## 3 Modeling of Laser Echo Signals Under Multi-Channel Continuous-Wave Lasers  
To improve the target detection performance of the laser fuze, the amplitude of the target echo signal needs to be increased. The actual number of photons penetrating the smoke particle environment needs to increase, and the collision probability of the emission particles with the smoke particles needs to decrease. In low-visibility smoke environments, it is not meaningful to simply increase the total number \(N_{photon}\) of particles emitted in the spatial collision probability from Equation (3), because the collision cross-section is objective. The particle collision angles \(\theta_{ps1}\), \(\theta_{ps1}\) and \(\theta_{ps}\) are difficult to change, and this will not affect the subsequent direction of scattering transmission. Before the transmission direction of the photons is changed, there is a possibility that the 3D collision cross-sections corresponding to different smoke particles overlap with each other, and the cross-section merging area is shown in Figure 4. Thus, it is reasonable to reduce the collision probability \(p_{CN}\) by keeping the total number of emission particles \(N_{photon}\) constant and decreasing the number of particles emitted \(n_{photon}\) within the localization. We use simultaneous emission of particles into multiple initial spatial locations between the localized irradiation area of the laser source and the outside of the collision cross-section which corresponds to the multi-channel laser emission structure. When the number of optical paths is \(N_{Laser}\), the collision probability can be re-expressed as  
\[p^{\prime}_{CN}(t_{0}\to t_{0}+\Delta t)=\frac{1}{N_{Laser}}\times p_{CN}(t_{0} \to t_{0}+\Delta t) \tag{9}\]  
During laser fuze emission of laser light into a dense smoke environment, the photon energy can be expressed by the results of the change in weighting coefficients. Based on the smoke particle concentration and particle collision process, the photon transmission process is classified into photon collision with a single smoke particle, photon collision with any two smoke particles at the same time, and no collision. The initial weight is set to \(W_{0}=1\) and the particle energy weight after collision can be expressed as  
\[W_{N_{ca}}=(\prod_{n_{cc1}=0}^{n_{c1}}p_{c1}^{(sca1)}\times\eta_{s1}^{(sca1)}) \times(\prod_{n_{cc2}=0}^{n_{c2}}p_{cN}^{(sca2)}\times\eta_{sN}^{(sca2)})\times W _{0} \tag{10}\]  
where \(n_{c1}\) and \(n_{c2}\) are the maximum number of collisions for the two different types. \(\eta_{s1}\) and \(n_{sca1}\) are the energy attenuation coefficient and the number of collisions of photons with individual smoke particles. \(\eta_{sN}\) and \(n_{sca2}\) are the energy attenuation coefficient and the number of simultaneous collisions of photons with multiple smoke particles, respectively. Based on the Mie scattering model, it can be concluded that the laser intensity varies with the relative change in the scattering angle, which can be expressed by the scattering phase function as [19]. When the spatial distribution of smoke particles is uniform, the scattering phase function can be considered to only be related to the angle between the incidence and scattering. Based on the single scattering model [20], the echo power of a photon after collision with any two smoke particles can be expressed as  
\[P_{\delta_{v_{i}}}=P_{\delta_{v_{i}-1}}\times\frac{Q_{sca}}{\Omega_{T}|\vec{d} _{i}|^{2}}\times p(\theta_{ps1},\theta_{ps2})\times e^{-Q_{cat}\times|\vec{d}_{ i}|}\delta_{v_{j}} \tag{11}\]  
where \(\delta_{v}\) is the sum of the unit volumes of multiple smoke particles corresponding to the collision scattering that occurs for photons. \(P_{\delta_{v_{i}}}\) and \(P_{\delta_{v_{i}-1}}\) are the scattered power before and after the collision. \(\Omega_{T}\) is the stereo angle of the laser emission beam. \(Q_{sca}\) and \(Q_{ext}\) are the smoke scattering and attenuation coefficient. \(\vec{d}\) is the photon shift vector.