## 2 Three-Dimensional Collision and Transmission Processes of Photons in Smoke  
The transmission process of an emitting laser in smoke can be described by the interaction between shaped hypothetical particles based on wave-particle duality, collisional dynamics, and the law of momentum conservation [17]. Laser fuzes employ lasers with wavelengths in the range of 0.65 \(\upmu\)m to 1.65 \(\upmu\)m, and the large number of small suspended particles in a smoke scenario have sizes ranging from a few micrometers to over a dozen micrometers, which is close to the size of these two types of particles [9]. Thus, the elastic scattering of photons by smoke particles adheres to the Mie scattering model [18], and the collision between photons and individual smoke particles can be approximated as a fully elastic collision based on spherical particles. Under this condition, the intersection detection results indicate whether the backward scattering process exists for small-sized photons collided by large-sized smoke particles based on the particle motion trajectories at adjacent time intervals. Combined with the three-dimensional properties of particles, the specific process of backscattering depends on the probability of spatial collision between particles. In the case of photon incidence towards a single smoke particle, the particle collision probability can be determined from the maximum 3D collision cross-section between the particles, which is shown in Figure 1.  
For a determined single smoke particle, the collision probability \(p_{c1}\) is expressed as  
\[p_{c1}(t_{0}\to t_{0}+\Delta t)=\sum_{i=1}^{N_{photon}}\frac{n_{photon}}{N_{ photon}}\times\pi(r_{s}+r_{p})^{2}\times\Delta v_{i}\Delta t \tag{1}\]  
where \(n_{photon}\) and \(N_{photon}\) are the number of particles emitted in a localized area and the total number of emissions, respectively. The bottom surface of the localized area is the maximum three-dimensional collision cross-section \(\pi(r_{s}+r_{p})^{2}\) and the height is light range. \(r_{p}\) and \(r_{s}\) are the size of the photon and smoke particle, respectively. \(\Delta v\) is the relative movement speed of the particles. \(\Delta t\) is the time interval and \(t_{0}\) is the initial moment. Low-visibility smoke scenes are characterized by a dense positional distribution of smoke particles in space, due to the inclusion of a large number of freely suspended smoke particles. When the center-of-mass distance of the smoke particles exceeds the sum of the radius of the two smoke particles and the diameter of the photon particle, an emission photon is simultaneously affected by the collision of multiple smoke particles. The collision cross-section of any two particles is shown in Figure 2.  
Figure 1: Three-dimensional collision cross-section between a photon and a single smoke particle. Similarly to the case of a single smoke particle, the velocity vector of an emission photon is affected by a combination of the initial velocity distribution and the collision geometry. The former includes the velocity direction, magnitude, and relative position, while the latter is the contact position between the photon and multiple smoke particles. During multisphere elastic collisions with smooth particles, the velocity vector change is mainly expressed in the non-symmetric direction deflection, and the rotational motion of the particles can be neglected. Thus, the collision probability can be calculated by using the combined area of the collision cross-section. Based on Figure 2, the collision cross-section \(\sigma_{sc}\) can be expressed based on the intersecting circle equation, as shown:  
\[ \begin{split}\sigma_{sc}=&\pi(r_{s1}+r_{p})^{2}+ \pi(r_{s2}+r_{p})^{2}-\frac{1}{2}\theta_{1}(r_{s1}+r_{p})^{2}-\frac{1}{2} \theta_{2}(r_{s2}+r_{p})^{2}\\ &-\frac{1}{2}\sqrt{(-r_{d}+r_{s1}+r_{s1})(r_{d}+r_{s1}-r_{s1})(r_ {d}-r_{s1}+r_{s1})(r_{d}+r_{s1}+r_{s1})}\\ \end{split}  \tag{2}\]  
where the first term is the sum of the single collision cross-section areas.