Polarization Characteristics of Massive HVI Debris Clouds Using an Improved Monte Carlo Ray Tracing Method for Remote Sensing Applications

Abstract

As a signature phenomenon of massive hypervelocity impacts (HVIs) in space, debris clouds provide critical optical information for satellite remote sensing and the assessment of large-scale impacts. However, studies of the optical scattering properties of debris clouds remain limited, and existing vector radiative transfer (VRT) methods struggle to accurately simulate the optical characteristics of these complex scatterers. To address this gap, this paper presents an improved Monte Carlo VRT program (PGS–MC) for multicomponent polydisperse scatterers to precisely evaluate the radiation and polarization characteristics of complex scatterers. Based on the Monte Carlo ray tracing (MCRT) method, our program introduces a particle grouping strategy (PGS) to further emphasize the importance of accounting for optical property discrepancies between different materials and particle sizes, thus significantly improving the fidelity of VRT simulations. Moreover, our program, developed using the compute unified device architecture (CUDA), can be run parallelly on graphics processing units (GPUs), which effectively reduces the computational time. The validation results indicated that the developed PGS–MC program can accurately and efficiently simulate the polarization of complex 3D scatterers. A further investigation showed that the polarization characteristics of debris clouds are highly sensitive to parameters such as the angle between the incident and detection directions, number density, particle size distribution, debris material, and wavelength. In addition, the polarization imaging of debris clouds offers distinct advantages over intensity imaging. This study offers guidance for analyzing the VRT properties of massive HVI debris clouds. Additionally, it provides a practical tool and concrete ideas for modeling the polarization characteristics of various complex scatterers, such as aircraft contrails and clouds, etc.

1. Introduction

Debris clouds, a signature phenomenon of massive HVIs, typically result from unintentional satellite collisions or vehicle intercepts [1,2]. A debris cloud can be described as an ever-expanding cloud of particles propelled by the shockwaves associated with the impact [3,4]. Relevant studies have confirmed that a substantial amount of debris clouds, even after rapid dispersion and thermal equilibration in space, retains robust optical signatures that span the visible-to-infrared spectral regions [1]. This is primarily attributed to Mie scattering from micrometer-sized particles within clouds that interact with solar radiation [5,6]. However, when investigating debris clouds, relying solely on the radiative characteristics is insufficient for a precise detection and identification of targets. Hence, it is crucial to extend the dimensional information of the target by examining the representation and characteristics of the information across different dimensions.

Polarization provides multidimensional information about targets and is widely used in various fields. The scattering of light by debris clouds can generate or change the polarization state of light, thereby providing information about the targets. Hence, probing the polarization characteristics of HVI debris clouds is crucial for impact event recognition, scenario simulation, and sensor selection. However, the current literature lacks comprehensive studies on the polarization characteristics of debris clouds. High-fidelity simulations of large-scale HVI scenarios in a controlled laboratory setting remain challenging, with HVI apparatus incurring substantial costs [3]. Computational modeling is imperative to comprehend the polarization and radiative characteristics of debris clouds holistically.

A debris cloud can be considered a special type of scatterer. Its polarization is mainly studied by simulating the multiscattering behavior through the vector radiative transfer (VRT) method. In VRT simulations, the polarization state is represented by Stokes vectors, while a scattering phase matrix is employed to characterize the changes in the polarization due to scattering. Contemporary methods include differential equation resolution methods [7,8,9,10] and ray tracing-based methods [11,12,13,14].

Although the VRT method has been extensively studied, there remain challenges in modeling debris clouds. One of the main problems is the diversity of debris materials and sizes [3,15,16], which leads to errors in the calculation of the optical properties and simulation of scattering processes. To address these complexities, some studies have focused on streamlining the optical properties of scatterers. Representative methods include integrating optical properties over the entire range of particle sizes [17,18] or substituting the particle size range with an effective radius [19,20]. These methods approximate a polydisperse system as a monodisperse system, but tend to introduce substantial errors [21,22,23]. To mitigate the estimation errors caused by particle size diversity, a strategy of segmenting the particle size into distinct subregions [21] has been identified as particularly effective for radiative transfer. However, when considering the multicomponents of particles and vector radiation, estimation errors are further introduced.

Additionally, the complex 3D spatial distribution of the debris clouds requires meticulous consideration [24,25] during the scattering simulations. The Monte Carlo ray tracing (MCRT) method offers a simplified, efficient, and intuitive approach for solving complex VRT problems [26,27,28,29]. In this method, the concept of dividing the voxel space [30] is at the core of modeling complex 3D structures. Moreover, with advancements in computational capabilities in terms of the GPU, the conventionally time-consuming nature of MCRT calculations can be mitigated through parallel processing [31,32,33].

In summary, previous methods for simulating VRT have inadequate treatment of the influence of variations in microscopic particle properties on the macroscopic optical characteristics of scatterers. Moreover, there is a lack of an effective and highly precise simulation technique for VRT involving multicomponent polydisperse scatterers with intricate three-dimensional structures. Furthermore, there remains a dearth of a comprehensive and openly available investigation into the radiative and polarization attributes of massive HVI debris clouds. To bridge these gaps, this study endeavors to develop an innovative VRT software (PGS–MC v1.0.0) that can efficiently and precisely model the radiative and polarization properties of intricate scatterer systems. Additionally, we apply this procedure to analyze massive HVI debris clouds. Our main contributions are as follows:

• A novel particle grouping strategy (PGS) is employed to calculate the optical property parameters in the VRT of multicomponent polydisperse scatterers based on improvements to the strategy outlined in [21]. The case of multiple materials is further considered in the PGS by grouping particles with similar material properties and sizes into the same subregion. The optical property parameters are calculated independently for each sub-region. In particular, vector radiation associated with polarization is considered, with the calculation of the scattering phase matrix for each subregion delineated. Experimentally, it is demonstrated that PGS significantly refines the calculation of optical properties compared to the traditional integration method, thereby reducing the estimation error.
• A PGS-based Monte Carlo VRT program (PGS–MC) is proposed for modeling the VRT of complex scatterers. The program not only accurately calculates the polarization and radiation characteristics but also integrates the complex spatial distribution of the scatterer, nonuniformity of the particle size, and multicomponent mixing to ensure a highly accurate and realistic simulation. Several necessary changes are introduced to accommodate the simulation’s complexity. Firstly, the concept of spatial voxels is adopted with independent properties for each voxel. Secondly, during photon–particle interactions within a voxel, both the particle material and particle size range are probabilistically determined, guiding the subsequent photon scattering and absorption based on the respective optical properties of the designated subregion. Augmenting this, scattering phase matrix calculations for each subregion factor in the collision probabilities and particle size scattering probabilities can be made, thus realizing a near-exact simulation of continuous particle size distributions. Finally, to reduce the time cost of the simulation, two efficient computational strategies are used in PGS–MC. Specifically, we first calculated and organized the optical properties into look-up tables (LUTs) using a CPU and then exploited the computational power of GPUs to enable parallel simulations of photon emission and transmission.
• We further conduct simulation experiments on the scattering radiation and polarization properties of the large-scale HVI debris clouds using PGS–MC. Specifically, the effects of factors on the radiation and polarization characteristics of the debris clouds are discussed, such as the angle between the incidence and detection directions, number density, particle size distribution, and material type. Finally, an imaging simulation is performed.

The rest of this paper is organized as follows. In Section 2, the specific calculations for the PGS determination of optical properties are presented, along with the flow of the PGS–MC program. In Section 3, the accuracy and credibility of the proposed method are verified, and the selection of model parameters is provided. Section 4 describes several experiments designed for a comprehensive study of the radiative, polarization, and imaging properties of debris clouds. The conclusions and future work are presented in Section 5 and Section 6, respectively.

2. Theoretical Modeling and Method

In this study, we employed an improved MCRT program, namely PGS–MC, to simulate the polarization characteristics of a multicomponent polydisperse debris cloud system. We incorporated the 3D spatial configuration of the debris cloud and delineated the parameters for the system, such as the particle size distribution and refractive index, which are detailed in Section 2.1. In Section 2.2, we describe the calculation of the optical property parameters using the PGS, which considers the variances between different components and particle sizes. In Section 2.3, we elaborate the process of employing the PGS–MC program to simulate the radiation and polarization characteristics of debris clouds.

2.1. Debris Cloud Model and Particle System

The 3D structure of the debris cloud was modeled to ensure simulation fidelity. We adopted the method reported in [34] to establish a dual-helix model representing the internal and external contours of a debris cloud. The equations governing external contour ${\eta }_{ext}$ and the internal contour ${\eta }_{int}$ are as follows:

$${\eta }_{ext}=\varphi \left(\xi \right)=2\sqrt{-\left(2{\xi }^2+1\right)+\sqrt{8{\xi }^2+1}},\xi \in \left[{\xi }_{\min },1\right],$$

(1)

$${\eta }_{int}=k_s·\varphi (\xi /k_s),$$

(2)

where $\xi$ is the normalized coordinate in the axial direction; ${\eta }_{ext}$ is the normalized radial length for the external contour; the x-axis origin indicates the impact point, the negative semiaxis is the back–splash–cloud portion, and the positive semiaxis is the main body of the debris cloud. The x-axis normalization ranges from ${\xi }_{\min }$ to 1; ${\eta }_{int}$ is the normalized radial length for the internal contour; $k_s$ is a scaling factor representing the magnitude of the scaling of the inner contour with respect to the outer contour.

Figure 1a shows the normalized contour model of the debris cloud for $k_s=0.8$ and ${\xi }_{\min }=0.4$. The model is extended to 3D and scaled according to the given sizes. Figure 1b shows the 3D debris cloud model, with the axial and radial scaling factors set to 100 m and 60 m, respectively. Micron-sized debris are generally near-spherical particles condensed by the melting of colliding bodies, with different sizes and materials. Thus, debris clouds are typically multicomponent, polydisperse scatterers. According to the requirements of the Mie theory, the size of the debris must be similar to the wavelength. Since we are interested in infrared bands, this paper focuses mainly on particles with diameters below 10 µm. Notably, the debris cloud during diffusion consists of solid particles that gradually decrease in temperature, so we assume that the particle size does not change. A power-law distribution was used to describe the particle size distribution of the debris clouds [15,16]:

$$n\left(r\right)=a{\left({\displaystyle \frac{m\left(r\right)}{m_t}}\right)}^{-b},$$

(3)

where $n\left(r\right)$ denotes the number density of particles with radius r, $m\left(r\right)$ is the single-particle mass, and $m_t$ is the total mass of the debris. The scaling constant a adjusts the conservation of mass under the debris distribution curve, and the constant b is related to the relative kinetic energy per unit mass of the target.

2.2. Optical Properties Calculated by PGS

Particles with varying refractive indices and radii exhibit significant disparities in photon scattering, making methods that integrate entire particle systems prone to substantial estimation errors. In contrast, this study employed a PGS, which allocates particles with similar properties to specific subregions and calculates the optical parameters for each subregion accordingly.

First, we divided the particles into $n_c$ groups (components) based on the material, each group having a unique refractive index $m_j$ and particle size distribution $n_j\left(r\right)$, where j denotes the group number. Subsequently, considering the collision probabilities of photons with particles of different radii in the radiative transfer, $n_j\left(r\right)$ of each group is divided into $n_r$ subregions with the same cumulative collision probability [22]. The radius of the i-th region in the j-th group of the particle is denoted by $r_{i,j}$, and the range of $r_{i,j}$ can be expressed as

$${\displaystyle \frac{i-1}{n_r}}\le {\displaystyle \frac{{\int }_{r_{min,j}}^{r_{i,j}}{r}^2{n}_j\left(r\right)dr}{{\int }_{r_{min,j}}^{r_{max,j}}{r}^2{n}_j\left(r\right)dr}}\le {\displaystyle \frac{i}{n_r}},$$

(4)

where $r_{min,j}$ and $r_{max,j}$ are the minimum and maximum values of the particle size in group j, respectively. The particle system is thus divided into $n_r\times n_c$ subregions, each with similar properties.

The optical property parameters of each particle subregion are calculated individually. We used $p_{i,j}$ to denote the i-th particle size region in the j-th component. The attenuation ${\mu }_{ext}$, absorption coefficient ${\mu }_{abs}$, and scattering coefficient ${\mu }_{sca}$ of the region $p_{i,j}$ are given by

$${\mu }_{ext,i,j}\left(\lambda \right)={\mu }_{abs,i,j}\left(\lambda \right)+{\mu }_{sca,i,j}\left(\lambda \right),$$

(5)

$${\mu }_{abs,i,j}\left(\lambda \right)={\int }_{r_{min,i,j}}^{r_{max,i,j}}\pi r^2{Q}_{abs}\left(r,m_j,\lambda \right)n_j\left(r\right)dr,$$

(6)

$${\mu }_{sca,i,j}\left(\lambda \right)={\int }_{r_{min,i,j}}^{r_{max,i,j}}\pi r^2{Q}_{sca}\left(r,m_j,\lambda \right)n_j\left(r\right)dr.$$

(7)

The absorption probability $A_{i,j}$, albedo ${\omega }_{i,j}$, and asymmetry factor $g_{i,j}$ of the region $p_{i,j}$ are given by

$$A_{i,j}\left(\lambda \right)={\displaystyle \frac{{\mu }_{abs,i,j}\left(\lambda \right)}{{\mu }_{ext,i,j}\left(\lambda \right)}},$$

(8)

$${\omega }_{i,j}\left(\lambda \right)={\displaystyle \frac{{\mu }_{sca,i,j}\left(\lambda \right)}{{\mu }_{ext,i,j}\left(\lambda \right)}}=1-A_{i,j}\left(\lambda \right),$$

(9)

$$g_{i,j}\left(\lambda \right)={\displaystyle \frac{{\int }_{r_{min,i,j}}^{r_{max,i,j}}{r}^2{Q}_{sca}\left(r,m_j,\lambda \right)g\left(r,m_j,\lambda \right)n_j\left(r\right)dr}{{\int }_{r_{min,i,j}}^{r_{max,i,j}}{r}^2{Q}_{sca}\left(r,m_j,\lambda \right)n_j\left(r\right)dr}},$$

(10)

where $r_{max,i,j}$ and $r_{min,i,j}$ are the maximum and minimum values of the radius in the region $p_{i,j}$, respectively; $Q_{sca}$, $Q_{abs}$, and g denote the scattering efficiency, absorption efficiency, and asymmetry coefficient, respectively, of the radiation of wavelength $\lambda$ interacting with a particle of radius r and refractive index $m_j$, which can be calculated from the classical Mie theory [5,6].

The vector radiation is described by the Stokes vector $\mathbf{S}={[I,Q,U,V]}^T$, and the single-scattering process of the vector radiation with particles can be expressed as

$${\mathbf{S}}_{sca}={\mathbf{M}}_{i,j}\left(\alpha \right)·{\mathbf{S}}_{in},$$

(11)

where ${\mathbf{S}}_{sca}$ is the Stokes vector after single scattering, obtained by multiplying the scattering phase matrix ${\mathbf{M}}_{i,j}\left(\alpha \right)$ with the pre-scattering Stokes vector ${\mathbf{S}}_{in}$. For spherical particles, the scattering phase matrix of the region $p_{i,j}$ can be expressed as

$${\mathbf{M}}_{i,j}\left(\alpha \right)=\left[\begin{array}{cccc}{s_{11}}_{,i,j}\left(\alpha \right)& s_{12,i,j}\left(\alpha \right)& 0& 0\\ s_{12,i,j}\left(\alpha \right)& s_{11,i,j}\left(\alpha \right)& 0& 0\\ 0& 0& s_{33,i,j}\left(\alpha \right)& s_{34,i,j}\left(\alpha \right)\\ 0& 0& -s_{34,i,j}\left(\alpha \right)& s_{33,i,j}\left(\alpha \right)\end{array}\right],$$

(12)

where $\alpha$ is the scattering angle; $s_{mn,i,j}$ denote the scattering phase matrix coefficients of the region $p_{i,j}$. In estimating $s_{mn,i,j}$ for each subregion, it is imperative to account for the probability of scattering events occurring with different particles within the subregion. The probability of scattering with a given particle is the product of two probabilities: the likelihood of a photon colliding with the particle and the subsequent probability of the photon being scattered, both of which are directly proportional to the spherical particle’s cross-sectional area $\pi r^2{n}_j\left(r\right)$ and albedo $\omega$, respectively. Therefore, the integral weights $w_{sca}$ of the scattering phase matrix coefficients for each particle are represented as the product of the cross-sectional area and albedo. Finally, $s_{mn,i,j}$ of the region $p_{i,j}$ are given by

$$s_{mn,i,j}\left(\alpha \right)={\displaystyle \frac{{\int }_{r_{min,i,j}}^{r_{max,i,j}}{s}_{mn,norm}\left(r,m_j,\alpha \right)w_{sca}\left(r,m_j\right)dr}{{\int }_{r_{min,i,j}}^{r_{max,i,j}}{w}_{sca}\left(r,m_j\right)dr}},(m,n=[1,2,3,4]),$$

(13)

$$w_{sca}(r,m_j)=\pi r^2{n}_j\left(r\right)\omega \left(r,m_j\right),$$

(14)

$$s_{mn,norm}\left(r,m_j,\alpha \right)={\displaystyle \frac{s_{mn}\left(r,m_j,\alpha \right)}{s_{11,max}\left(r,m_j\right)}},$$

(15)

where $s_{mn,norm}$ denote the normalized scattering phase matrix coefficients for a single particle; $s_{11,max}$ is the maximum value of $s_{11}$ for the current particle. The scattering phase matrix coefficients $s_{mn}$ and albedo $\omega$ for the current particle can be calculated by the classical Mie theory [5,6]. Note that $\omega$ here corresponds to individual particles and should be distinguished from the estimated albedo ${\omega }_{i,j}$ for the subregions.

For Equation (15), the normalization of $s_{mn}$ considers the mutual independence of scattering across various particle sizes within the MC simulation phase. In VRT simulations, the normalization does not affect the change in vector radiation since photons exist in only two discrete states: survival or annihilation ($I=1$ or $I=0$), with no explicit concept of energy.

In summary, Equations (4)–(15) provide details for using the PGS to calculate the optical property parameters of multicomponent polydisperse systems. For comparison, the optical properties obtained by integrating the entire system are detailed in the Appendix A.

In this study, the two methods are referred to as the PGS and integration methods [21], respectively.

2.3. PGS–MC Vector Radiative Transfer

Based on the 3D scatterer model and optical property parameters, the polarization characteristics were calculated by tracking the interaction of the vector radiation with different debris particles using the PGS-based Monte Carlo (PGS–MC) VRT program. Compared with the conventional MC method, the present method adds the selection of particle types before simulating the interaction of photons with the particles. The PGS calculates the optical property parameters for the selected particles, considering the variations among different particles to refine the simulation outcomes. Additional modifications were implemented in the MC method to accommodate the complex 3D structure of the debris cloud scatterer. Figure 2 shows the PGS–MC process.

Figure 3 shows a schematic of the polarization simulation of the debris cloud obtained by the PGS–MC program. Figure 3a shows the principle of the simulation process. Figure 3b shows a cross-sectional view of photon tracking, illustrating both the single-scattering and multi-scattering processes of photons in a multicomponent polydisperse scatterer. The dashed gridlines represent the voxel space, whereas the spheres of varying colors and diameters depict debris of different materials and sizes. Figure 3c shows the transmission cases for photons within a single voxel, including single scattering, multiscattering, absorption, and penetration. The process of modeling the VRT using PGS–MC is described in detail below.

2.3.1. Target Area Setting and Photon Initialization

First, the 3D target region and coordinate system are established, as shown in Figure 3a. The target region is denoted by $\mathbf{G}(X_{min},X_{max},Y_{min},Y_{max},Z_{min},Z_{max})$, where $(X_{min},Y_{min},Z_{min})$ and $(X_{max},Y_{max},Z_{max})$ are the minimum and maximum boundaries on the x, y, and z axes, respectively. We divide the entire target area into a grid space comprising square voxels according to the set resolution. The optical properties of the spatial voxels are independent of each other.

Light was incident from one side of the area. The incident solar projection surface was assumed to be a rectangular surface covering the target, denoted by $\mathbf{A}(X_{min},Y_{min},w,l)$, where w and l are the sizes of the projection surface on the x and y axes, respectively.

The incident photons were initialized according to the aforementioned settings. For the initial position coordinates ${\mathbf{p}}_{\mathbf{0}}=(x_0,y_0,z_0)$, we generally set the mesh area so that $z_0=Z_{min}$, and $x_0$, $y_0$ are given by

$$x_0=x_{min}+R_{x}w,$$

(16)

$$y_0=y_{min}+R_{y}l,$$

(17)

where $R_x$ and $R_y$ are uniformly distributed random numbers between 0 and 1. The initial incidence direction ${\mathbf{u}}_{\mathbf{0}}=(u_x,u_y,u_z)$ is computed from the incident zenith angle ${\theta }_0$ and azimuth angle ${\phi }_0$ by

$$u_x=sin{\theta }_{0}cos{\phi }_0,$$

(18)

$$u_y=sin{\theta }_{0}sin{\phi }_0,$$

(19)

$$u_z=cos{\theta }_0.$$

(20)

For natural light incidence, the initial Stokes vector was ${\mathbf{S}}_{\mathbf{0}}={[1,0,0,0]}^T$. The unique weight method [35] is used for the initialization of photons, i.e., only 1 and 0 are used to represent the survival and annihilation of photons, respectively. This approach facilitates the counting of annihilated photons compared to describing the attenuation of the photon transport process with consecutive weights between 0 and 1.

2.3.2. Photon Movement within Voxels

In this study, photons followed random optical trajectories conforming to the Lambert–Beer law in a specified region [36]. A ballistic approach was used to determine the optical path and location of the interaction. When the cumulative optical depth of the photon movement exceeds a set optical depth threshold, the photon is said to collide with the particle, leading to scattering or absorption [24]. This critical collision optical depth is denoted by $L_{max}$, and is given by

$$L_{max}=ln\left({\displaystyle \frac{1}{R_{\sigma }}}\right),$$

(21)

where $R_{\sigma }$ is a random number uniformly distributed between 0 and 1. The distance of the photon along the transmission direction to the boundary of the current voxel is denoted by $l_b$ and is given by

$$l_b=min\left({\displaystyle \frac{x_b}{u_x}},{\displaystyle \frac{y_b}{u_y}},{\displaystyle \frac{z_b}{u_z}}\right),$$

(22)

$${axe}_b=\left\{\begin{array}{c}⌊axe+1⌋-axe\left(u_{axe}>0\right),\hfill \\ ⌈axe-1⌉-axe\left(u_{axe}<0\right),\hfill \end{array}axe\in \{x,y,z\}\right.,$$

(23)

where $u_x$, $u_y$, and $u_z$ are the cosines of the incidence directions with respect to the x, y, and z axes, respectively. ${axe}_b$($x_b$, $y_b$, or $z_b$) are the distances from the starting point to the three faces of the voxel along the directions of $u_{axe}$($u_x$, $u_y$, or $u_z$), respectively. This is calculated by rounding down (or up when $u_{axe}<0$) after moving one voxel unit in the direction of transmission, which ensures the continuous transmission of the ray, effectively circumventing potential computational disruptions that might otherwise result in program failure.

The optical depth of the photon movement starts to increase after the incident light enters the target area or is scattered.

$$L_{t+1}=L_t+\Delta L_{t+1},$$

(24)

$$\Delta L_{t+1}=l_{b,t}{\mu }_{ext,i,j},$$

(25)

where $L_t$ is the t-th cumulative optical depth, and $\Delta L_{t+1}$ is the optical depth of the ($t+1$)-th penetrating voxel.

Next, a judgment was made regarding whether the photon collides with a particle. If the cumulative optical depth $L_t$ satisfies $L_t\le L_{max}$, there is no collision. Subsequently, the ray penetrates this voxel and moves into the next voxel, as shown in Figure 3c, Case 4. The direction of transmission does not change, and the coordinates of the location of the ray are updated by

$${axe}_{new}=axe+u_{axe}{l}_{b,t},axe\in \{x,y,z\},$$

(26)

where ${axe}_{new}$ is the updated coordinate.

If $L_t\ge L_{max}$, the photon collides with the particle in this voxel. In this case, $L_t$ is cleared to zero ($L_0=0$). The moving distance $l_b$ of the photon in this voxel can be expressed as follows:

$$l_c={\displaystyle \frac{\left(L_{max}-L_j\right)}{{\mu }_{ext,i,j}}}.$$

(27)

The location at which the ray interacts with the particle is then determined, and the position coordinates are updated as follows:

$${axe}_{new}=axe+u_{axe}{l}_c,axe\in \{x,y,z\}.$$

(28)

2.3.3. Collisions and Interactions between Photons and Particles

Collisions between a photon and a particle are accompanied by interactions. These two behaviors are sequential and do not occur simultaneously. Therefore, the collision probability equations for particles of different sizes are only positively related to r and $n\left(r\right)$.

The probability of a photon colliding with a particle in the j-th component can be expressed as follows:

$$P_j={\displaystyle \frac{{\int }_{r_{min,j}}^{r_{max,j}}{r}^2{n}_j\left(r\right)dr}{{\sum }_{q=1}^{n_c}{\int }_{r_{min,q}}^{r_{max,q}}{r}^2{n}_q\left(r\right)dr}}.$$

(29)

The selection of component j is determined by

$${\sum }_{q=1}^{j-1}{P}_q\le R_c\le {\sum }_{q=1}^j{P}_q,$$

(30)

where $R_c$ is a random number between 0 and 1. With component j determined, similarly, the probability of a photon colliding with a particle in the i-th subregion can be expressed as follows:

$$P_{i,j}={\displaystyle \frac{{\int }_{r_{min,i,j}}^{r_{max,i,j}}{r}^2{n}_j\left(r\right)dr}{{\int }_{r_{min,j}}^{r_{max,j}}{r}^2{n}_j\left(r\right)dr}}.$$

(31)

The selection of the particle size region i is determined by

$${\sum }_{q=1}^{i-1}{P}_{q,j}\le R_r\le {\sum }_{q=1}^i{P}_{q,j},$$

(32)

where $R_r$ is a random number between 0 and 1.

The interaction of photons with the particles may involve absorption or scattering. We used an albedo ${\omega }_{i,j}$ with a random number $R_{abs}$ between 0 and 1 to determine the absorption. If $R_{abs}\ge {\omega }_{i,j}$, the photon is absorbed and the transmission process ends, as shown in Figure 3c, Case 3. Otherwise, the photon scatters, as in Case 2. The scattering changes the movement direction of the photon, which can be described by the scattering angle $\theta$ and rotation angle $\phi$.

For the Stokes vector representation of the vector radiation, the scattering angle $\theta$ and the azimuthal angle $\phi$ are chosen using the rejection method [19,37], which is detailed in the Appendix B.

Based on the scattering and azimuth angles, the after-scattered Stokes vector is given by [19]

$${\mathbf{S}}_{new}=\mathbf{R}(-\gamma ){\mathbf{M}}_{i,j}\left(\alpha \right)\mathbf{R}\left(\beta \right)\mathbf{S},$$

(33)

where the matrix form of $\mathbf{R}\left(\beta \right)$ is as follows:

$$\mathbf{R}\left(\beta \right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& cos\left(2\beta \right)& sin\left(2\beta \right)& 0\\ 0& -sin\left(2\beta \right)& cos\left(2\beta \right)& 0\\ 0& 0& 0& 1\end{array}\right).$$

(34)

${\mathbf{M}}_{i,j}\left(\alpha \right)$ is the scattering phase matrix, which corresponds to the scattering process of the Stokes vector and can be calculated from Equations (12)–(14). The matrix form of $\mathbf{R}(-\gamma )$ is the same as $\mathbf{R}\left(\beta \right)$, and the parameter $\gamma$ is given by

$$cos\gamma ={\displaystyle \frac{-u_z+{\widehat{u}}_{z}cos\alpha }{\pm \sqrt{\left(1-{cos}^2\alpha \right)\left(1-{\widehat{u}}_z^2\right)}}}.$$

(35)

Finally, the direction cosine is updated by $\theta$ and $\phi$ [19].

2.3.4. Boundary Conditions and Photon Statistics

Based on the space set described in Section 2.3.1, we determined whether a photon was ejected from the boundary. If ejected, information regarding the current photon $n_p$ is recorded, including the outgoing coordinates, direction, Stokes vector, and scattering counts. In addition, the effective number of photons incident into the debris cloud is counted by $N_{eff}=N_{in}-N_{invalid}$, where $N_{in}$ is the total number of incident photons, and $N_{invalid}$ is the number of photons not incident into the debris cloud. The Stokes vector statistics of the outgoing light within a particular stereo angle ${\Omega }_{out}=({\theta }_{out},{\phi }_{out})$ is given by

$${\mathbf{S}}_{out,\Omega }={\displaystyle \frac{I_{0}cos{\theta }_{in}}{\left|{cos}_{{\theta }_{out}}\right|d{\Omega }_{out}}}·{\displaystyle \frac{{\sum }_{n_p=0}^{N_{out,\Omega }}{\mathbf{S}}_{out,n_p}}{N_{eff}}},$$

(36)

where $I_{0}cos{\theta }_{in}$ represents the total energy incident perpendicular to the surface, and $N_{out,\Omega }$ is the total number of emission photons in ${\Omega }_{out}$. The outgoing Stokes vector is denoted by ${\mathbf{S}}_{out,\Omega }={[I_{out,\Omega },Q_{out,\Omega },U_{out,\Omega },V_{out,\Omega }]}^T$.

The degree of linear polarization ($\mathit{DoLP}$) within the stereoscopic angle ${\Omega }_{out}$ is given by

$$DoL{P}_{out,\Omega }={\displaystyle \frac{\sqrt{Q_{out,\Omega }^2+U_{out,\Omega }^2}}{I_{out,\Omega }}}.$$

(37)

2.3.5. Efficient Acceleration Strategies

Two efficient acceleration strategies were used to solve the time–cost problem. The optical properties of each component, particle size range, and wavelength were calculated before the simulation and converted into an LUT as input to the MC procedure. On the other hand, considering the mutual independence of each photon tracking process, the VRT program was designed to run on a GPU. The CUDA was used for data management and parallel implementation. Multithreading is first divided into multiple grids, each containing many thread blocks with multiple threads that share memory; each thread uses independent and efficient registers for a single simulation.

Taken together, the LUT strategy has some memory constraints, but eliminates the need to compute the optical properties in radiative transfer, thus dramatically reducing the time required for a single simulation. The parallel implementation of MCRT solves the time–cost problem associated with a large number of simulations, allowing the MCRT to obtain highly accurate results quickly.

The pseudocode of PGS–MC is described in the Appendix C.

It is essential to emphasize that, due to the limitations of traditional methods in simulating complex scatterers in VRT, we have exclusively utilized the “integration method” for comparison with our PGS–MC model. To accommodate the intricate three-dimensional structure of the scatterers and ensure the validity of the comparison, the integration method in this study employs the same MCRT framework as PGS–MC. The key distinctions between these two approaches are as follows: Firstly, the integration method calculates the optical properties using Equations (A1)–(A7) in Appendix A, with the entire system sharing unified optical property parameters, while in PGS–MC, optical properties are determined for each group separately. Furthermore, the integration method oversimplifies the diverse particle properties by assuming uniform microscopic particle properties throughout the VRT process. In this approach, ${\mu }_{ext,i,j}$ in Equations (25) and (27) are replaced by a single ${\overline{\mu }}_{ext}$ value derived from Equation (A1), and particle selection steps in Section 2.3.3 are omitted. Additionally, ${\omega }_{i,j}$ used for absorption evaluation is replaced by $\overline{\omega }$ from Equation (A5), and the coefficients $s_{mn,i,j}$ of ${\mathbf{M}}_{i,j}$ in Equation (33) are replaced by ${\overline{s}}_{mn}$ obtained from Equation (A7). The comparison of simulation results between PGS–MC and the integration method will be extensively deliberated in the forthcoming experiments.

3. Method Validation and Performance Analysis

Prior to deploying the PGS–MC for analyzing the radiation and polarization characteristics of the debris clouds, all the aspects of the procedure were thoroughly validated and analyzed, including the accuracy of the VRT process, necessity of introducing the PGS, convergence of the results with respect to the predefined parameters, and selection of these parameters.

3.1. Validation of VRT Process

We rigorously validated the accuracy of the VRT process using the PGS–MC simulation program. First, the accuracy of our method in the 3D Rayleigh scattering polarization mode was verified using the experiments of R.Vaillon et al. [29]. We used PGS–MC to calculate the Stokes I and Q of the forward and backward scattering processes and compared them with the results reported in [29], as shown in Figure 4. The pertinent details can be found in Appendix D. Our results overlapped almost exactly with the reference, proving the accuracy of the VRT process in our method for Rayleigh scattering simulations.

We then verified the accuracy of our method for simulating the Mie scattering polarization properties using the two-layer medium model [38]. The polarization results for different optical depths were simulated by adjusting the thicknesses of the upper and lower layers. The Stokes I and Q curves of backscattering corresponding to the surface layer from 0 to 10 mm were obtained using PGS–MC and compared with the results reported in [38], as shown in Figure 5. Our method matches the results, thus proving the accuracy of the VRT process in the PGS–MC for Mie scattering polarization.

3.2. Comparison of PGS with Previous Method

The central idea of the PGS involves grouping particles with identical materials and similar radii into the same subregion, thereby reflecting the differences in the optical properties between subregions within the VRT simulations. Theoretically, an increase in the number of subregions ($n_r$) should yield simulations that more closely approximate reality. Specifically, with increasing $n_r$, the polarization results are expected to converge to a stable value.

Herein, we present a representative case to prove the above inference. The scatterer is the 3D debris cloud introduced in Section 2.1.

Table 1. Input parameters for Section 3.2.

Material	mt (kg)	Range of r (μm)	b	λ (μm)	m	Np	(${\mathit{\theta }}_0$, ${\mathit{\phi }}_0$)
Al	10.0	1.0∼10.0	1.5	4.0	6.7717 ± 38.679 i	${10}^8$	($0^{°},0^{°}$)

presents the input parameters, including the material, total debris mass ($m_t$), radius range, power-law distribution coefficient of the particle size (b), wavelength ($\lambda$), refractive index (m), number of input photons ($N_p$), and angle of incidence (${\theta }_0,{\phi }_0$). The results of the polarization distributions for different $n_r$ ($n_r=2$, 4, 6, 8, or 10) are calculated using the PGS–MC program and compared with the result obtained using the conventional integration method, as shown in Figure 6.

The results shown in Figure 6 demonstrate that the polarization distribution converges to a stable value for $n_r\ge 6$, thus validating the previous inference. Further, there is a significant discrepancy between the polarizations calculated using the conventional integration method and the PGS–MC. Given that the integration method is equivalent to simulations with $n_r=1$, it is evident that the PGS effectively reduces the estimation errors introduced by integration.

To further explain the distinction between the PGS and the integration method, we computed the scattering phase matrix coefficients [39] for the 1st, 5th, and 8th sub-regions using Equation (13) at $n_r=8$. These are subsequently juxtaposed with the outcomes derived from the integration method, as shown in Figure 7.

As shown in the figure, there is a pronounced disparity between the outcomes from individual subregions and those derived from full-region integration. Notably, the $s_{12}/s_{11}$ in Subregion 1 consistently presents a negative value, while both Subregions 5 and 8 show positive values in some angular ranges and exhibit pronounced oscillations. In contrast, the results obtained using the integration method exhibit relatively smoother curves between the results of the subregions due to the integration of the parameters over the entire size region. This disparity becomes increasingly evident when observing the changes in the scattering direction and polarization, resulting in an overall statistical bias.

3.3. Convergence of Results and Parameter Selection

Within the model framework, two important parameters are identified: $N_p$ and $n_r$. Their selection is intrinsically linked to both the accuracy of the simulation outcomes and the computational throughput. Theoretically, increasing the values of these two parameters improves the accuracy of the results and gradually converges to the true value.

We compared the bidirectional distribution of natural light scattered by the debris cloud for four incident photon numbers ($N_p=2\times {10}^7$, $5\times {10}^7$, $1\times {10}^8$, and $2\times {10}^8$) to verify the convergence of the simulation results with respect to $N_p$ at more refined spatial angles, as shown in Figure 8. For visibility, the polar plots are used to describe the bidirectional distributions. The angular and radial scales represent the azimuth angles (0°∼360°) and zenith angles (0°∼180°), respectively. The $\mathit{DoLP}$ is depicted through a color gradient. In this case, we adopt the debris cloud model described in Section 2.1, with the input parameters given in

Table 2. Input parameters for Section 3.3.

Material	mt (kg)	Range of r (μm)	b	λ (μm)	m	Nr	(${\mathit{\theta }}_0$, ${\mathit{\phi }}_0$)
Al	20.0	1.0∼10.0	1.0	4.0	6.7717 ± 38.679 i	8	($0^{°},0^{°}$)

.

As illustrated in Figure 8, there is a noticeable fluctuation in the statistical outcomes when $N_p$ is minimal. However, as $N_p$ increases, the bidirectional distribution of $\mathit{DoLP}$ becomes smoother and converges gradually. In this case, for $N_p$ exceeding $5\times {10}^7$, a further increase in the photon count ceases to induce significant changes in the distribution.

Table 3. Time efficiencies for different $N_p$ values.

Np	$2\times {10}^7$	$5\times {10}^7$	$1\times {10}^8$	$2\times {10}^8$
Time (s)	14.342	25.416	58.708	114.56

gives the time efficiency of PGS–MC program with different $N_p$. Balancing the considerations of the statistical error against the computational time, a count of ${10}^8$ photons emerged as optimal for this scenario.

Next, the convergence of the statistical results for the number of particle size subregions was verified. We compared the $\mathit{DoLP}$ distributions at four different $n_r$ ($n_r=6$, 7, 8, and 9), as shown in Figure 9. The other parameters remained the same as in the previous case, with $N_p={10}^8$.

As shown in Figure 9, the significant differences between the results for $n_r=6$ and $n_r>6$ are mainly in the distributions and peak values. As $n_r$ increases, the overall $\mathit{DoLP}$ value increases and approaches the convergence at $n_r=8$, with the peak value converging at 0.10. We can conclude that the greater division of regions will bring the statistical results closer to the real value. Yet, increasing $n_r$, while enhancing the accuracy in the simulation, augments the system’s complexity. Consequently, a higher photon count is mandated for the results to converge, which increases the computational overhead. Therefore, $n_r=8$ is chosen.

4. Experiments and Discussion

The radiation and polarization characteristics of the debris cloud system were further investigated. The effects of the simulation parameters, such as the incidence angle, observation angle, particle density, particle size distribution, and debris material, on the $\mathit{DoLP}$ and radiation were analyzed, and the polarization imaging of the debris clouds was simulated. The spatial distribution of the debris cloud given in Section 2.1 was used as the baseline model in each of the following studies. The complex refractive indices, $m=n+ki$ of the debris used in the following simulations are documented in

Table 4. Complex refractive indices of different debris materials at different wavelengths.

Debris Material	2 µm		3 µm		4 µm		5 µm
	n	k	n	k	n	k	n	k
Fe	3.4830	6.8790	3.9950	9.5287	4.1710	12.111	4.2250	14.823
Cu	0.3127	14.274	0.7041	21.555	1.2539	28.774	1.9587	35.908
Al	2.3493	20.309	4.4865	29.824	6.7717	38.679	9.1528	47.199

. These values were taken from the literature [40,41,42].

Table 5. Input parameters for Section 4.1.

Case	Material	mt (kg)	Range of r (μm)	b	$\mathit{\lambda }$ (μm)	nr	Np	${\mathit{\theta }}_{\mathit{0}}$	${\mathit{\phi }}_{\mathit{0}}$	${\mathit{\phi }}_{\mathit{out}}$
1	Al	5.0	1.0∼10.0	1.5	2.0/4.0	8	${10}^8$	-	$0^{°}$	-
2	Al	-	1.0∼10.0	1.0	4.0	8	${10}^8$	$0^{°}$	$0^{°}$	${90}^{°}$
3	Al	5.0	-	-	2.0∼5.0	8	${10}^8$	$0^{°}$	$0^{°}$	${90}^{°}$
4	-	5.0	1.0∼10.0	1.5	2.0∼5.0	8	${10}^8$	$0^{°}$	$0^{°}$	${90}^{°}$

presents the input parameters for the cases used in this study. Notably, in order to minimize interference from atmospheric and terrestrial backgrounds, wavelengths exceeding 2 µm were employed in the experiments. Subsequent studies have demonstrated that shorter wavelengths correlate with increased frequency of fluctuations in the optical properties curve concerning scattering angle, posing challenges for the detailed analysis of scatterer properties.

4.1. Influence of Parameters on Simulation Results

4.1.1. Angles of Incidence and Observation

The effects of the incidence and observation angles on the $\mathit{DoLP}$ were evaluated. Table 5 presents the input parameters for this case (Case 1).

We subsequently probed the polarization distribution of the debris cloud at three distinct angles of incidence: 0°, 30°, and 60°, as shown in Figure 10. To eliminate wavelength-specific influences, we executed simulations at two different wavelengths (${\lambda }_1=2.0$µm and ${\lambda }_2=4.0$µm).

The figure shows the clear polarization modes for both the wavelengths. At an angle of incidence of 0°, the bidirectional distribution is shown as concentric circles centered at the polar coordinate origin, which suggests a near-uniformity of the polarization distribution across different observation azimuths. Moreover, shifting the angle of incidence caused a congruent shift in these concentric circles; however, the intrinsic polarization distributions remained largely consistent. For the angles of incidence of 30° and 60°, the centroid was also at approximately 30° and 60°, respectively, indicating that the angle of incidence determined the offset of the centroid. These observations emphasized that the polarization of debris clouds was inherently tied to the angular relationship between the incidence and observation directions.

The $\mathit{DoLP}$ at $\lambda =4.0$µm in Figure 10 spans a range of approximately 0∼0.4, which is significantly higher than the range of 0∼0.1 seen in Figure 8 and Figure 9. This discrepancy is likely attributable to variations in the number density $\rho$ and particle size distribution $n\left(r\right)$ of the debris, which will be further investigated.

4.1.2. Number Density of Debris

The number density $\rho$, which is directly related to the optical depth of the debris cloud, is found to be a crucial parameter in VRT analyses. To elucidate the effect of the number densities on the $\mathit{DoLP}$ of the debris cloud, we varied the number densities by adjusting the total debris mass $m_t$, with the input parameters listed in Table 5, Case 2. Figure 11 shows the $\mathit{DoLP}$ profiles corresponding to five different $m_t$($m_t=1$, 5, 20, 50, and 100 kg). From an analysis in the previous section, the $\mathit{DoLP}$ distributions for different azimuths remained largely the same for an angle of incidence of 0°. Therefore, the results for ${\phi }_{out}={90}^{°}$ can correctly reflect the $\mathit{DoLP}$ distribution for any other azimuth angle.

The figure clearly shows that the distribution of the $\mathit{DoLP}$ as a function of the scattering angle remains relatively consistent across different $m_t$, with a minor peak at 24° and a major peak at 70°. The $\mathit{DoLP}$ exhibited a pronounced decline with increasing $m_t$. Specifically, the peak value decreased from 0.154 at $m_t=1$ to 0.053 when $m_t$ reached 100 kg, marking a substantial reduction of 65.58%.

Table 6. Statistics of photon scattering counts corresponding to different $\rho$.

mt (kg)	$\mathit{\rho }$ ($\times {10}^{-10}$μm−1)	A (%)	BS (%)	FS (%)	P (%)	ASC	BASC	FASC
1	3.477	0.086	1.764	3.336	94.81	1.040	1.053	1.033
5	17.39	0.426	7.853	13.84	77.88	1.205	1.262	1.173
20	69.55	1.652	22.19	31.25	44.91	1.877	2.025	1.772
50	173.9	4.016	35.16	36.96	23.87	3.288	3.467	3.117
100	347.7	7.853	44.03	35.26	12.85	5.512	5.622	5.375

presents the tabulated photon scattering statistics corresponding to the different debris qualities, which includes parameters such as the number density ($\rho$), absorption rate (A), backward scattering rate (BS), forward scattering rate (FS), penetration rate (P), average scattering counts (ASC), backward average scattering counts (BASC), and forward average scattering counts (FASC). The juxtaposition of the above metrics revealed a notable increase in the ASC with increasing $\rho$. This accentuated the depolarization effect attributable to multi-scattering, thus explaining the observed attenuation in the $\mathit{DoLP}$ with increasing $\rho$ (or $m_t$). Meanwhile, both absorption and scattering phenomena are significantly enhanced with an increase in $\rho$, particularly for backward scattering, which exceeds forward scattering at $m_t=100$ kg.

Notably, at $\lambda =4.0$µm, the $\mathit{DoLP}$ peaks at a scattering angle of approximately 70°, which coincides with the peak of $s_{12}/s_{11}$ for Subregion 1 shown in Figure 7. A plausible inference is that this distribution pattern is largely due to first-order scattering from finer particles, with a strong correlation with $s_{12}$ in the smaller-particle-size region.

4.1.3. Particle Size Distribution

The particle size distribution $n\left(r\right)$ is a particularly important parameter in debris cloud systems. In this section, we evaluate the impact of both the particle size range and the attenuation coefficient b on the $\mathit{DoLP}$ distribution. To ensure a congruent comparison, we maintained $m_t=5$ kg. Table 5 presents the input parameters (Case 3).

Figure 12 shows the $\mathit{DoLP}$ distributions corresponding to three particle diameter regions: d1 (1∼4 µm), d2 (4∼10 µm), and d3 (1∼10 µm). Specifically, d1 accentuates the small-particle-size region, d2 accentuates the large-particle-size region, and d3 provides a wide range of particle size distributions. For a comprehensive evaluation, simulations were performed at four different wavelengths ($\lambda =2.0$, 3.0, 4.0, and 5.0 µm).

From the figure, it is evident that the $\mathit{DoLP}$ distributions of d1 and d3 are similar, distinctly contrasting with that of d2, which contains only large debris. Specifically, both d1 and d3 exhibit two peaks at $\lambda =2.0$µm and a single peak at the other three wavelengths, whereas d2 demonstrates pronounced fluctuations, particularly within the region $\theta <{90}^{°}$. Further, with an increase in the wavelength, there was a noticeable broadening of the $\mathit{DoLP}$ distribution corresponding to each particle size range, with this trend being particularly pronounced for the larger-particle-size range, d2. Moreover, the $\mathit{DoLP}$s corresponding to d1 and d3 both exhibited a general increase with increasing wavelength, whereas that corresponding to d2 showed a slight decrease. This emphasizes that the polarization magnitude in the smaller regions is more sensitive to wavelength variations.

The attenuation coefficient b is related to the kinetic energy of the impact. For a fixed $m_t$, b changes $n\left(r\right)$, which affects the polarization pattern. A higher value of b indicates a greater proportion of small particles. We further simulated the $\mathit{DoLP}$ distributions of three b values (b1 = 1.0, b2 = 1.5, b3 = 2.0) for a particle size range of 1∼10 µm, which are shown in Figure 13. We analyzed the simulation results at different $\lambda$ separately.

The $\mathit{DoLP}$ corresponding to different b showed similarity at various $\lambda$. Evidently, as b increases, the overall $\mathit{DoLP}$ exhibited a significant increase, which is attributed to the increasing predominance of smaller particles that govern the polarization. This conclusion may be attributed to the inverse relationship between the probability of photon-particle collisions and particle size, given a constant mass. Notably, at $\lambda =2.0$µm and $\lambda =3.0$µm, a peak in the DoLP curve for b1 emerged at forward scattering angles less than 30°. This can be interpreted as a pronounced polarization effect displayed by the greater number of larger particles associated with b1. As $\lambda$ increases, this peak gradually diminished and eventually disappeared. According to the prior analysis, this can be attributed to the decreasing polarization of larger particles with increasing $\lambda$. Moreover, we observed a general shift in the $\mathit{DoLP}$ curves with changing $\lambda$. As $\lambda$ transitioned from 2.0 µm to 3.0 µm and eventually to 4.0 µm, the forward scattering peak angle increased from 40° at $\lambda =2.0$µm to 60° at $\lambda =3.0$µm, stabilizing at 75° for $\lambda =4.0$µm.

4.1.4. Debris Materials

The influence of debris materials on the $\mathit{DoLP}$ was investigated. In particular, differences in the complex refractive index m between materials lead to variations in their respective scattering phase matrices and inherent optical properties. Three common spacecraft metallic materials—Fe, Cu, and Al—were selected for this computational study. Table 4 presents the complex refractive indices. Table 5 presents the other parameters for Case 4. Figure 14 shows the derived comparative results.

We simulated scenarios with mixed components, calculating the polarization distribution using both the PGS–MC and integration methods, as indicated by the blue solid line and the purple dashed line in Figure 14. The $\mathit{DoLP}$ distribution obtained from PGS–MC lies between those of the individual components. This phenomenon can be explained by the similarity of $\mathit{DoLP}$ distributions of the independent metal components (differing primarily in their $\mathit{DoLP}$ amplitudes). Consequently, the $\mathit{DoLP}$ distribution of the mixed scatterer mirrors a weighted combination of the polarization from these independent components.

Table 7. PFSP and PFSA derived from two methods at different wavelengths for a multicomponent debris cloud system.

Parameters	Method	2 µm	3 µm	4 µm	5 µm
PFSP	PGS–MC	0.110	0.177	0.339	0.468
	Integration	0.177	0.276	0.387	0.468
PFSA (deg)	PGS–MC	38.8	58.0	74.2	75.0
	Integration	42.5	61.0	69.8	73.0

presents the peak forward scattering polarization (PFSP) and peak forward scattering angle (PFSA) results derived from the two methods, demonstrating significant differences.

Table 8. PFSP error and PFSA errors of the integration method relative to the PGS–MC.

Parameters	2 µm	3 µm	4 µm	5 µm
PFSP Error (%)	60.91	55.93	14.16	0.00
PFSA Error (%)	9.54	5.17	5.93	2.67

presents the relative errors of the PFSP and PFSA derived from the data listed in Table 7. Clearly, at $\lambda =5.0$µm, the PFSP values from both the methods are in agreement, with a minimal PFSA error of just 2.67%. However, with decreasing $\lambda$, both the errors rapidly increase, reaching a PFSP error of 60.91% and a PFSA error of 9.54% at $\lambda =2.0$µm. This suggests that compared with PGS–MC, the integration method suffers from significant accuracy problems, particularly at $\lambda \le 5.0$µm.

4.2. Polarization Imaging Simulation

Finally, a polarization imaging simulation of the debris clouds was performed; Figure 3 shows a schematic of the simulation. The experimental parameters align with those of the mixed-component experiment detailed in Section 4.1.4. The detector had a resolution of 320 × 256 pixels, with an angular deviation of 90° between the directions of incidence and detection. The detector was positioned at a substantial distance from the debris cloud, allowing the received light beams to be approximated as parallel beams. The spatial resolution of the debris cloud location was set to 1 m. By quantifying the positional information and polarization of the photons emitted within a fixed solid angle, the number of photons received by each pixel and their corresponding polarizations can be computed for the simulated imaging.

Figure 15 shows the imaging simulation results. The range and trend of the $\mathit{DoLP}$ with respect to the wavelength in our polarization imaging simulation aligned seamlessly with the results detailed in Section 4.1.4, underscoring the accuracy of our simulation approach. Further, we conducted a comparative analysis of the polarization and intensity images, wherein the intensity images were normalized. Notably, in contrast to polarization images, intensity images exhibited minimal sensitivity to wavelength variations, showing a subtle decrease in the intensity with increasing wavelength. In addition, the brightness of the regions close to the internal portion decreased in the intensity images. This phenomenon is attributed to the fact that the optical depth is thicker in the external part, and scattering events are more prevalent, resulting in a concentration of brightness in these regions. Conversely, polarization images showed an inverse relationship with the optical depth, as explained in Section 4.1.2.

These inferences offer a theoretical foundation and practical guidance for the application of multispectral polarization imaging devices in large-scale hypervelocity impact detection and identification.

5. Conclusions

In this study, we developed a CUDA-implemented program, namely PGS–MC, tailored for a precise quantification of the radiation and polarization characteristics in multicomponent polydisperse debris cloud systems. The accuracy of the VRT process in our program was validated through typical Rayleigh and Mie scattering cases. The necessity of accounting for the optical properties of different types of particles is emphasized by comparing the scattering phase matrix and $\mathit{DoLP}$ obtained using the PGS–MC and the integration method. Subsequently, the convergence of the results was demonstrated by studying the bidirectional polarization distribution for different $N_p$ and $n_r$, and appropriate simulation parameters were selected. Finally, the effects of various factors on the radiation and polarization were investigated. Further, simulations of the imaging were performed.

The key conclusions drawn from our simulations are as follows:

• Under proper simulation parameters, the proposed PGS–MC program effectively reduced the error caused by the estimation of the optical properties of particles by up to 60.91% compared with conventional integration methods. In addition, the program can easily simulate the VRT of more complex scatterers by operating in the 3D voxel space.
• The polarization was sensitive to various parameters. First, the $\mathit{DoLP}$ depended on the angle between the incident and observed directions. Second, with the decrease in the particle density from $3.477\times {10}^{-8}$µm⁻¹ to $3.477\times {10}^{-10}$µm⁻¹, the polarization increased by 190.5%, corresponding to the diffusion process of the debris cloud. Further, the dominant polarization could be primarily attributed to Mie scattering induced by smaller particles approximately 1 µm in size. In addition, debris clouds with different metallic materials have similar polarizations, and for multicomponent mixtures, the $\mathit{DoLP}$ is an intermediate between the components. The polarization of the debris cloud also increased with wavelength, and the $\mathit{DoLP}$ could reach more than 0.4 at $\lambda =5.0$µm.
• Polarization imaging has unique advantages over conventional infrared imaging for debris cloud detection. In the field of multispectral analyses, polarization increases the sensitivity to wavelength variations compared with the intensity, providing a significant advantage for polarimetric detection across different spectral bands. In addition, as the debris clouds diffuses, the polarization level remains at a relatively high plateau, although the intensity drops dramatically over a range of optical depths. We advocate the use of the proposed PGS–MC program, which was designed for a quantitative assessment of the VRT in complex scatterers. This program integrates compositional diversity, continuous particle-size distributions, and complex 3D configurations.

A comprehensive study of the polarization characteristics of debris clouds can provide a basis for the creation of a database of the optical properties of debris clouds and contribute to subsequent detection and identification efforts. Beyond debris clouds, the program is applicable to VRT calculations for other similar types of complex scatterers, including aircraft contrails, tail flames, clouds, and atmosphere, supporting the military, space exploration, and atmospheric remote sensing fields.

6. Future Work

Regarding the engineering applications of our program, despite the high fidelity and parallel computation capabilities of the proposed PGS–MC in this paper, there remains considerable room for improvement. Firstly, selecting appropriate parameters, such as $N_p$ and $n_r$, is essential for different simulations. However, determining these parameters often requires additional test experiments, which can be tedious. Secondly, accurate simulations for specific scatterers, such as multiple thin hierarchies, necessitate smaller spatial voxels, significantly increasing computational effort. Furthermore, enhancing particle grouping and spatial resolution can improve simulation accuracy but also considerably elevate time cost, requiring a balance between accuracy and speed. To address these challenges, a more efficient particle grouping method and dynamic voxel size adjustment can be considered to further optimize the PGS–MC.

For engineering applications of debris cloud simulations, a smoothed particle hydrodynamics (SPH) model of the debris cloud will be utilized in the future as an input to our PGS–MC program to obtain higher-fidelity results. In addition, the polarization characteristics of the spontaneous emission in long-wavelength infrared (LWIR) light will be considered for more comprehensive explorations.