## 3 Modeling of Laser Echo Signals Under Multi-Channel Continuous-Wave Lasers
\(p(\theta_{ps1},\theta_{ps2})\) is the scattering phase function based on the collision angles of multiple particles. After the photons have been received, the laser echo power signal is expressed as  
\[ \begin{split} S_{cM}(t)=\int\cdots\int_{v_{1}\cdots v_{M}}& P_{T}\frac{A_{R}\eta_{atm}\eta_{sys}Q_{sca}^{M}\text{cos}\theta_{r}}{ \Omega_{T}}\times\prod_{i=1}^{M}\frac{p(\theta_{ps1},\theta_{ps2})}{|\vec{d}_{ i}|^{2}}\\ &\times e^{(-\sigma_{cat}\times\sum_{i=1}^{M})|\vec{d}_{i}|}\times \delta_{v_{i}}\cdots\delta_{v_{M}}\times S_{T}(t-\tau_{M})\end{split}  \tag{12}\]  
where \(P_{T}\) is the laser emission power; \(A_{R}\) is the laser receiving optical area; \(\eta_{atm}\) and \(\eta_{sys}\) are the atmospheric transmittance and the optical transmittance of the laser detection system, respectively; \(\varphi_{r}\) is the angle between the received light and the axis of the received field of view; \(S_{T}\) is the FMCW laser emission power signal; and \(\tau_{M}\) is the delay time under multiple collisions. In particular, when photons only collide with individual smoke particles, the scattering phase function \(p(\theta_{ps1},\theta_{ps2})\) also only needs to be expressed by the individual particle collision angle \(\theta_{ps}\), and \(p(\theta_{ps1},\theta_{ps2})\) in Equation (12) can be transformed into \(\theta_{ps}\). On this basis, when a photon is received by the detector after collision with a stationary target at distance \(R\), the laser target echo power signal under smoke conditions can be expressed as  
\[ \begin{split} S_{cN}(t)=\int\cdots\int_{v_{1}\cdots v_{M}\cdots v _{N}}& P_{T}\frac{A_{R}\eta_{atm}\eta_{sys}Q_{sca}^{M}\rho_{t} \text{cos}\varphi_{r}\text{cos}\varphi_{t}}{\Omega_{T}}\times\prod_{i=1}^{M} \frac{p(\theta_{ps1},\theta_{ps2})}{|\vec{d}_{i}|^{2}}\\ &\times\prod_{j=N-M}^{N}\frac{p(\theta_{ps})}{|\vec{d}_{j}|^{2}} \times e^{(-\sigma_{cat}\times(\sum_{i=1}^{M}|\vec{d}_{i}|+\sum_{j=N-M}^{M}| \vec{d}_{j}|))}\\ &\times\delta_{v_{i}}\cdots\delta_{v_{M}}\times S_{T}(t-\tau_{N}) \end{split}  \tag{13}\]where \(\rho_{t}\) is target reflection coefficient; \(\varphi_{t}\) is the incidence angle at which the photon collides with the target; \(\tau_{N}\) is the total delay time. When photons are not in collision with smoke particles and are received by the detector, or in a smoke-free environment, the laser target echo power signal can be expressed as  
\[S_{c0}(t)=P_{T}\times\frac{A_{R}\eta_{atm}\eta_{sys}\rho_{t}\text{cos}\varphi_{ t}}{R^{2}}e^{-2\sigma_{ext}R}\times S_{T}(t-\tau_{T}) \tag{14}\]  
where the time delay is \(\tau=\frac{2R}{c}\), and \(c\) is the speed of light. For a multi-channel continuous-wave parallel laser emission system, as shown in Figure 4(a), the collision probability is at this point directly affected not only by the number of particles emitted into the space in which they are located, but also by the spatial position of the emitted laser, as shown in Figure 4(b), based on Equation (9). The collision probability based on the emission of a multi-channel CW laser can be expressed as  
\[p_{N_{laser}}(t_{0}\to t_{0}+\Delta t)=\left\{ \begin{array}{l}\sum_{i=1}^{N_ {laser}}\sum_{j=1}^{N_{photon}}\frac{n_{photon}^{(ij)}}{N_{photon}}\times \sigma_{sc}(r_{s1},r_{s2})\times\Delta v_{ij}\Delta t,N_{c}\
eq 1\\ \sum_{i=1}^{N_{laser}}\sum_{j=1}^{N_{photon}}\frac{n_{photon}^{(ij)}}{N_{ photon}}\times\pi(r_{s}+r_{p})^{2}\times\Delta v_{ij}\Delta t,N_{c}=1\end{array} \right. \tag{15}\]  
where \(N_{c}\) is the number of smoke particles when collisions occur with the emission particles. \(N_{c}=1\) is the type of collision based on a single smoke particle, and \(N_{c}\
eq 1\) is the type of simultaneous collision of multiple smoke particles. \(n_{photon}^{(ij)}\) is the actual number of particles emitted in each channel in the space created by the light path and the collision cross-section, which is related to the beam divergence angle and positional layout of the emission laser.