- Email: [email protected]
Particle sizing and velocity measurement of microspheres from the analysis of polarization of the scattered light
Optics and Lasers in Engineering 50 (2012) 57–63
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Particle sizing and velocity measurement of microspheres from the analysis of polarization of the scattered light X.Q. Huang a, V. Lepiller a, Y. Bailly a,n, F. Guermeur a, P. Herve´ b a b
´, Belfort, France FEMTO-ST, Dept. ENISYS, Universite´ de Franche Comte LEME EA4416, Universite´ Paris Ouest Nanterre La De´fense, France
a r t i c l e i n f o
abstract
Available online 27 August 2011
A brief review of the existing particle sizing methods is presented. An optical method under development is introduced from the analysis of the polarization ratio of the light scattered by the particles based on Lorenz–Mie theory. The theoretical background is summarized with the numerical calculation presented. A photogrammteric system has been set up to perform the measurements. Calibration of the experimental setup has been carried out on polystyrene microspheres of different size. The experimental values of the polarization ratio have been obtained by analyzing the particle images taken by the CCD to render the particle size under investigation. Several experiments and their results are demonstrated to illustrate the application fields of the optical method presented in the current study. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Particle sizing Granulometry Microsphere Polarization Mie theory
1. Introduction Granulometry measurement has been widely investigated and developed in a variety of industries and research laboratories, which aims at characterizing the form and the diameter distribution of the particles in a targeted group ranging from several nanometers to several millimeters. Techniques have been developed corresponding to various application fields. The past several decades has seen the advancement of the optical methods attribute to the application of laser technique. The optical methods evidently show their advantages by providing non-intrusive and real-time measurements as well as the possibility of the combination with velocimetry [1]. In this article, an optical particle sizing method based on the Lorenz–Mie theory by analyzing the polarized light scattered by the targeted particles will be introduced, with the demonstration of the experimental setup and the calibration work. The current work has been developed from a previous study on the interaction between light and particles [2–4]. The application of this method will also be discussed. 1.1. Particle sizing methods There has been a variety of research work on the optical methods developed based on the analysis of the interaction between the particles and the incident light. A summary of the
available optical techniques frequently appeared in the literature is presented by Onofri and Fdida et al. shown in Table 1 [6,7]. In fluid mechanics, we are interested in integrating the particle size measurement with that of the velocity. The techniques listed in Table 1 are classified into three categories including the single particle counters, the integral detection and the field measurements. The single particle counters, as characterized, take the measurements by observing the interaction between the light and a single particle as well as its movement. The size distribution of the whole field can be obtained through integration over a series of observations. By counting the particle number, the flow flux can be determined. Another category called as integral methods signifies the measurements on a certain volume, which provides the granulometric distribution as well as the concentration. A common experimental system of this category developed in industry is Malvern particlesize analyzers, which functions by the analysis of the diffraction of the scattered light. Attribute to the development of the photogrammetric systems, the techniques which implement the measurements over a certain field such as the holography and ombroscopy systems can also provide the detailed characterization of the particles in the field, including their size, position and velocity. Due to the different theoretical basis on which each of them has been developed, the measurable size range of different method is also indicated in the table. 1.2. Interaction between laser light and micro particles
n
Correspondence to: Parc Technologique, 2, avenue Jean Moulin, 90000 Belfort, France. Tel.: þ33 03 84 57 82 08; fax: þ 33 03 84 57 00 32. E-mail address: [email protected] (Y. Bailly). 0143-8166/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2011.07.016
Mathematical approaches have been developed for describing the light scattering phenomenon by the particles. Generally
58
X.Q. Huang et al. / Optics and Lasers in Engineering 50 (2012) 57–63
Table 1 Techniques developed or under-developing in the literature [1,5–7].
speaking, the concentration of the particles in the medium is considered as low in fluid mechanics (the maximum volume fraction occupied by particles is under 3%), so the interaction between the incident light and the particles is influenced mainly by two parameters, the particle dimension compared with the incident wavelength and the particle shape [5,7], with the complex refractive index of the particles assumed to be known. The sphericity of the particles under investigation and the relative dimension between the particle size and the wavelength of the incident light will decide the theoretical approach that we can employ. For the spheres or the particles with a regular form (such as spheroid and cylinder), the Lorenz–Mie theory (LMT) is widely adopted for the mathematical analysis and calculation. In the recent study of Gre´han and Gouesbet [8], generalized the Lorenz– Mie theory (GLMT) and its validity has been investigated for the particles of a size comparable to the dimension of laser beam. In principle, LMT can be applied to particle size of several nanometers to a few kilometers. However, considering the calculation expense and the current computer technique, the major application domain of LMT is limited to the particles smaller than a few microns [7]. In this article, we aim at developing a new method for the size determination of the sub-microscale and microscale spherical particles based on the application of LMT combined with the particle velocity measurement, characterized as the method of polarization ratio. The Lorenz–Mie theory provides a set of mathematical functions based on the Maxwell electromagnetic equations, which describe the behavior of the electromagnetic field in a linear, isotropic and homogeneous medium, when an incident monochromatic plane wave is scattered by a spherical particle. Derivations of these relations can be found in the books of Isbimaru and Van de Hulst [9,10]. A more detailed illustration has been provided by Bohren and Huffman, as well as Born and Wolf [11,12]. Historically, the first two essential papers on this subject are those of Mie [13] and Debye [14] in the early 20th century. The scattered far-field is converted to the direction of propagation. The traditional spherical polar coordinates adopted and the propagation given by the radial direction er, the components of the field can be transformed along the zenithal direction eh and the azimuthal direction eu. This property of transformation is applicable not only for homogeneous spheres but also for an arbitrary particle shape and homogeneity [12]. The basic interaction of the incident light and the scattering fields is illustrated in Fig. 1, which have been selected after Bohren and Huffman [11].
Fig. 1. Illustration of light scattering by spheres.
Considering a monochromatic plane wave polarized in X direction ! ! ! ! ! ! E i ¼ E0 X eiðkzotÞ ¼ ðE==0 e ==i þ E?0 e ?i ÞeiðkzotÞ ¼ E==i e ==i þ E?i e ?i ð1:1Þ illuminating a spherical particle centered at O, the scattered electric field ! ! ! E s ¼ E== e == þ E? e ? at the point P is given by the following relation: ! ! ! E==i E== eikðrzÞ S2 0 ¼ E?i 0 S1 E? ikr
ð1:2Þ
ð1:3Þ
where E==0 andE?0 are the amplitudes of the incident electric field, respectively, parallel and perpendicular to the scattering plane. The scattering plane is formed by the direction of the incident light and the scattered light whose normal vector is given by eu. E== andE? are the corresponding components of the electric fields of the scattered light, while y is the scattering angle, k the wave number of the incident light, r the distance to the observation point and z the usual Cartesian coordinate. S1 ðyÞand S2 ðyÞare dimensionless amplitude functions given by S1 ðx,m, yÞ ¼
1 X 2n þ 1 an ðx,mÞpn ðyÞ þbn ðx,mÞtn ðyÞ nðn þ 1Þ n¼1
ð1:4aÞ
S2 ðx,m, yÞ ¼
1 X 2n þ 1 an ðx,mÞtn ðyÞ þ bn ðx,mÞpn ðyÞ nðn þ 1Þ n¼1
ð1:4bÞ
X.Q. Huang et al. / Optics and Lasers in Engineering 50 (2012) 57–63
x is the size parameter defined as x¼ pd/l, which integrates the particle diameter d and the incident wavelength l. m¼nr þini is the complex refractive index, while pn(y) and tn(y) are angular functions given respectively by P1n(cos y)/sin y and d P1n(cos y)/dy, in which P1n is the associated first kind Legendre function of degree n and order 1. The scattering coefficient an and bn are defined as an ðx,mÞ ¼
c0n ðmxÞcn ðxÞmcn ðmxÞc0n ðxÞ c0n ðmxÞxn ðxÞmcn ðmxÞx0n ðxÞ
bn ðx,mÞ ¼
mcn ðmxÞcn ðxÞcn ðmxÞcn ðxÞ 0 0 mcn ðmxÞxn ðxÞcn ðmxÞxn ðxÞ
0
ð1:5aÞ
0
ð1:5bÞ
with cn(z)¼zjn(z) and xn(z)¼zh1n(z). The functions appearing in the previous expressions are, respectively, a spherical Bessel function and a spherical Bessel function of the third kind, also called as Hankel function. They are given by jn(z)¼(p/2z)1/ 2 Jn þ 1/2(z), with Jn þ 1/2(z) Bessel function of the first kind and h1n(z) ¼jn(z)þ iyn(z), with yn(z)¼(p/2z)1/2Yn þ 1/2(z) and Yn þ 1/2(z) Bessel function of the second kind. The prime in the expressions of the scattering coefficient indicates differentiation with respect to the variable in parentheses i.e. mx and x. We denote by polarization ratio P the proportion between the components polarized perpendicular and parallel to the scattering plane. When the incident light is unpolarized, linearly polarized at 451, or circularly polarized, the amplitude of the incident electric field parallel and perpendicular (E==i andE?i ) are equivalent. Thus, the scattered intensity component can be represented by the scattered irradiance per unit incident irradiance. It follows from Ref. [3] that P is given by
59
the theoretical work of Wiscombe [16] and Fu et al. [17]. As a result, the relation between polarization ratio P and the particle diameter d as well as that between the phase function of the intensity polarized parallel are presented in Fig. 2. The curves in Fig. 2a demonstrate the variation of the polarization ratio with the particle diameter of polystyrene (m ¼1.588 þ0.001i) under different incident wavelengths at a diffusion angle of 801. The calculation results for the white-light laser are obtained by integrating the polarization ratio value corresponding to different diameter over the monochromatic wavelength ranging from 0.4 to 1 mm, as indicated by the following equation: P_whiteðd,m, yÞ ¼
Z
1
0:4
Pl ðl,d,m, yÞdl
The relation between the polarization ratio and the particle diameter provides an approach for particle sizing. By performing an inverse calculation based on the experimental values of polarization ratio, the particle diameter can be found. As shown in Fig. 2a, for particles with a diameter less than 0.5 mm, when illuminated by a monochromatic laser beam, the curves are generally monotone decreasing, which renders us the possibility for investigation of the particle size with the polarization ratio value obtained from the experiments. But when the particles are larger, the values of polarization ratio vibrate intensively which may bring significant errors for the determination of the particle size. On the other hand, the theoretical values from the white-light laser are much more stable. Based on this fact, to enhance the certainty of the experiments,
2
Pðl,d,m, yÞ ¼
9S1 ðl,d,m, yÞ9 i? ¼ 2 i== 9S2 ðl,d,m, yÞ9
ð1:6Þ
where i? is the scattered irradiance per unit incident irradiance given the incident light is polarized perpendicular to the scattering plane, while i== is its counterpart when the incident light is polarized parallel. Since S1 and S2, as illustrated in Eq. (1.4a) and Eq. (1.4b), are determined by the scattering angle (observation angle) y, the size parameter x, and the complex refractive index m, for the particles with known refractive index, S1 and S2 are decided by the particle size when measuring at a certain angle with a monochromatic laser beam of a fixed wavelength. Thus, by comparing the experimental values of P with the calculated value, we will be able to find the size of the particles under investigation. As indicated in Eq. (1.6), the scattering angle y is an important parameter for the calculation of polarization ratio. When measurements are taken over a certain observation area, the scattering angles for different particles in the area can differ from each other according to their relative positions to the center of the area. Given this, the solid angle and the position of the particles should be taken into account for theoretical calculation when the size of the surface area is comparable with the distance from the observation point to the area center. However, in the current study, the influence of the solid angle can be neglected since the observed area is rather small compared with the distance between the particles and the lens that collects the scattered intensity. The scattering angle thus is represented by the solid angle of the particle at the center of the observed surface. 1.3. Numerical calculation For the theoretical calculation, a code has been developed in Mathematica based on the original work of Lompado [15], which offers precise and detailed numerical calculation with the help of
Fig. 2. (a) Variation of polarization ratio with particle size under different incident wavelength and (b) variation of scattered intensity polarized parallel with particle size under different incident wavelength; (y ¼ 801, m¼ 1.588 þ 0.001i).
60
X.Q. Huang et al. / Optics and Lasers in Engineering 50 (2012) 57–63
measurements with white-light laser provides a better reference of the relation between the particle diameter and the polarization ratio. We can also refer to the variation of the intensity parallel polarized since the curves are generally monotone increasing, especially for the white-light laser. Moreover, measurements at different diffusion angles with different incident laser light can be carried out to provide more experimental results for a certain particle size. The particle diameter is the value which matches the best the results given by each inverse calculation under different experimental condition. By establishing an experimental setup that enables us to obtain the proportion of the perpendicular to parallel polarized component of the scattered light intensity, i.e. the polarization ratio P, at selected angles with different incident laser beams, we can find the diameter of the particles through an inverse calculation with the experimental polarization ratio values.
2. Experimental setup and calibration Here an experimental system has been set up for the determination of the particle size based on the scenario introduced above, as illustrated in Fig. 3. As indicated in Fig. 3, the cylindrical lens generates a fine light sheet from the single-line beam coming out of the laser device, which illuminates the particle samples and generates an observation section. In the current study, we have employed a double pulsed Q-switched Nd:YAG laser which provided a wavelength of 532 nm (pulse duration is 9 ns, time interval between pluses is 400 ms), as well as an Ion laser Stabilite 2017 with a laser-line filter at 514 nm. The incident lights are circularly polarized. The scattered light is collected by a convex lens at a certain scattering angle, and then the two components of polarization are separated into two orthogonal directions by a cube polarizer. Via the reflection onto two mirrors and a prism, the two parts of scattered light are simultaneously received by the digital camera (TSI 630062: 4008 2672 pixels, spectral range of 340–900 nm). The two images of the targeted particles, polarized parallel and perpendicular, respectively, are projected on the CCD simultaneously, each occupying a specific area of the CCD. Thus the proportion of the perpendicular to parallel components of the scattered light intensity can be calculated by correlating and comparing the two images. At the same time, when double pulsed Nd:YAG laser is employed, the two laser pulses with an interval of 20 ms illuminate the targeted particles successively, which provides the information of the particle movement during Dt¼20 ms. Correlation between the images generated by the first and second pulse renders the particle velocity. Polystyrene microspheres of different diameters have been applied to the current study for the calibration of the experimental
Fig. 3. Experimental setup of the particle size measurement.
setup, which were dispersed in pure water in small quartz tanks as experimental samples. The diameters of the spherical particles produced by Spherotech Inc. have been well calibrated. Here we have selected the diameter of 0.61 mm, 1.09 mm, 2.07 mm, 3.43 mm and 4.45 mm to represent a certain range of particle size. The complex refractive index m of the polystyrene particles is known as 1.588þ0.0001i according to Ma et al. [18]. For each incident wavelength of 514 nm and 532 nm, we have selected two scattering angles (y ¼641 and y ¼721) to perform the measurements. Fig. 4 presents the photos taken by the high-speed CCD camera during the experiments with a high solution, which shows at the same time on the CCD two images of the particles illuminated by the laser sheet, with the left one polarized parallel and the right one perpendicular (mirror symmetry), as well as the comparison of the images of the particles illuminated by two successive pulses. Through a correlation program in Matlab, the corresponding particles on the two images can be found so that the ratio of the two polarized light intensity components indicated by the images on the camera CCD can be calculated. As stated above in Section 1.3, measurements under different experimental conditions will enhance the certainty of the particle size determination. To calibrate the experimental setup, measurements have been performed at two different diffusion angles under two different incident wavelengths of 532 nm and 514 nm, respectively. Thus 4 pairs of images are provided with four values of polarization ratio for particles of a single size (Fig. 5). Besides, the variation of the parallel polarized intensity can be obtained from the response of the CCD. The comparison of the experimental polarization ratio values and those of the Lorenz–Mie theory are shown as below. By assigning an error value (n%) to restrict the inverse calculation, a series of particle diameters were found according to the polarization ratio values provided by the experiments. In the current study, the relative deviation of the polarization ratio value n% has been set to be 10%. As indicated by the theoretical curves in Fig. 2a, each experimental polarization ratio value may correspond to several possible diameters due to the fluctuation of the calculation results. For each particle sample investigated, by analyzing the four sets of polarization ratio data obtained under different experimental condition, those possible diameter values
Fig. 4. Images of spherical polystyrene particles (1.09 mm, laser Nd:YAG). I:(1st pulse) and I? (1st pulse) correlated for granulometry; I:(1st pulse) and I:(2nd pulse) correlated for velocimetry.
X.Q. Huang et al. / Optics and Lasers in Engineering 50 (2012) 57–63
61
Fig. 5. Comparison of the polarization value between theory and experiment: (a) l ¼ 532 nm, y ¼ 641; (b) l ¼ 532 nm, y ¼721; (c) l ¼514 nm, y ¼ 641 and (d) l ¼514 nm, y ¼721.
Table 2 Comparison of particle size between the experimental value and the calibrated value. Diameter calibrated (mm) Diameter experimental (mm) Root mean square (RMS) Absolute deviation of particle size (mm) Relative deviation of particle size (%)
0.61 0.62 0.0920 0.01 1.60
given by all the four sets of data were listed out. Then the theoretical polarized parallel intensity corresponding to each of these diameters (according to Fig. 2b) is compared with the experimental value, to render the final result of the particle diameter. The results obtained from the experiments for all the five samples of particle are listed in Table 2, compared with the calibrated values provided by the fabricant. The RMS errors from the inverse calculation process are presented as well. As shown in the table, the accuracy of the method of polarization ratio can be assured.
3. Applications 3.1. Application to particle sizing of soot The method of polarization ratio has been applied to the size determination of soot yielded by a gas flame [19]. The experiment followed the scheme demonstrated in Fig. 3. As indicated in Section (1.2), the theoretical polarization ratio values based on the Lorenz–Mie theory for the white-light laser are much more stable, providing a monotone relation between the polarization ratio and the particle diameter, which greatly enhances the accuracy of the experiments. Therefore, in the
1.09 0.96 0.1617 0.13 11.9
2.07 2.05 0.0937 0.02 0.97
3.43 2.75 0.1427 0.69 20.1
4.45 4.20 0.1954 0.25 5.62
measurements of the soot particle size, a white-light laser (SuperK Versa, KOHERAS, spectral range of 450–2500 nm) was employed, as well as a double pulsed Nd:YAG laser. Fig. 6 shows the images of a single soot particle, with the two components polarized parallel and perpendicular taken by the camera. In Fig. 6(a), the images are not clear since in this case the particle was illuminated by a white-light laser of continuous wavelength and the exposure time was decided by the camera to be 1 ms. While in Fig. 6(b), the particle was illuminated by a Nd:YAG laser of double pulses with each lasting for 5 ns. By calculating the proportion of the two components, i.e. the polarization ratio, we can find the corresponding diameter of the soot particle. Fig. 7(a) and (b) demonstrates the results of the inverse calculation which render the diameter range of the soot particles in the gas flame. The experimental results with different incident laser lights have proved to be consistent. The size range of the soot produced by the flame is determined by the experiments to be 0.38–0.5 mm. 3.2. Combination with velocimetry The authors have worked on the simultaneous determination of granulometry and velocity distribution of oil droplets at the
62
X.Q. Huang et al. / Optics and Lasers in Engineering 50 (2012) 57–63
Fig. 6. Images of the soot particle with two polarized components taken simultaneously by the CCD (20 20 pixels): (a) Illuminated by a continuous white-light laser (0.4–1 mm, exposure time 1 ms) and (b) illuminated by a double pulsed Nd:YAG laser (0.532 mm, exposure time 5 ns). Fig. 8. Scheme of the propeller.
Fig. 7. Determination of the diameter of the soot particle: (a) illuminated by white-light laser and (b) illuminated by Nd:YAG laser.
exit of a propeller with the same experimental setup based on the method of polarization ratio. Experiments were carried out to investigate the swirl movement in a transparent tube after a propeller exit with four blades, each one composing a quarter of the turn (shown in Fig. 8, D ¼26 mm). A light sheet from the double Nd:YAG laser illuminated the particles on the cross section of the flow flux(Z), i.e. a surface vertical to the flowing direction. The measurements were taken under a flow flux of 30 l/min with a pressure difference of 5 mb between the entrance and the exit of the transparent tube. The correlations between the two images of polarized components as well as that between the two images from successive laser pulses were carried out by a program in Matlab. The determination of the particle size has been performed by referring
Fig. 9. (a) Particle size distribution at 36 mm from propeller exit and (b) particle size distribution along radial position (cross section A, z ¼36 mm).
X.Q. Huang et al. / Optics and Lasers in Engineering 50 (2012) 57–63
63
the diameter. The method of the polarization ratio is applicable for the granulometry measurement of the particles ranging from submicron to a few microns. In addition, given that the double pulsed Nd:YAG laser has been employed, the experimental device can also be applied to a simultaneous study of the size and the velocity of particles under investigation in the field of fluid mechanics.
Acknowledgements The authors would like to express their thanks to the research project of SIMBA (Contract no. 072906524) of ‘‘l’Etat et la Communaute´ d’Agglome´ration du Pays de Montbe´liard’’ for the financial support. References
Fig. 10. (a) Tangential velocity distribution in the tube at different distance from the propeller exit and (b) tangential velocity profile at different distances from propeller exit.
to the variation of the parallel intensity component received by the camera. The particle size distribution along the cross section at z¼36 mm is demonstrated in Fig. 9, with the velocity distribution on the cross section along the tube shown in Fig. 10.
4. Conclusion Based on the two real-time measurements demonstrated above, the method of polarization ratio provides non-intrusive simultaneous study of granulometry and velocimetry with a good repeatability. The application of the experimental method to the characterization of particles in fluid mechanics is promising. The sensitivity of the theoretical calculation to the particle size d, the diffusion angle y and the wavelength of the incident light l can be the major reason of the experimental error. As can be concluded from the experimental results shown in Table 2, the accuracy of the method presented in the current study can be assured by taking the measurements under different experimental conditions (different y and l). Moreover, when a white-light laser is employed, a generally monotone relation between the polarization ratio and the diameter can be obtained. The variation of the intensity polarized parallel or perpendicular can also provide a reference for the determination of
[1] Black DL, Mc Quay MQ, Bonin MP. Laser-based techniques for particle-size measurement: a review of sizing methods and their industrial applications. Prog Energy Combust Sci 1996;22:267–306. [2] Hou F, Herve´ P, Riguet A. Particles characteristics determination from the polarization of their scattered radiation. In: Proceedings of the sixth international aerosol Conference IAC, Taipei; 2002. [3] Herve´ P, Kleitw A, Hamouda A, Leporcq B. Determination of submicronic particle size distribution by means of polarized light: application to condensation nuclei and turboreactors. J AeroSci 1994;25:525–6. [4] Herve´ P. Proce´de´ et dispositif pour de´terminer par le taux de polarisation de la lumie re diffuse´e, la taille des particules microniques ou submicroniques, French Patent; 1995. [5] Jones AR. Light scattering for particle characterization. Prog Energy Combust Sci 1999;25:1–53. [6] Fdida N, Blaisot JB, Floch A, Dechaume D. Drop size measurement techniques applied to gasoline sprays: determination of the relevant parameters for application to spray combustion computations. In: proceedings of the tenth international conference on liquid atomization and spray systems: ICLASS, Kyoto, Japan; 2006. p. 7–105. [7] Onofri F. Etat de l’art de la Granulome trie Laser en Me´canique des Fluides. In: 9e me Congre s Francophone de Ve´locime´trie Laser. Brussel, Belgium; 2004. p. CF.2.1–13. [8] Gre´han G, Gouesbet G G, Guilloteau F, Chevaillier JP. Comparison of the diffraction theory and the generalized Lorenz–Mie theory for a sphere arbitrarily located into a laser beam. Opt Commun 1992;90:1–6. [9] Isbimaru A. Wave propagation and scattering in random media. IEEE Press; 1997. [10] van de Hulst HC. Light scattering by small particles. New York: John Wiley and Sons; 1957. [11] Bohren CF, Huffman DR. Absorption and scattering of light by small particles. John Wiley & Sons Inc.; 1983. [12] Born M, Wolf E. Principles of optics. Cambridge University Press; 1999. 7th version. ¨ ¨ ¨ [13] Mie G. Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen. Ann. d. Phys. 1908;25:377–445. [14] Debye P. Der Lichtdruck auf Kugeln von beliebigem Material. Ann. D. Phys. 1909;30:57–136. [15] Lompado A. Light scattering by a spherical particle. Mathematica code, University of Alabama in Huntsville; 2002. [16] Wiscombe WJ. Mie scattering calculations: advances in technique and fast, vector-speed computer codes. National Center for Atmospheric Research Library; 1988. [17] Fu Q, Sun WB. Mie theory for light scattering by a spherical particle in an absorbing medium. Appl Opt 2001;40:1354–61. [18] Ma X, Lu JQ, Brock RS, Jacobs KM, Yang P, Hu XH. Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm. Phys Med Biol 2003;48:4165–72. [19] Huang XQ, Herve´ P, Bailly Y. Caracte´risation de la granulome´trie des particules dans une combustion. In: FLUVISU13-13e me Congre s Franc- ais de Visualisation et de Traitement d’Images en Me´canique des Fluides, Reims, France; 2009.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Particle sizing and velocity measurement of microspheres from the analysis of polarization of the scattered light X.Q. Huang a, V. Lepiller a, Y. Bailly a,n, F. Guermeur a, P. Herve´ b a b
´, Belfort, France FEMTO-ST, Dept. ENISYS, Universite´ de Franche Comte LEME EA4416, Universite´ Paris Ouest Nanterre La De´fense, France
a r t i c l e i n f o
abstract
Available online 27 August 2011
A brief review of the existing particle sizing methods is presented. An optical method under development is introduced from the analysis of the polarization ratio of the light scattered by the particles based on Lorenz–Mie theory. The theoretical background is summarized with the numerical calculation presented. A photogrammteric system has been set up to perform the measurements. Calibration of the experimental setup has been carried out on polystyrene microspheres of different size. The experimental values of the polarization ratio have been obtained by analyzing the particle images taken by the CCD to render the particle size under investigation. Several experiments and their results are demonstrated to illustrate the application fields of the optical method presented in the current study. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Particle sizing Granulometry Microsphere Polarization Mie theory
1. Introduction Granulometry measurement has been widely investigated and developed in a variety of industries and research laboratories, which aims at characterizing the form and the diameter distribution of the particles in a targeted group ranging from several nanometers to several millimeters. Techniques have been developed corresponding to various application fields. The past several decades has seen the advancement of the optical methods attribute to the application of laser technique. The optical methods evidently show their advantages by providing non-intrusive and real-time measurements as well as the possibility of the combination with velocimetry [1]. In this article, an optical particle sizing method based on the Lorenz–Mie theory by analyzing the polarized light scattered by the targeted particles will be introduced, with the demonstration of the experimental setup and the calibration work. The current work has been developed from a previous study on the interaction between light and particles [2–4]. The application of this method will also be discussed. 1.1. Particle sizing methods There has been a variety of research work on the optical methods developed based on the analysis of the interaction between the particles and the incident light. A summary of the
available optical techniques frequently appeared in the literature is presented by Onofri and Fdida et al. shown in Table 1 [6,7]. In fluid mechanics, we are interested in integrating the particle size measurement with that of the velocity. The techniques listed in Table 1 are classified into three categories including the single particle counters, the integral detection and the field measurements. The single particle counters, as characterized, take the measurements by observing the interaction between the light and a single particle as well as its movement. The size distribution of the whole field can be obtained through integration over a series of observations. By counting the particle number, the flow flux can be determined. Another category called as integral methods signifies the measurements on a certain volume, which provides the granulometric distribution as well as the concentration. A common experimental system of this category developed in industry is Malvern particlesize analyzers, which functions by the analysis of the diffraction of the scattered light. Attribute to the development of the photogrammetric systems, the techniques which implement the measurements over a certain field such as the holography and ombroscopy systems can also provide the detailed characterization of the particles in the field, including their size, position and velocity. Due to the different theoretical basis on which each of them has been developed, the measurable size range of different method is also indicated in the table. 1.2. Interaction between laser light and micro particles
n
Correspondence to: Parc Technologique, 2, avenue Jean Moulin, 90000 Belfort, France. Tel.: þ33 03 84 57 82 08; fax: þ 33 03 84 57 00 32. E-mail address: [email protected] (Y. Bailly). 0143-8166/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2011.07.016
Mathematical approaches have been developed for describing the light scattering phenomenon by the particles. Generally
58
X.Q. Huang et al. / Optics and Lasers in Engineering 50 (2012) 57–63
Table 1 Techniques developed or under-developing in the literature [1,5–7].
speaking, the concentration of the particles in the medium is considered as low in fluid mechanics (the maximum volume fraction occupied by particles is under 3%), so the interaction between the incident light and the particles is influenced mainly by two parameters, the particle dimension compared with the incident wavelength and the particle shape [5,7], with the complex refractive index of the particles assumed to be known. The sphericity of the particles under investigation and the relative dimension between the particle size and the wavelength of the incident light will decide the theoretical approach that we can employ. For the spheres or the particles with a regular form (such as spheroid and cylinder), the Lorenz–Mie theory (LMT) is widely adopted for the mathematical analysis and calculation. In the recent study of Gre´han and Gouesbet [8], generalized the Lorenz– Mie theory (GLMT) and its validity has been investigated for the particles of a size comparable to the dimension of laser beam. In principle, LMT can be applied to particle size of several nanometers to a few kilometers. However, considering the calculation expense and the current computer technique, the major application domain of LMT is limited to the particles smaller than a few microns [7]. In this article, we aim at developing a new method for the size determination of the sub-microscale and microscale spherical particles based on the application of LMT combined with the particle velocity measurement, characterized as the method of polarization ratio. The Lorenz–Mie theory provides a set of mathematical functions based on the Maxwell electromagnetic equations, which describe the behavior of the electromagnetic field in a linear, isotropic and homogeneous medium, when an incident monochromatic plane wave is scattered by a spherical particle. Derivations of these relations can be found in the books of Isbimaru and Van de Hulst [9,10]. A more detailed illustration has been provided by Bohren and Huffman, as well as Born and Wolf [11,12]. Historically, the first two essential papers on this subject are those of Mie [13] and Debye [14] in the early 20th century. The scattered far-field is converted to the direction of propagation. The traditional spherical polar coordinates adopted and the propagation given by the radial direction er, the components of the field can be transformed along the zenithal direction eh and the azimuthal direction eu. This property of transformation is applicable not only for homogeneous spheres but also for an arbitrary particle shape and homogeneity [12]. The basic interaction of the incident light and the scattering fields is illustrated in Fig. 1, which have been selected after Bohren and Huffman [11].
Fig. 1. Illustration of light scattering by spheres.
Considering a monochromatic plane wave polarized in X direction ! ! ! ! ! ! E i ¼ E0 X eiðkzotÞ ¼ ðE==0 e ==i þ E?0 e ?i ÞeiðkzotÞ ¼ E==i e ==i þ E?i e ?i ð1:1Þ illuminating a spherical particle centered at O, the scattered electric field ! ! ! E s ¼ E== e == þ E? e ? at the point P is given by the following relation: ! ! ! E==i E== eikðrzÞ S2 0 ¼ E?i 0 S1 E? ikr
ð1:2Þ
ð1:3Þ
where E==0 andE?0 are the amplitudes of the incident electric field, respectively, parallel and perpendicular to the scattering plane. The scattering plane is formed by the direction of the incident light and the scattered light whose normal vector is given by eu. E== andE? are the corresponding components of the electric fields of the scattered light, while y is the scattering angle, k the wave number of the incident light, r the distance to the observation point and z the usual Cartesian coordinate. S1 ðyÞand S2 ðyÞare dimensionless amplitude functions given by S1 ðx,m, yÞ ¼
1 X 2n þ 1 an ðx,mÞpn ðyÞ þbn ðx,mÞtn ðyÞ nðn þ 1Þ n¼1
ð1:4aÞ
S2 ðx,m, yÞ ¼
1 X 2n þ 1 an ðx,mÞtn ðyÞ þ bn ðx,mÞpn ðyÞ nðn þ 1Þ n¼1
ð1:4bÞ
X.Q. Huang et al. / Optics and Lasers in Engineering 50 (2012) 57–63
x is the size parameter defined as x¼ pd/l, which integrates the particle diameter d and the incident wavelength l. m¼nr þini is the complex refractive index, while pn(y) and tn(y) are angular functions given respectively by P1n(cos y)/sin y and d P1n(cos y)/dy, in which P1n is the associated first kind Legendre function of degree n and order 1. The scattering coefficient an and bn are defined as an ðx,mÞ ¼
c0n ðmxÞcn ðxÞmcn ðmxÞc0n ðxÞ c0n ðmxÞxn ðxÞmcn ðmxÞx0n ðxÞ
bn ðx,mÞ ¼
mcn ðmxÞcn ðxÞcn ðmxÞcn ðxÞ 0 0 mcn ðmxÞxn ðxÞcn ðmxÞxn ðxÞ
0
ð1:5aÞ
0
ð1:5bÞ
with cn(z)¼zjn(z) and xn(z)¼zh1n(z). The functions appearing in the previous expressions are, respectively, a spherical Bessel function and a spherical Bessel function of the third kind, also called as Hankel function. They are given by jn(z)¼(p/2z)1/ 2 Jn þ 1/2(z), with Jn þ 1/2(z) Bessel function of the first kind and h1n(z) ¼jn(z)þ iyn(z), with yn(z)¼(p/2z)1/2Yn þ 1/2(z) and Yn þ 1/2(z) Bessel function of the second kind. The prime in the expressions of the scattering coefficient indicates differentiation with respect to the variable in parentheses i.e. mx and x. We denote by polarization ratio P the proportion between the components polarized perpendicular and parallel to the scattering plane. When the incident light is unpolarized, linearly polarized at 451, or circularly polarized, the amplitude of the incident electric field parallel and perpendicular (E==i andE?i ) are equivalent. Thus, the scattered intensity component can be represented by the scattered irradiance per unit incident irradiance. It follows from Ref. [3] that P is given by
59
the theoretical work of Wiscombe [16] and Fu et al. [17]. As a result, the relation between polarization ratio P and the particle diameter d as well as that between the phase function of the intensity polarized parallel are presented in Fig. 2. The curves in Fig. 2a demonstrate the variation of the polarization ratio with the particle diameter of polystyrene (m ¼1.588 þ0.001i) under different incident wavelengths at a diffusion angle of 801. The calculation results for the white-light laser are obtained by integrating the polarization ratio value corresponding to different diameter over the monochromatic wavelength ranging from 0.4 to 1 mm, as indicated by the following equation: P_whiteðd,m, yÞ ¼
Z
1
0:4
Pl ðl,d,m, yÞdl
The relation between the polarization ratio and the particle diameter provides an approach for particle sizing. By performing an inverse calculation based on the experimental values of polarization ratio, the particle diameter can be found. As shown in Fig. 2a, for particles with a diameter less than 0.5 mm, when illuminated by a monochromatic laser beam, the curves are generally monotone decreasing, which renders us the possibility for investigation of the particle size with the polarization ratio value obtained from the experiments. But when the particles are larger, the values of polarization ratio vibrate intensively which may bring significant errors for the determination of the particle size. On the other hand, the theoretical values from the white-light laser are much more stable. Based on this fact, to enhance the certainty of the experiments,
2
Pðl,d,m, yÞ ¼
9S1 ðl,d,m, yÞ9 i? ¼ 2 i== 9S2 ðl,d,m, yÞ9
ð1:6Þ
where i? is the scattered irradiance per unit incident irradiance given the incident light is polarized perpendicular to the scattering plane, while i== is its counterpart when the incident light is polarized parallel. Since S1 and S2, as illustrated in Eq. (1.4a) and Eq. (1.4b), are determined by the scattering angle (observation angle) y, the size parameter x, and the complex refractive index m, for the particles with known refractive index, S1 and S2 are decided by the particle size when measuring at a certain angle with a monochromatic laser beam of a fixed wavelength. Thus, by comparing the experimental values of P with the calculated value, we will be able to find the size of the particles under investigation. As indicated in Eq. (1.6), the scattering angle y is an important parameter for the calculation of polarization ratio. When measurements are taken over a certain observation area, the scattering angles for different particles in the area can differ from each other according to their relative positions to the center of the area. Given this, the solid angle and the position of the particles should be taken into account for theoretical calculation when the size of the surface area is comparable with the distance from the observation point to the area center. However, in the current study, the influence of the solid angle can be neglected since the observed area is rather small compared with the distance between the particles and the lens that collects the scattered intensity. The scattering angle thus is represented by the solid angle of the particle at the center of the observed surface. 1.3. Numerical calculation For the theoretical calculation, a code has been developed in Mathematica based on the original work of Lompado [15], which offers precise and detailed numerical calculation with the help of
Fig. 2. (a) Variation of polarization ratio with particle size under different incident wavelength and (b) variation of scattered intensity polarized parallel with particle size under different incident wavelength; (y ¼ 801, m¼ 1.588 þ 0.001i).
60
X.Q. Huang et al. / Optics and Lasers in Engineering 50 (2012) 57–63
measurements with white-light laser provides a better reference of the relation between the particle diameter and the polarization ratio. We can also refer to the variation of the intensity parallel polarized since the curves are generally monotone increasing, especially for the white-light laser. Moreover, measurements at different diffusion angles with different incident laser light can be carried out to provide more experimental results for a certain particle size. The particle diameter is the value which matches the best the results given by each inverse calculation under different experimental condition. By establishing an experimental setup that enables us to obtain the proportion of the perpendicular to parallel polarized component of the scattered light intensity, i.e. the polarization ratio P, at selected angles with different incident laser beams, we can find the diameter of the particles through an inverse calculation with the experimental polarization ratio values.
2. Experimental setup and calibration Here an experimental system has been set up for the determination of the particle size based on the scenario introduced above, as illustrated in Fig. 3. As indicated in Fig. 3, the cylindrical lens generates a fine light sheet from the single-line beam coming out of the laser device, which illuminates the particle samples and generates an observation section. In the current study, we have employed a double pulsed Q-switched Nd:YAG laser which provided a wavelength of 532 nm (pulse duration is 9 ns, time interval between pluses is 400 ms), as well as an Ion laser Stabilite 2017 with a laser-line filter at 514 nm. The incident lights are circularly polarized. The scattered light is collected by a convex lens at a certain scattering angle, and then the two components of polarization are separated into two orthogonal directions by a cube polarizer. Via the reflection onto two mirrors and a prism, the two parts of scattered light are simultaneously received by the digital camera (TSI 630062: 4008 2672 pixels, spectral range of 340–900 nm). The two images of the targeted particles, polarized parallel and perpendicular, respectively, are projected on the CCD simultaneously, each occupying a specific area of the CCD. Thus the proportion of the perpendicular to parallel components of the scattered light intensity can be calculated by correlating and comparing the two images. At the same time, when double pulsed Nd:YAG laser is employed, the two laser pulses with an interval of 20 ms illuminate the targeted particles successively, which provides the information of the particle movement during Dt¼20 ms. Correlation between the images generated by the first and second pulse renders the particle velocity. Polystyrene microspheres of different diameters have been applied to the current study for the calibration of the experimental
Fig. 3. Experimental setup of the particle size measurement.
setup, which were dispersed in pure water in small quartz tanks as experimental samples. The diameters of the spherical particles produced by Spherotech Inc. have been well calibrated. Here we have selected the diameter of 0.61 mm, 1.09 mm, 2.07 mm, 3.43 mm and 4.45 mm to represent a certain range of particle size. The complex refractive index m of the polystyrene particles is known as 1.588þ0.0001i according to Ma et al. [18]. For each incident wavelength of 514 nm and 532 nm, we have selected two scattering angles (y ¼641 and y ¼721) to perform the measurements. Fig. 4 presents the photos taken by the high-speed CCD camera during the experiments with a high solution, which shows at the same time on the CCD two images of the particles illuminated by the laser sheet, with the left one polarized parallel and the right one perpendicular (mirror symmetry), as well as the comparison of the images of the particles illuminated by two successive pulses. Through a correlation program in Matlab, the corresponding particles on the two images can be found so that the ratio of the two polarized light intensity components indicated by the images on the camera CCD can be calculated. As stated above in Section 1.3, measurements under different experimental conditions will enhance the certainty of the particle size determination. To calibrate the experimental setup, measurements have been performed at two different diffusion angles under two different incident wavelengths of 532 nm and 514 nm, respectively. Thus 4 pairs of images are provided with four values of polarization ratio for particles of a single size (Fig. 5). Besides, the variation of the parallel polarized intensity can be obtained from the response of the CCD. The comparison of the experimental polarization ratio values and those of the Lorenz–Mie theory are shown as below. By assigning an error value (n%) to restrict the inverse calculation, a series of particle diameters were found according to the polarization ratio values provided by the experiments. In the current study, the relative deviation of the polarization ratio value n% has been set to be 10%. As indicated by the theoretical curves in Fig. 2a, each experimental polarization ratio value may correspond to several possible diameters due to the fluctuation of the calculation results. For each particle sample investigated, by analyzing the four sets of polarization ratio data obtained under different experimental condition, those possible diameter values
Fig. 4. Images of spherical polystyrene particles (1.09 mm, laser Nd:YAG). I:(1st pulse) and I? (1st pulse) correlated for granulometry; I:(1st pulse) and I:(2nd pulse) correlated for velocimetry.
X.Q. Huang et al. / Optics and Lasers in Engineering 50 (2012) 57–63
61
Fig. 5. Comparison of the polarization value between theory and experiment: (a) l ¼ 532 nm, y ¼ 641; (b) l ¼ 532 nm, y ¼721; (c) l ¼514 nm, y ¼ 641 and (d) l ¼514 nm, y ¼721.
Table 2 Comparison of particle size between the experimental value and the calibrated value. Diameter calibrated (mm) Diameter experimental (mm) Root mean square (RMS) Absolute deviation of particle size (mm) Relative deviation of particle size (%)
0.61 0.62 0.0920 0.01 1.60
given by all the four sets of data were listed out. Then the theoretical polarized parallel intensity corresponding to each of these diameters (according to Fig. 2b) is compared with the experimental value, to render the final result of the particle diameter. The results obtained from the experiments for all the five samples of particle are listed in Table 2, compared with the calibrated values provided by the fabricant. The RMS errors from the inverse calculation process are presented as well. As shown in the table, the accuracy of the method of polarization ratio can be assured.
3. Applications 3.1. Application to particle sizing of soot The method of polarization ratio has been applied to the size determination of soot yielded by a gas flame [19]. The experiment followed the scheme demonstrated in Fig. 3. As indicated in Section (1.2), the theoretical polarization ratio values based on the Lorenz–Mie theory for the white-light laser are much more stable, providing a monotone relation between the polarization ratio and the particle diameter, which greatly enhances the accuracy of the experiments. Therefore, in the
1.09 0.96 0.1617 0.13 11.9
2.07 2.05 0.0937 0.02 0.97
3.43 2.75 0.1427 0.69 20.1
4.45 4.20 0.1954 0.25 5.62
measurements of the soot particle size, a white-light laser (SuperK Versa, KOHERAS, spectral range of 450–2500 nm) was employed, as well as a double pulsed Nd:YAG laser. Fig. 6 shows the images of a single soot particle, with the two components polarized parallel and perpendicular taken by the camera. In Fig. 6(a), the images are not clear since in this case the particle was illuminated by a white-light laser of continuous wavelength and the exposure time was decided by the camera to be 1 ms. While in Fig. 6(b), the particle was illuminated by a Nd:YAG laser of double pulses with each lasting for 5 ns. By calculating the proportion of the two components, i.e. the polarization ratio, we can find the corresponding diameter of the soot particle. Fig. 7(a) and (b) demonstrates the results of the inverse calculation which render the diameter range of the soot particles in the gas flame. The experimental results with different incident laser lights have proved to be consistent. The size range of the soot produced by the flame is determined by the experiments to be 0.38–0.5 mm. 3.2. Combination with velocimetry The authors have worked on the simultaneous determination of granulometry and velocity distribution of oil droplets at the
62
X.Q. Huang et al. / Optics and Lasers in Engineering 50 (2012) 57–63
Fig. 6. Images of the soot particle with two polarized components taken simultaneously by the CCD (20 20 pixels): (a) Illuminated by a continuous white-light laser (0.4–1 mm, exposure time 1 ms) and (b) illuminated by a double pulsed Nd:YAG laser (0.532 mm, exposure time 5 ns). Fig. 8. Scheme of the propeller.
Fig. 7. Determination of the diameter of the soot particle: (a) illuminated by white-light laser and (b) illuminated by Nd:YAG laser.
exit of a propeller with the same experimental setup based on the method of polarization ratio. Experiments were carried out to investigate the swirl movement in a transparent tube after a propeller exit with four blades, each one composing a quarter of the turn (shown in Fig. 8, D ¼26 mm). A light sheet from the double Nd:YAG laser illuminated the particles on the cross section of the flow flux(Z), i.e. a surface vertical to the flowing direction. The measurements were taken under a flow flux of 30 l/min with a pressure difference of 5 mb between the entrance and the exit of the transparent tube. The correlations between the two images of polarized components as well as that between the two images from successive laser pulses were carried out by a program in Matlab. The determination of the particle size has been performed by referring
Fig. 9. (a) Particle size distribution at 36 mm from propeller exit and (b) particle size distribution along radial position (cross section A, z ¼36 mm).
X.Q. Huang et al. / Optics and Lasers in Engineering 50 (2012) 57–63
63
the diameter. The method of the polarization ratio is applicable for the granulometry measurement of the particles ranging from submicron to a few microns. In addition, given that the double pulsed Nd:YAG laser has been employed, the experimental device can also be applied to a simultaneous study of the size and the velocity of particles under investigation in the field of fluid mechanics.
Acknowledgements The authors would like to express their thanks to the research project of SIMBA (Contract no. 072906524) of ‘‘l’Etat et la Communaute´ d’Agglome´ration du Pays de Montbe´liard’’ for the financial support. References
Fig. 10. (a) Tangential velocity distribution in the tube at different distance from the propeller exit and (b) tangential velocity profile at different distances from propeller exit.
to the variation of the parallel intensity component received by the camera. The particle size distribution along the cross section at z¼36 mm is demonstrated in Fig. 9, with the velocity distribution on the cross section along the tube shown in Fig. 10.
4. Conclusion Based on the two real-time measurements demonstrated above, the method of polarization ratio provides non-intrusive simultaneous study of granulometry and velocimetry with a good repeatability. The application of the experimental method to the characterization of particles in fluid mechanics is promising. The sensitivity of the theoretical calculation to the particle size d, the diffusion angle y and the wavelength of the incident light l can be the major reason of the experimental error. As can be concluded from the experimental results shown in Table 2, the accuracy of the method presented in the current study can be assured by taking the measurements under different experimental conditions (different y and l). Moreover, when a white-light laser is employed, a generally monotone relation between the polarization ratio and the diameter can be obtained. The variation of the intensity polarized parallel or perpendicular can also provide a reference for the determination of
[1] Black DL, Mc Quay MQ, Bonin MP. Laser-based techniques for particle-size measurement: a review of sizing methods and their industrial applications. Prog Energy Combust Sci 1996;22:267–306. [2] Hou F, Herve´ P, Riguet A. Particles characteristics determination from the polarization of their scattered radiation. In: Proceedings of the sixth international aerosol Conference IAC, Taipei; 2002. [3] Herve´ P, Kleitw A, Hamouda A, Leporcq B. Determination of submicronic particle size distribution by means of polarized light: application to condensation nuclei and turboreactors. J AeroSci 1994;25:525–6. [4] Herve´ P. Proce´de´ et dispositif pour de´terminer par le taux de polarisation de la lumie re diffuse´e, la taille des particules microniques ou submicroniques, French Patent; 1995. [5] Jones AR. Light scattering for particle characterization. Prog Energy Combust Sci 1999;25:1–53. [6] Fdida N, Blaisot JB, Floch A, Dechaume D. Drop size measurement techniques applied to gasoline sprays: determination of the relevant parameters for application to spray combustion computations. In: proceedings of the tenth international conference on liquid atomization and spray systems: ICLASS, Kyoto, Japan; 2006. p. 7–105. [7] Onofri F. Etat de l’art de la Granulome trie Laser en Me´canique des Fluides. In: 9e me Congre s Francophone de Ve´locime´trie Laser. Brussel, Belgium; 2004. p. CF.2.1–13. [8] Gre´han G, Gouesbet G G, Guilloteau F, Chevaillier JP. Comparison of the diffraction theory and the generalized Lorenz–Mie theory for a sphere arbitrarily located into a laser beam. Opt Commun 1992;90:1–6. [9] Isbimaru A. Wave propagation and scattering in random media. IEEE Press; 1997. [10] van de Hulst HC. Light scattering by small particles. New York: John Wiley and Sons; 1957. [11] Bohren CF, Huffman DR. Absorption and scattering of light by small particles. John Wiley & Sons Inc.; 1983. [12] Born M, Wolf E. Principles of optics. Cambridge University Press; 1999. 7th version. ¨ ¨ ¨ [13] Mie G. Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen. Ann. d. Phys. 1908;25:377–445. [14] Debye P. Der Lichtdruck auf Kugeln von beliebigem Material. Ann. D. Phys. 1909;30:57–136. [15] Lompado A. Light scattering by a spherical particle. Mathematica code, University of Alabama in Huntsville; 2002. [16] Wiscombe WJ. Mie scattering calculations: advances in technique and fast, vector-speed computer codes. National Center for Atmospheric Research Library; 1988. [17] Fu Q, Sun WB. Mie theory for light scattering by a spherical particle in an absorbing medium. Appl Opt 2001;40:1354–61. [18] Ma X, Lu JQ, Brock RS, Jacobs KM, Yang P, Hu XH. Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm. Phys Med Biol 2003;48:4165–72. [19] Huang XQ, Herve´ P, Bailly Y. Caracte´risation de la granulome´trie des particules dans une combustion. In: FLUVISU13-13e me Congre s Franc- ais de Visualisation et de Traitement d’Images en Me´canique des Fluides, Reims, France; 2009.