Analyzing Polarization Transmission Characteristics in Foggy Environments Based on the Indices of Polarimetric Purity

In this paper, we investigated depolarization performance of polarized light in fog scattering system using the indices of polarimetric purity (IPPs) based on the Monte Carlo (MC) algorithm. We compared and analyzed the performances of degree of polarization (DoP) and IPPs in mono-disperse and poly-disperse scattering systems. The depolarization performance of mono-disperse scattering system is dependent on incident infrared wavelength. For the poly-disperse scattering system, the depolarization performance is significantly dependent on the particle-size distributions and the proportion of small particles. These results demonstrate that the IPPs can describe the depolarization performances of disperse systems effectively. It is of great practical significance because it can transmit information in high fidelity better than the DoP.


I. INTRODUCTION
Fog is a common phenomenon in our life. In foggy environments, light will be significantly scattered and absorbed by water droplets, and the light intensity information will be attenuated seriously, affecting the transmission efficiency of information [1]. When light transmits in turbid medium, photons will be strongly scattered, decreasing light intensity and making it impossible to obtain useful information through intensity. In contrast, polarization can describe the information since the polarization state of photons could preserve better after scattering. Besides amplitude, phase, spatial distribution [2], [3], polarization is another basic characteristic of electromagnetic wave, which can be utilized to transport information. When light interacts with objects, the polarization state of light will change accordingly, carrying the information of objects. Therefore, it can be used to characterize the target object [4]. In addition, the polarization of light has better transmission characteristics when passing a scattering system. Thus, it is beneficial to explore The associate editor coordinating the review of this manuscript and approving it for publication was Xinxing Zhou . the polarization transmission characteristics of light in the foggy system, achieving high-fidelity transmission. In recent years, polarization state of light has been widely concerned for its great potential applications in communication [5], [6], navigation [7], [8], detection [9], and imaging [10], [11].
Compared with the traditional imaging method, polarization imaging is more effective in highly scattering system, improving the imaging quality significantly. Schechner et al. proposed a physical model and algorithm using polarized light defogging, which achieved good results and has been successfully applied to underwater polarization restoration [12]- [16]. Later, this method has been modified and improved by some other researchers in different scenarios [17]- [21]. Hu et al. improved this model and polarization information processing algorithm to effectively solve the following problems: (1) The restoration of objects with high degree of polarization (DoP) (low depolarization) in the scene cannot be realized [17], [19]; (2) Restoration fails in high-concentration scattering environment [18]; (3) The restoration effect is poor in a non-uniform light field environment [20]. Based on the dependence of light scattering on wavelength and the optical correlation theory, Shao et al.
proposed a polarization imaging method through highly turbidity water [21]. Xu et al. investigated the transmission characteristics of polarization information in different disperse systems in details, such as mono-disperse and polydisperse scattering systems [22]- [25]. Wang et al. studied the polarization characteristics in the cloud disperse system [26], [27] and Zhang et al. explored the imaging quality of biological tissues [28], [29].
However, the structure of the atmospheric environment is very complicated, and various particles affect the measurement of polarization parameters. Therefore, researchers are trying to investigate polarization parameters to describe specific microstructures. Among them, Mueller matrix (MM) [22]- [28] has attracted more and more attentions as a characterization method since it can comprehensively reflect the polarization characteristics of media. In the past several years, MM has been preliminarily used in optical communication [22]- [24], polarization imaging [25], [26], and detection of cancer tissues [28]. It has been demonstrated that MM of a medium can be further decomposed into parameters and matrices with physical significance, helping us understand the properties of the MM [30]- [38]. Among them, a set of parameters calculated by the covariance matrix of the Muller matrix is called the indices of polarization purity (IPPs) [31]- [33], which is a comprehensive parameter describing the degree of depolarization of the scattering medium [35]- [37], such as biological tissues [38]. Overall, it has been demonstrated that the IPPs can well describe the depolarization characteristics of the scattering system in different scenarios.
In this work, we have numerically studied the depolarization performances of mono-and poly-disperse scattering systems in the infrared band by using the IPPs, and compared the results with traditional method based on the DoP. In mono-disperse system, the dependence of depolarization performance on the incident wavelengths is studied. In polydisperse scattering system, we investigate the dependence of depolarization property on the particle size distribution, including the mean values and standard deviations of the scattering particle size.

A. SIMULATION METHOD
We performed all numerical simulations by using polarized Monte Carlo (MC) algorithm, which has been implemented in calculation of propagation of polarized light in scattering media [39]. A flow-chart including both the main steps of both standard and polarized MC programs is shown in Fig. 1. The steps 1, 2, 3, 4, and 5 are carried out only in the polarized MC program and the other steps should be included in both cases. In step 1, a reference plane is defined to describe the polarization state of light. In step 2, the polarization state of launched photons is defined by Stokes vector. In step 3, the scattering angle θ and azimuthal angle ψ are chosen based on the phase function of the considered scatters and a rejection method. In steps 4 and 5, the reference plane, Stokes vector and meridian plane should be updated for each scattering event owing to the randomness of scattering. Further details about the polarized MC method used in the paper can be found in [39].
Stokes vector (S=[I, Q, U, V] T ) is used to describe the polarization of incident light. The DoP notation is usually used in this field and can be expressed as The degrees of linear polarization (DoLP) and degrees of circular polarization (DoCP) can be expressed by the following formulas The MM can completely describe the optical properties of a scattering medium. Therefore, it is important to decompose the MM. According to the concept of parallel decomposition of Stokes vectors, we can consider the emitting light as a convex linear combination of several incoherent totally polarized states. The Mueller-Jones matrix can be used to describe a pure non-depolarizing deterministic system, in which the completely polarized incident light will result in emitting light with complete polarization. Due to the one-toone relation between MM and Hermitian matrix H, any parallel decomposition expressed in terms of H can be directly translated into the corresponding expression in terms of MM, and vice versa. H can be expressed as the following convex linear combination by four coherency matrices that represent respective pure systems [34], [35] where λ 0 ≥ λ 1 ≥ λ 2 ≥ λ 3 ≥ 0 and µ I (i = 0,1,2,3) are mutually orthogonal feature vectors, and then the IPPs can be defined by the following equation The combination of P 1 , P 2 , and P 3 forms a threedimension space and each point in this space can represent depolarization characteristic of a media. For example, the coordinate (0, 0, 0) and (1, 1, 1) correspond to ideal depolarization and non-depolarizing samples, respectively. For other point, the values of the P 1 , P 2 , and P 3 could be utilized to study the intrinsic depolarizing mechanism. The following quadratic relation between the depolarization index (P) and the three indices of purity (P 1 , P 2 and P 3 ) can be obtained as: where the greater the absolute value of P is, the better the polarization protection ability of the display medium is.

III. RESULTS AND DISCUSSION
The scattering and absorption of mist particles will not only weaken the intensity of light, but also change the polarization state of incident light. For depicting the photon scattering, we established a realistic fog environment model to simulate the propagation behavior of photon. The incident polarized light travels along the z-axis, and the incident light is a point source, in which the incident wavelength are 3.8µm-4.6µm and 10µm-12µm. We simulated the transmission of light in the scattering medium by emitting photons with number of 10 7 for both ensuring the simulation accuracy and saving time, as schematically shown in Fig. 2.
Considering that the main component of fog is water, thus the refractive index of mist particles is set as n = 1.33. We ignored the dispersion of water since the slightly variation on refractive index of mist particles has no significant influence on the performances of DoP and IPPs. The refractive index of atmosphere is set as n = 1 because fog usually forms in the atmospheric surface layer with refractive index approaching n = 1. We use a semi-infinitely wide detection plane to receive scattered photons from different paths. The transmission distance is L.

A. MONO-DISPERSE SCATTERING SYSTEM
A mono-disperse scattering system is simulated to explore the depolarization response of the scattering system. The particle radius is r = 1 µm and the particle number density is 2.1 × 10 −11 /µm 3 . For comparison, the scattering system is illuminated by both circularly polarized light (S=[1,0,0,1]) and linearly polarized light (S=[1,1,0,0]) with wavelengths of 3.8µm-4.6µm and 10µm-12µm. The simulation results are shown in Fig. 3. In the mono-disperse scattering system, as transmission distance increases, both DoCPs and DoLPs decrease in a similar trend. The reductions of DoCPs and DoLPs could be attributed that the photons undergo more collisions in a longer transmission. In addition, the DoCPs and DoLPs increase as the incident wavelength increases. According to the Mie scattering theory, the wavelength increases may result in the decreasing scattering coefficient, so more forward scattered photons could be collected by the detector. Nevertheless, Figs. 3 (a) and 3(b) show that although the DoCPs increases with the increasement of incident wavelength, the change is not significant. Meanwhile, DoCPs and DoLPs show small variations at wavelengths of λ = 10µm and λ = 12µm. Therefore, it is difficult to describe the transmission performance of the scattering system at different wavelengths.
Regarding to this, we use IPPs to study the dependences of the depolarization characteristics of mono-disperse medium on the incident light wavelength and transmission distance. For this purpose, the scattering system is separately VOLUME 8, 2020  Fig. 4.
The polarization purity indices P 1 , P 2 , P 3 and depolarization index P have upward tendency with the increasing incident wavelength. According to the Mie scattering theory, when the incident wavelength increases, the scattering coefficient decreases, and more photons reach the forward detection surface, so the polarization purity of the system increases, and the corresponding system depolarization performance is less. As the transmission distance increases, the photons will collide with more particles, so the number of photons received in the forward direction will decrease. Therefore, the calculated depolarization index P of scattering system will be reduced, and the corresponding depolarization performance will increase. It can be observed that the polarization purity indices P 1 , P 2 , P 3 and depolarization index P can well describe the depolarization characteristics of the scattering medium. Moreover, a comparison between Fig. 3 and Fig. 4 shows that IPPs can distinguish the depolarization performance of scattering media at different transmission distances and different incident wavelengths better than the DoP. Regarding to this result, we hereafter use IPPs to analyze the depolarization characteristics of poly-disperse scattering media.

B. POLY-DISPERSE SCATTERING SYSTEM
In order to better simulate the real fog environment, we need to construct a scattering medium system composed of a mixture of different particles. We at first mix particles with two different sizes of r = 1µm and r = 3µm, and the concentration of the particles is 2.1 * 10 −11 /µm 3 . The ratio of the number of small particles to the total particle numbers is defined as the mixing ratio. The scattering system is separately illuminated by light source with wavelength of λ = 4.2µm and λ = 10µm. The simulation results are shown in Fig. 5.
In this scattering medium, as shown in Figs. 5 (a) and (d), we can see that DoCP and DoLP increase with increasing mixing ratio, however their changes are smoothly, and cannot describe the depolarization performance of the scattering system under different mixing ratios. In Figs. 5(b) and 5(e), the polarization purity indices P 1 , P 2 , and P 3 increase as the mixing ratio increases, and they can describe the depolarization performance of the scattering system under different mixing ratios, indicating that the depolarization ability of the scattering medium is gradually reduced with increased mixing ratio. Then we calculated the corresponding depolarization index P and plotted the results in Figs. 5(c) and (f), it can be observed that the value of P also increases as the mixing ratio increases, the larger its value, the smaller the depolarization effect of the scattering system. In other words, the depolarization ability of the scattering medium decreases as the mixing ratio increases. It is because when the number of small particles in the scattering system gradually increases, the total scattering coefficient decreases and more photons can be received in the forward detection. As a result, the incident polarized light cannot be quickly depolarized into unpolarized light, leading to the decreased depolarization ability of scattering system. The above simulation results show that the depolarization performance of the two-particles mixing system under forward-detection positively depends on the mixing ratio. Figure 5 shows that IPPs can describe the depolarization performance of scattering media under different mixing ratios better than DoP.
In fact, the particle size of a real fog environment follows a certain distribution, which has a certain influence on the depolarization performance of the scattering system. Lognormal distribution is applicable to all random processes and can well reflect the distribution characteristic of the fog particles. Therefore, we investigated a poly-disperse scattering system, where the distribution of particle size can be expressed as the following lognormal distribution [40] where R and σ is the mean value and standard deviation of the distribution inside scattering system, respectively. Equation (7) shows that the sizes and densities of particles depend on the standard deviations σ . We set the mean radius of particles as R = 1.5µm and the standard deviation as σ = 0.3µm, σ = 0.5µm and σ = 0.7µm. As the standard deviation σ increases, the size distribution of the particles will become wider and the particle spectral density will change accordingly, as shown in Fig. 6. We performed numerical simulations of the poly-disperse scattering system, where the mean value of particles radius is R = 1.5µm and the standard deviations are σ = 0.3µm, σ = 0.5µm and σ = 0.7µm. The simulation results are plotted in Fig. 7. We can see from Figs. 7 (a) and (d) that DoLPs and DoCPs have minor changes even coincide as the standard deviation σ increases. Therefore, it is difficult to use DoLPs and DoCPs to describe the dependence of depolarization effect of the scattering system on standard deviations. From Figs. 7(b), 7(c), 7(e) and 7(f), we can see that the polarization purity indices P 1 , P 2 , P 3 and depolarization index P of the scattering system will all become larger when the standard deviation σ increases. This phenomenon shows that in a poly-disperse scattering system, the depolarization performance of scattering systems with same average radius is positively correlated with the standard deviation. When the standard deviation is large, the size of the particles included in the scattering system will increase, as shown in Fig. 6. As a result, the probability of forward scattering increases and more photons could be received by the forward detector. Figure 7 demonstrates that IPPs can describe the dependence of depolarization performance of the scattering system on standard deviation better than DoP.
Overall, the results above show that the depolarization performances of the poly-disperse scattering system significantly depend on the scattering particle radius and the standard deviations of particle size distributions. The most important is that the depolarization performances of the scattering system can be characterized by the IPPs effectively compared to the DoP.

IV. CONCLUSION
In this paper, we numerically investigated the evolution of depolarization performance of polarized light in fogscattering system by using MC algorithm. We compared VOLUME 8, 2020 and analyzed the change of DoP and IPPs in different scattering systems, including mono-disperse and poly-disperse scattering systems. For the mono-disperse scattering system, both DoP and IPPs decrease when the transmission distance becomes larger. However, IPPs could distinguish the depolarization responses at different wavelengths better than DoP. In addition, two poly-disperse systems were studied: if there are only two kinds of particles, the depolarization performance depends on the mixing ratio of the particles; if the system has more than two kinds of particles, the depolarization performance will be affected by the standard deviation of the particle size distribution. The above results demonstrate that the IPPs can describe the depolarization performances of disperse systems more effectively. Therefore, the IPPs may be efficient in other polarization technologies, such as the polarization detection, polarization imaging polarization communications and so on.