1. Introduction

For light waves, most material surfaces in nature can be regarded as rough surfaces satisfying a particular probability distribution. When polarized light waves incident on the surface of an object, they will be scattered, and the polarization state of the scattered light contains a lot of surface information. Studying the polarization scattering characteristic of rough surfaces helps us understand the essence of polarization scattering from the root.

Polarization scattering is of great value in various fields, such as the Mueller matrix imaging technology in biomedicine, which can diagnose cell fibrosis and cancer through polarization characteristics [1–3]. In the field of detection and recognition, polarization complements intensity detection in another dimension. In complex environments such as clouds [4,5], underwater [6,7], and other turbid media [8], extraction and conversion of polarization parameters can greatly enhance detection efficiency, which is of great value. Regarding material detection, the polarization characteristics of materials can be described by the Mueller matrix, which can be applied to nondestructive material detection, etc [9].

For the scattering problem of rough surfaces, the bidirectional reflection distribution function (BRDF) method was the earliest to study the unpolarized scattering characteristics [10]. Adding the polarization factor, BRDF can be extended to the polarization bidirectional reflection distribution function (p-BRDF) [11], and the Mueller matrix is used to describe the light field scattered by rough surfaces. After that, the p-BRDF method is supplemented by parameters such as the masking function [12,13] so that the BRDF model remains bounded at large angles. However, the p-BRDF model depends on the experimental measurement and is greatly affected by the external environment in the actual measurement.

It is difficult to calculate the scattered light field of rough surfaces accurately. For analytical methods, approximate methods are usually adopted, such as the Kirchhoff approximation [14–16] when the roughness is large. Rayleigh perturbation theory can be used when the rough surface is similar to the smooth surface [17]. This approximate method dramatically reduces the calculation amount of optical scattering, but it can only be applied if the approximate condition is satisfied. In principle, the numerical calculation method does not need any approximation. The finite-difference time-domain method discretizes the Maxwell equation directly and performs the difference calculation, realizing the effective calculation of the wideband problem. In addition, it has high computational efficiency due to the use of a cubic grid and difference operation [18,19].

The measurement of the Mueller matrix relies on the interplay between the polarization state generator (PSG) and the polarization state analyzer (PSA). The PSG generates specific polarization states, while the PSA receives the scattering polarization states. These states are typically chosen as a combination of linear and circularly polarized light. A system of equations is constructed to measure the Mueller matrix by generating multiple measurement data sets. By combining the incident polarization states with the scattering polarization states, an equation can be derived to calculate the Mueller matrix [1–3,6,8].

A minimum of six independent polarization states is required to determine the Mueller matrix fully. This paper realizes the simplification of PSG and PSA through the model, which can realize the calculation of the Mueller matrix. This theory of calculating the Mueller matrix using six-polarization incident light is described in more detail in the second part.

To clarify the physical meaning of each component of the Mueller matrix, Lu and Chipman proposed to decompose the Mueller matrix into a series combination of a diattenuator, a retarder, and a depolarizer [20]. Some properties of polarization scattering can be characterized by the decomposition parameters of the Mueller matrix [21,22]. All information about the polarization transformation that light undergoes as it scatters from a rough surface is contained in the Mueller matrix [23].

However, due to the extensive computation amount of light scattering, calculating the Mueller matrix of two-dimensional randomly rough surfaces is rarely carried out by numerical method. In this work, we use the finite-difference time-domain method to calculate the scattering result of randomly rough surfaces. We use the incident light of six polarization states to obtain the numerical calculation results of scattered light and calculate the surface Mueller matrix of the rough surface. The above method is used to calculate the Mueller matrix. The measurement accuracy of the Mueller matrix and the symmetry of the elements of the Mueller matrix are verified. The polarization vector, diattenuation vector, and depolarization parameters are obtained by the polarization decomposition of the Mueller matrix. On this basis, the effects of different incident angles on the elements of the Mueller matrix and their derived parameters are analyzed.

2. Theory and methods

2.1 Measure the Mueller matrix

The Stokes vector describes the intensity and polarization state of light with the four measurable parameters, as shown in Eq. (1).

In Eq. (1), ${S_0}$ represents the total light intensity. ${S_1}$ and ${S_2}$ represent the linearly polarized part. ${S_3}$ represents the circularly polarized part of the total light intensity. ${E_x}$ and ${E_y}$ represent the electric field intensity in the x direction and y direction, respectively. φ is the phase difference between the y direction and the x direction. $\langle \rangle $ represents a temporal average on the measurement time. The polarization information such as linear polarization degree, circular polarization degree, and polarization angle can be obtained by operation between them.

Soleillet related a linear function with 16 parameters to the Stokes vector. On this basis, Mueller et al. proposed a new method to describe the polarization state of light waves, called the Stokes-Mueller system. The Mueller matrix is a 4 × 4 real matrix, which represents the transformation process. The correlation between the Stokes parameter and the Mueller matrix can explain the polarization state change in the optical system during light transmission. The Mueller matrix is defined as a linear process of the polarization state change relationship:

In Eq. (2), ${S^{out}}$ is the reflected or transmitted Stokes vector. ${S^{in}}$ is the Stokes vector of the incident light.

By expanding Eq. (2), the transformation process between the Mueller matrix and Stokes parameter can be obtained, as shown in Eq. (3).

The superscript ‘ stands for the Stokes vector of scattered light. The Mueller matrix $M$ characterizes the polarization state variation of the scattering surface.

The Mueller matrix is usually measured by PSG and PSA together. Since the Mueller matrix has sixteen elements, its solution requires at least sixteen equations. PSG produces incident light with different polarization states, while PSA enables the detection of several specific scattering polarization states. This paper uses six specific incident polarization states instead of PSG and numerical results instead of PSA. As shown in Fig. 1, the polarization state of the incident light was changed successively: 1:0° linearly polarized light, 2:45° linearly polarized light, 3:90° linearly polarized light, 4:135° linearly polarized light, 5: left hand circularly polarized light, 6: right hand circularly polarized light. The polarization states of the scattered light are calculated numerically by the finite-difference time-domain method.

 

Fig. 1. Six incident polarization states are incident on the rough surface successively and the scattered polarization states are calculated separately.

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The light wave is decomposed into the x direction and y direction. Different incident polarization states can be obtained by the superposition of polarized light vectors in the x direction and y direction. The x and y polarization settings of the six polarization states are as follows:

1: The 0° linearly polarized light has an intensity of 1 in the x direction and 0 in the y direction. The Stokes vector is ${\left[ {\begin{array}{{cccc}} 1&1&0&0 \end{array}} \right]^T}$.

2: The 90° linearly polarized light has an intensity of 0 in the x direction and 1 in the y direction. The Stokes vector is ${\left[ {\begin{array}{{cccc}} 1&{ - 1}&0&0 \end{array}} \right]^T}$.

3: The 45° linearly polarized light has the same intensity of 0.7071 in the x and y direction. Yet the y and x direction phase difference is 0. The Stokes vector is ${\left[ {\begin{array}{{cccc}} 1&0&1&0 \end{array}} \right]^T}$.

4: The 135° linearly polarized light has the same intensity of 0.7071 in the x and y direction. Yet the y and x direction phase difference is π. The Stokes vector is ${\left[ {\begin{array}{{cccc}} 1&0&{ - 1}&0 \end{array}} \right]^T}$.

5: The left-hand circularly polarized light has the same intensity of 0.7071 in the x and y direction. The y and x direction phase difference is $\frac{\pi }{2}$. The Stokes vector is ${\left[ {\begin{array}{{cccc}} 1&0&0&1 \end{array}} \right]^T}$.

6: The right-hand circularly polarized light has the same intensity of 0.7071 in the x and y direction. The y and x direction phase difference is $\frac{{3\pi }}{2}$. The Stokes vector is ${\left[ {\begin{array}{{cccc}} 1&0&0&{ - 1} \end{array}} \right]^T}$.

The superscript T stands for the transpose of the matrix.

In this process, the normalized intensity of the light source is set. Under these conditions, the element ${m_{00}}$ of the Mueller matrix can characterize the scattering intensity of the rough surface.

The incident intensity and phase are substituted into Eq. (4). The incident polarization states 1 and 3 are combined. The incident Stokes vector ${S_2}$ and ${S_3}$ are 0. We can get the first column element and the second column element of the Mueller matrix, as shown in Eq. (4).

In Eq. (4), ${E_x}$ and ${E_y}$ are the intensity of the electric field in the x direction and y direction of scattered light respectively. Superscripts 0 and 90 indicate scattering when the incident light is 0° linearly polarized light and 90° linearly polarized light.

After the combination of incident light 2 and 4, two columns of Mueller matrix elements are also obtained: ${m^{\prime}_{00}}$, ${m^{\prime}_{10}}$, ${m^{\prime}_{20}}$, ${m^{\prime}_{30}}$ and ${m_{02}}$, ${m_{12}}$, ${m_{22}}$, ${m_{32}}$. The combination of incident light 5 and 6 can get ${m^{\prime\prime}_{00}}$, ${m^{\prime\prime}_{10}}$, ${m^{\prime\prime}_{20}}$, ${m^{\prime\prime}_{30}}$ and ${m_{03}}$, ${m_{13}}$, ${m_{23}}$, ${m_{33}}$. The upper corner notation $\mathrm{\prime }$ and $\mathrm{\prime\prime }$ are used to distinguish the first column of Mueller matrix elements obtained by different combinations.

In this way, we can get three different columns of elements of the Mueller matrix, all of which have a common first column. These Mueller matrix elements are reduced to an all-element Mueller matrix with the help of the first column.

Take the mean of three groups of first column elements as the first-column elements of the final calculation of the Mueller matrix. To ensure that the elements of the Mueller matrix are coupled into a complete Mueller matrix, we need to process the column elements of the obtained different Mueller matrix.

The first column of the Mueller matrix consists entirely of weight coefficients. The weight coefficients of other column elements can be obtained as shown in Eq. (5).

By taking the weight coefficients of the elements in the second, third, and fourth columns respectively, the full Mueller matrix can be obtained, as shown in Eq. (6).

In the above process of obtaining the Mueller matrix, the intensity of incident light has been normalized.

2.2 Decompositions of the Mueller matrix

The polarization scattering characteristics of rough surfaces can be described by the Mueller matrix, which contains complete information about polarization and depolarization characteristics compared with the Jones matrix form and Poincare sphere method. The decomposition of the Mueller matrix can make the parameters of polarization information more clearly expressed.

The decomposition method proposed by Lu and Chipman decomposes the measured Mueller matrix into the product of the component matrices, which represent different properties [20]. On this basis, there are many ways to decompose the Mueller matrix, such as the symmetric depolarized Mueller matrix decomposition method which takes singular value decomposition as the basic step. In this decomposition method, the depolarization Mueller matrix is decomposed into five terms representing basic polarization devices, namely diagonal depolarizers stacked between two pairs of retarders and attenuators.

The characteristic parameters of the Mueller matrix can be interpreted and parameterized directly [24]. In addition, the differential decomposition of the Mueller matrix obtains the differential Mueller matrix from the macroscopic matrix by characteristic analysis. It is decomposed into a complete 16 differential matrices corresponding to the basic type of optical behavior of depolarized anisotropic media.

The decomposition of the Mueller matrix can be divided into serial and parallel modes [21]. A serial decomposition represents a general Mueller matrix as a product of a particular Mueller matrix. The physical significance is that the whole system is considered a cascade of polarized components so that incident light interacts with them in turn. This arrangement of components constitutes a “serial equivalent system”. A parallel decomposition is a representation of a Mueller matrix as a convex sum of a Mueller matrix. The physical meaning of parallel decomposition is that the incident beam interacts with multiple optical components simultaneously and is eventually incoherently recombined into a single output beam.

Any Mueller matrix can be decomposed in series and parallel, and the resulting model is consistent with the mathematical structure and reciprocity property of the Mueller matrix [25]. Therefore, the polarization decomposition of the Mueller matrix is of general significance, and its decomposition-derived parameters are of great significance as the characterization of polarization scattering.

By generalized polar decomposition, the Mueller matrix can be expressed as:

The superscript T is the transpose of the matrix.

The matrix ${M_D}$ is related to pure light attenuation. The matrix ${M_R}$ represents the pure retarder. The depolarization matrix ${M_\Delta }$ exhibits both depolarization and attenuation characteristics.

The bidirectional diattenuation vector $\vec{D}$ can attenuate linearly polarized light and circularly polarized light [20]. The bidirectional diattenuation vector $\vec{D}$ is expressed as:

The polarization vector $\vec{P}$ can polarize natural light. The polarization vector $\vec{P}$ is expressed as:

Depolarization index ${P_\Delta }$ gives an average measure of the depolarization power of an optical system [26].

Mueller matrix describes a non-depolarizing system if and only if when ${P_\Delta }$=1. The value of ${P_\Delta }$ ranges from 0 (fully depolarized) to 1 (non-depolarized).

In order to ensure the physical realizability of the measured Mueller matrix, non-over polarization conditions should be satisfied [27].

The non-over polarization condition is shown in Eq. (12), which means that the polarization of the scattered light after the incident light passing through the passive optical system is less than or equal to 1.

To ensure the physical realizability of the Mueller matrix [26,27], the following requirement should be met:

3. Results and discussion

Our work in this section is as follows: first, we use the method described in Section 2 to calculate the Mueller matrix. To verify the accuracy of our calculation results, we select any elliptically polarized light as incident light to calculate the Stokes vector of the scattered polarization state. This result is compared with that obtained by the linear transmission formula ${S^{out}} = M \cdot {S^{in}}$.

Then we decompose the Mueller matrix by the polarization decomposition method to get the derived parameters. The derived parameters were compared with the data in the Ref. [28].

Finally, we calculate the Mueller matrix of different materials at different incidence angles and describe the polarization scattering characteristics by derived parameters.

We established a rough surface model with root mean square height $\sigma = 0.5\mathrm{\mu}\textrm{m}$. And the size of the model is $6\mathrm{\mu}\textrm{m} \times \textrm{6}\mathrm{\mu}\textrm{m}$. The model was set with the refractive index of 1.4, 1.8, and 2.2. Here we have chosen specific refractive indices 1.4, 1.8, and 2.2. These values are commonly encountered in optics and materials science.

At the wavelength of 0.6328µm, some polymers and optical glasses often have refractive indices between 1.4 and 1.8. These polymers and optical glasses are widely used in the field of optics. We used three refractive index models, representing polymers and optical glasses, which may not be universally applicable to all materials but have important implications for the theoretical methods used to calculate the Mueller matrix.

Only two refractive indices are insufficient to fully distinguish the polarization scattering properties caused by this refractive index difference. In addition, we also select a larger refractive index, 2.2. We use three ideal refractive index models, compared with the two metals and themselves, and can more clearly compare the influence of refractive index on the parameters of the Mueller matrix.

Then the scattering polarization states are numerically calculated by using six incident polarization states with center wavelength $\lambda = 0.6328\mathrm{\mu}\textrm{m}$. The full width at half maximum is 50 nm.

Using the method described in Section 2, we calculate the surface Mueller matrix of the rough surface through six incident polarization states and their scattering polarization states. The Mueller matrix obtained by calculation is shown in Fig. 2. The Mueller matrix is the strength graph with normalization processing, namely all elements of the Mueller matrix divided by the element m₀₀.

 

Fig. 2. The Mueller matrix element strength diagram. The length of the rough surface is $6\mathrm{\mu}\textrm{m}$ in both x and y directions. The number of rows and columns corresponds to the subscript of the elements of the Mueller matrix. The first row and the first column correspond to the Mueller matrix element m₀₀. The first row and second column correspond to m₀₁, etc.

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Figure 2 is divided into 16 small images corresponding to 16 elements of the Mueller matrix. The length of the image in the x and y directions is $6\mathrm{\mu}\textrm{m}$ respectively, that is, the area directly above the rough surface. The horizontal and vertical coordinates represent its position, and the colors represent its intensity. The small image contains 200 by 200 pixels. Each pixel has a Mueller matrix. The obtained Mueller matrix of each point is verified by Eq. (13).

The results shown in Fig. 2 are obtained at the incidence angle $({\theta _i},{\varphi _i}) = (45^\circ ,90^\circ )$ and the scattering angle $({\theta _s},{\varphi _s}) = (0^\circ ,0^\circ )$. ${\theta _i}$, ${\varphi _i}$ are the incident zenith angle and azimuth angles, respectively. ${\theta _r}$, ${\varphi _r}$ are the reflected zenith and azimuth angles, respectively.

There is a strong symmetry between the elements of the Mueller matrix: m₀₀= m₁₁; m₀₁= m₁₀; m₂₂= m₃₃; m₂₃=-m₃₂. The same result exists at different angles of incidence. This result is also mentioned in the [29].

The symmetry of the Mueller matrix is affected by the characteristics of the medium, which is rarely affected by the incidence angle and the reflection angle. Different materials exhibit different symmetry properties [15,29,30]. The calculating result of the Mueller matrix in this paper is carried out on the surface of isotropic media.

After obtaining the Mueller matrix, we verify the result with an arbitrarily incident elliptically polarized light. As the incident light, the elliptically polarized light interacts with the rough surface to obtain its scattered light field ${E_x}$, ${E_y}$ and phase $\varphi $, which is converted into Stokes vector ${\left[ {\begin{array}{{cccc}} {{S_0}}&{{S_1}}&{{S_2}}&{{S_3}} \end{array}} \right]^T}$. The Stokes vector is shown as the first row in Fig. 3.

 

Fig. 3. Stokes vector in numerical simulation and Stokes vector in linear transmission formula calculation.

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When the incident elliptically polarized light is determined, we get its Stokes vector is ${\left[ {\begin{array}{{cccc}} 5&3&2&{2\sqrt 3 } \end{array}} \right]^T}$. Using the Mueller matrix obtained and the linear transfer formula of the Mueller matrix ${S^{out}} = M \cdot {S^{in}}$, the Stokes vector ${\left[ {\begin{array}{{cccc}} I&Q&U&V \end{array}} \right]^T}$ is obtained. This vector is represented as the second row in Fig. 3.

The maximum relative errors of Stokes vector elements in the strength matrix in Fig. 3 are 0.0562, 0.0635, 0.0443, and 0.0461, respectively. The Stokes vector of rough surface scattering obtained by these two methods is highly similar, which proves that the Mueller matrix obtained by this method has high accuracy.

The Mueller matrix obtained by the above method is the surface Mueller matrix. The average value of the Mueller matrix elements at each point on the surface can be obtained.

According to the decomposition characteristics of the Mueller matrix, we can get that m₀₀ represents the scattering intensity of natural light in rough surfaces. The bidirectional diattenuation vector $\vec{D}$ shows the attenuation capacity of the incident polarized light. And the polarization vector $\vec{P}$ represents the polarization ability of rough surface on unpolarized light. $\vec{m}$ shows the phase delay and depolarization ability of rough surface. The modulus of bidirectional diattenuation vector $|D|$, polarization vector $|P|$ and depolarization parameter ${P_\Delta }$ of the Mueller matrix can be obtained by the polarization decomposition method.

After obtaining this derived parameter, We compare the results obtained by this method with the result in [28], as shown in Fig. 4.

 

Fig. 4. The module of polarization vector from black paint coating. The olive-green solid line is the derived result of the Mueller matrix obtained by measuring the rough surface model with the refractive index n = 1.46 and k = 1.32. The black dotted line is the measurement result in Ref. [28]. The red dotted line is the prediction result of the p-BRDF model; The blue dotted line is the result obtained when the reference refractive index n = 1.405 and k = 0.2289.

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Figure 4 shows the degree of polarization at different incident angles when the scattering angle is (60°,0°).

The experiment curve is obtained with the source of a tungsten filament lamp. After getting the Mueller matrix, we treat the incident light unpolarized. $|\mathrm{{\rm P}} |$ represents the degree of polarization here.

The experiment result in Fig. 4 is from the Electro-Optical Research Laboratory (EORL) at New Mexico State University [28]. The estimate result is based on the p-BRDF model proposed by Priest and Meier [31] that describe the scattering of reflective surfaces by various target materials. The blue dotted reference line is obtained when the refractive index is n = 1.405 and k = 0.2289 [32].

The blue reference line applies a different refractive index. This curve has the similar changing trend as the other three curves, which justifies the calculation. While this large difference shows that accurate optical parameters are essential for calculation.

The roughness of the rough surface is $2.99\mathrm{\mu}\textrm{m}$. The Mueller matrix was measured with an incident center wavelength of 650 nm.

It can be seen from Fig. 4 that the degree of polarization obtained by measuring the Mueller matrix is consistent with the experiment and estimate results [28]. The method of measuring the Mueller matrix can realize the extraction of the degree of polarization.

The derived parameters obtained by this method are verified with the measure and estimate results [28]. The maximum relative error between p-BRDF model prediction and measurement is 0.091, while that between Mueller matrix prediction and measurement is 0.086. The two good verification results prove the accuracy of this method for measuring the Mueller matrix.

After verifying the accuracy of the Mueller matrix by the derived parameters. We use this method to measure the Mueller matrix at different incident angles and obtain the derived parameters of the Mueller matrix by polarization decomposition method.

The polarization scattering characteristics of rough surfaces are influenced by various factors, with the incidence angle playing a crucial role [12,29]. There are few studies on the influence of incident angle on the decomposition parameters of the Mueller matrix. Here we simulate and analyze the polarization scattering characteristics of the same scattering angle at different incidence angles. We can gain a comprehensive understanding of the scattering process by focusing on the polarization scattering properties at various incidence angles for a consistent scattering angle. This investigation allows us to gather valuable insights into the interaction between light and the encountered material or surface, specifically how the polarization state of the scattered light changes with the incidence angle.

The plane directly above the rough surface serves as the detection surface at scattering angle $({\theta _s},{\varphi _s}) = (0^\circ ,0^\circ )$. When the incident azimuth is constant, the incident angle and the incident medium are changed. The dielectric with a refractive index of 1.4, 1.8, and 2.2 and two metallic media were selected for the rough surface. When the wavelength is 0.6328um, the refractive index of Cu is n = 0.27 and k = 3.41 and that of Fe is n = 2.90 and k = 3.07 [33].

Under the above conditions, the Mueller matrix was measured and decomposed to obtain the Mueller matrix elements, the module of polarization vector, the module of bidirectional diattenuation vector and depolarization parameters. The results are shown in Fig. 5.

 

Fig. 5. When the scattered angle $({\theta _s},{\varphi _s}) = (0^\circ ,0^\circ )$, the derived parameters are obtained from the average Mueller matrix at different incident angles ${\theta _s}$. (a)The Mueller matrix element ${m_{00}}$; (b)Depolarization parameter ${P_\Delta }$; (c)The module of polarization vector $\vec{P}$; (d)The module of bidirectional diattenuation vector $\vec{D}$.

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Figure 5 shows the variation of Mueller matrix parameters with incident angles in five different media. The parameters are obtained from the polarization decomposition of the Mueller matrix described in Section 2. The black, red, and blue dotted lines represent the dielectric with a refractive index of 1.4, 1.8, and 2.2, respectively. Hollow box black lines represent metal Fe, and solid triangular olive- green lines represent metal Cu.

In Fig. 5(a), m₀₀ represents the scattering of natural light from rough surfaces. For the rough surface, the maximum scattering intensity is located at the specular reflection angle. As the incidence angle increases, the specular reflection angle shifts and the scattering intensity decreases at $({\theta _s},{\varphi _s}) = (0^\circ ,0^\circ )$. In the case of a medium with a low refractive index, the incidence angle has minimal impact on the scattering intensity. However, the incidence angle significantly influences the scattering intensity for a medium with a higher refractive index. Interestingly, the half-peak values are consistently observed at approximately 60° incidence angles regardless of the refractive index.

For metals, normal incidence results in higher absorption, which manifests as lower scattering intensity. At large incidence angle, the diffuse reflection intensity is higher. The result shows that the intensity increases gradually at scattering angle $({\theta _s},{\varphi _s}) = (0^\circ ,0^\circ )$ with the increase of incidence angle.

Figure 5(b) shows the depolarization parameter obtained by the polarization decomposition of the Mueller matrix. The depolarization parameter ${P_\Delta }$ gives an average measure of the depolarizing power of the optical system [26]. ${P_\Delta }$ of all kinds of media show the same trend with the change of incidence angle. The depolarization parameter ${P_\Delta }$ maintains a stable value within a certain incident angle, which is affected by the material. There is a minimum as the incidence angle increases, and then it increases sharply.

The depolarization parameter curves exhibit similar patterns across different media. Notably, there is a peak at an incidence angle of 0°. Around the 0° incidence angle, a stable range of depolarization parameters can be observed.

While the difference between the extreme values is relatively small for metals, dielectrics are influenced by their refractive index. In different media, ${P_\Delta }$ reaches its maximum value at a 0° incidence angle, decreasing as the incidence angle increases. When ${P_\Delta }$ reaches minimum value, it will increase rapidly. For metals, this minimum corresponds to a smaller incidence angle than for dielectrics.

In Fig. 5(c), $|\textrm{P} |$ represents the module of polarization vector. Here the polarization vector $\vec{P}$ is modelled and quantized to describe the polarization scattering characteristics of rough surfaces.

For the non-polarized light, the scattering light is more likely to exhibit polarization characteristics at a large angle of incidence. This implies that the scattering of natural light is minimally affected by polarization under normal incidence. A consistent trend can be observed in the $|\textrm{P} |$ across different media, with a minimum value occurring at 0° incidence angle.

As the incidence angle increases, the curves for the two dielectrics with a high refractive index exhibit slow growth. However, when the incidence angle reaches approximately 60°, there is a rapid increase for $|\textrm{P} |$. The dielectric with a low refractive index shows a similar trend to the two metals, with the rate of increase gradually intensifying as the incidence angle increases.

In Fig. 5(d), the bidirectional diattenuation vector $\vec{D}$ represents the attenuation capacity of the rough face to linearly polarized light and circularly polarized light. $|\textrm{D} |$ of metals and dielectrics exhibit entirely different properties. $|\textrm{D} |$ of the two metals are almost unaffected by the incidence angle and always show low attenuation to the polarized light. $|\textrm{D} |$ of the dielectric with a low refractive index are greatly different due to the influence of the incident angle.

When calculating the derived parameters of the Mueller matrix, it is found that there are significantly different bidirectional diattenuation vectors and polarization vectors between the dielectric and the metal. At the same time, $|\textrm{D} |$ and $|\textrm{P} |$ are observed, and there is a certain relationship between them. For the dielectric, $|\textrm{D} |$ and $|\textrm{P} |$ have very similar trends, and the value error is also very small. Metals exhibit very different properties. $|\textrm{D} |$ of the two metals are almost not affected by the incidence angle, while $|\textrm{P} |$ is greatly affected by the incidence angle. $|\textrm{P} |$ increases obviously with the increase of the incidence angle.

We extract the parameters of Mueller matrix at different incidence angles. The polarization characteristics of rough surfaces can be characterized clearly by measuring the Mueller matrix and extracting the parameters from it. Different kinds of media at different angles show greater scattering characteristics. The polarization scattering characteristics of rough surface can be studied by this method.

4. Conclusion

In this work, we use six polarized lights with different polarization states as incident light to numerically calculate the scattered polarization states and realize the calculation of the rough surface Mueller matrix. Using the calculated Mueller matrix as an example to predict the scattering polarization state, the maximum relative error between the simulation and expected results is 0.0635, which verifies the accuracy of measuring the Mueller matrix. In addition, we also demonstrate the strong symmetry of the elements of the Mueller matrix by this calculation method.

On this basis, the Mueller matrix is decomposed by the polarization decomposition method, and the bidirectional diattenuation vector, polarization vector, and depolarization parameter are obtained to characterize polarization scattering. In the case that the incident light is unpolarized, the obtained polarization vector is converted to polarization degree, and the results are compared with the measured and estimated results of the p-BRDF model. The average relative error between the result obtained from the Mueller matrix and the measurement result is 0.086, which is lower than the 0.091 predicted by the p-BRDF model, verifying the effectiveness of the Mueller matrix measurement method.

Finally, we use this method to realize the calculation of the Mueller matrix under various angles. The derived parameters characterizing polarization scattering properties are obtained by the polarization decomposition method. The results show that the bidirectional diattenuation vector of metal is not sensitive to angle. In a certain range of angles, the depolarization parameters of the metal remain stable. These are properties that the dielectric does not have. The polarization scattering characteristics of dielectric indicate that both the polarization vector and the bidirectional diattenuation vector of dielectric are affected by the incidence angle, and there is little difference between them. Both of them increase with the increase of the incidence angle. This recognition using polarization scattering properties can be effectively achieved with the help of the Mueller matrix.

The combination of the Mueller matrix parameters and degree of polarization is more conducive to the analysis of the polarization characteristics of the rough surfaces. In addition, the relationship between the parameters of the Mueller matrix obtained by the polarization decomposition, such as depolarization parameter ${P_\Delta }$ and polarization vector $\vec{P}$, still needs to be further studied.

In conclusion, we propose an approach to calculating the Mueller matrix using the finite-difference time-domain method of six polarization states of incident light and scattered light. The results show that the method can calculate the Mueller matrix of the rough surface accurately, which can effectively supplement the deficiency of the experimental measurement. The method and its results are beneficial in exploring the polarization scattering characteristics of rough surfaces and have potential applications in target polarization recognition and measurement.