particles,

C_{a λ}

and

C_{s λ}

can be calculated using Mie theory [33]. D is the diameter of the particles. N(D) represents the number density of particles with diameter D.

In this study, three commonly used monomodal PSD functions, i.e. the Rosin-Rammler (R-R), standard Normal (S-N), and Logarithmic Normal (L-N) distribution, were considered. The mathematical representations of the monomodal volume frequency distribution functions are expressed as follows [34]:

f_{R-R} (D) = \frac{σ}{\bar{D}} \times {(\frac{D}{\bar{D}})}^{σ - 1} \times \exp [- {(\frac{D}{\bar{D}})}^{σ}],

f_{S-N} (D) = \frac{1}{\sqrt{2 π} σ} \times \exp [- \frac{{(D - \bar{D})}^{2}}{2 σ^{2}}],

f_{L-N} (D) = \frac{1}{\sqrt{2 π} D \ln σ} \times \exp [- \frac{{(\ln D - \ln \bar{D})}^{2}}{2 {(\ln σ)}^{2}}],

where

\bar{D}

represents the characteristic diameter parameter and

σ

is the dispersion ratio. The volume frequency distribution is denoted by f(D). The concentration of particles are set as constant value and assumed to be known.

As mentioned, to obtain κ_λ and σ _s _λ, the absorption and scattering cross-section, C _a _λ and C _s _λ, are calculated using the Mie theory, in which the complex refractive index m needs to be known. And m can be expressed as n – ik, in which n and k represent the real and imaginary part of the complex refractive index, respectively. In the present work, the complex refractive index m is assumed unknown and needs to be estimated. The absorption and scattering cross-section, C _a _λ and C _s _λ can be calculated using a random guessed value in the inverse process. Then, it updates with the proceeding of the inverse process until the objective function reaches a small value.

2.2 The inverse model

For the inverse problem involved in this work, six parameters needed to be obtained, i.e. two sets of complex refractive indices (n ₁, k ₁, and n ₂, k ₂) which correspond to different wavelengths, and the PSD parameters $\bar{D}$ [μm] and $σ$ . In retrieving the PSDs with n and k known beforehand, the multi-wavelength method was applied to obtain more information about the particle systems to get more accurate retrieved results [35]. However, in this work, the spectral complex refractive index is unknown. Therefore, if multi-wavelength method is applied, it means more spectral refractive indices need to be obtained. To minimize the number of wavelengths without losing useful information, the samples of the same particle system with two different thicknesses were considered. The two-wavelength method was applied.

The inverse procedure can be concluded as follows: 1) Measure the diffuse reflectance R _mea and diffuse transmittance signals T _mea. 2) Use the inverse techniques to retrieve m and the PSDs which make the calculated diffuse reflectance R _est and diffuse transmittance signals T _est reach the best agreement with R _mea and T _mea In this study, R _mea and T _mea were calculated using the direct finite volume method instead of measurement. The inverse technique used in the present work is a bio-based swarm intelligence algorithm QPSO which is proved to be an effective and robust optimization algorithm. The detailed principle of QPSO was introduced in [31] and will not be repeated here.

The inverse problem is proceeded by minimizing the objective function which is expressed as follows:

F_{obj} = \sum_{i = 1}^{2} \sum_{j = 1}^{2} {{[\frac{R_{est} (L_{i}, λ_{j}) - R_{mea} (L_{i}, λ_{j})}{R_{mea} (L_{i}, λ_{j})}]}^{2} + {[\frac{T_{est} (L_{i}, λ_{j}) - T_{mea} (L_{i}, λ_{j})}{T_{mea} (L_{i}, λ_{j})}]}^{2}},

where the subscripts i and j represent the radiative signals for different thicknesses of the samples and incident wavelengths and i = 1, 2, and j = 1, 2.

3. Results and discussion

This section shows the numerical tests of the inverse problem in retrieving the complex refractive index and PSDs. The diffuse reflectance R _mea(L_i, λ_j) and transmittance signals T _mea(L_i, λ_j) used in the tests were calculated using Eqs. (2) and (3) . R _mea(L_i, λ_j) and T _mea(L_i, λ_j) served as the measurement values. The two incident wavelengths were set to 436 nm and 690 nm. The corresponding original values of the complex refractive index were set as m ₁ = 1.352-0.00485i and m ₂ = 1.358-0.00249i according to the actual properties of microalgae illustrated in [36]. The original values of the three above mentioned PSDs are listed in Table 1 and the control parameters of QPSO are listed in Table 2 where N _s stands for the swarm size, N _c is the maximum number of iterations and t is the number of the current iteration. The contraction-expansion coefficient β is the control parameter of the QPSO, which can be tuned to control the convergence speed of the QPSO [31]. The searching spaces for n and k are set as (1.3, 1.5) and (0.001, 0.1), respectively, for $\bar{D}$ and $σ$ are set as (0, 10). And the diameters of the particles are ranging from 0 to 10 μm. PSO algorithm is a stochastic optimization method which has certain randomness. Therefore, each case was repeated 20 times to reduce the effect of the randomness.

Table 1. Original Value of PSDs

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Table 2. Control Parameters of QPSO

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For the parameter estimation problem, a detailed examination of the sensitivity coefficients can provide considerable insight into the inverse problem. These coefficients can show possible areas of difficulty and also lead to improved experimental design [37]. Thus, initially the sensitivity of the radiative signal to the inverse parameters needs to be analyzed. The sensitivity coefficient is one of the most important characteristic parameters in the sensitivity analysis, which is the first derivative of the radiative signals to a specific inverse parameter. The sensitivity coefficient is defined as:

S_{m_{i}} (ρ) = {\frac{\partial ρ}{\partial m_{i}} |}_{m_{i} = m_{0}} = \frac{ρ (m_{0} + m_{0} Δ) - ρ (m_{0} - m_{0} Δ)}{2 m_{0} Δ},

where

m_{i}

denotes the independent variable which stands for the complex refractive index and PSDs in this research;

Δ

represents a tiny change which is set to 0.5%, and

ρ

represents the transmittance or reflectance signal. Furthermore, it is anticipated that the radiative signals are related to the thickness of the samples which may affect the estimation results. Thus, the sensitivity of T and R to the complex refractive index and PSDs were calculated for different thicknesses L (see Fig. 2 ).

Fig. 2 The sensitivity coefficients of (a) n, (b) k, (c) $\bar{D}$ , and (d) σ under different thicknesses. The PSDs for R-R, S-N, and L-N distributions are (8.3, 5.6), (5.2, 4.5), and (6.1, 3.4), respectively. Note that the sensitivity coefficients of the PSDs are much lower than those of the complex refractive index.

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From Fig. 2, it can be seen that, for the same situation, the sensitivity coefficients of the transmittance signals are much higher than those of the reflectance in an optically thin region. Furthermore, the sensitivity of transmittance and reflectance to n and k is much higher than those to $\bar{D}$ and σ, and the difference can increase by several orders of magnitude. This phenomenon leads to inaccurate retrieved results of $\bar{D}$ and σ which are discussed later. Meanwhile, the optimal sample thickness can be determined according to the sensitivity analysis. As shown in Fig. 2, for the R-R distribution, the sensitivity is relatively high when the thickness is in the interval (0, 0.06 m). Thus, the thicknesses of the samples for R-R distribution are chosen as 0.025 m and 0.05 m. Similarly, the thicknesses for S-N distribution are 0.05 m and 0.1 m, for L-N distribution are 0.1 m and 0.2 m. Moreover, it is important to know that the comparison of sensitivity coefficients value is meaningful only when they are in the same problem, for example, the sensitivity coefficients $S_{σ} (T)$ , $S_{\bar{D}} (T)$ , $S_{n} (T)$ , and $S_{k} (T)$ in the L-N distribution. The comparison of $S_{n} (T)$ for S-N distribution and $S_{n} (T)$ for L-N distribution is meaningless as they are in the different problems.

The relative error proposed to evaluate the estimated results is defined as:

ε_{rel} = \frac{| z_{est} - z_{ori} |}{z_{ori}} \times 100 %,

where ε _rel denotes the relative error. The original value of the parameter is denoted by Z_ori, and the estimated value is represented by Z _est.

The retrieved results and calculation time are shown in Table 3 . As can be seen, the complex refractive index can be retrieved with a tolerable error (less than 2.5%). However, the average retrieved results of the PSDs, i.e. $\bar{D}$ and $σ$ , are not tolerable, and can reach 17%. In addition, the retrieved results of $\bar{D}$ and $σ$ have a relatively large range which can be seen in Fig. 3 . As mentioned previously, the sensitivity coefficient represents the change in the reflectance or transmittance when there is a tiny change in the parameters $\bar{D}$ , $σ$ , n, or k. The reason the relative errors of the PSDs are so much larger than those of n and k is that the sensitivity coefficients of $\bar{D}$ and $σ$ are much lower than those of n and k. In other words, the radiative signals are insensitive to the PSDs, which means even when the relative errors of the retrieved values of $\bar{D}$ and $σ$ are very large, the deviation in the calculated reflectance and transmittance signals can be ignored. This is verified in Fig. 3 where the obtained values of $\bar{D}$ and σ varied over a large range but the objective function reduces to the predetermined value. Meanwhile, it is interesting to note that the retrieved results of $\bar{D}$ for the R-R distribution, $\bar{D}$ and σ for the S-N distribution, and σ for the L-N distribution were relatively accurate, which is because the sensitivity of the radiative signals to these parameters was relatively higher compared with other PSD parameters (see Fig. 2).

Table 3. Retrieved Results of the Complex Refractive Index and PSDs

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Fig. 3 The retrieved results of (a) n, (b) k, (c) $\bar{D}$ , and (d) σ for each independent run.

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It can be concluded that the influences of the PSDs on the diffused reflectance and transmittance signals are less significant than that of the complex refractive index. Even though the retrieved results of the PSDs were not accurate, those of the complex refractive index were relatively satisfactory. To improve the accuracy of the PSDs, more information about the properties of the particle system should be used in the inverse procedure. At the same time, the new information should be more sensitive to PSDs and less sensitive to m. It was found that the collimated transmittance was more sensitive to $\bar{D}$ and σ, and less sensitive to n and k (see Fig. 4 ). Meanwhile, the sensitivity coefficients $S_{σ} (T_{c})$ , $S_{\bar{D}} (T_{c})$ , $S_{n} (T_{c})$ , and $S_{k} (T_{c})$ for the same type of particle size distribution are basically in the same order. Therefore, the collimated transmittance signal were applied to improve the inversion accuracy of the PSDs. Based on the above analysis, a secondary optimization was performed in which $\bar{D}$ and σ were parameters that needed to be retrieved and the complex refractive indices retrieved by the first optimization were applied as the original values. The objective function in the secondary optimization is expressed as:

{F^{'}}_{obj} = \sum_{i = 1}^{2} \sum_{j = 1}^{2} {{[\frac{T_{c, est} (L_{i}, λ_{j}) - T_{c, mea} (L_{i}, λ_{j})}{T_{c, mea} (L_{i}, λ_{j})}]}^{2}},

where the measured and estimated collimated transmittance are denoted by

T_{c, mea}

and

T_{c, est}

. The measured collimated transmittance was calculated using the direct model. The inverse procedure is shown in Fig. 5 .

Fig. 4 The sensitivity coefficients of the collimated transmittance to (a) n and k, and (b) PSDs for different thicknesses.

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Fig. 5 The flowchart for the whole optimization procedure.

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The results of the secondary optimization are shown in Table 4 . It was evident that the estimated results of the secondary optimization were much more accurate than those of the first optimization, which proved that the proposed inversion method was suitable for the simultaneous estimation of the complex refractive indices and PSDs. The original distributions, retrieved PSDs, and corresponding relative errors are shown in Fig. 6 . It was evident that the results of the secondary optimization were much more accurate than the first optimization, which proves the proposed method was effective.

Table 4. Retrieved Results for the PSDs

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Fig. 6 The retrieved results of the PSDs: (a) volume frequency distribution, and (b) relative error. The legends ‘1st’ and ‘2nd’ correspond to the results of the first and second optimization. The description ‘Ori’ in (a) denotes the original distribution of the particle size.

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4. Conclusion

In this study, a secondary optimization method to simultaneously retrieve the complex refractive index and PSDs was proposed. The applicability of this method was shown by numerical simulations for spherical particle systems. The sensitivities of the radiative signals to the complex refractive index and PSDs were analyzed. It was found that the PSDs were relatively insensitive to the diffuse reflectance and transmittance, which was taken advantage of in the inverse process. The basic principle of the proposed method is to introduce the collimated transmittance into the secondary optimization process to increase the accuracy of the retrieved PSDs. The secondary optimization method was proved to be fast and accurate and was easy to implement. As the diffuse reflectance and transmittance, and collimated transmittance can be measured by the double-integrating-sphere system, the proposed method is very promising as a simple technique to determine the complex refractive index and PSDs simultaneously. Further research will focus on the experimental study to validate the present numerical optimization method.

Acknowledgments

The supports of this work by the National Natural Science Foundation of China (No. 51476043), the Major National Scientific Instruments and Equipment Development Special Foundation of China (No. 51327803), and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 51421063) are gratefully acknowledged. A very special acknowledgment is also made to the editors and referees who make important comments to improve this paper.

References and links

1. Y. Takano and K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46(1), 3–19 (1989). [CrossRef]

2. K. D. Kanniah, J. Beringer, P. North, and L. Hutley, “Control of atmospheric particles on diffuse radiation and terrestrial plant productivity A review,” Prog. Phys. Geogr. 36(2), 209–237 (2012). [CrossRef]

3. L. Pilon, H. Berberoğlu, and R. Kandilian, “Radiation transfer in photobiological carbon dioxide fixation and fuel production by microalgae,” J. Quant. Spectrosc. Ra. 112(17), 2639–2660 (2011). [CrossRef]

4. M. Su, M. Xue, X. Cai, Z. Shang, and F. Xu, “Particle size characterization by ultrasonic attenuation spectra,” Particuology 6(4), 276–281 (2008). [CrossRef]

5. D. Doxaran, J. Ehn, S. Bélanger, A. Matsuoka, S. Hooker, and M. Babin, “Optical characterisation of suspended particles in the Mackenzie River plume (Canadian Arctic Ocean) and implications for ocean colour remote sensing,” Biogeosciences 9(8), 3213–3229 (2012). [CrossRef]

6. S. Pfeifer, W. Birmili, A. Schladitz, T. Müller, A. Nowak, and A. Wiedensohler, “A fast and easy-to-implement inversion algorithm for mobility particle size spectrometers considering particle number size distribution information outside of the detection range,” Atmos. Meas. Tech. 7(1), 95–105 (2014). [CrossRef]

7. X. Sun, H. Tang, and J. Dai, “Retrieval of particle size distribution in the dependent model using the moment method,” Opt. Express 15(18), 11507–11516 (2007). [CrossRef] [PubMed]

8. K. J. Zarzana, C. D. Cappa, and M. A. Tolbert, “Sensitivity of aerosol refractive index retrievals using optical spectroscopy,” Aerosol Sci. Technol. 48(11), 1133–1144 (2014). [CrossRef]

9. X. Wu, Y. Wu, J. Yang, Z. Wang, B. Zhou, G. Gréhan, and K. Cen, “Modified convolution method to reconstruct particle hologram with an elliptical Gaussian Beam illumination,” Opt. Express 21(10), 12803–12814 (2013). [CrossRef] [PubMed]

10. 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(12), 2677–2685 (1999). [CrossRef] [PubMed]

11. L. M. Ruan, H. Qi, W. An, and H. P. Tan, “Inverse radiation problem for determination of optical constants of fly-ash particles,” Int. J. Thermophys. 28(4), 1322–1341 (2007). [CrossRef]

12. X. Zhang, Y. Huang, R. Rao, and Z. Wang, “Retrieval of effective complex refractive index from intensive measurements of characteristics of ambient aerosols in the boundary layer,” Opt. Express 21(15), 17849–17862 (2013). [CrossRef] [PubMed]

13. H. Tang and J. Lin, “Inversion of visible optical extinction data for spheroid particle size distribution based on PCA,” Optik 125(19), 5494–5507 (2014). [CrossRef]

14. M. Su, F. Xu, X. Cai, K. Ren, and J. Shen, “Optimization of regularization parameter of inversion in particle sizing using light extinction method,” China Particuology 5(4), 295–299 (2007). [CrossRef]

15. M. Tanaka, T. Nakajima, and T. Takamura, “Simultaneous determination of complex refractive index and size distribution of airborne and water-suspended particles from light scattering measurements,” J. Meteorol. Soc. Jpn. 60(6), 1259–1271 (1982).

16. B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid® phantoms in the 500-2250 nm range,” Opt. Express 21(26), 32450–32467 (2013). [CrossRef] [PubMed]

17. B. Zhang, H. Qi, Y. T. Ren, S. C. Sun, and L. M. Ruan, “Application of homogenous continuous Ant Colony Optimization algorithm to inverse problem of one-dimensional coupled radiation and conduction heat transfer,” Int. J. Heat Mass Trans. 66, 507–516 (2013). [CrossRef]

18. G. Masiello and C. Serio, “Simultaneous physical retrieval of surface emissivity spectrum and atmospheric parameters from infrared atmospheric sounder interferometer spectral radiances,” Appl. Opt. 52(11), 2428–2446 (2013). [CrossRef] [PubMed]

19. H. Mao and D. Zhao, “Alternative phase-diverse phase retrieval algorithm based on Levenberg-Marquardt nonlinear optimization,” Opt. Express 17(6), 4540–4552 (2009). [CrossRef] [PubMed]

20. C. H. Huang and C. H. Wang, “The design of uniform tube flow rates for Z-type compact parallel flow heat exchangers,” Int. J. Heat Mass Trans. 57(2), 608–622 (2013). [CrossRef]

21. Z. Tao, N. J. McCormick, and R. Sanchez, “Ocean source and optical property estimation from explicit and implicit algorithms,” Appl. Opt. 33(15), 3265–3275 (1994). [CrossRef] [PubMed]

22. K. W. Kim and S. W. Baek, “Efficient inverse radiation analysis in a cylindrical geometry using a combined method of hybrid genetic algorithm and finite difference Newton method,” J. Quant. Spectrosc. Ra. 108(3), 423–439 (2007). [CrossRef]

23. R. Das, S. C. Mishra, M. Ajith, and R. Uppaluri, “An inverse analysis of a transient 2-D conduction–radiation problem using the lattice Boltzmann method and the finite volume method coupled with the genetic algorithm,” J. Quant. Spectrosc. Ra. 109(11), 2060–2077 (2008). [CrossRef]

24. R. Das, S. C. Mishra, T. B. P. Kumar, and R. Uppaluri, “An inverse analysis for parameter estimation applied to a non-fourier conduction–radiation problem,” Heat Transf. Eng. 32(6), 455–466 (2011). [CrossRef]

25. H. Qi, L. M. Ruan, H. C. Zhang, Y. M. Wang, and H. P. Tan, “Inverse radiation analysis of a one-dimensional participating slab by stochastic particle swarm optimizer algorithm,” Int. J. Therm. Sci. 46(7), 649–661 (2007). [CrossRef]

26. H. Qi, D. L. Wang, S. G. Wang, and L. M. Ruan, “Inverse transient radiation analysis in one-dimensional non-homogeneous participating slabs using particle swarm optimization algorithms,” J. Quant. Spectrosc. Ra. 112(15), 2507–2519 (2011). [CrossRef]

27. Y. P. Sun, C. Lou, and H. C. Zhou, “Estimating soot volume fraction and temperature in flames using stochastic particle swarm optimization algorithm, Int. J. Heat Mass Trans. 54(1–3), 217–224 (2011). [CrossRef]

28. S. Stephany, J. C. Becceneri, R. P. Souto, H. F. de Campos Velho, and A. J. Silva Neto, “A preregularization scheme for the reconstruction of a spatial dependent scattering albedo using a hybrid ant colony optimization implementation,” Appl. Math. Model. 34(3), 561–572 (2010). [CrossRef]

29. M. A. Behrang, E. Assareh, M. R. Assari, and A. Ghanbarzadeh, “Assessment of electricity demand in Iran’s industrial sector using different intelligent optimization techniques,” Appl. Artif. Intell. 25(4), 292–304 (2011). [CrossRef]

30. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3(1), 1–122 (2010). [CrossRef]

31. B. Zhang, H. Qi, S. C. Sun, L. M. Ruan, and H. P. Tan, “Solving inverse problems of radiative heat transfer and phase change in semitransparent medium by using Improved Quantum Particle Swarm Optimization,” Int. J. Heat Mass Trans. 85, 300–310 (2015). [CrossRef]

32. M. F. Modest, Radiative Heat Transfer (Elsevier, 2003).

33. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 2008)

34. H. Qi, L. M. Ruan, S. G. Wang, M. Shi, and H. Zhao, “Application of multi-phase particle swarm optimization technique to retrieve the particle size distribution,” Chin. Opt. Lett. 6, 346–349 (2008). [CrossRef]

35. Z. Z. He, H. Qi, Y. C. Yao, and L. M. Ruan, “Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm,” Appl. Therm. Eng.in press.

36. E. Lee, R. L. Heng, and L. Pilon, “Spectral optical properties of selected photo synthetic microalgae producing biofuels,” J. Quant. Spectrosc. Ra. 114, 122–135 (2013). [CrossRef]

37. H. Qi, C.-Y. Niu, S. Gong, Y.-T. Ren, and L.-M. Ruan, “Application of the hybrid particle swarm optimization algorithms for simultaneous estimation of multi-parameters in a transient conduction-radiation problem,” Int. J. Heat Mass Trans. 83, 428–440 (2015). [CrossRef]

Functions	(n, k)	$ε_{rel}$ [%]	( $\bar{D}$ , $σ$ )	$ε_{rel}$ [%]	Time [s]
R-R	(1.359, 0.00493)	(0.54, 1.58)	(8.165, 6.123)	(1.63, 9.34)	1258
R-R	(1.363, 0.00253)	(0.40, 1.77)	(8.165, 6.123)	(1.63, 9.34)	1258
S-N	(1.352, 0.00488)	(0.03, 0.69)	(5.410, 4.364)	(4.04, 3.03)	1468
S-N	(1.355, 0.00250)	(0.22, 0.53)	(5.410, 4.364)	(4.04, 3.03)	1468
L-N	(1.352, 0.00497)	(0.01, 2.49)	(5.029, 3.298)	(17.56, 2.99)	1742
L-N	(1.353, 0.00254)	(0.41, 1.97)	(5.029, 3.298)	(17.56, 2.99)	1742

Function	Original values ( $\bar{D}$ , $σ$ )	Estimated values ( $\bar{D}$ , $σ$ )	$ε_{rel}$ [%]
R-R	(8.3, 5.6)	(8.30, 5.72)	(0.04, 2.13)
S-N	(5.2, 4.5)	(5.21, 4.48)	(0.23, 0.43)
L-N	(6.1, 3.4)	(6.13, 3.47)	(0.48, 1.98)

Functions	(n, k)	$ε_{rel}$ [%]	( $\bar{D}$ , $σ$ )	$ε_{rel}$ [%]	Time [s]
R-R	(1.359, 0.00493)	(0.54, 1.58)	(8.165, 6.123)	(1.63, 9.34)	1258
R-R	(1.363, 0.00253)	(0.40, 1.77)	(8.165, 6.123)	(1.63, 9.34)	1258
S-N	(1.352, 0.00488)	(0.03, 0.69)	(5.410, 4.364)	(4.04, 3.03)	1468
S-N	(1.355, 0.00250)	(0.22, 0.53)	(5.410, 4.364)	(4.04, 3.03)	1468
L-N	(1.352, 0.00497)	(0.01, 2.49)	(5.029, 3.298)	(17.56, 2.99)	1742
L-N	(1.353, 0.00254)	(0.41, 1.97)	(5.029, 3.298)	(17.56, 2.99)	1742

Function	Original values ( $\bar{D}$ , $σ$ )	Estimated values ( $\bar{D}$ , $σ$ )	$ε_{rel}$ [%]
R-R	(8.3, 5.6)	(8.30, 5.72)	(0.04, 2.13)
S-N	(5.2, 4.5)	(5.21, 4.48)	(0.23, 0.43)
L-N	(6.1, 3.4)	(6.13, 3.47)	(0.48, 1.98)

Simultaneous retrieval of the complex refractive index and particle size distribution

Abstract

1. Introduction

2. Theory

2.1 The direct model

2.2 The inverse model

3. Results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Tables (4)

Equations (10)

Optics Express