# Four-Channel Polarimetric-Spectral Intensity Modulation Imager
## 2 Optical Layout and Principle  
The optical layout of FCPS is depicted in Figure 1. L1, S, and L2 denote the objective lens, field stop, and collimating lens, respectively. R1, R2, R3, and R4 denote the phase retarders used to modulate incident light, and they are placed behind the collimating lens, where the thickness ratio of R1 and R2 is 1:2 and that of R3 and R4 is 2:1. The fast axes of R1 and R3 are 0\({}^{\circ}\), and that of R2 and R4 are 45\({}^{\circ}\). Linear polarizers P1, P2, P3, and P4 are placed behind the phase retarder so that the fast axes of P1, P2, P3, and P4 are 0\({}^{\circ}\), 90\({}^{\circ}\), 45\({}^{\circ}\), and 0\({}^{\circ}\), respectively. Following the linear polarizer is a dispersion prism for light splitting. L denotes an imaging lens, and the CCD is placed on the back focal plane of L.  
The light from a scene is collected and collimated by the front telescopic system (L1, S, and L2), and then it passes through phase retarders R1, R2, R3, and R4. Next, the light is modulated and then passed through P1, P2, P3, and P4, respectively. In this way, four beams of modulated light can be obtained, and they are imaged on the CCD after being  
Figure 1: The optical layout of FCPS. dispersed through the dispersion prism. Finally, four modulation curves can be acquired on the CCD. According to the definition of the Stokes parameters as expressed Equation (1), six polarimetric intensities of a scene should be measured for the determination of full Stokes parameters. \[S(\lambda)= \begin{bmatrix}S_{0}(\lambda)\\ S_{1}(\lambda)\\ S_{2}(\lambda)\\ S_{3}(\lambda)\end{bmatrix} = \begin{bmatrix}I_{0}(\lambda)+I_{90}(\lambda)\\ I_{0}(\lambda)-I_{90}(\lambda)\\ I_{45}(\lambda)-I_{135}(\lambda)\\ I_{R}(\lambda)-I_{L}(\lambda)\end{bmatrix}  \tag{1}\]  
where \(S_{0}\) and \(S_{1}\) present the total power of the light and the preference for the linear \(0^{\circ}\) over \(90^{\circ}\) polarization, \(S_{2}\) indicates the difference between the linear \(45^{\circ}\) and \(135^{\circ}\) polarization, while \(S_{3}\) for the right circular over left circular polarization states. The Muller matrix of phase retarder and linear polarizer is shown in Equations (2) and (3). \[M_{R}(\varphi(\lambda),\alpha)= \begin{bmatrix}1&0&0&0\\ 0&\cos^{2}(2\alpha)+\cos(\varphi(\lambda))\sin^{2}(2\alpha)&(1-\cos(\varphi( \lambda)))\sin(2\alpha)\cos(2\alpha)&-\sin(\varphi(\lambda))\sin(2\alpha)\\ 0&(1-\cos(\varphi(\lambda)))\sin(2\alpha)\cos(2\alpha)&\sin^{2}(2\alpha)+ \cos(\varphi(\lambda))\cos^{2}(2\alpha)&\sin(\varphi(\lambda))\cos(2\alpha)\\ 0&\sin(\varphi(\lambda))\sin(2\alpha)&-\sin(\varphi(\lambda))\cos(2\alpha)& \cos(\varphi(\lambda))\end{bmatrix}  \tag{2}\]  
\[M_{P}(\beta)= \begin{bmatrix}1&\cos(2\beta)&\sin(2\beta)&0\\ \cos(2\beta)&\cos^{2}(2\beta)&\sin(2\beta)\cos(2\beta)&0\\ \sin(2\beta)&\sin(2\beta)\cos(2\beta)&\sin^{2}(2\beta)&0\\ 0&0&0&0\end{bmatrix}  \tag{3}\]  
In Equation (2), \(\alpha\) is the fast axis direction of the phase retarder and \(\varphi(\lambda)\) is the phase delay generated by the phase retarder. The \(\varphi(\lambda)\) can be expressed as \(\varphi=\frac{2\pi}{\lambda}\Delta nd\) where \(\Delta n\) and \(d\) is the birefringence and thickness of the phase retarder. In Equation (3), \(\beta\) is the direction of the linear polarizer. In FCPS, we defined that \(d\) is the thickness of R2 and R3 and 2d is the thickness of R1 and R4. Therefore, the phase delay generated by R2 and R3 is \(\varphi_{1}(\lambda)=\frac{2\pi}{\lambda}\Delta nd\) and the phase delay generated by R1 and R4 is \(\varphi_{2}(\lambda)=\frac{4\pi}{\lambda}\Delta nd\). We let \(\varphi(\lambda)=\varphi_{1}(\lambda)=\frac{1}{2}\varphi_{2}(\lambda)\).