# Detection of Underwater Targets Using Polarization Laser Assisted Echo Detection Technique
## 2 Underwater Target Polarization Detection Mechanism
In polarization measurement systems, the polarizer (usually a line polarizer) and the phase delay (usually a quarter wave plate is used) are very important basic optical elements, as shown in Figure 1.  
A beam of light propagates \(E_{in}\) along the Z-axis direction and passes through the phase delays (the fast-axis direction is at an angle \(\theta\) to the X-axis direction) and a line polarizer (the polarization direction is at an angle \(\varphi\) to the X-axis) in turn. The Stokes parameters can be obtained by rotating the phase delayers and polarizers, and this rotational measurement parameter method is highlighted below. Through the above introduction, we know that  
Figure 1: Principle diagram of polarization measurement. the Stokes vector has four basic parameters, and to solve them completely, at least four equations are needed. If a quarter wavelet (a kind of phase delay) and a line polarizer are placed on the measurement line, where when the direction of the fast axis of the quarter wavelet is at an angle \(\theta\) to the X- axis, the M matrix of the wavelet can be expressed by Equation (3). \[M_{b}= \begin{bmatrix}1&0&0&0\\ 0&\cos^{2}2\theta&\cos 2\theta\sin 2\theta&-\sin 2\theta\\ 0&\cos 2\theta\sin 2\theta&\sin^{2}2\theta&\cos 2\theta\\ 0&\sin 2\theta&-\cos 2\theta&0\end{bmatrix}  \tag{3}\]  
When the polarization direction of the linear polarizer is at an angle \(\varphi\) to the X-axis, Equation (4) can express the M-matrix of the polarizer. \[M_{P}= \begin{bmatrix}1&\cos 2\phi&\sin 2\phi&0\\ \cos 2\phi&\cos^{2}2\phi&\sin 2\phi\cos 2\phi&0\\ \sin 2\phi&\sin 2\phi\cos 2\phi&\sin^{2}2\phi&0\\ 0&0&0&0\end{bmatrix}  \tag{4}\]  
Then the Stokes vector \(S_{out}\) of the incident light \(E_{out}\) can be expressed in Equation (5), which \(I\) indicates the total intensity of the light, the intensity \(Q\) difference between the horizontal (X-axis) polarization and the vertical (Y-axis) polarization, the intensity U difference between 45 degrees and 135 degrees in the direction of the polarized part of the light, and the intensity V difference between the left and right circular components of the light. \[S_{\text{out}}= \begin{bmatrix}I^{\prime}\\ Q^{\prime}\\ U^{\prime}\\ V^{\prime}\end{bmatrix} =M_{2}M_{1}S_{\text{in}}=M_{2}M_{1} \begin{bmatrix}I\\ Q\\ U\\ V\end{bmatrix}  \tag{5}\]  
The light intensity expression is shown in Equation (6):  
\[ \begin{array}{l}I_{(\varphi,\theta)}=\frac{1}{2}I+\frac{1}{2}\cos 2\varphi(Q \times\cos^{2}2\theta+U\times\cos 2\theta\sin 2\theta-V\times\sin 2\theta)\\ +\frac{1}{2}\sin 2\varphi(Q\times\cos 2\theta\sin 2\theta+U\times\sin^{2}2 \theta+V\times\cos^{2\theta})\end{array}  \tag{6}\]  
If four angles (\(0^{\circ}\), \(45^{\circ}\), \(90^{\circ}\) and \(135^{\circ}\)) are given when the phasor is horizontal, and the intensities measured at these four angles are noted as \(I_{1}\), \(I_{2}\), \(I_{3}\), \(I_{4}\), then a system of equations consisting of the three equations represented by Equation (7) is obtained. \[ \begin{bmatrix}I\\ Q\\ U\end{bmatrix} = \begin{bmatrix}\frac{1}{2}(I_{1}+I_{2}+I_{3}+I_{4})\\ I_{1}-I_{3}\\ I_{2}-I_{4}\end{bmatrix} = \begin{bmatrix}I_{1}+I_{3}\\ I_{1}-I_{3}\\ I_{2}-I_{4}\end{bmatrix}  \tag{7}\]  
When the phaser is at a \(45^{\circ}\) angle to the horizontal direction, \(\varphi\) take 0 degrees, and the light intensity \(I_{5}\) is recorded as:  
\[I_{5}=\frac{1}{2}(I-V) \tag{8}\]  
The parameters derived from Equations (5)-(7) are [PERSON], 1 st Baronet vector parameters, and the degree of polarization DOP can be obtained by substituting these parameters into Equation (9) [13]. \[DOP=\frac{\sqrt{Q^{2}+U^{2}+V^{2}}}{I} \tag{9}\]  
In many practical remote sensing applications, the parameter \(V\) is usually so small that the error of the experimental system can be ignored, i.e., treated as zero.