# Research on Laser Dual-Mode Fusion Detection Method of Ship Wake Bubbles
## 3 Simulation of Ship's Wake Bubble Laser Dual-Mode Detection Based on Monte Carlo
### Monte Carlo Simulation Model  
In this study of the propagation of lasers in water, water is regarded as a discrete scattering medium, and the laser beam is regarded as a photon beam composed of many photons. Therefore, the problem of laser propagation in water can be transformed into the problem of photon beam propagation in water. The distribution of particles in seawater is random. When photons propagate in water, they will randomly collide with the medium. The Monte Carlo method is based on the computer simulation of the photon motion trajectory. The basic process is that the photon enters the medium in a specific direction to determine the free path of the photon motion when the collision occurs. If scattering occurs, the corresponding scattering phase function can be selected. Then, the new direction of motion of the photon is determined. If absorption occurs, the photon disappears. These processes are repeated randomly until the photon is received by the receiver. The statistics of photon motion reflect the propagation law of photons in water. The Monte Carlo simulation is based on the statistical experimental method of the computer, which can simulate the motion process of a large number of photons and record the state of photons.  
Figure 2: Annular wake confrontation form: (1) Wake junction.  
It does not need too much approximation, and can simulate the absorption and scattering characteristics of the water body under different attenuation coefficients.  
In this paper, we consider the case of multiple scattering of the water medium, that is, the scattering particles in water are dense, and we consider the single, secondary, and higher-level scattering and attenuation on the scattering path. To simulate the propagation of photons in water, three basic conditions need to be determined: the motion distance of photons, the motion direction of photons, and the decision of photon survival. The process can be divided into six steps:  
* the emission wavelength and energy determination;
* determination of free path;
* scattering angle calculation;
* new direction of motion after collision;
* new free path;
* receive or die.  
The free path formula:  
\[l_{m+1}=-\frac{\ln\xi_{m+1}}{N^{1/3}\ln(1-\tau r^{2}\cdot N^{1/3})} \tag{1}\]  
\(l_{m+1}\) is the new beam propagation distance after \(m\) collisions; \(p_{m}\) is the probability of collision of motion; \(p_{n}\) is the probability that the motion does not collide; \(r\) is the particle radius; \(N\) is the logarithmic density about \(c\); \(\xi_{m+1}\) is a random number between \((0,1)\).  
After the \(m\)th collision, the emission direction of the photon is \((a_{m},\beta_{m},\gamma_{m})\) relative to the previous coordinate system. After the \(m+1\)th collision, the emission direction is \((\alpha_{m+1},\beta_{m+1},\gamma_{m+1})\) relative to the \(m\)th coordinate system. Among them:  
\[\left\{\begin{array}{l}\cos\alpha_{m+1}=\frac{\sin\theta_{m+1}(\cos\alpha_{m }\cos\gamma_{m}\cos\phi_{m+1}-\sin\phi_{m+1}\cdot\cos\beta_{m})}{\cos\phi^{2} \alpha_{m}+\cos^{2}\beta_{m}+\cos^{2}\alpha_{m}\cos\theta_{m+1}}\\ \cos\beta_{m+1}=\frac{\sin\theta_{m+1}(\cos\beta_{m}\cos\gamma_{m}\cos\phi_{m +1}-\sin\phi_{m+1}\cdot\cos\alpha_{m})}{\sqrt{\cos\alpha_{m}+\cos^{2}\beta_{m} +\cos^{2}\beta_{m}\cos\theta_{m+1}}}\\ \cos\gamma_{m+1}=-\sin\beta_{m}\cdot\cos\phi_{m}\cdot\sqrt{\cos^{2}\alpha_{m} +\cos^{2}\beta_{m}\cdot\cos\phi_{m+1}}+\cos\gamma_{m}\cdot\cos\theta_{m}\end{array}\right. \tag{2}\]  
The Monte Carlo photon simulation motion program is shown in Figure 3.  
Figure 3: Program diagram of photon-simulated motion.