# On-Orbit Polarization Calibration for Multichannel Polarimetric Camera
## 2 The Multichannel Polarimetric Camera and Main Polarization Elements
### On-Orbit Rotary Analyzer Polarization Calibration Model Based on Polarization Optics Theory  
The polarization parameters of the polarizing effects in the optical system are the diattenuation \(\epsilon_{1}\) and its angle of linear polarization \(\theta_{1}\). According to the polarization optics theory [19], if all of the depolarization, phase retardance and circular polarized component are ignored, the Mueller matrices [13] for the reflective optical system can be given by  
\[\mathbf{M}_{\mathrm{R}}=\mathbf{A}(\theta_{1})*\frac{1}{2}\left[ \begin{array} []{ccc}R_{S}+R_{P}&R_{S}-R_{P}&0\\ R_{S}-R_{P}&R_{S}+R_{P}&0\\ 0&0&2\sqrt{R_{S}R_{P}}\end{array} \right]*\mathbf{A}(\theta_{1}), \tag{3}\]  
Figure 4: The on-orbit polarization calibration assembly for the Earth Observing Scanning Polarimeter (EOSP) (**left**) and the Aerosol Polarimetry Sensor (APS) (**right**). where \(\mathbf{A}\) (\(\theta_{1}\)) is the rotation matrices [21]. Here we use \(\varepsilon_{1}\) to replace \(R_{s}\) and \(R_{p}\). The Mueller matrices of the optical system can be rewritten as  
\[\mathbf{M}_{\mathrm{R}}(\theta_{1},\varepsilon_{1},\omega,\varphi)=\frac{R_{ \mathrm{S}}+R_{p}}{2}\left[ \begin{array}{ccc}1&\varepsilon_{1}\cos 2\theta_{1}& \varepsilon_{1}\sin 2\theta_{1}\\ \varepsilon_{1}\cos 2\theta_{1}&\cos^{2}2\theta_{1}+\sqrt{1-\varepsilon_{1}^{2}} \sin^{2}2\theta_{1}&(1-\sqrt{1-\varepsilon_{1}^{2}})\cos 2\theta_{1}\sin 2 \theta_{1}\\ \varepsilon_{1}\sin 2\theta_{1}&(1-\sqrt{1-\varepsilon_{1}^{2}})\cos 2\theta_{1} \sin 2\theta_{1}&\sin^{2}2\theta_{1}+\sqrt{1-\varepsilon_{1}^{2}}\cos^{2}2 \theta_{1}\end{array} \right], \tag{4}\]  
where (\(\omega\), \(\varphi\)) represents the fields of view (or pixels). Each pixel owns independent Mueller matrices \(\mathbf{M}_{\mathrm{R}}\) (\(\theta_{1}\), \(\varepsilon_{1}\), \(\omega\), \(\varphi\)) as the polarizing effects change as field of view. For the linear analyzer of the polarimetric camera, the extinction ratio of each polarization analyzer channel is \(E_{i}\) (\(0\leq E_{i}<1\), for an ideal analyzer, \(E_{i}=0\)), where \(i=1\), \(2\), \(3\) represents the three polarization analyzer channels. The polarizer axis angle of the linear analyzer is \(\theta\), which has been depicted in Figure 1. Its Mueller matrices are given by [19]  
\[\mathbf{M}_{\mathrm{P}}(E_{i},\theta)=\frac{T_{i}}{2}\left[ \begin{array}{ccc} 1+E_{i}&(1-E_{i})\cos 2\theta&(1-E_{i})\sin 2\theta\\ (1-E_{i})\cos 2\theta&(1-E_{i})\cos^{2}2\theta+2\sqrt{E_{i}}\sin^{2}2\theta&(1- \sqrt{E_{i}})^{2}\cos 2\theta\sin 2\theta\\ (1-E_{i})\sin 2\theta&(1-\sqrt{E_{i}})^{2}\cos 2\theta\sin 2\theta&(1-\sqrt{E_{i}}) ^{2}\sin^{2}2\theta+2\sqrt{E_{i}}\end{array} \right]. \tag{5}\]  
As the collimating lens, the beam splitters and converging lens are designed and optimized into approximately unpolarized elements. The polarizing effects can be ignored when their total transmittance in each polarization analyzer channel is \(Ti\) (\(I=1\), \(2\), \(3\)). According to the polarization optics theory, when the Stokes vector of the incent light is \(\mathbf{S}_{in}=[IQU]^{T}\), the Stokes vector of the light received by the detector is \(\mathbf{S}_{out}=T_{i}\mathbf{M}_{\mathrm{P}}(E_{i},\theta)\times\mathbf{M}_{ \mathrm{R}}(\theta_{1},\varepsilon_{1},\omega,\varphi)\times\mathbf{S}_{in}.\) The schematic of polarization radiation transmission is shown in Figure 5.  
\(T_{sys}(i)\) is used to describe the polarization radiation transmission factor of each polarization analyzer channel, which is defined by \(T_{i}T_{x}(R_{s}+R_{p})=T_{sys}(i)\). The polarization radiation transmission factors of each channel  
The radiance intensity of the received light is  
\[