# Polarization Transmission of Visible Light in Inhomogeneous Sea Fog Particle Environment
## 2 Establishment of an Inhomogeneous Particle Scattering Model
### Model Building Based on Polarization Monte Carlo Method  
The radius \(a\) of the inhomogeneous sea fog particles is close to the incident wavelength \(\lambda\). Thus, Mie scattering theory is used to solve the optical parameters that can characterize the optical properties [18]. According to Mie scattering theory, the calculation formulas of extinction efficiency \(Q_{ext}\), scattering efficiency \(Q_{sca}\), and absorption efficiency \(Q_{abs}\) are as follows:  
\[Q_{ext}=\frac{2}{x^{2}}\sum_{n=1}^{\infty}{(2n+1)\text{Re}(a_{n}(\psi_{n},m^{ \prime},\xi_{n})+b_{n}(\psi_{n},m^{\prime},\xi_{n}))} \tag{9}\]  
\[Q_{sca}=\frac{2}{x^{2}}\sum_{n=1}^{\infty}{(2n+1)(\left|a_{n}(\psi_{n},m^{ \prime},\xi_{n})\right|^{2}+\left|b_{n}(\psi_{n},m^{\prime},\xi_{n})\right|^{2})} \tag{10}\]  
\[Q_{abs}=Q_{ext}-Q_{sca} \tag{11}\]  
where \(m^{\prime}\) is the refractive index of inhomogeneous particle, \(x\) is the scale parameter of the equivalent particle. Given that \(\psi_{n}\) and \(\xi_{n}\) are Bessel functions, the calculation formulas of \(a_{n}\) and \(b_{n}\) in Equations (9) and (10) are as follows:  
\[a_{n}(\psi_{n},m^{\prime},\xi_{n})=\frac{\psi_{n}^{\prime}(m^{\prime}x)\psi_{ n}(x)-m^{\prime}\psi_{n}(m^{\prime}x)\psi_{n}^{\prime}(x)}{\psi_{n}^{\prime}(m^{ \prime}x)\xi_{n}(x)-m^{\prime}\psi_{n}(m^{\prime}x)\xi_{n}^{\prime}(x)} \tag{12}\]  
\[b_{n}(\psi_{n},m^{\prime},\xi_{n})=\frac{m^{\prime}\psi_{n}^{\prime}(m^{\prime }x)\psi_{n}(x)-\psi_{n}(m^{\prime}x)\psi_{n}^{\prime}(x)}{m^{\prime}\psi_{n} ^{\prime}(m^{\prime}x)\xi_{n}(x)-\psi_{n}(m^{\prime}x)\xi_{n}^{\prime}(x)} \tag{13}\] \begin{table}
\begin{tabular}{c c c c c c} \hline
**Relative humidity** & 45\% & 55\% & 65\% & 75\% & 85\% \\ \hline
**Equivalent Refractive Index** & 1.4585+ & 1.4518+ & 1.4459+ & 1.4405+ & 1.4355+ \\  & \(i0.0001\) & \(i0.0001\) & \(i0.0001\) & \(i0.0001\) & \(i0.0001\) \\ \hline \end{tabular}
\end{table} 
Table 3: Equivalent refractive index under different relative humidity conditions. The magnitude of the scattering amplitude can be calculated using the obtained \(a_{n}\), \(b_{n}\) and associated Legendre functions \(\pi_{n}\), \(\tau_{n}\):  
\[S_{1}=\sum_{n}\frac{2n+1}{n(n+1)}(a_{n}(\psi_{n},m^{\prime},\xi_{n})\pi_{n}+b_{n }(\psi_{n},m^{\prime},\xi_{n})\tau_{n}) \tag{14}\]  
\[S_{2}=\sum_{n}\frac{2n+1}{n(n+1)}(a_{n}(\psi_{n},m^{\prime},\xi_{n})\tau_{n}+b_ {n}(\psi_{n},m^{\prime},\xi_{n})\pi_{n}) \tag{15}\]  
The Mueller matrix is used to relate the magnitude of the scattering amplitude to the Stokes vector:  
\[S^{\prime}=MS,M= \begin{bmatrix}m_{11}&m_{12}&0&0\\ m_{12}&m_{11}&0&0\\ 0&0&m_{33}&m_{34}\\ 0&0&-m_{34}&m_{33}\end{bmatrix}  \tag{16}\]  
where:  
\[ \begin{array}{l}m_{11}=\frac{1}{2}(|S_{1}|^{2}+|S_{2}|^{2}),m_{12}=\frac{1}{ 2}(|S_{1}|^{2}-|S_{2}|^{2})\\ m_{33}=\frac{1}{2}(S_{1}S_{2}^{*}-S_{1}^{*}S_{2}),m_{34}=\frac{1}{2}(S_{1}S_{2 }^{*}-S_{2}^{*}S_{1})\end{array}  \tag{17}\]  
Finally, the Stokes vector is used to calculate the DOP of the polarized light after the action of sea fog particles:  
\[\text{DOP}=\frac{\sqrt{Q^{2}+U^{2}+V^{2}}}{I} \tag{18}\]  
Monte Carlo is a simulation method based on Mie scattering. This method uses repeated statistical experiments to simulate the scattering of photons during transmission. This method is chosen as the problem-solving tool, because it has fast convergence speed, high precision and the ability to simulate the original problem directly. However, the classical Monte Carlo method is only suitable for homogeneous particles. It cannot simulate the photon scattering in the transmission medium composed of inhomogeneous particles. Because this method does not consider the nonuniqueness of the particles' refractive index; that is, the refractive index setting is just a simple input module. In response to this problem, we change the previous simple input module of refractive index to the input module of refractive index calculation. First, we set incident wavelength and relative humidity.