# Method for Detecting Underwater Microbubbles Using Dual-Mode Fusion of Laser Polarization
## 2 Dual-Mode Fusion Detection Mechanism
### Ideal Bubble State Simulation  
This section establishes a dynamic time-varying simulation model of underwater microbubbles based on the vector Monte Carlo method, and uses the vector Monte Carlo method to simulate the polarization state of the laser backscattered echo signal of the particles in the underwater bubble cluster. Compared with the traditional Monte Carlo method, this method adds the link of outputting the polarization state of each photon, and at the same time increases the preprocessing neighborhood screening work, counting the number of photons in the backscattered echo photons that are similar to the preprocessing polarization reference value, and the specific flow is shown in Figure 2.  
To solve the Stokes vector before and after photon scattering using the meridian plane method, the coordinate system as shown in Figure 3 needs to be established. In Figure 3, the 3D coordinate system \(\mathrm{X^{\prime}Y^{\prime}Z^{\prime}}\) represents the local coordinate system with the photon scattering point as the origin O\({}^{\prime}\). The vectors \(\overrightarrow{\mathrm{O^{\prime}A}}\) and \(\overrightarrow{\mathrm{O^{\prime}B}}\) represent the motion direction of the photon before and after scattering, respectively, and can also be represented by the unit vectors \(\mathrm{\hat{k}_{i}}\) and \(\mathrm{\hat{k}_{s}}\); points A and B represent the intersection of the photon's motion direction before and after scattering with the unit sphere. AO\({}^{\prime}\)B represents the scattering plane; \(\alpha\) represents the scattering angle, which is the angle between \(\overrightarrow{\mathrm{O^{\prime}A}}\) and \(\overrightarrow{\mathrm{O^{\prime}B}}\); \(\beta\) represents the angle at which the plane CO\({}^{\prime}\)A rotates clockwise to the plane AO\({}^{\prime}\)B as seen from the opposite direction of \(\overrightarrow{\mathrm{O^{\prime}A}}\) and \(\overrightarrow{\mathrm{O^{\prime}B}}\); \(\gamma\) represents the angle from which the plane CO\({}^{\prime}\)B rotates clockwise to the plane AO\({}^{\prime}\)B as seen in the reverse direction of \(\overrightarrow{\mathrm{O^{\prime}B}}\); \(\phi\) represents the angle at which \(\overrightarrow{\mathrm{O^{\prime}X}}\) rotates counterclockwise to \(\overrightarrow{\mathrm{O^{\prime}E}}\) as viewed from the reverse of \(\overrightarrow{\mathrm{O^{\prime}C}}\); and \(\overrightarrow{\mathrm{O^{\prime}E}}\) and \(\overrightarrow{\mathrm{O^{\prime}D}}\) are, respectively, the projections of \(\overrightarrow{\mathrm{O^{\prime}A}}\) and \(\overrightarrow{\mathrm{O^{\prime}B}}\) on the plane \(\mathrm{X^{\prime}O^{\prime}Y^{\prime}}\)[16]. The polarization state of photons is represented by the Stokes vector:  
Figure 2: Simulation algorithm flow chart. The Stokes vector is relative to a reference plane, and the selection of the reference plane is not carried out at the same time. The Stokes vector photon is endless and also stays the same, so we select principle of the prior specification reference. The meridional plane method stipulates that for any direction vector, the meridional plane formed by the vector and the axis of the coordinate system \(\overrightarrow{O^{\prime}Z^{\prime}}\) should be selected as its reference plane. Therefore, for the photon incidence and scattering direction vectors \(\hat{\mathbf{k}}_{\mathrm{i}}\) and \(\hat{\mathbf{k}}_{\mathrm{s}}\), the corresponding Stokes vectors should be selected as their reference plane, CO\({}^{\prime}\)A and CO\({}^{\prime}\)B, respectively [17]. For tracking the photon polarization state, the vector Monte Carlo method is referred to. The conditional probability distribution function of the sampling method to obtain the photon scattering angle \(\alpha\) and azimuth \(\beta\) requires the renewal of the Stokes vector [18]; the concrete steps are as follows:  
1. The Stokes vector \(S_{\mathrm{a}}\) of the incident photon in the \(\hat{\mathbf{k}}_{\mathrm{i}}\) direction is converted to the Stokes vector \(S_{\mathrm{b}}\) of the scattering plane AO\({}^{\prime}\)B as the reference plane.