1. Introduction

It is well-known that the spherical homogeneous particle is characterized by two parameters: diameter and refractive index, on which the light-scattering distribution of a particle depends. Thus, the accurate determination of the refractive index and the size of particles is of major interest in spray, atmospheric aerosol, environmental monitoring, cell studies and other things [1–6]. A number of publications have focused on the estimating of the refractive indices and the size of particles, and considerable studies concentrate on the assessment of the size or the refractive index with the condition that a priori information about the other one is known beforehand [1–15]. However, in the heating process or for the chemical reactions within the droplets, the refractive index may change, and the size of particle is a function of particle refractive index, such as evaporation and combustion, the refractive index should be measured in situ for more accurate sizing. Consequently, a method intended for the simultaneous measurement of the refractive index and the size distribution of particulates is urgently essential for understanding the physical mechanism of those phenomena.

Several techniques have been developed for getting this information from light-scattering data with the help of additional detector/light sources and algorithm [5–8]. For example, Hu et.al. used a 60°+90° dual-scattering-angle optical particle counter to measure both the refractive index and the size distribution simultaneously, applied to atmospheric aerosol field measurement in Beijing and Hefei, China [5]. Recently, the time-shift technique, namely, the pulsed-displacement technique, was exploited by Schäfer et.al, and the diameter of a spherical transparent particle is given by the time separation between peaks (scattering orders) and the relative refractive index can be estimated by first estimating the incident angles of the first and second modes of the second-order refraction scattering order [6]. Wu et.al. proposed phase rainbow refractometry to measure the transient refractive index and droplet size (micron scale) accurately as well as the tiny diameter change (nanoscale) by the phase shift between the ripple structures [9].

The focus of such work aims to develop a simple and accurate method of simultaneously retrieving the refractive index and size of a particle with extended interferometric particle imaging (IPI) technique. The proposed method, in fact, is similar to phase rainbow refractometry in terms of the basic principle of measurement, in which the defocused image is utilized, but optical paths are different, eventually leading to that the interferogram is from different scattering orders, and recorded at different angles. Phase rainbow refractometry results from the external reflection (p = 0 in Debye series) and the refracted light (p = 2) at rainbow angle, the ripple fringe superimposes on the Airy peaks. One interferogram of our method superimposes from two external reflections (p = 0 and p = 0) at the scattering angles of θ=90° with opposite two-sheet illumination, independent of the refractive index, which is very beneficial for the case when the size of particle is a function of particle refractive index. The other is from the external reflections (p = 0) and the refracted light (p = 1) with one of two-sheet illumination in the side scattering angle region. Phase shift is similarly capable of measuring tiny changes in the particle diameter and the refractive index. The method is demonstrated through experiments, and higher measuring accuracy is obtained.

2. Theoretical analysis

Figure 1(a) schematically shows the experimental setup. A transparent spherical particle with refractive index n in a surrounding medium n₀ (m = n/n₀>1) is illuminated by two laser sheets oppositely from two sides (see red lines). For each laser sheet, the light scattered from a single particle can be interpreted as the sum of all the scattering orders present at the location of the detector. This decomposition of scattered light into various scattering orders is well described by the Debye series [16,17], or by using a geometric optics [18] approach to compute the scattered field. The reflected ray at the particle surface (p = 0) and the first-order refracted ray within the particle (p = 1) are dominant, compared with higher order refraction (p = 2,3,…) at 30°∼80° forward scatter region in Mie scattering, two bright spots called glare points appear on the particle [see fuchsia dots in Fig. 1(b)]. Those glare points are imaged and form two glare-point images on the focus image plane and the interference fringe patterns on the out-of-focus image plane. The diameter of the particle is related to the separation of the glare-point pair or the fringe spacing of the interferogram. The relationship between particle diameter d and the number of fringes N can be expressed by the following [19,20]

 

Fig. 1. (a) Schematic of extended IPI set-up; (b) Ray and glare point geometry for reflection and first-order refraction at different scattering angles.

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where $\theta$ is the scattering angle, $\alpha = 2{\tan ^{ - 1}}({{{d_a}} \mathord{\left/ {\vphantom {{{d_a}} {2{z_\textrm{o}})}}} \right.} {2{z_\textrm{o}})}}$ is the collecting angle, $N = {\alpha \mathord{\left/ {\vphantom {\alpha {\Delta \theta }}} \right.} {\Delta \theta }}$ is the number of fringes, $\Delta \theta$ is the fringe angular spacing, m = n/n₀ is the relative refractive index, $\lambda$ is the light wavelength of the laser sheet, d_a is the aperture diameter of the imaging lens, M = z_i/z_o is magnification of the optical system, z_i and z_o are image distance and object distance (see Fig. 2), respectively, and $\Delta l = ML$, and L is glare-point separation.

 

Fig. 2. Illustration of the experimental configuration of extended IPI set-up.

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At θ=90°, the relative intensity of p = 0 ray is much higher than that of p = 1 ray for perpendicular polarized light; as a result, the brightness of two glare points is different, which makes it difficult to identify and extract the glare point separation and the fringe spacing with high accuracy in the image processing of IPI. Therefore, the two counter-propagating sheets with the same intensity are used to illumine particles, and the two glare points with the same brightness are created, resulting from two lights reflected at the droplet surface (p = 0) [see red dots Fig. 1(b)]. Similarly, those two scattered lights of p = 0 rays form two glare-point images on the focus image plane and the interference fringe patterns on the out-of-focus image plane, independent of the particle refractive index n, as no refraction is involved. The method is proposed by Valero et.al to measure the size and velocity using digital image plane holography, in which a mirror located after the fluid provides a second light sheet propagating in the opposite direction to the incident light [21]. The relationship between particle diameter d and the two externally reflected glare points in the image plane Δl is given by

3. Experiment and results

3.1 Experimental setup

To validate the presented method of simultaneously measuring both the refractive index and the size of a particle experimentally, the experimental system setup shown as in Fig. 1 is assembled. A 532 nm CW semiconductor laser with the maximum power of 5W (MGL-N) is used as a light source, and the laser beam, compressed by a pair of cylindrical lenses, becomes approximately a vertical sheet with a thickness of 0.2 mm. The vertically linearly polarized light, which is s-polarized light, is divided into two light beams of the same intensity by a splitter and then illuminate particles from two counter-propagating directions. The imaging lens is an AF 85 mm f/2.8D lens. The CCD 1 (JAIGO-5000-USB) has 2560 × 2048 pixels and the pixel size is 5 µm × 5 µm, the CCD 2 (GRAS-14S5M/C camera) has 2448 pixels × 2048 pixels and pixel size of 3.45 µm × 3.45 µm. In experiment, we observe interferometric out-of-focus images of particles. Using Eq. (1), the refractive index n is given by

3.2 Water droplet measurement

We first complete the experiment of sample 1, i.e. the experiment of water droplet, which is done in air and so the relative refractive index is m = 1.33. For refractive index measurement system, the scattered angle for laser 1 is set at θ=74° for the observation of the clear fringe images, where the intensities of the reflected (p = 0) and refracted (p = 1) rays are equal (for laser beam 2, the scatting angle is (π-θ)). The measuring magnification and the distance between the CCD plane and lens plane are M = 0.6 and z_r=228.44 mm for θ=90°, and M = 0.879 and z_r=182.68 mm for θ=74°, respectively. Figure 3 is the captured 16-group interference fringe patterns; each column is a group experiment and the top and bottom rows are the fringe patterns of the same droplet at θ=90° and θ=74°, respectively. The fringe angular spacing/the number of fringes is extracted with sub-pixel accuracy using Fourier transformation and the improved Rife algorithm [22], then the particle diameter is calculated using Eq. (4), and the refractive index is obtained by using Eq. (5) with the combination of droplet size determined. For the first group fringe pattern pairs, the estimated frequency of the top interferogram is 0.0475 lp/pix, N = 4.2865 for θ=90°, the estimated frequency of the bottom interferogram is 0.0336 lp/pix, N = 5.4697 for θ=74°, the particle diameter is calculated using Eq. (4) and d = 19.62 µm, the refractive index is calculated using Eq. (5) and n = 1.3649. Each group fringe pattern pair in Fig. 3 is analyzed and the result is shown as in Fig. 4. The average diameter is d = 19.53 µm and the refractive index is 1.3447 ± 0.0131. The standard deviation of the refractive index is 0.0524.

 

Fig. 3. The interferogram images of water droplet at θ=90° (top row) and θ=74° (bottom row), each column is the fringe patterns of the same droplet.

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Fig. 4. Measurement results of water droplet.

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3.3 Polystyrene microsphere measurement

The size and refractive index measurement of polystyrene microsphere is the same as those of water droplet and the experimental setup system shown in Fig. 2 is also used and the parameter of the experiment is set as those in water droplet. In the experiment, the particles are immersed in deionized water, the scatting angle is set to θ=45° in air for refractive index measurement system, and the visibility of the fringes should be a maximum in the defocused image. The measuring magnification and the distance are M = 0.42 and z_r=124.65 mm for θ=90°, and M = 0.2484 and z_r=108.98 mm for θ=45°, respectively. Figures 5(a) and 5(b) are the captured interference fringe pattern at θ of 90° and 45°, respectively, there are six particles in the field of view. The first and most important step in the image analysis of Fig. 5 is to accurately identify in Fig. 5(a) with its corresponding particle in Fig. 5(b). The center of each fringe pattern in Fig. 5 is determined with high accuracy by a unidirectional gradient-matched algorithm [22], denoted with red asterisks, as also shown in Fig. 5. Then we match each particle image in Fig. 5(a) and 5(b) using the center coordinate with combination of the camera calibration. The matched particle image pair of the same particle is labeled with the same serial number, as shown in Fig. 5; namely, the fringe pattern of a particle in Fig. 5(a) is represented by No. 1, then its matched fringe pattern in Fig. 5(b) is also represented by No. 1, and so on. After matching each particle image in the two CCDs, each particle image in Fig. 5(a) and 5(b) are processed as follows. Similarly, the fringe angular spacing/the number of fringes with sub-pixel accuracy is extracted using Fourier transformation and the improved Rife algorithm; then the diameter and the refractive index of particles is calculated when combined with Eq. (4) and Eq. (5), and the results are shown as in Table 1. Note that the result in Table 1 is the geometric correction. In Fig. 2, with the camera at 45°, the laser sheet is not perpendicular to the imaging axis, and thus the defocusing distance and magnification are not uniform across the image, and correction is required. Many experiments have been done and 27 particle image pairs are successfully matched and processed and the corresponding diameter and the refractive index of particles are shown in Fig. 6, respectively. The particle diameter and the refractive index are 50.68 ± 0.13µm and 1.5844 ± 0.0057, and the standard deviation of the diameter and the refraction index are 0.67 µm and 0.0294, respectively.

 

Fig. 5. The interferogram images and the result of the center detection at θ of 90° (left) and 45° (right).

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Fig. 6. Measurement results of the standard particles.

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Table 1. Measurement results of the polystyrene microspheres in Fig. 4

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We have carried out a lot of experiments with different samples and the same sample with different sizes; the overall results of the interferogram detection and frequency estimation are similar to the one presented in Fig. 3 and Fig. 5, which is thus very satisfactory globally. An example of spray field of water and polystyrene microsphere is shown in Fig. 7, using the experimental setup system shown in Fig. 2. The scatting angle is set to θ=70° for refractive index measurement system, M = 0.1898 and z_r=102.28 mm for θ=90°, and M = 0.2950 and z_r=112.15 mm for θ=70°, respectively. Figure 7 is processed by positioning, matching, the fringe angular spacing estimating, and then the diameter and the refractive index of particles is calculated. The result is shown in Fig. 8. Similarly, the result in Fig. 8 is also the geometric correction. The particle diameter and the refractive index of water spray are 61.41 ± 2.41 µm and 1.3319 ± 0.0069, respectively. The standard deviation of the refraction index is 0.0074. The size of three polystyrene microspheres are 44.92 µm, 44.80 µm and 90.87 µm, respectively, and the refractive index of polystyrene microspheres is 1.5850 ± 0.0015.

 

Fig. 7. The interferometric images and the result of the center detection at θ of 90° (left) and 70° (right).

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Fig. 8. Measurement results of water and polystyrene microsphere spray.

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Finally, we simply discuss measuring error in our experiment. From the experimental result, we can find that, the relatively low measuring accuracy occur in water droplet with the diameter of 19.7 µm. The lager measuring error is from the lager error of the extracted frequency, owing to that the error of fringe frequency estimation is mainly from the number of the fringes N. The maximum error of the extracted frequency is obtained for the smallest diameter due to the small number of the fringes N. The smallest amount of visible fringes, with N = 1, the error of the extracted frequency is the maximum value for a given experimental system, i.e.θ, m, λ, and α given. So, the water droplet has a lower measuring accuracy. As far as the refractive index is considered, the main reason is that, as we take it, on one hand, the fringe pattern of water droplet from two CCDs is likely not to match because the water droplet moves relatively fast. In our experiment, we do not control the two CCDs for synchronization, CCD 1 records the fringe patterns of a droplet, and CCD 2 maybe collects the fringe patterns of another droplet, the diameters of those two droplets maybe are different, and eventually leads to relatively large error. On the other hand, the error is from the diameter measurement error and droplet non-sphericity. Water droplet from a single droplet generator may be an ellipsoid after leaving the nozzle due to airflow, which causes error. Additionally, we do not take account of other parameter errors (such as z_r, z₀,θ,α) and sources of noise present (diffraction effect, the laser sheet thickness, light scattering, particle non-sphericity, etc.), the detailed error estimation of IPI has been analyzed in [12] and [23]. The final measuring precision of the particle diameter and the refractive index has been improved in our experiment, and the method presented is very powerful in particle sizing and the refractive index measurement.

4. Conclusion

In this paper, we investigate the method of simultaneously measuring both the refractive index and the size of a particle with extended IPI technique. The size of a particle is detected from the diameter measurement system, which is independent of the refractive index, and the refractive index of particle can be extracted from the refractive index measurement system when combined with droplet size determined. Experiments are done for polystyrene microsphere, water droplet, glass bead, and spray for IPI defocused image and the relative error of the measured refractive index n and the measured particle diameter d is less than 2%. The method presented in this paper can be applied to spray and atmospheric aerosol, particularly beneficial for evaporation and combustion in spray. Furthermore, with extended IPI technique, we may observe two focus images of particles or a focus image and an interferometric out-of-focus image of particles, and simultaneously measuring the particle size and refractive index of combustion field is still under our further exploration.