# Multi-Band Polarization Imaging in a Harsh Sea Fog Environment
## 2 Multi-Band Polarization Imaging Principle  
### The Stokes Vector Method  
The Stokes vector is characterized by three independent parameters (amplitude \(E_{x}\), amplitude \(E_{y}\), and phase difference \(\delta\)) and is the most often measured macroscopic quantity. In order to determine the polarization parameters in the solution of the interaction between the scattering medium and the polarized light, the Stokes vector is used. The Stokes vector consists of four parameters, shown by the matrix below. \[s= \begin{bmatrix}S_{0}\\ S_{1}\\ S_{2}\\ S_{3}\end{bmatrix} = \begin{bmatrix}\left\langle\left|E_{x}\right|^{2}\right\rangle +\left\langle\left|E_{y}\right|^{2}\right\rangle\\ \left\langle\left|E_{x}\right|^{2}\right\rangle-\left\langle\left|E_{y}\right| ^{2}\right\rangle\\ \left\langle 2E_{x}E_{y}\cos\delta\right\rangle\\ \left\langle 2E_{x}E_{y}\sin\delta\right\rangle\end{bmatrix}  \tag{1}\]  
where \(E_{x}\) and \(E_{y}\) are the components of the electric vector in the selected coordinate system in the \(x\) and \(y\) directions, \(\delta\) is the difference in the bit phase of the two vibrational components at the time of study, and the \"\(\left\langle\right\rangle\)\" sign denotes the time-averaged value [10]. \(S_{0}\) represents the sum of the intensities in the \(X\) and \(Y\) directions, \(S_{1}\) represents the difference between the intensities in the \(X\) and \(Y\) directions, \(S_{2}\) represents the difference between the intensities in the \(+45\) and \(-45\) degree directions, and \(S_{3}\) indicates whether the light is right (or left) polarized. This collection of four-dimensional vectors may be used to represent the state and degree of polarization of any polarized light. The following is the relationship for totally polarized light:  
\[S_{0}^{2}=S_{1}^{2}+S_{2}^{2}+S_{3}^{2} \tag{2}\]  
The above equation demonstrates that these four factors are not entirely independent. The following relationship is determined for partly polarized light:  
\[S_{0}^{2}>S_{1}^{2}+S_{2}^{2}+S_{3}^{2} \tag{3}\]The degree of polarization DOP (Degree of Depolarization), as the proportion of fully polarized light in the whole intensity, is expressed by the following equation:  
\[P=\frac{\sqrt{S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}}{S_{0}} \tag{4}\]  
In the optical path to be measured, the Stokes vector is produced by introducing a polarizer and phase-delay device (1/4 waveplate) and measuring the modulated light intensity. If the polarization state of the input light is represented by the Stokes vector \(S_{in}=\left[I,Q,U,V\right]^{T}\)[11] and the polarization state of the corresponding outgoing light after a series of optical devices is represented by the Stokes vector \(S_{out}\), then \(S_{out}\) may be determined by matrix operations as follows:  
\[S_{out}= \begin{bmatrix}S_{0}\\ S_{1}\\ S_{2}\\ S_{3}\end{bmatrix} =M\times S_{in}= \begin{bmatrix}m_{11}&m_{12}&m_{13}&m_{14} \\ m_{21}&m_{22}&m_{23}&m_{24}\\ m_{31}&m_{32}&m_{33}&m_{34}\\ m_{41}&m_{42}&m_{43}&m_{44}\end{bmatrix} \times \begin{bmatrix}S_{0}\\ S_{1}\\ S_{2}\\ S_{3}\end{bmatrix} _{in} \tag{5}\]  
where the 4 \(\times\) 4 matrix \(M\) is known as the Mueller matrix and may be expressed as a cascade matrix when several optical polarization components are present. In the system, the Mueller matrix of the variable liquid crystal phase delay slice is  
\[M_{LCVR}= \begin{bmatrix}1&0&0&0\\ 0&\cos^{2}2\beta+\sin^{2}2\beta\cos\delta&\cos 2\beta\sin 2\beta(1-\cos\delta)&- \sin 2\beta\sin\delta\\ 0&\cos 2\beta\sin 2\beta(1-\cos\delta)&\sin^{2}2\beta+\cos^{2}2\beta\cos \delta&\cos 2\beta\sin\delta\\ 0&\sin 2\beta\cos\delta&-\cos 2\beta\sin\delta&\cos\delta\end{bmatrix}  \tag{6}\]  
where \(\beta\) is the angle between the fast axis and the horizontal direction, and \(\delta\) is the magnitude of the phase delay. If the \(X\)-directional detector is used, the polarizer's Mueller matrix is  
\[M_{90}=\frac{1}{2} \begin{bmatrix}1&1&0&0\\ 1&1&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}