The size and refractive index of particles can be analyzed through the measurement of polarization state of scattered light. The change of polarization state in Mie scattering has been represented by ellipsometric parameters, Ψ and Δ, like the reflection ellipsometry. The analysis method is called Mie-scattering ellipsometry. By in-process Mie-scattering ellipsometry, the growth processes of carbon particles in argon plasma and in methane plasma were analyzed. It was found that carbon particles grow by coagulation in argon plasma, while they grow by carbon coating in methane plasma. It is also shown that imaging Mie-scattering ellipsometry has the potential for the easier confirmation of optical adjustment from a long distance, as well as for the analysis of spatial distribution of particle size.
- Mie-scattering ellipsometry
- Mie scattering
- fine particles
- imaging ellipsometry
- imaging Mie-scattering ellipsometry
- dusty plasma
- Coulomb crystal
In the same way as in the commonly-used ellipsometry for thin-film analysis by the reflection of polarized light, the size and refractive index of particles can be analyzed through the measurement of polarization state of the scattered light. The change of polarization state in the size range of Mie scattering is able to be defined by using ellipsometric parameters,
The evaluation of size and size distribution of fine particles is important for their research and production. Since 1994, it has become possible to observe fine particles stationarily suspended in plasmas forming Coulomb crystals [3–5]. In such states, the position and movement of individual fine particles are easily analyzed.
Mie-scattering ellipsometry, including polarization-sensitive light scattering measurement method, was applied for the analyses of the growth of fine particles of amorphous carbon [2, 3, 6–9] and amorphous silicon . Lately, image sensors were used for the detection of scattered light instead of photodetector [11, 12]. It is a kind of imaging ellipsometry of Mie scattering. Imaging Mie-scattering ellipsometry has the potential for the analysis of spatial distribution of particle size and the easy confirmation of optical adjustment from a long distance .
Mie-scattering ellipsometry in
In this chapter, the principium of Mie-scattering ellipsometry, its application, imaging method, and their systems are presented.
2. Principium of Mie-scattering ellipsometry
In the same way as in the reflection ellipsometry, ellipsometric parameters
When particles are polydisperse in the size range of Mie scattering, the scattered light generally results in some depolarization even if the incident light is fully polarized . In this case, the Stokes vector and the Mueller matrix are required for the calculation of polarization state. When incident light is linearly polarized with the azimuth at 45° to the scattering plane, the Stokes vector of scattered light from a particle “
Generally, the two polarization components of the light scattered from polydisperse particles are imperfectly coherent. When the degree of coherence is expressed by
Thus, the ellipsometric parameters
3. Monitoring and analyses by Mie-scattering ellipsometry
Ψ- Δ trajectory in growth of polydisperse fine particles
The ellipsometric parameters of polydisperse particles are calculated as functions of mean size and size distribution spread. For the calculation, the lognormal size distribution function, which was verified as appropriate to apply to the coagulation process of particles , is expressed as
Figure 1 shows the evolution of the ellipsometric parameters in the increase of geometric mean diameter with the refractive index of 2.5–0.75
3.2. System layout for Mie-scattering ellipsometry
Figure 2 shows the system layout for the Mie-scattering ellipsometry. The incidence light beam from an argon-ion laser (wavelength: 488 nm, output power: 100 mW) was linearly polarized at an azimuth of 45° from the plane of scattering through a Glan-Taylor prism polarizer. Its polarization state is expressed as
Thus, all four Stokes parameters are determined through the Fourier integral of Eqs. (7a) and (7b). Then, the ellipsometric parameters
A parallel-plate rf (radio frequency) discharge reactor (rf electrode diameter: 5 cm, grounded electrode size: 22 cm × 6.5 cm, and separation of the electrodes: 2.0 cm) was used in the experiments (Figure 3). Argon or methane gas was introduced into the chamber at a position far from the plasma region so as not to disturb or evacuate the suspended particles with its flow. After the generation of plasma at a pressure of 40 Pa and an rf power of 2 or 8 W, the ultra-fine particles of carbon were supplied all at once into the plasma. Then the ellipsometric measurement was started. The observation point was fixed at a position 1.2 cm below the rf electrode on the center axis, which was located in the plasma bulk.
System configuration for the measurement of Mie-scattering ellipsometry during the growth of carbon fine particles in the rf plasma reactor is shown in Figure 4.
In situ measurement by Mie-scattering ellipsometry
3.3.1. Carbon particle growth by coagulation in argon plasma
Argon gas was introduced into the chamber at a position far from the plasma region so as not to disturb or evacuate the suspended particles with its flow. After the generation of argon plasma at a pressure of 40 Pa and an rf power of 2 W, the particles of carbon soot were supplied all at once into the plasma. Then the ellipsometric measurement was started .
The particles were observed from the scattering light to be located at both sides of the measurement position on the laser beam line in the early period. At about 700 seconds after the injection, the light intensity gradually increased at the measurement position. The induction period depended on the experimental conditions. The evolution of the ellipsometric parameters after the initial period is shown on the
In order to fit the calculated trajectory to the experimental data, the simulation was carried out changing parameters; in steps of 0.1 for the real part of the refractive index, 0.05 for the imaginary part, 1 degree for the scattering angle, and 0.1 for the geometric standard deviation along with the assumption of lognormal size distribution of polydisperse particles. The best-fit trajectory was obtained at the refractive index of 2.3–0.35
From the correspondence to the best-fit trajectory, the geometric mean diameter was determined for each experimental data point. Then the particle density was evaluated with the known diameter and the measured scattered intensity. Figure 6 gives the results and shows that the density gradually increases until the particle begins to grow and then rapidly decreases.
The bright region in the plasma formed by Mie scattering, where particles were distributed, was observed by the naked eye under the same plasma conditions. The region was initially dome-shaped. However, it became concentrated at the center axis and moved toward the rf electrode at the same time . The transition may be due to the time evolution of plasma potential interacting with and interacted by negatively charged particles. The particle transfer and concentration causes coagulation, resulting in the increase of the diameter. Then, the plasma potential distribution is affected and enhances the movement. The abrupt decrease of the density at around 800 s in Figure 6 supports this speculation.
3.3.2. Carbon particle growth by coating in methane plasma
The inlet port of methane gas was located near a pumping port so as not to disperse particles trapped in methane plasma, which was maintained under the conditions of 40 Pa gas pressure, 16 sccm gas flow rate, and 8 W rf power. Ultra-fine carbon particles were injected instantaneously into the plasma through another gas inlet port .
After the injection of ultra-fine carbon particles in the plasma, the bright region of Mie scattering was observed visually. The region existed only around the sheath-plasma boundary near the rf electrode for the first 300 second, and then extended into the plasma bulk in the shape of a dome, keeping the almost same shape for about 2000 seconds.
The evolution of the ellipsometric parameters during the period is shown by points on the
The calculation of Mie-scattering ellipsometry of spherical polydisperse particles  was performed for the analysis of the experimental data. Figure 8(a) shows the results of the simulation for the growth of an ensemble of particles with a constant geometric standard deviation 1.5 for the lognormal size distribution, of which three different values of refractive index; 2.3–0.35
The calculated parameters converge to certain values while the experimental ones oscillate remarkably. Then, the calculation with a smaller geometric standard deviation of 1.1 was executed for refractive indexes of 1.7, 1.5, and 1.4 as shown in Figure 8(b). It is seen that the trajectories draw loops. The inclination of the first loop of the experimental data is close to that with the index of 1.5.
Figure 8(c) shows the ellipsometric parameters in further evolution calculated in the case of a fixed refractive index of 1.5 with different geometric standard deviations of 1.15, 1.1, and 1.05. Comparing the results with the experimental trajectory, it can be seen that the standard deviation decreases with the increase of particle size, that is, the size distribution became a rather monodisperse one.
When fine particles grow by coating, size distribution function changes to that of translation-operated lognormal on a linear size scale as
Figure 9(a) shows the variation of the value of
The particles suspended in plasma were observed after the growth using a scanning electron microscope (SEM). All particles were single spheres. Figure 12 shows a SEM micrograph for the particles suspended in plasma for 3 h. It is seen that the particles are nearly spherical and equal in size with the diameter of 3.0–3.1 μm. The spherical shape supports the former discussion on Mie-scattering ellipsometry and coating growth mechanism.
From these results, it can be speculated about the growth of the carbon particles as follows. Hydrogenated amorphous carbon coated the ultra-fine carbon particles in methane plasma. About the smaller value of refractive index than that of a hydrogenated amorphous carbon, it can be understood that the void of 25% volume fraction was included in the coated carbon. Without the homogeneous nucleation of carbon particles in the plasma, which was confirmed by the fact that the Mie-scattered light is not observed in methane plasma without the injection of carbon ultra-fine particles under the same plasma conditions, the particles were equally coated independent of their initial size. From the spherical shape of the grown particles, it is imagined that the particles were coated isotropically. These grown particles with diameter of more than 3 μm can be suspended in methane plasma.
4. Imaging Mie-scattering ellipsometry
Imaging Mie-scattering ellipsometry, like the imaging ellipsometry for thin film analysis, enables the analysis of spatial distribution of particle size, as well as the easier confirmation of optical adjustment from a long distance. The difference between imaging Mie-scattering ellipsometer and above-mentioned Mie-scattering ellipsometer is the imaging of the distribution of fine particles using a 2D image sensor instead of a photodetector.
4.1. Experimental setup for imaging Mie-scattering ellipsometry
Ellipsometric measurement by a system with a rotating compensator has an advantage of the determination of all four Stokes parameters in one rotation. The light intensity changes by the rotation of compensator, whose azimuth is
When the azimuth angle of the polarizer (P) and that of the analyzer (A) are 90 and ±45°, respectively, and the rotating compensator is put between the polarizer and particles, Fourier coefficients indicated in Eq. (9) are related with
The system for the experimental example of imaging Mie-scattering ellipsometry is shown in Figure 13 . The ellipsometer consists of polarizer and analyzer modules and a 2D image sensor. The polarizer module includes a 532 nm wavelength laser, a polarizer (P = 90°), and a rotating compensator. The analyzer module includes a wire grid polarizer (A = −45°) and a mirror. A digital video camera containing a 2D image sensor for the detection of 2D distributed light intensity was placed 1.5 m apart from the analyzer module. An optical band-pass filter with 1 nm FWHM at 532 nm is attached in front of the video camera. The images of distribution of fine particles by scattered light are expanded and focused using built-in functions and an extra lens. Thirty 2D images were captured every second. Although a rotating compensator is set in analyzer modules in conventional rotating-compensator ellipsometers [31, 32], it was set in the polarizer module in this case for the analyzer module able to be installed in a vacuum vessel.
The polarizer module and the digital video camera can be attached to view windows of a large plasma device in the atmosphere. The analyzer module is prepared to be fixed in a vacuum vessel . It was ascertained that an image reflected by the mirror in the analyzed module could be viewed and recorded with the video camera out of the vacuum vessel of the large helical devise (LHD), a nuclear fusion experimental devise in Toki in Japan, when laser light was scattered by a small test material placed at the point of dust particle measurement.
4.2. Experiments of imaging Mie-scattering ellipsometry
In order to confirm the functional capability of the ellipsometric system, a preliminary experiment was carried out using a small dust plasma cubic chamber, 15 cm on a side . Spherical divinylbenzene polymer particles of 2.27 μm in diameter with a dispersion degree of 0.1 μm were injected and suspended in argon plasma under a pressure of 50 Pa.
Light intensity data for 1600 pixels in the area covering the light scattering image were less than 75 of the maximum value of 255. Average values, i.e., total summed values of the data divided by pixel number, were calculated for images every 6° of compensator azimuth angle. The average light intensity (
These values are plotted in
Using the same system of imaging Mie-scattering ellipsometry, the spatial size distribution of parallelly layered fine particles in argon plasma was analyzed . Figure 15 shows a video image of light scattered by spherical divinylbenzene particles of 2.74±0.09 μm in diameter suspended in argon plasma with the pressure of 65 Pa. It is seen that the particles form vertical strings in the lower region and horizontal layers parallelly in the upper region. Fine particles more than 10 in each parallel layers were analyzed by imaging Mie-scattering ellipsometry for each area of 1320 pixels in video images. By the comparison of the determined values
The principium of Mie-scattering ellipsometry and its analytical method were explained in detail. Then, the results of analyses by in-process Mie-scattering ellipsometry for carbon particle growth in argon plasma, as well as in methane plasma, were shown.
In argon plasma, it was suggested from the simulated trajectory best fit to the experimental data that the carbon particles included voids and the size distribution was that of polydisperse particles, such as the diameter ranging mostly from 70 to 150 nm for the mean size of 100 nm. The mean diameter and the density of particles were evaluated, and their time dependence suggests that the size increase of the particles is mostly due to coagulation.
In the case of methane plasma, the trajectory obtained by the model of translation-operated lognormal size distribution on a linear scale agreed with experimental result better than that of lognormal of a constant geometric standard deviation. It was found that ultra-fine carbon particles were coated by hydrogenated amorphous carbon during suspension in methane plasma. The particles were equally coated independent of their initial size without homogeneous nucleation in the plasma. The spherical shape of carbon particles observed in SEM image supports that the particles were coated isotropically with carbon.
An imaging Mie-scattering ellipsometry system, consisting a laser, a polarizer, a rotating compensator, an analyzer and a 2D image sensor, was developed. It was shown that the use of a 2D image sensor instead of a photodetector enables easy confirmation of optical adjustment from a long distance by the image of fine particles without any problems for measurement. The size of distributed fine particles was determined for each separated area. The developed system of imaging Mie-scattering ellipsometer has the potential for the easier confirmation of optical adjustment from a long distance, as well as for the analysis of spatial distribution of particle size.
The authors thank Mr. Masahiro Kawano and Mr. Toshihiko Arayama for experimental assistance, and Prof. Suguru Masuzaki and Prof. Kunihide Tachibana for useful discussions. This work was partly supported by the Ministry of Education, Science, Sports and Culture of Japan under Grant Number 24244094.
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