# Four-Channel Polarimetric-Spectral Intensity Modulation Imager
## 4 Simulation of Alignment Errors in FCPS  
According to Section 3, the system matrix calibration requires \(0^{\circ}\) linearly polarized light, \(45^{\circ}\) linearly polarized light, and left-handed circularly polarized light. Left-handed circular light can be obtained by the combination of \(45^{\circ}\) linearly polarized light and a quarter-wave plate (QWP). A linear polarizer is placed behind the left-rotated circular light, and the detector receives the light intensity passing through the polarizer. If the received light intensity remains unchanged when the line polarizer is rotavated, it proves that the light produced is via left-handed circular polarization.  
The FCPS can demodulate the Stokes vector well. However, this is based on the premise that the calibration of the system matrix is accurate. During the installation of the PSIM, the fast axis direction of the linear polarizer \(P\) is taken as the reference. During the installation process, there will be phase retarder angle error, as shown in Figure 12.  
In order to verify the influence of phase retarder angle error, the Monte Carlo method was adopted for random selection. There are 440 random error combinations. The rank of the system matrix and the demodulated DOLP are shown in Figures 13 and 14.  
It can be seen from Figure 13 that the rank of the matrix remains at 4, which proves that all system matrices are of full rank. The maximum DOLP error of demodulation is much less than 0.1%. The RMS of 440 randomly error combinations' DOLP is shown in Figure 15. The small value of RMS proves that DOLP inversion is effective. Through the above analysis, it can be concluded that the angle error of the phase retarders has little effect on data inversion.  
The calibration light is accurate and there is an angle error between the fast axis direction of the linear polarizer and the calibrated light. The angle error of the linear polarizer is 0.2\({}^{\circ}\), 0.3\({}^{\circ}\), 0.4\({}^{\circ}\), and 0.5\({}^{\circ}\), respectively, and the data demodulation is shown in Figure 16.  
In Figure 16b, the angle error of the linear polarizer is 0.3\({}^{\circ}\) and the spectral curve of the reconstructed \(S_{3}\) begins to separate from the simulation input curve. In Figure 16c, the linear polarizer angle error is 0.4\({}^{\circ}\), the reconstructed \(S_{2}\) begins to separate from the input \(S_{2}\), and the separation angle between the spectral curve of the reconstructed \(S_{3}\) and the  
Figure 14: The demodulated DOLP.  
Figure 15: The RMS of DOLP.  
simulation input curve gradually increases. In Figure 16d, the reconstructed \(S_{2}\) and \(S_{3}\) are separated from the input \(S_{2}\) and \(S_{3}\) at a greater angle, when the linear polarizer angle error is 0.5\({}^{\circ}\). The maximum deviation between the reconstructed and the input is calculated in Table 2. The maximum deviation of reconstructed \(S_{3}\) increases by one order of magnitude when the linear polarizer angle error is 0.4, which shows that the inversion effect decreases. In summary, we obtain that the maximum calibration light angle error allowed by the linear polarizer cannot exceed 0.4\({}^{\circ}\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Linear Polarizer Angle Error** & **R-\(S_{2}\) and \(S_{2}\)** & **R-\(S_{3}\) and \(S_{3}\)** \\ \hline
0.2 & 0.0097 & 0.0057 \\ \hline
0.3 & 0.0145 & 0.0086 \\ \hline
0.4 & 0.0193 & 0.0115 \\ \hline
0.5 & 0.0241 & 0.0145 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The maximum deviation between the reconstructed and the input.  
Figure 16: The linear polarization angle error. (**a**) Linear polarization angle error is 0.2\({}^{\circ}\). (**b**) Linear polarization angle error is 0.3\({}^{\circ}\). (**c**) Linear polarization angle error is 0.4\({}^{\circ}\). (**d**) Linear polarization angle error is 0.5\({}^{\circ}\).