## 2 Three-Dimensional Collision and Transmission Processes of Photons in Smoke
\(r_{s1}\) and \(r_{s2}\) are the particle sizes of different spherical smoke particles, and \(\theta_{1}\) and \(\theta_{2}\) correspond to the maximum collision circumferential angles, respectively. The second and third terms are expressed as the area of the combined collision cross-section, which is related to the particle size, as well as the distance from the center of the sphere. On this basis, the collision probability \(p_{cN}\) of a photon with multiple smoke particles can be expressed as  
\[p_{cN}(t_{0}\to t_{0}+\Delta t)=\sum_{i=1}^{N_{photon}}\frac{n_{photon}}{N_{ photon}}\times\sigma_{sc}(r_{s1},r_{s2})\times\Delta v_{i}\Delta t \tag{3}\]  
Based on Equations (1) and (3), the photon collision probability depends on the 3D collision cross-section. The scattering direction of particles after a collision can only be determined if the collision probability is known. Otherwise, the direction of transmission always remains the same. Figure 2: Three-dimensional collision cross-section between a photon and any two smoke particles. For the actual direction of particle transport under perfect elastic collision conditions, the constant vector sum of the system momentum before and after the collision is used, which is shown in Equation (4):  
\[\vec{p}_{p}+\vec{p}_{s1}+\vec{p}_{s2}=\vec{p}_{p}^{\prime}+\vec{p}_{s1}+\vec{p}_{ s2}^{\prime} \tag{4}\]  
where \(\vec{p}_{p}\) is the average momentum vector of a large number of emission particles, which are used to represent the laser beam before the collision. \(\vec{p}_{s1}\) and \(\vec{p}_{s2}\) are the momentum vectors of the two smoke particles. \(\vec{p}_{p}^{\prime}\), \(\vec{p}_{s1}^{\prime}\), and \(\vec{p}_{s2}^{\prime}\) are the momentum vectors after the collision, respectively. When photons are incident on any two smoke particles, the simultaneous collision interaction that occurs instantaneously can be regarded as a two-body collision process within a brief time interval. As the photon movement speed is much higher than the velocity of the smoke particles, the collision process between these two particles is exceedingly short, which can be neglected for the collision process of particle deformation and internal structure alteration. Under these fast collision conditions, the particle can be considered to remain rigid before and after the collision. Moreover, the wavelength of the emitted laser is usually close to \(1\)\(\upmu\)m, which can ensure that the size of the photon particles is smaller than the size of the smoke particles. Thus, in the case of perfect elastic collisions based on the particles of a rigid sphere, the angle between \(\vec{p}_{p}\) and \(\vec{p}_{p}^{\prime}\) is expressed as the scattering angle \(\theta_{s}\), which is shown in Figure 3. \(r_{r_{1}}\) and \(r_{r2}\) are the radius of a rigid-body particle. Depending on the relative spatial positions of the two smoke particles, the scattering direction of the scattered particles is determined by the angle \(\theta_{vc}\) between the center-of-mass line of the smoke particles and the direction of incident velocity. The particle scattering angle \(\theta_{s}\) can be expressed as  
\[\theta_{s}= \begin{cases}\quad 2\theta_{vc},\quad 0<\theta_{vc}\leqslant\frac{ \pi}{2}\\ 2\pi-2\theta_{vc},\frac{\pi}{2}<\theta_{vc}\leqslant\pi\end{cases}  \tag{5}\]  
The collision angles \(\theta_{ps1}\) and \(\theta_{ps2}\) are the angles between the center-of-mass line at the contact position of the particle and the direction of the initial velocity, respectively. When \(\theta_{ps1}<\theta_{ps2}\), the particle scattering angle \(\theta_{s}\) can be quantitatively expressed as  
\[\theta_{s}= \begin{cases}\frac{7}{2}\pi-4\theta_{ps1}-3\theta_{ps2},\frac{\pi }{4}<\theta_{ps1}<\frac{\pi}{2}\\ \frac{5}{2}\pi-4\theta_{ps1}-3\theta_{ps2},\ 0<\theta_{ps1}<\frac{\pi}{4}\\ \frac{3}{2}\pi-2\theta_{ps2},\theta_{ps1}=\frac{\pi}{4}\end{cases}  \tag{6}\]  
Figure 3: Scattering angle and collision angle based on rigid-body sphere collision.