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Abstract
A novel particle sizing method based on diffraction glare points is proposed, which is independent of the refractive index of particle. With geometric optics approximation, the relationship between distance of diffraction glare points and particle size is obtained. In analysis of measurement parameters, we find that 4° is the optimal central scattering angle for measuring wide size range particles based on diffraction glare points. With an experimental system at this angle, diffraction glare points of four kinds of standard particles were imaged. The relative deviations between measured and nominal particle size were not greater than 2%, less than that using reflection glare points, demonstrating validity and advantage of the particle sizing method based on distance of diffraction glare points.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Glare points are the bright spots on a spherical particle when it is illuminated by light, which have been widely used in particle field measurement such as flash boiling spray and bubbly flows [1–4]. Particle measurements using glare points have advantages of high-speed, non-contact and multi-particles simultaneous measurement. Generally, particle size is measured based on the distance between glare points of reflected and various orders transmitted light. For example, reflection glare point and third order transmission glare point at scattering angle of 96° are used for sizing bubbles [4]; reflection glare point and first order transmission glare point at scattering angle of 69° and 90° are used for measuring water droplets and SiO2 particles respectively [5]. Positions of various orders transmission glare points relate to refractive index of particle, so that the refractive index of particle should be known in the particle sizing method based on transmission glare points [6]. If the refractive index is unknown, particle diameter could be measured using intensity ratio between first transmission glare point and reflection glare point [7]. Whereas, when particle is absorptive, the position of transmission glare point could be difficult to obtain because of low intensity [8]. In this case, the distance of two reflection glare points illuminated by two mutual incident laser sheets can be using for particle sizing [9,10]. Besides, polarization in light scattering should also be considered to keep the glare points intense [11]. In general, particle size measurements based on reflection and various orders transmission glare points could be affected by refractive index of particle and polarization in light scattering.
With the intention to avoid these influences, we propose a novel particle sizing method based on the distance of two diffraction glare points of a spherical particle. Firstly, the diffraction of a spherical particle is considered with geometric optics approximation, and the amplitude distribution on conjugated plane and distance of the two diffraction glare points are given. Then the effects of central scattering angle and aperture angle on diffraction glare points are analyzed, giving optimal system parameters to obtain diffraction glare points. At last, an experimental system was set up for verification, and the measurement results based on diffraction glare points and reflection glare points are compared and discussed.
2. Principle of diffraction glare points
From geometric optics perspective, scattered light of a spherical particle can be decomposed into lights of diffraction, reflection, and various orders transmission [12–15]. With an approximation for θ>2.5/α, the scattering amplitude function of diffraction can be expressed as [13]:
where x = ka = 2πa/λ is dimensionless size parameter of the particle, and k, λ, a are wave number, wave length of incident laser and radius of spherical particle. In Eq. (1), the term out of the square bracket means amplitude of the function, and the two terms in the square bracket are two phases, corresponding to the diffraction phases caused by lower and upper boundaries of the particle, i.e. A and B denoted in Fig. 1. Equation (1) indicates that the two diffractions caused by two edges of a particle have the same amplitude, and are different in phase. Figure 1 shows the imaging process of the two diffractions, in which r and s are object distance and image distance respectively, θ0 is the central scattering angle, m is the refractive index of the particle, and 2b is the aperture of imaging lens, causing aperture angle of α = 2b/r in far-field approximation. After the imaging lens, the two diffractions form amplitude distribution on the conjugated plane of y, where A' and B' are two maximums, namely, two diffraction glare points, the distance of which is L. The w = yr/(as) means dimensionless position on y plane, and w = ± 1 correspond to lower and upper boundaries of the particle respectively.In forward scattering angle, diffraction, reflection, and first order transmission are the main parts of overall scattered light. Therefore, not only diffraction, but also reflection (p = 0) and first order transmission (p = 1) would be imaged, of which the scattering amplitude functions S(p = 0) and S(p = 1) can also be obtained using geometric optics approximation [14]. Then, based on Huygens principle, the amplitudes of diffraction, reflection and first order transmission on imaging plane can be obtained from respective scattering amplitude functions, giving [16]:
in which σ is residual phase caused by approximation, having no effects on the intensity distribution on imaging plane, i.e. |E(w)|2. With Sdiff, S(p = 0), S(p = 1) and Eq. (2), all of which are based on geometric optics approximation, an example of intensity distributions on imaging plane is shown in Fig. 2. Figure 2(a) contains three distributions of diffraction, reflection and first order transmission, and Fig. 2(b) shows distribution of sum of the three terms and the distribution based on Mie solution, which is overall and restricted. The parameters of Fig. 2 are θ0 = 4°, α = 6°, r = s = 100mm, d = 2a = 30μm, λ = 650nm, m = 1.33. It can be seen in Fig. 2(a) that, at near forward scattering angle of θ0 = 4°, the diffraction is most intense in general and forms two obvious peaks near the positions of w = ± 1, corresponding to the upper and lower boundaries of particle. The intensity distribution of first order transmission has maximum at the position near w = 0, corresponding to center of particle. The intensity distribution of reflection is so low that could be nearly ignored. Figure 2(b) shows that sum of the three terms cause the three peaks corresponding to diffraction and first order transmission, and the overall intensity distribution approaches that based on Mie solution.
Fig. 2 (a) Intensity distributions of diffraction, reflection and first order transmission, (b) Intensity distributions of sum of the three terms and Mie. The parameters are θ0 = 4°, α = 6°, r = s = 100mm, d = 2a = 30μm, λ = 650nm, m = 1.33.
It can be seen in Fig. 2 that the diffraction glare points are located at the positions near w = ± 1. In fact, exact positions of diffraction glare points can be obtained from analytic expression derived from Eqs. (1) and (2) with approximation. Ignoring the change of amplitude of the function in Eq. (1) within small aperture angle α, the amplitude term could be extracted from integral in Eq. (2) and only the phase terms remain in integral. Then the intensity distribution on imaging plane was derived as
In Eq. (3), the two sinc functions correspond to two intensity distributions caused by the two diffractions. Sinc function has maximum when the argument is zero, so the positions of the maximums in Eq. (3) are:which are the positions of diffraction glare points, and the distance of them is:where d is the diameter of particle, s/r is the magnification of imaging system. Equations (4) and (5) indicate that diffraction glare points are not exactly on the edges of particle. The distance of diffraction glare points is somewhat longer than particle diameter [17], while the increment is relatively small, i.e. 2/x = 2/150 with the parameters in Fig. 2. Equation (5) shows that, particle size can be easily obtained from magnification of the imaging system and distance of diffraction glare points.3. Analysis of particle measurement parameters
It can be seen in Eq. (3) that, overall intensity distribution of diffraction on imaging plane attenuates greatly with increment of central scattering angle θ0, which means small θ0 benefits to obtaining high intense diffraction glare points. To study the effects of θ0 in detail, Fig. 3 shows intensity distributions of diffraction, reflection and first order transmission under different central scattering angles, with the same aperture angle of 6°. The other parameters are the same as Fig. 2. Figure 3(a) shows that diffraction glare points are much more intense than the reflection and first order transmission ones at central scattering angle of 3.1°, which means two diffraction glare points appear obviously in this case. With increment of θ0, intensity of diffraction glare point gradually decreases, while intensities of reflection and first transmission glare points nearly keep the same value. At θ0 = 5°, as shown in Fig. 3(c), intensity of diffraction glare point is lower than that of first order transmission, which means three glare points could be seen in this case. In Fig. 3(d) with θ0 = 10°, the intensity of diffraction glare point is so low that equals to intensity of reflection glare point, indicating diffraction glare points are difficult to be distinguished.
Fig. 3 Intensity distributions of diffraction, reflection and first order transmission under different central scattering angles of (a) θ0 = 3.1°, (b) θ0 = 4°, (c) θ0 = 5°, (d) θ0 = 10°, with the same aperture angle of α = 6°. The other parameters are r = s = 100mm, d = 2a = 30μm, λ = 650nm, m = 1.33.
Figure 3 shows important effect of central scattering angle on intensity distribution of diffraction on imaging plane. To find appropriate central scattering angle for obtaining obvious diffraction glare points, Fig. 4(a) shows the variation of glare point intensity with θ0. Intensity of glare point is the maximum of intensity distribution, including diffraction, reflection and first order transmission. The other parameters are the same as Fig. 2. It can be seen in Fig. 4(a) that diffraction glare points are only obvious at small θ0 and the intensities of reflection and first order transmission change little with θ0, which agree with the analysis in Fig. 3. We denote θe as the central scattering angle that intensity of diffraction glare point equals to that of first order transmission glare point, indicating that diffraction glare points can be easily distinguished when central scattering angle is smaller than θe. Figure 4(b) shows the variation of θe with particle size. It is obvious that θe varies greatly when particle size is small, while θe keeps at 4° when particle size is larger than 100μm. It means that, at central scattering angle smaller than 4°, diffraction glare points can be seen obviously for arbitrary particle size. Considering that small central scattering angle limits the aperture angle, which is harmful to obtaining sharp glare points with small broadening, 4° is the optimal central scattering angle.
Fig. 4 (a) Variation of glare point intensity with θ0, including diffraction, reflection and first order transmission, (b) Variation of θe with particle size. The parameters are α = 6°, r = s = 100mm, d = 2a = 30μm, λ = 650nm, m = 1.33.
In particle measurement using glare points, the effects of aperture angle should also be considered. Small aperture angle leads to the broadening of a glare point, which could be regarded as diffraction of the aperture. Figure 5 gives the intensity distributions of diffraction, reflection and first order transmission with different aperture angles, under the same central scattering angle of 4°. The other parameters are the same to Fig. 2. Comparing Fig. 5(a) and 5(b), it can be seen that intensities of diffraction glare points under two aperture angles have great difference, while aperture angle has little effects on intensities of the reflection and first order transmission glare points. Whereas, when the aperture angle keeps reducing, as shown in Fig. 5(c) and 5(d), aperture angle has little effects on intensities of all the glare points. It means that large aperture angle contributes to high intense diffraction glare points. Besides, from Fig. 5(a) to Fig. 5(d), it can be seen that the broadening of glare points increases with the decrement of aperture angle, which could make glare points more difficult to be distinguished.
Fig. 5 Intensity distributions of diffraction, reflection and first order transmission with different aperture angles of (a) α = 3.9°, (b) α = 2°, (c) α = 1°, (d) α = 0.5°, under the same central scattering angle of θ0 = 4°. The other parameters are r = s = 100mm, d = 2a = 30μm, λ = 650nm, m = 1.33.
Actually, it is difficult to set the half aperture angle close to central scattering angle, i.e. that in Fig. 5(a), because incident laser has width. Therefore, the first order transmission glare point is generally visible or even brighter than diffraction glare points. It can be seen in Fig. 5 that diffraction and first order transmission glare points are both broadened with a limited aperture angle. The broadening, i.e., the radius of glare point, can be easily derived from the zero and maximum positions of the sinc functions in Eq. (3), as wb = 2π/(αx). In order to distinguish diffraction and first order transmission glare points, considering that the first order transmission glare point is located at the position of w = 0 approximately, combing with the position of diffraction glare points and Rayleigh criteria, the limitation of aperture angle is:
When d = 30μm, with λ = 532nm, the lower limit of half aperture is 1.2°. If optimal central scattering angle of 4° is chosen, half aperture angle would not be larger than 4°. At this situation, the range of aperture size of lens is about 4.2mm to 14mm with r = 100mm, leading to lower limit of measurable size about 15μm with Eq. (6). Whereas, for measuring small particles, Fig. 4(b) shows that central scattering angle can be chosen larger than 4°, which means half aperture angle can also be chosen larger to decrease broadening of glare points and benefit to distinguishing.4. Experimental results and discussions
An experimental system was set up to image glare points, as shown in Fig. 6, in which the central scattering angle is adjustable. The 532nm wavelength laser gone in turn through an elevator, a pair of lens for expanding, a reflector, a pair of cylindrical lens for converging, and then illuminated at the standard particles dispersed in sample cell, which is placed on a rotary table. The aperture, imaging lens (Phenix f = 85mm) and CCD (Point Grey GRAS-50S5M-C 2448 × 2048) were fixed on a rail, the end of which were connected to a rotary table so that the central scattering angle of imaging system could be rotated at any angle from 0° to 90° precisely.
Fig. 6 Photo of the experimental system and imaged glare points, including a transmission glare point in the middle and two diffraction glare points at sides.
The system is firstly used to measure particle size based on diffraction glare points, so the central scattering angle and half aperture angle of the system were set at 4° and 3° respectively, and the refraction of sample cell was considered. The images captured by CCD are shown in the right of Fig. 6, including images of 30μm and 51μm standard polystyrene latex particles, 36.9μm and 47.8μm standard glass bead particles. Although many particles are dispersed in the sample, the glare points from a single particle was easily distinguished because the number concentration of particles is very low. It can be seen in Fig. 6 that three glare points appear in each image, including a transmission glare point in the middle and two diffraction glare points at sides.
In fact, diffraction and reflection glare points are combined in the Debye decomposition of Mie solution, which is strict, so that they cannot be separated extremely in experiment. Whereas, intensities of these two glare points have big difference at different conditions, i.e. diffraction is the main part at near forward scattering angle, while reflection is the main part at side scattering angle of 90°. Hence, we can approximately speak of diffraction glare point at the former condition and speak of reflection glare point at the latter condition.
The captured images shown in Fig. 6 were processed with autocorrelation algorithm to obtain positions of the diffraction glare points, during which the first order transmission glare point in the middle was ignored. Then the extraction accuracy was improved to sub-pixel level by Gaussian interpolation. Then the distance between glare points (L) and diameter of particle were obtained based on Eq. (5), with λ = 532nm, s/r = 1.685, the pixel size of 3.45μm, and the number of pixels between diffraction glare points. Ten images of each kind of standard particles were processed, and the mean values of L and particle diameter are shown in Table 1. The relative deviations between measured and nominal diameter of four kinds of standard particles are not greater than 2%, demonstrating effectiveness of particle sizing method based on diffraction points.
Table 1. Nominal and experimental diameters of the four kinds of standard particles, based on distance of diffraction glare points.
There is an interesting phenomenon that the optimal central scattering angle θe keeps almost unchanged at 4° for particles larger than 100μm, as shown in Fig. 4, which is based on geometric optics approximation. To verify that the scattering angle of 4° is applicable for large particles, the experimental system described above is adopted to image the glare points of standard PSL particles with particle sizes of 70, 90, and 110 microns, as shown in Fig. 7. Although the intensity of the diffraction glare points on the left and right sides is slightly less than that of the refraction glare point in the middle, the intensity of the diffraction glare points is high enough to be resolved.
Currently, common particle sizing methods based on glare points are mainly categorized into two kinds, one is based on distance of two reflection glare points that illuminated by two mutually incident lasers, the other is based on distance of reflection and first order transmission glare points. Measuring the results of the latter method is influenced by reflective index of particle. The former method is independent of the reflective index, the same as the method based on diffraction glare points proposed in this paper. To access particle sizing accuracy of the method based reflection glare points, an experimental system was set up as shown in Fig. 8, in which Fig. 8(a) is the photo of the system and Fig. 8(b) is the corresponding glare point image of a 30μm standard PSL particle. In Fig. 8(a), locations of the aperture, imaging lens and CCD are all kept the same as that in Fig. 6, without changing the magnification and aperture angle. The central scattering angle was set to 90° to obtain equally intense reflection glare points.
Fig. 8 (a) Photo of particle sizing system based on two reflection glare points; (b) Corresponding glare point image of 30μm standard PSL particles.
Ten glare point images of 30μm standard PSL particles as Fig. 8(b) are processed and the mean value of L was obtained. Then the particle size was obtained with [10]:
Results are shown in Table 2, comparing with the measured results using diffraction glare points. It can be seen in Table 2 that the relative deviation of the method based on diffraction glare points is smaller than that using reflection glare points, indicating that diffraction glare points have advantage over reflection glare points in particle sizing accuracy.Table 2. Experimental results of 30μm standard PSL particles using reflection and diffraction glare points.
In fact, the difference of relative deviations between the two methods shown in Table 2 can be analyzed by the distance of glare points. In the particle size measurement using distance of glare points, any glare point would be broadened caused by aperture angle, which means long distance of glare points benefits to distinguishing. In Eqs. (5) and (7), it is obvious that distance of diffraction glare points is longer than that of reflection glare points. Therefore, the diffraction glare points can be used to measure smaller particles with the same aperture angle. Besides, because extraction of glare point position has error, longer distance makes smaller relative deviation and higher precision of particle sizing.
To date, measurement of spherical particles, such as boiling spray and bubbly flows has been developed into several techniques. For example, Phase Doppler method measure the velocity by the Doppler shift of scattering signals by moving particles, and measure the size by the signals’ phase difference received in two fixed directions [18]. For the limitations of system structure of Phase Doppler method, it’s difficult to measure high concentration of large size particles. The Time-shift method is developed to solve that problem, which measure the particle size by the time interval between the reflected and refracted light signals, when the particles fly across a thin laser sheet [19]. The Interferometric Particle Sizing (ILIDS) measure the size by the frequency of reflected and refracted light interference fringes, which measures multiple particles simultaneously [20]. But the particle concentration is limited by the overlapping of the interference fringes. The glare points method can measure multiple particles simultaneously of higher concentrations than ILIDS. Taking this advantage, spherical particle sizing method based on diffraction glare points would benefit to measuring absorbing and concentrated particles.
5. Conclusions
Based on diffraction of a spherical particle with geometric optics approximation, amplitude and intensity distribution of diffraction on imaging plane is derived, and the relationship between distance of diffraction glare points and particle size is obtained. A novel particle sizing method based on distance of diffraction glare points is proposed, which is independent of refractive index of particle. Effects of central scattering angle and aperture angle on various glare points, i.e. diffraction, reflection, first order transmission, are simulated and analyzed. We find that 4° is the optimal central scattering angle for measuring wide size range particles based on diffraction glare points. With this central scattering angle, diffraction glare points of four kinds of standard particles were imaged in experiment, and the particle size was obtained. The relative deviations between experimental and nominal particle size were not more than 2%, less than that using reflection glare points, demonstrating validity and advantage of the particle sizing method based on diffraction glare points.
Funding
National Natural Science Foundation of China (Grant 61275012).
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