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Uncertainty evaluation method of the scattering matrix measurements for the polarized scanning nephelometer

Minwang Chen, Zhenwei Qiu, Qiang Hu, and Jin Hong

Open Access Open Access

Abstract

The polarized scanning nephelometer (PSN) is an instrument for measuring the multiangle scattering matrix of aerosol particles. Prior to being used, its measurement accuracy should be first verified. However, due to the complexity of the errors involved, it is very difficult to evaluate the measurement accuracy of the instrument reasonably, which limits the application of the PSN. In this work, taking di-ethyl-hexyl-sebacate droplets as an example, an uncertainty evaluation method of the scattering matrix measurements for the PSN is proposed. According to the measurement characteristics, the sources of uncertainty are divided into systematic effects and random effects. The corresponding evaluation method is then proposed for these two uncertainty categories, and the uncertainty introduced by systematic effects is investigated in depth via the Monte Carlo method. The evaluation results clearly show the variation trend of each uncertainty component as a function of the scattering angle in different matrix elements. The combined uncertainty is calculated and analyzed, and the result shows that the measurement uncertainty of the instrument is less than 0.0185, which verifies the accuracy of the instrument. This method can provide a reference for the accuracy evaluation of similar instruments.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The scattering matrix is a useful parameter to obtain information about the physical characteristics of particles, since it is sensitive to the size distribution, shape, and refractive index of particles [1]. Research on the scattering matrix measurement technology has been ongoing since 1949 [2]. Hunt et al. [3] first introduced the polarization phase modulation to attain the scattering matrix measurement technology, which was then extended by Thompson et al. [4]. Volten et al. [5] developed an instrument to investigate the scattering matrix at wavelengths of 441.6 nm and 632.8 nm and for scattering angles from 5° to 173°. Muñoz et al. [6] built an improved device covering a range of scattering angles of 3°–177°. In addition, many other research groups have carried out extensive investigations in this field [7–13].

As is well known, once an instrument is developed, its accuracy or performance needs to be verified. In order to verify the accuracy of the instrument, researchers have used specific instruments to measure homogeneous spherical particles; the measurement results are then compared with fitted values to obtain the relative deviation. For example, in Muñoz’s work, the reliability of the measurements was tested by comparing the results of water droplet measurements to the results obtained from Mie fitting calculations for a distribution of homogeneous spherical particles [6]. Despite being very convenient, this method is not fully convincing, since the aerosol parameters used in the fitting measurement results may be quite different from the actual parameters. Several research groups have compared manufacturer's data with measurement results. For example, Jean-Luc et al. used polystyrene spheres with an index of refraction of 1.58 at a wavelength of 632.8 nm to test their system. They found that Mie theory calculations are in good agreement with most experimental results except for the discrepancies occurring when the positions of some peaks were shifted from the predicted values. They speculated that the reason for this result may be the slight differences between the modal-diameter value used for the calculations and the actual distributions [14]. This method is more convincing since it has a traceable reference standard; however, it is difficult to determine whether the difference between the measured value and the theoretical value is caused by instrument errors or is due to the fact that the scattering characteristics of the measured object are calculated incorrectly. Thus far, there have been only few studies on the influence of analytical instrument errors on the measurement results of the scattering matrix.

The abovementioned method of comparing the measured value with the reference value (Mie fitting or theory calculations) is essentially a method for evaluating the accuracy based on error theory [15]. As research progresses, uncertainty theory develops on the basis of error theory. Measurement uncertainty is a non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand based on the information used and is an important index of the quality of the measurement results [16, 17]. According to the classification of uncertainty sources, the measurement uncertainty of the instrument is one of the most important components of the uncertainty of the measurand, which represents the impact of the instrument performance on the measurement results. Therefore, this component can be used to quantitatively characterize the accuracy of the instrument. However, due to the influence of various systematic and/or random errors in the measurement process, it is difficult to evaluate the measurement uncertainty of the instrument independently. Therefore, this study estimates the accuracy of the instrument by evaluating the measurement uncertainty of the scattering matrix. However, unlike the uncertainty evaluation of a single output value, the uncertainty evaluation of the matrix is more complex in form. In addition, the measurement process of the scattering matrix is cumbersome and has many sources of uncertainty; this may be a reason why the uncertainty theory has not been applied to the scattering matrix measurement.

In a previous work, in which di-ethyl-hexyl-sebacate (DEHS) droplets were used as the measurand, Hu calculated the accuracy of the instrument quantitatively by calculating the relative deviation between the measured value and the Mie fitting value. In addition, the random part of the uncertainty was considered, but the systematic components related to the measurement uncertainty were not discussed [18]. The present work proposes to redesign the measurement experiment of the DEHS droplets based on the causal analysis of measurement uncertainty. In addition to considering the random factors contributing to measurement uncertainty, the influence of systematic factors on the measurement results of the scattering matrix is also studied, and the measurement accuracy of the instrument is evaluated based on the uncertainty of the measurement results. This study provides a practical method for modeling and evaluating the uncertainty of the PSN scattering matrix measurement. This method makes use of both experimental measurements and Monte Carlo method (MCM) simulations, which can be easily understood and expanded. The validity and practicability of the method can be demonstrated through an evaluation example. It is anticipated that this research will provide a reference for evaluating the uncertainty of the measurement results of similar instruments.

2. Theory

2.1 Basic theory

As shown in Eq. (1), the scattering matrix F is a 4 × 4 matrix connecting the Stokes vector of incident light and scattered light [1]:

$$\left( {\begin{array}{c} {{I_{sca}}}\\ {{Q_{sca}}}\\ {{U_{sca}}}\\ {{V_{sca}}} \end{array}} \right) = \frac{{{\lambda ^2}}}{{4\pi {D^2}}}\mathbf{F}(\theta )\left( {\begin{array}{c} {{I_{in}}}\\ {{Q_{in}}}\\ {{U_{in}}}\\ {{V_{in}}} \end{array}} \right). $$
In Eq. (1), the Stokes parameters I, Q, U, and V describe the intensity and state of the polarization of the light beam, the subscripts in and sca indicate the incident and scattered light, respectively, D is the distance between the ensemble of particles and the detector, $\lambda$ is the light beam wavelength, and θ is the scattering angle, i.e., the angle between the direction of propagation of the incident and the scattered beams.

However, as shown in Eq. (2), if each particulate is randomly oriented and is accompanied by its mirror counterpart, the original form of the matrix, which contains 16 elements, can be simplified to a block-diagonal form, which consists of only six non-zero independent elements [19]. Thus, for a group consisting of a large number of particles, this assumption is reasonable.

$$F = \left[ {\begin{array}{{cccc}} {{F_{11}}(\theta )}&{{F_{12}}(\theta )}&{{F_{13}}(\theta )}&{{F_{14}}(\theta )}\\ {{F_{21}}(\theta )}&{{F_{22}}(\theta )}&{{F_{23}}(\theta )}&{{F_{24}}(\theta )}\\ {{F_{31}}(\theta )}&{{F_{32}}(\theta )}&{{F_{33}}(\theta )}&{{F_{34}}(\theta )}\\ {{F_{41}}(\theta )}&{{F_{42}}(\theta )}&{{F_{43}}(\theta )}&{{F_{44}}(\theta )} \end{array}} \right] = \left[ {\begin{array}{{cccc}} {{F_{11}}(\theta )}&{{F_{12}}(\theta )}&0&0\\ {{F_{12}}(\theta )}&{{F_{22}}(\theta )}&0&0\\ 0&0&{{F_{33}}(\theta )}&{{F_{34}}(\theta )}\\ 0&0&{ - {F_{34}}(\theta )}&{{F_{44}}(\theta )} \end{array}} \right]. $$

2.2 Instrumentation

The PSN can measure the scattering matrix with scattering angles from 3.5° to 170° and at wavelengths of 445 nm and 633 nm. The schematic diagram of the instrument is shown in Fig. 1 [18]; it includes three main modules: a light source module, a scanning polarimeter module, and a sample cell module. The light source module is composed of a laser, a Glan-laser prism (G), a filter wheel (F), and a retarder wheel (R). The filter wheel is equipped with six gray filters of different densities (1%, 3.2%, 5%, 10%, 32%, and 50%), the retarder wheel is equipped with two half-wave plates and one quarter-wave plate for each laser. The scanning polarimeter module includes a total-reflection prism (P), an analyzer wheel (A), a receiving telescope, and a single photon counting module (SPCM). The analyzer wheel is equipped with four analyzers and a quarter-wave plate for each laser. The sample cell module is used to store the measured particles.

 figure: Fig. 1.

Fig. 1. Schematic diagram of the PSN. G: Glan-laser Prism; M: Rotating motor; F: Filter wheel; R: Retarder wheel; P: Total-reflection prism; A: Analyzer wheel; SPCM: Single photon counter module.

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Firstly, the incident laser light is polarized by the Glan-laser prism (G) and modulated by the wave plates. The modulated light then enters the sample cell and is scattered by the particles. Finally, after passing through the total-reflection prism and analyzer wheel, the scattered light is received by the SPCM. The scattering intensity at a specific angle can be obtained through the rotation of the reflecting prism and the counting of the SPCM. The scattering matrix can then be solved by combining the Mueller matrix of the light source and analyzer. The following is a general introduction to the measurement of the scattering matrix.

The Mueller matrix for the retarder is:

$$M({\psi ,{\varphi_1}} )= \left[ {\begin{array}{{cccc}} 1&0&0&0\\ 0&{1 - (1 - \cos \psi ){{\sin }^2}2{\varphi_1}}&{(1 - \cos \psi )\sin 2{\varphi_1}\cos 2{\varphi_1}}&{ - \sin \psi \sin 2{\varphi_1}}\\ 0&{(1 - \cos \psi )\sin 2{\varphi_1}\cos 2{\varphi_1}}&{1 - (1 - \cos \psi ){{\cos }^2}2{\varphi_1}}&{\sin \psi \cos 2{\varphi_1}}\\ 0&{\sin \psi \sin 2{\varphi_1}}&{ - \sin \psi \cos 2{\varphi_1}}&{\cos \psi } \end{array}} \right], $$
where φ1 is the azimuthal angle of the fast-axis, and ψ is the retardance of the retarder. The Mueller matrix of the analyzer or Glan-laser prism is:
$$M({{\varphi_2}} )= \frac{1}{2}\left[ {\begin{array}{cccc} 1&{\cos 2{\varphi_2}}&{\sin 2{\varphi_2}}&0\\ {\cos 2{\varphi_2}}&{{{\cos }^2}2{\varphi_2}}&{\sin 2{\varphi_2}\cos 2{\varphi_2}}&0\\ {\sin 2{\varphi_2}}&{\sin 2{\varphi_2}\cos 2{\varphi_2}}&{{{\sin }^2}2{\varphi_2}}&0\\ 0&0&0&0 \end{array}} \right], $$
where φ2 is the azimuthal angle of the polarization direction of the polarizer. The Stokes vector reaching the SPCM is obtained by multiplying the Stokes vector of the laser beam by all the Mueller matrices of the components in the optical train. Here, only the intensity reaching the SPCM is important, since the SPCM is not sensitive to polarization. By repeating the measurement for these 4 × 4 different combinations of the orientation angles of the optical polarizing elements, 16 different signal intensities can be obtained. Thus, the 16 scattering matrix elements of the aerosol particles can be determined.

The measurement function of the scattering matrix F is then introduced. F can be expressed according to:

$$\mathbf{DN} = {k_1}{k_2}{\mathbf{M}_{sca}}({T_{sca}},r,{\delta _l})\mathbf{F}{\mathbf{M}_{light}}({T_{light}},u), $$
where DN is the response of the SPCM, which represents the intensity of the scattered light after passing through the scanning polarimeter module. Mlight is the light source matrix, which is the combination of the Stokes vector after the incident light passes through four different polarization channels, and Msca is known as the measurement matrix of the scanning polarimeter module. After determining Mlight and Msca, the scattering matrix F can be solved. The specific values and uncertainties of the above parameters are obtained from calibration experiments, and the meaning of each letter and its specific value are shown in Table 1 and Table 2. The filter transmittance given in Table 1 is the actual transmittance at a set value of 5%, as this value is used when measuring DEHS. More details on how to calculate the scattering matrix were mentioned in a previous work [18].

Tables Icon

Table 1. Meaning, value, and uncertainty of each parameter in the measurement equation.

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Table 2. Configurations of the orientation and transmittance of the light source module and scanning polarimeter module.

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Regarding data processing, a commonly used method is here referred to; that is, all elements of the general scattering matrix, except for F11, were normalized by F11 (θ) at same angle, and F11 (θ) was normalized to 1 at 30° [1]. Thus, the F matrix becomes ${\tilde{\mathbf F}}$ in Eq. (6), which represents the normalized scattering matrix:

$${\tilde{\mathbf{F}}} = \left[ {\begin{array}{{cccc}} {{F_{11}}/{F_{11}}\left( {{{30}^ \circ }} \right)}&{{F_{12}}/{F_{11}}}&0&0\\ {{F_{12}}/{F_{11}}}&{{F_{22}}/{F_{11}}}&0&0\\ 0&0&{{F_{33}}/{F_{11}}}&{{F_{34}}/{F_{11}}}\\ 0&0&{ - {F_{34}}/{F_{11}}}&{{F_{44}}/{F_{11}}} \end{array}} \right]$$
F11 is the transformation of the total intensity of the incident light; for unpolarized incident light, this element is proportional to the flux of the scattered light and is also called the phase function. In addition, for unpolarized incident light, the ratio –F12/F11 equals the degree of linear polarization of the scattered light. For further details, the reader can refer to Ref. [20].

3. Uncertainty of the scattering matrix

3.1 Source analysis of the uncertainty

Due to the introduction of polarization, the measurement equation of the PSN is more complex than that of intensity measurement in the same configuration. Since the uncertainty of each parameter is coupled with each other, it is difficult to analyze each of them separately. In addition, the influence of changes in the aerosol itself, external environment, and other factors on the measurement results cannot be independently quantified. To simplify the analysis, a causal analysis method is used to classify the sources of uncertainty.

According to the measurement equation, it can be known that the measurement of the scattering matrix is essentially a measurement based on the model, which is an inverse solution problem. The accuracy of the DN value and of the parameters in the measurement equation is a prerequisite for an accurate calculation of the scattering matrix. Therefore, the sources of uncertainty of the scattering matrix can be divided into the factors affecting the DN value and the uncertainty of the calibration parameters in the measurement equation.

The former may include factors that may change during the measurement process, such as environmental conditions, physical and chemical properties of aerosols, and other factors outside the instrument, as well as instrument factors, such as detector noise and power fluctuations of the light source. In general, these are all random factors. The latter is a systematic factor, and the uncertainty of the measurement equation parameters will be transferred to the scattering matrix along with the measurement equation, resulting in an unknown bias. In most measurement experiments, the uncertainty considered only includes the influence of random factors. In this work, the influence of systematic factors is obtained through Monte Carlo simulations.

Broadly speaking, the sources of uncertainty are divided into two relatively independent categories: random factors and systematic factors.

3.2 Uncertainty introduced by random factors

In order to study the influence of random effects, experiments were carried out using DEHS droplets. The experiments were conducted under controlled conditions to minimize the influence of random factors, and the scattering matrix was solved using the real obtained DN value and the measurement equation. Figure 2 shows the measurement results of all 16 elements of the scattering matrix. The data points and error bars in the figure correspond to the average values and standard errors of 20 measurements, respectively.

 figure: Fig. 2.

Fig. 2. Scattering matrix measurement results of DEHS droplets. Uncertainties are indicated by error bars, unless obscured by symbols.

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Furthermore, according to the A-type evaluation formula of uncertainty, the standard error is used to characterize the uncertainty introduced by random factors [16].

It can be seen that the overall aerosol satisfies the assumption of a random uniform orientation. The uncertainty of the six independent matrix elements is plotted in Fig. 3. In this case, the uncertainty is denoted as uran. The uncertainty curves in Fig. 3 show the sensitivity of the scattering matrix to random factors at various scattering angles.

 figure: Fig. 3.

Fig. 3. Uncertainty of each matrix element introduced by random effects for: (a) F11/F11 (30°), (b) F12/F11, (c) F22/F11, (d) F33/F11, (e) F34/F11, and (f) F44/F11.

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3.3 Uncertainty introduced by systematic factors

In order to evaluate the measurement uncertainty introduced by systematic factors, a MCM specifically developed for uncertainty evaluation is used. The MCM uses probability distribution propagation [17], which is particularly suitable for the uncertainty evaluation of this nonlinear model similar to the measurement of the scattering matrix.

Through simulation measurements, the DN value of the instrument response can be obtained, and the scattering matrix and its uncertainty can then be inversely solved from the measurement equation. Since the DN value obtained through simulations does not contain random factors, but is the only possible actual response after passing through the instrument system, it can be used to characterize the influence of systematic factors on the measurement results. The equation to obtain the DN value via simulations is:

$$\mathbf{DN} = {k_1}{k_2}{\mathbf{M}_{sca}}({T_{sca}},r,{\delta _l}){\bar{\mathbf F}}{\mathbf{M}_{light}}({T_{light}},u). $$
In Eq. (7), ${\bar{\mathbf F}}$ is the average value of the scattering matrix (unnormalized) measured 20 times in the experiment. A random number is generated according to the probability density function (PDF) of each parameter to obtain the DN value. We assume that the PDF here is a normal distribution, and the average and standard deviation of each parameter correspond to the values and uncertainties in Table 1 and Table 2. After obtaining the simulation results for DN, the scattering matrix under the influence of systematic factors can be solved through the simulation of Eq. (8):
$${\mathbf{F}_{sim}}\textrm{ = }\frac{1}{{{k_1}^{\prime}{k_2}^{\prime}}}{\mathbf{A}^{\textrm{ - }1}}({T_A}^{\prime},{r^{\prime}},{\varphi ^{\prime}}_{QWP},{\delta _l}^{\prime}) \cdot \mathbf{DN} \cdot {\mathbf{G}^{\textrm{ - }1}}({T_G}^{\prime},{u^{\prime}}), $$
where the parameters of the instrument with superscripts denote the calibration values. After obtaining Fsim, Eq. (8) can be normalized in the form of Eq. (4). Given a number of simulations m = 106, the corresponding 106 calculation results of the scattering matrix can be obtained, from which the standard deviation of each matrix element can be calculated and used as the uncertainty. In this case, the uncertainty is denoted as usys. Finally, the variation of the uncertainty introduced by the instrument parameters can be obtained as a function of the scattering angle, as shown in Fig. 4. This figure illustrates the uncertainty results of six independent matrix elements. The uncertainty curves show the sensitivity of the scattering matrix to the instrument parameters at various scattering angles.

 figure: Fig. 4.

Fig. 4. Uncertainty of each matrix element introduced by the systematic effects corresponding to: (a) F11/F11_(30°), (b) F12/F11, (c) F22/F11, (d) F33/F11, (e) F34/F11, and (f) F44/F11.

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3.4 Analysis of the uncertainty evaluation results

Since the two types of uncertainty are uncorrelated, for each matrix element, the root-sum-square method can be used to calculate the combined uncertainty:

$$u = \sqrt {{u^2}_{ran} + {u^2}_{sys}}. $$
The variation of the combined standard uncertainty as a function of the scattering angle is plotted in Fig. 5.

 figure: Fig. 5.

Fig. 5. Combined standard uncertainty corresponding to: (a) F11/F11 (30°), (b) F12/F11, (c) F22/F11, (d) F33/F11, (e) F34/F11, and (f) F44/F11.

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According to Eqs. (10) and (11), the average uncertainty of each matrix element and of the whole matrix can be defined, with n = 47 being the number of measured scattering angles. The results are shown in Table 3.

$${\overline u _{ij}} = \frac{1}{n}\sum {{u_{ij}}({\theta _n})}$$
$$u = \overline {\overline {u_{ij}} } $$

Tables Icon

Table 3. Average uncertainty of each matrix element and of the whole matrix.

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As shown in Table 3, the uncertainty of the six matrix elements is less than 0.04, while the uncertainty of the whole matrix is 0.0185; this shows that the measurement uncertainty of the instrument must be less than 0.0185. This is why the experiment is carried out under controlled conditions, since it is necessary to suppress the influence of other factors, not only the instrument, on the measurement results as much as possible. In general, the result of the uncertainty evaluation can be considered to be satisfactory (the measurement uncertainty index is 0.02). Therefore, it can be concluded that the scattering matrix measured via the PSN for different particles is reliable.

4. Conclusions

Taking DEHS particles as an example, an uncertainty evaluation method of the scattering matrix measurements for the PSN is proposed, and the uncertainty evaluation result is used to characterize the measurement accuracy of the instrument. The established uncertainty analysis model can be easily understood and expanded to different instruments or measurands; the errors that affect the measurement can be changed according to the desired scenario. The proposed method is based on the application of the uncertainty theory and is thus a highly scientific approach. In particular, the influence of the instrument parameters was investigated in detail, and the errors sampling was simulated based on the MCM, which is more in line with the actual influence of errors. Furthermore, this method can be easily implemented through computer programming and can thus improve the evaluation efficiency and attain an intelligent evaluation.

The evaluation example shows that the proposed method of using the uncertainty evaluation to verify the accuracy of the instrument is effective and feasible, thus providing a reference for the uncertainty evaluation of related instruments.

Funding

Chinese Academy of Sciences (YZ201664).

Acknowledgments

We thank the anonymous reviewers for their helpful suggestions on the paper.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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2. M. Kerker, “Light Scattering Instrumentation for Aerosol Studies: An Historical Overview,” Aerosol Sci. Technol. 27, 522–540 (1997). [CrossRef]  

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4. R. C. Thompson, J. R. Bottiger, and E. S. Fry, “Measurement of polarized light interactions via the Mueller matrix,” 19, 1323–1332 (1980).

5. H. Volten, O. Muñoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res.: Atmos. 106, 17375–17401 (2001). [CrossRef]  

6. O. Munoz, F. Moreno, D. Guirado, J. L. Ramos, A. Lopez, F. Girela, J. M. Jeronimo, L. P. Costillo, and I. Bustamante, “Experimental determination of scattering matrices of dust particles at visible wavelengths: The IAA light scattering apparatus,” J. Quant. Spectrosc. Radiat. Transfer 111, 187–196 (2010). [CrossRef]  

7. Q. Hu, Z. Qiu, W. Cui, and J. Hong, “Optimum angles for a full-Stokes scanning polarimeter,” 58, 4277 (2019). [CrossRef]  

8. Q. Hu, Z. Qiu, J. Hong, and D. Chen, “New light trap design for stray light reduction for a polarized scanning nephelometer,” 90(2019).

9. Y. Wang, A. Chakrabarti, and C. M. Sorensen, “A light-scattering study of the scattering matrix elements of Arizona Road Dust,” 163, 72–79 (2015).

10. B. Barkey, S. E. Paulson, and A. Chung, “Genetic Algorithm Inversion of Dual Polarization Polar Nephelometer Data to Determine Aerosol Refractive Index,” 41, 751–760 (2007).

11. L. Hespel, A. Delfour, and B. Guillame, “Mie Light-Scattering Granulometer with an Adaptive Numerical Filtering Method. II. Experiments,” Appl. Opt 40, 974–985 (2001). [CrossRef]  

12. A. Ghabbach, M. Zerrad, G. Soriano, and C. Amra, “Accurate metrology of polarization curves measured at the speckle size of visible light scattering,” Opt. Express 22, 14594–14609 (2014). [CrossRef]  

13. O. V. Angelsky, S. G. Hanson, C. Y. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, “On polarization metrology (estimation) of the degree of coherence of optical waves,” Opt. Express 17, 15623–15634 (2009). [CrossRef]  

14. C. Jean-Luc and B. I. J, “Polar nephelometer based on a rotational confocal imaging setup.,” 45(2006).

15. R. Taylor John, “An Introduction To Error Analysis,” (1982).

16. Bipm, Iec, Ifcc, Ilac, Iso, Iupac, Iupap, and Oiml, Evaluation of measurement data – Guide to the expression of uncertainty in measurement (2008).

17. Bipm, Iec, Ifcc, Ilac, Iso, Iupac, Iupap, and Oiml, “Evaluation of measurement data-Supplement 1 to the “Guide to the expression of uncertainty in measurement"-propagation of distributions using a Monte Carlo Method,” Joint Committee for Guides in Metrology, JCGM (2011).

18. Q. Hu, Z. Qiu, J. Hong, and D. Chen, “A polarized scanning nephelometer for measurement of light scattering of an ensemble-averaged matrix of aerosol particles,” Journal of Quantitative Spectroscopy and Radiative Transfer 261, 107497 (2021). [CrossRef]  

19. M. I. Mishchenko and M. A. Yurkin, “On the concept of random orientation in far-field electromagnetic scattering by nonspherical particles,” Opt. Lett. 42, 494–497 (2017). [CrossRef]  

20. M. S. Quinby-Hurt, P. G. Hull, and A. J. Hunt, Polarized Light Scattering in the Marine Environment (Light Scattering by Nonspherical Particles : Theory, Measurements, and Applications, 2000).

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. Schematic diagram of the PSN. G: Glan-laser Prism; M: Rotating motor; F: Filter wheel; R: Retarder wheel; P: Total-reflection prism; A: Analyzer wheel; SPCM: Single photon counter module.

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Fig. 2.
Fig. 2. Scattering matrix measurement results of DEHS droplets. Uncertainties are indicated by error bars, unless obscured by symbols.

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Fig. 3.
Fig. 3. Uncertainty of each matrix element introduced by random effects for: (a) F11/F11 (30°), (b) F12/F11, (c) F22/F11, (d) F33/F11, (e) F34/F11, and (f) F44/F11.

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Fig. 4.
Fig. 4. Uncertainty of each matrix element introduced by the systematic effects corresponding to: (a) F11/F11_(30°), (b) F12/F11, (c) F22/F11, (d) F33/F11, (e) F34/F11, and (f) F44/F11.

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Fig. 5.
Fig. 5. Combined standard uncertainty corresponding to: (a) F11/F11 (30°), (b) F12/F11, (c) F22/F11, (d) F33/F11, (e) F34/F11, and (f) F44/F11.

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Tables (3)

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Table 1. Meaning, value, and uncertainty of each parameter in the measurement equation.

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Table 2. Configurations of the orientation and transmittance of the light source module and scanning polarimeter module.

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Table 3. Average uncertainty of each matrix element and of the whole matrix.

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Equations (11)

Equations on this page are rendered with MathJax. Learn more.

( I s c a Q s c a U s c a V s c a ) = λ 2 4 π D 2 F ( θ ) ( I i n Q i n U i n V i n ) .
F = [ F 11 ( θ ) F 12 ( θ ) F 13 ( θ ) F 14 ( θ ) F 21 ( θ ) F 22 ( θ ) F 23 ( θ ) F 24 ( θ ) F 31 ( θ ) F 32 ( θ ) F 33 ( θ ) F 34 ( θ ) F 41 ( θ ) F 42 ( θ ) F 43 ( θ ) F 44 ( θ ) ] = [ F 11 ( θ ) F 12 ( θ ) 0 0 F 12 ( θ ) F 22 ( θ ) 0 0 0 0 F 33 ( θ ) F 34 ( θ ) 0 0 − F 34 ( θ ) F 44 ( θ ) ] .
M ( ψ , φ 1 ) = [ 1 0 0 0 0 1 − ( 1 − cos ⁡ ψ ) sin 2 2 φ 1 ( 1 − cos ⁡ ψ ) sin ⁡ 2 φ 1 cos ⁡ 2 φ 1 − sin ⁡ ψ sin ⁡ 2 φ 1 0 ( 1 − cos ⁡ ψ ) sin ⁡ 2 φ 1 cos ⁡ 2 φ 1 1 − ( 1 − cos ⁡ ψ ) cos 2 2 φ 1 sin ⁡ ψ cos ⁡ 2 φ 1 0 sin ⁡ ψ sin ⁡ 2 φ 1 − sin ⁡ ψ cos ⁡ 2 φ 1 cos ⁡ ψ ] ,
M ( φ 2 ) = 1 2 [ 1 cos ⁡ 2 φ 2 sin ⁡ 2 φ 2 0 cos ⁡ 2 φ 2 cos 2 2 φ 2 sin ⁡ 2 φ 2 cos ⁡ 2 φ 2 0 sin ⁡ 2 φ 2 sin ⁡ 2 φ 2 cos ⁡ 2 φ 2 sin 2 2 φ 2 0 0 0 0 0 ] ,
D N = k 1 k 2 M s c a ( T s c a , r , δ l ) F M l i g h t ( T l i g h t , u ) ,
F ~ = [ F 11 / F 11 ( 30 ∘ ) F 12 / F 11 0 0 F 12 / F 11 F 22 / F 11 0 0 0 0 F 33 / F 11 F 34 / F 11 0 0 − F 34 / F 11 F 44 / F 11 ]
D N = k 1 k 2 M s c a ( T s c a , r , δ l ) F ¯ M l i g h t ( T l i g h t , u ) .
F s i m  =  1 k 1 ′ k 2 ′ A  -  1 ( T A ′ , r ′ , φ ′ Q W P , δ l ′ ) ⋅ D N ⋅ G  -  1 ( T G ′ , u ′ ) ,
u = u 2 r a n + u 2 s y s .
u ¯ i j = 1 n ∑ u i j ( θ n )
u = u i j ¯ ¯
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