# Method for Detecting Underwater Microbubbles Using Dual-Mode Fusion of Laser Polarization
## 2 Dual-Mode Fusion Detection Mechanism
### Ideal Bubble State Simulation
The rotation matrix of the Stokes vector from the meridional plane CO\({}^{\prime}\)A to the scattering plane AO\({}^{\prime}\)B is as follows: \[R(\beta)= \begin{bmatrix}1&0&0&0\\ 0&\cos(2\beta)&\sin(2\beta)&0\\ 0&-\sin(2\beta)&\cos(2\beta)&0\\ 0&0&0&1\end{bmatrix} \] (2) where \(\beta\) represents the angle of clockwise rotation of CO\({}^{\prime}\)A to AO\({}^{\prime}\)B along \(\overrightarrow{O^{\prime}\mathrm{A}}\). By substituting Equation (2) into Equation (3), we can calculate the Stokes vector defined by the incident light on the scattering plane AO\({}^{\prime}\)B before scattering occurs: \[S_{\mathrm{b}}=R(\beta)S_{\mathrm{a}}\] (3) where \(S_{\mathrm{a}}\) represents the Stokes vector in the incident direction \(\hat{\mathbf{k}}_{\mathrm{i}}\) with respect to the meridional plane CO\({}^{\prime}\)A. \(S_{\mathrm{b}}\) represents the Stokes vector of \(\hat{\mathbf{k}}_{\mathrm{i}}\) relative to the scattering plane AO\({}^{\prime}\)B. 2. On the scattering plane AO\({}^{\prime}\)B, the Stokes vector \(S_{\mathrm{c}}\) of photons in the \(\hat{\mathbf{k}}_{\mathrm{s}}\) direction after scattering is calculated using the bubble Mueller matrix \(M(\alpha)\). The Mueller matrix can describe the polarization characteristics of the scattering process. Since the Stokes vector is relative to a scattering plane, it is necessary to transform the Stokes vector \(S_{\mathrm{a}}\) into the scattering plane AO\({}^{\prime}\)B, and then use the incident Stokes vector to multiply the scattering matrix \(M(\alpha)\) to solve the Stokes vector after photon scattering, namely: \[S_{\mathrm{c}}=M(\alpha)S_{\mathrm{b}}=M(\alpha)R(\beta)S_{\mathrm{a}}\] (4)  
Figure 3: The meridian plane solves the photon scattering coordinate system. where \(S_{\rm c}\) represents the Stokes vector of the scattering photon in the \(\hat{\rm k}_{\rm s}\) direction with respect to the scattering plane AO\({}^{\prime}\)B. 3. The Stokes vector \(S_{\rm c}\) of the photon in the direction of \(\hat{\rm k}_{\rm s}\) is transformed from the scattering plane AO\({}^{\prime}\)B to the meridian plane CO\({}^{\prime}\)B of \(\hat{\rm k}_{\rm s}\). The Stokes vector of the scattered photon relative to the new meridional CO\({}^{\prime}\)B is as follows: \[S_{\rm d}=R(\pi-\gamma)S_{\rm c}=R(\pi-\gamma)M(\alpha)R(\beta)S_{\rm a}\] (5)  
In the simulation, we use the line polarized light S = [1,1,0,0] in any direction to perform the polarization simulation for different sizes and numbers of bubble groups, and obtain the bubble group receding polarization curves as shown in Figure 4.  
The depolarization of bubble groups of varying sizes as the amount distribution changes is depicted in Figure 4. The polarization characteristics of bubbles are weak in the simulation of small-particle-size and sparse discrete bubbles due to the loss of polarization information caused by water turbidity attenuation/distortion and multiple scattering depolarization of bubble particles. As a result, the polarization degree of simulated bubbles appears to have a strong fluctuation and be non-monotonic. Generally speaking, there is still a positive correlation between the depolarization and the size and quantity of bubbles. The depolarization impact caused by repeated scattering is more evident the larger the size and number of bubbles and the smaller the linear polarization degree characterized by the bubbles. Meanwhile, through the simulation, we can know that the polarization degree of 100-600 \(\upmu\)m size bubbles under the number density of 100-600 \(\rm cm^{-3}\) is in the interval of [0.85, 0.98], which provides a discriminating threshold for the subsequent bubble image identification and feature extraction.