Depolarization Characteristics of Different Reflective Interfaces Indicated by Indices of Polarimetric Purity (IPPs)

Compared with the standard depolarization index, indices of polarimetric purity (IPPs) have better performances to describe depolarization characteristics of targets with different roughnesses of interfaces under different incident angles, which allow us a further analysis of the depolarizing properties of samples. Here, we use IPPs obtained from different reflective interfaces as a criterion of depolarization property to characterize and classify targets covered by organic paint layers with different roughness. We select point-light source as radiation source with wavelength as 632.8 nm, and four samples, including Cu, Au, Al and Al2O3, covered by an organic paint layer with refractive index of n = 1.46 and Gaussian roughness of α = 0.05~0.25. Under different incident angles, the values of P1, P2, P3 at divided 90 × 360 grid points and their mean values in upper hemisphere have been obtained and discussed in the IPPs space. The results show that the depolarization performances of the different reflective interfaces (materials, incident angles and surface roughness) are unique in IPPs space, providing us with a new avenue to analyze and characterize different targets.


Introduction
Reflecting properties of different surfaces can be applied in many areas, such as 3D graphics simulation [1], biological sensing [2,3] and prediction of vegetation [4,5], which has been investigated extensively in recent years. Polarization as a powerful tool plays a great role in Mie ellipsometry [6,7] or full radiative transfer simulations [8]. In addition, polarization information of reflected light, such as degree of polarization (DoP), angle of polarization (AoP), Stokes vector et al., can provide more additional information about the targets in new aspects. When light interacts with targets, the polarization of the incident light may undergo a modification, carrying structural information of the targets. Therefore, it is important to characterize the process of the reflective polarization for exploring the polarization distributions and revealing the features of the surface, which can be used in polarization remote sensing [9,10], target detection and recognition [11][12][13][14], especially in the hidden-target identification. When the sample is not an ideal smooth surface, the reflected light will have different propagating directions, namely scattering. Reflection from a rough surface can be typically modeled using a bidirectional reflectance distribution function (BRDF) [15][16][17][18][19][20][21][22][23]. As a famous geometrical optics model, the Torrance-Sparrow (T-S) model [15] has been widely investigated and expanded to the polarized BRDF (pBRDF) models [16][17][18][19][20][21]. The Monte Carlo (MC) method has also been introduced to solve the composition of samples in fusion devices or reactive plasmas and a series of cases about polarization information [24][25][26][27][28][29][30][31][32][33], especially in photons tracking [24,25,29,30]. Wang et al. have combined the MC model with BRDF, giving birth to a flexible method to acquire the reflective polarization information from a rough surface [34]. However, these models cannot represent depolarization of the samples, which is important in real applications [35][36][37][38][39][40][41][42].

The IPPs of Material Media
To start, we focused on a set of three parameters, called IPPs, which provide an extended way to characterize, analyze and classify samples with different depolarization property. It is because IPPs can represent the relative statistical weight of the decomposed nondepolarizing components, providing a more accurate description of the depolarization properties of the sample. IPPs are defined as relative differences among the four eigenvalues (taken in decreasing order λ 0 ≥ λ 1 ≥ λ 2 ≥ λ 3 ≥ 0) of the covariance matrix H, which can be obtained from MM by the following equation [43]: In theory, four non-negative eigenvalues (λ 0 ≥ λ 1 ≥ λ 2 ≥ λ 3 ≥ 0) can be obtained from the covariance matrix H, which can be used to represent the relative statistical weights of the nondepolarizing components, from which the IPPs can be defined by the following equations: The depolarization index (P ∆ ) can be calculated from IPPs as follows: In general, nondepolarizing samples are characterized by P ∆ = P 1 = P 2 = P 3 = 1. The samples with P ∆ = P 1 = P 2 = P 3 = 0 are corresponding to ideal depolarizers, and there will be MM = diag(m 00 , 0, 0, 0), where m 00 is the mean intensity coefficient. Note that, the values of IPPs will be restricted by the following inequalities:

Microfacet Theory
In general, the height field of rough surface satisfies the Gaussian distribution with variance α. The larger the variance, the rougher the surface. In addition, rough surface is assumed to be made of many microfacets, called microfacets theory, and the normal vector of each microfacet can be determined by θ and σ shown in Figure 1b, which can be calculated by sampling [34]. Then, the reflective light and refractive light of microfacets can be calculated by the Fresnel formula and normal vector of microfacet [48,49].
The depolarization index ( ) can be calculated from IPPs as follows: ues of IPPs will be restricted by the following inequalities:

Microfacet Theory
In general, the height field of rough surface satisfies the Gaussian distribution variance α. The larger the variance, the rougher the surface. In addition, rough surf assumed to be made of many microfacets, called microfacets theory, and the norma tor of each microfacet can be determined by θ and shown in Figure 1b, which c calculated by sampling [34]. Then, the reflective light and refractive light of micro can be calculated by the Fresnel formula and normal vector of microfacet [48,49]. In the tracking process of the polarized light, each beam of light propagates in g coordinate system shown in Figure 1a, and Fresnel's law is used in the local coord system defined by normal vector of microfacets shown in Figure 1b. Thus, it is nece to translate two kinds of coordinate system. The coordinate transformation can b cessed by rotating an angle rot θ [48]. The rotation matrix is ( ) rot R θ , and the correspon can be expressed as follows [48]. In the tracking process of the polarized light, each beam of light propagates in global coordinate system shown in Figure 1a, and Fresnel's law is used in the local coordinate system defined by normal vector of microfacets shown in Figure 1b. Thus, it is necessary to translate two kinds of coordinate system. The coordinate transformation can be accessed by rotating an angle θrot [48]. The rotation matrix is R(θ) rot , and the corresponding Mueller matrix M(θ) rot can be expressed as follows [48].

Monte Carlo Simulation
As shown in Figure 2a, a sample is covered by some coating layers. The point-light source is emitted in the upper hemisphere. When a beam of light reaches to the coating layer with an angle of θ, refraction and reflection will happen. The reflected light will be collected in the upper hemisphere immediately. In contrast, the refracted light will go through a series of reflections and refractions, and can be collected in the upper hemisphere eventually, in which the upper hemisphere is divided into 90 × 360 grids with the step of 1 • (in both zenithal and azimuthal directions) to collect the reflective and refractive photons. The top view of the upper hemisphere is shown in Figure 2b, in which the upper hemisphere can be divided into 90 rings according to the zenith angle (0 •~9 0 • ), and combining azimuth angle (0 •~3 60 • ), the detection grid can be fixed. The exiting light from the coating layers can be collected according to their concrete positions defined by the azimuth and zenith in the corresponding grid, from which their Stokes vectors in each grid can be obtained by counting and summing the received photons' Stokes vectors.

Monte Carlo Simulation
As shown in Figure 2a, a sample is covered by some coating layers. The point-light source is emitted in the upper hemisphere. When a beam of light reaches to the coating layer with an angle of , refraction and reflection will happen. The reflected light will be collected in the upper hemisphere immediately. In contrast, the refracted light will go through a series of reflections and refractions, and can be collected in the upper hemisphere eventually, in which the upper hemisphere is divided into 90 × 360 grids with the step of 1° (in both zenithal and azimuthal directions) to collect the reflective and refractive photons. The top view of the upper hemisphere is shown in Figure 2b, in which the upper hemisphere can be divided into 90 rings according to the zenith angle (0°~90°), and combining azimuth angle (0°~360°), the detection grid can be fixed. The exiting light from the coating layers can be collected according to their concrete positions defined by the azimuth and zenith in the corresponding grid, from which their Stokes vectors in each grid can be obtained by counting and summing the received photons' Stokes vectors. The MC method is adopted to perform the simulation [34]. In order to explore the depolarization of samples covered by organic layers, we needed to get the MM of each grid under conditions of different incident angles and roughnesses of organic layers. Thus, we have defined the direction of the incident light as 30°, 40°, 50°, 60° and 70°, four kinds of incident lights with different polarization states as  The MC method is adopted to perform the simulation [34]. In order to explore the depolarization of samples covered by organic layers, we needed to get the MM of each grid under conditions of different incident angles and roughnesses of organic layers. Thus Calculating the next layer j (usually i +1 or i −1) to be scattered based on the number of current layer i and the direction of propagation.

2.
Sampling the normal vector of the j th layer according α x and α y [34].

3.
Transforming polarized light S from the global coordinate to the local coordinate by S l = M(θ) rot S.

4.
Calculating the direction of reflected and refracted light according to the Fresnel formula and normal vector on the selected microfacet.

5.
Obtaining reflected and refracted light from Fresnel's formula and MM, respectively, by S rl = M r S l , S tl = M t S l , where M r and M t are the reflective Muller matrix and the transmitting Muller matrix, respectively [34]. 6.
Translating them from the local coordinate to the global coordinate, respectively, by S r = M(θ) rot S rl ,S t = M(θ) rot S tl . Calculating MM and covariance matrix of each grid in the upper hemisphere. 9.
The process of photons tracking by MC is shown in Figure 3. The process of photons tracking by MC is shown in Figure 3. It should be noted that our simulation was based on two assumptions: (1) the scale of microfacet is much larger than the incident wavelength, which means the geometry optics can be applied; (2) the coating is so thin that the absorption can be ignored.

Comparing with BRDF Model
To demonstrate the accuracy and validity of the simulation model, we compared the results of reflection from a copper surface obtained by MC simulation and experimentbased BRDF model included in the SCATMECH [50]. Here, it should be noted that the SCATMECH is a light scattering library and published by the NIST in 2017 [50], which has been verified in many experiments. In both simulation schemes, the refractive index of copper and the surface roughness parameters are set as 0.27 + 3.40i and αx = αy = 0.2, respectively, and the incident nonpolarized light (with the wavelength of 632.8nm and the incident angle of 40°) is set as Sin = [1000] with 10 million emitted photons every time. The results of our MC model and the experiment-based BRDF model are plotted in the top and bottom panels in Figure 4, respectively. It is obvious that the reflective polarization patterns of I, Q, U, V, AoP and DoP obtained by our MC model agree well with those obtained by analytical BRDF model, which can verify the accuracy and validity of our model. It should be noted that our simulation was based on two assumptions: (1) the scale of microfacet is much larger than the incident wavelength, which means the geometry optics can be applied; (2) the coating is so thin that the absorption can be ignored.

Comparing with BRDF Model
To demonstrate the accuracy and validity of the simulation model, we compared the results of reflection from a copper surface obtained by MC simulation and experimentbased BRDF model included in the SCATMECH [50]. Here, it should be noted that the SCATMECH is a light scattering library and published by the NIST in 2017 [50], which has been verified in many experiments. In both simulation schemes, the refractive index of copper and the surface roughness parameters are set as 0.27 + 3.40i and α x = α y = 0. The process of photons tracking by MC is shown in Figure 3. It should be noted that our simulation was based on two assumptions: (1) the scale of microfacet is much larger than the incident wavelength, which means the geometry optics can be applied; (2) the coating is so thin that the absorption can be ignored.

Comparing with BRDF Model
To demonstrate the accuracy and validity of the simulation model, we compared the results of reflection from a copper surface obtained by MC simulation and experimentbased BRDF model included in the SCATMECH [50]. Here, it should be noted that the SCATMECH is a light scattering library and published by the NIST in 2017 [50], which has been verified in many experiments. In both simulation schemes, the refractive index of copper and the surface roughness parameters are set as 0.27 + 3.40i and αx = αy = 0.

Influence of Roughness
First, we chose samples (Cu, Al, Al 2 O 3 , Au) covered by organic paint layers as different reflective interfaces, in which the organic paint layer is a common paint and we assume it is a pure substance whose refractive index is 1.46 under the incident wavelength of Sensors 2021, 21, 1221 6 of 10 632.8 nm. We would investigate the dependence of P 1 , P 2 , and P 3 on the roughness of the organic paint layer. In the simulation, the roughness α x = α y ranges from 0.05 to 0.25 with step of 0.05. The incident angle and wavelength of light are set as 632.8 nm and 50 • in the upper hemisphere, respectively. We have investigated four samples, including Cu (n = 0.27 + 3.40i), Al (n = 1.4482 + 7.53i), Au (n = 0.18 + 3.43i), and Al 2 O 3 (n = 1.77), which are covered by an organic paint layer. The number of emitted photons is 10 million to ensure the accuracy of our MC simulation.
To study the overall depolarization property, we calculated the average values of P 1 , P 2 , and P 3 at all physically feasible points [46] in the upper hemisphere of reflective interface, as shown in Figure 5. It can be observed that P 1 , P 2 , and P 3 form IPPs space, in which the point (1, 1, 1) represents nondepolarizing samples and the other points represent depolarizing samples. In other words, the intrinsic depolarizing mechanisms can be demonstrated according to the coordinate in the IPPs space. Figure 5 shows that the values of P 1 , P 2 , and P 3 gradually decrease with increasing roughness, indicating the depolarization of the samples increases with the increasing roughness. It is because light will be scattered rather than reflected at a rough surface, leading to depolarization of light. In addition, we can see that the calculated results of Cu in IPPs space are closer to that of Au, while far away from those of Al and Al 2 O 3 . This phenomenon could be attributed to the refractive index of the samples. As shown above, the refractive index of Cu is similar to that of Au, but quite different from those of Al and Al 2 O 3 . Therefore, we may get different distributions in the IPPs space for different samples, making it possible to classify the samples.

Influence of Roughness
First, we chose samples (Cu, Al, Al2O3, Au) covered by organic paint layers as different reflective interfaces, in which the organic paint layer is a common paint and we assume it is a pure substance whose refractive index is 1.46 under the incident wavelength of 632.8nm. We would investigate the dependence of P1, P2, and P3 on the roughness of the organic paint layer. In the simulation, the roughness αx = αy ranges from 0.05 to 0.25 with step of 0.05. The incident angle and wavelength of light are set as 632.8 nm and 50° in the upper hemisphere, respectively. We have investigated four samples, including Cu (n = 0.27 + 3.40i), Al (n = 1.4482 + 7.53i), Au (n = 0.18 + 3.43i), and Al2O3 (n = 1.77), which are covered by an organic paint layer. The number of emitted photons is 10 million to ensure the accuracy of our MC simulation.
To study the overall depolarization property, we calculated the average values of P1, P2, and P3 at all physically feasible points [46] in the upper hemisphere of reflective interface, as shown in Figure 5. It can be observed that P1, P2, and P3 form IPPs space, in which the point (1, 1, 1) represents nondepolarizing samples and the other points represent depolarizing samples. In other words, the intrinsic depolarizing mechanisms can be demonstrated according to the coordinate in the IPPs space. Figure 5 shows that the values of P1, P2, and P3 gradually decrease with increasing roughness, indicating the depolarization of the samples increases with the increasing roughness. It is because light will be scattered rather than reflected at a rough surface, leading to depolarization of light. In addition, we can see that the calculated results of Cu in IPPs space are closer to that of Au, while far away from those of Al and Al2O3. This phenomenon could be attributed to the refractive index of the samples. As shown above, the refractive index of Cu is similar to that of Au, but quite different from those of Al and Al2O3. Therefore, we may get different distributions in the IPPs space for different samples, making it possible to classify the samples. In order to further demonstrate the advantages of IPPs, we calculated the values of P1, P2, and P3 at each grid in the upper hemisphere when the surface roughness of organic paint layer is αx = αy = 0.05, 0.10, 0.15, 0.20, and 0.25, as shown in Figure 6. Here, we take the sample of Cu as an example. The points at which the values of P1, P2, and P3 equal zero means that eigenvalues derived from the covariance matrix H are negative, called physically unfeasible points [46]. The number of these feasible points increases but the values are decreasing with increasing roughness, which means the average value is decreasing for all physically feasible points when the roughness increases. It is consistent with Figure  5. From the simulation results, we can obtain particular depolarization information of the sample from the physically feasible points. For example, the values of P1, P2, and P3 in the In order to further demonstrate the advantages of IPPs, we calculated the values of P 1 , P 2 , and P 3 at each grid in the upper hemisphere when the surface roughness of organic paint layer is α x = α y = 0.05, 0.10, 0.15, 0.20, and 0.25, as shown in Figure 6. Here, we take the sample of Cu as an example. The points at which the values of P 1 , P 2 , and P 3 equal zero means that eigenvalues derived from the covariance matrix H are negative, called physically unfeasible points [46]. The number of these feasible points increases but the values are decreasing with increasing roughness, which means the average value is decreasing for all physically feasible points when the roughness increases. It is consistent with Figure 5. From the simulation results, we can obtain particular depolarization information of the sample from the physically feasible points. For example, the values of P 1 , P 2 , and P 3 in the point of (60, 0) decrease with increasing roughness, but are always bigger than those in the point of (60, 60). It means that the MM for the latter case has more depolarization components in characteristic decomposition, which reflects the different points in the upper hemisphere having different depolarization components. In other words, light received at different points in the upper hemisphere undergoes various coding by the sample. This characteristic makes it more difficult for us to classify the samples by using the values of P 1 , P 2 , and P 3 at each grid in the upper hemisphere than by using their average values' distributions in IPPs space.
Sensors 2020, 20, x FOR PEER REVIEW 7 of 10 point of (60, 0) decrease with increasing roughness, but are always bigger than those in the point of (60, 60). It means that the MM for the latter case has more depolarization components in characteristic decomposition, which reflects the different points in the upper hemisphere having different depolarization components. In other words, light received at different points in the upper hemisphere undergoes various coding by the sample. This characteristic makes it more difficult for us to classify the samples by using the values of P1, P2, and P3 at each grid in the upper hemisphere than by using their average values' distributions in IPPs space.

Influence of Incident Angle
It is well known that the scattering of light at an irregular structure is highly dependent on the incident angle. Therefore, the dependences of depolarization of samples on the incident angles were investigated. Here, we chose the same reflective interface, but the surface roughness of the organic paint layer was fixed as αx = αy = 0.2. The incident angles are 30°, 40°, 50°, 60° and 70°. The calculated overall distributions of P1, P2, and P3 in IPPs space are shown in Figure 7. For metals, the results show that their depolarization properties decrease with the increasing incident angles. It is because light collected at most grids has smaller scattering components with increasing incident angles. On the contrary, oxides, such as Al2O3, hold opposite results that increasing incident angles result in more depolarization. These results illustrate that metals and oxides have different dependence of depolarization characteristics on incident angles, which makes it possible for us to classify samples.

Influence of Incident Angle
It is well known that the scattering of light at an irregular structure is highly dependent on the incident angle. Therefore, the dependences of depolarization of samples on the incident angles were investigated. Here, we chose the same reflective interface, but the surface roughness of the organic paint layer was fixed as α x = α y = 0.2. The incident angles are 30 • , 40 • , 50 • , 60 • and 70 • . The calculated overall distributions of P 1 , P 2 , and P 3 in IPPs space are shown in Figure 7.
Sensors 2020, 20, x FOR PEER REVIEW 7 of 10 point of (60, 0) decrease with increasing roughness, but are always bigger than those in the point of (60, 60). It means that the MM for the latter case has more depolarization components in characteristic decomposition, which reflects the different points in the upper hemisphere having different depolarization components. In other words, light received at different points in the upper hemisphere undergoes various coding by the sample. This characteristic makes it more difficult for us to classify the samples by using the values of P1, P2, and P3 at each grid in the upper hemisphere than by using their average values' distributions in IPPs space.

Influence of Incident Angle
It is well known that the scattering of light at an irregular structure is highly dependent on the incident angle. Therefore, the dependences of depolarization of samples on the incident angles were investigated. Here, we chose the same reflective interface, but the surface roughness of the organic paint layer was fixed as αx = αy = 0.2. The incident angles are 30°, 40°, 50°, 60° and 70°. The calculated overall distributions of P1, P2, and P3 in IPPs space are shown in Figure 7. For metals, the results show that their depolarization properties decrease with the increasing incident angles. It is because light collected at most grids has smaller scattering components with increasing incident angles. On the contrary, oxides, such as Al2O3, hold opposite results that increasing incident angles result in more depolarization. These results illustrate that metals and oxides have different dependence of depolarization characteristics on incident angles, which makes it possible for us to classify samples. For metals, the results show that their depolarization properties decrease with the increasing incident angles. It is because light collected at most grids has smaller scattering components with increasing incident angles. On the contrary, oxides, such as Al 2 O 3 , hold opposite results that increasing incident angles result in more depolarization. These results illustrate that metals and oxides have different dependence of depolarization characteristics on incident angles, which makes it possible for us to classify samples.
Similarly, the values of P 1 , P 2 , and P 3 at each grid in the upper hemisphere is not significantly dependent on the incident angles, as shown in Figure 8. Here, we still take the sample of Cu as an example. It can be seen that the number of physically unfeasible points slightly decreases with the increasing incident angles, and the physically feasible points have a tendency of spreading towards the center of the circle under a large incident angle. The depolarization performances at different grids are different. In addition, the P 1 , P 2 , P 3 of the same grid are different, which is because the P 1 , P 2 , P 3 as the relative differences of different pure systems mapped from the reflective interface depend on the inherent attribute of reflective interface, which can be used for analyzing IPPs decomposition of reflective interface and exploring the composition of reflective interface. Combining the distribution patterns of P 1 , P 2 , P 3 and the IPPs space has significant advantages in classifying the depolarization characteristics of samples. Similarly, the values of P1, P2, and P3 at each grid in the upper hemisphere is not significantly dependent on the incident angles, as shown in Figure 8. Here, we still take the sample of Cu as an example. It can be seen that the number of physically unfeasible points slightly decreases with the increasing incident angles, and the physically feasible points have a tendency of spreading towards the center of the circle under a large incident angle. The depolarization performances at different grids are different. In addition, the P1, P2, P3 of the same grid are different, which is because the P1, P2, P3 as the relative differences of different pure systems mapped from the reflective interface depend on the inherent attribute of reflective interface, which can be used for analyzing IPPs decomposition of reflective interface and exploring the composition of reflective interface. Combining the distribution patterns of P1, P2, P3 and the IPPs space has significant advantages in classifying the depolarization characteristics of samples.

Conclusions
In this paper, we have emphasized the interest of using IPPs as a criterion for characterization and classification of samples covered by organic paint layers. On one hand, the IPPs carry unique depolarization information of samples, thus leading the unique distributions of overall depolarization for different samples in IPPs space. The distributions of Cu, Al, Au, Al2O3 with different incident angles and roughnesses of organic paint layers were investigated and discussed. On the other hand, the IPPs of each grid vary, which represents that the light coded by samples vary in different directions. These have exhibited the significant potential of using IPPs for target detection and remote sensing, especially the identification of the hidden target.