# 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
\begin{array}{l}I_{out}=\frac{T_{sys}}{4}[(1+E_{i})F_{1}+(1-E_{i})\cos 2\theta F_{ 2}+(1-E_{i})\sin 2\theta F_{3}],\\ F_{1}=I+\varepsilon_{1}\cos 2\theta_{1}Q+\varepsilon_{1}\sin 2\theta_{1}U,\\ F_{2}=\varepsilon_{1}\cos 2\theta_{1}I+(\cos^{2}2\theta_{1}+\sqrt{1- \varepsilon_{1}^{2}}\sin^{2}2\theta_{1})Q+(1-\sqrt{1-\varepsilon_{1}^{2}}) \cos 2\theta_{1}\sin 2\theta_{1}U,\\ F_{3}=\varepsilon_{1}\sin 2\theta_{1}I+(1-\sqrt{1-\varepsilon_{1}^{2}})\cos 2 \theta_{1}\sin 2\theta_{1}Q+(\sin^{2}2\theta_{1}+\sqrt{1-\varepsilon_{1}^{2}} \cos^{2}2\theta_{1})U,\end{array}  \tag{7}\]  
where the polarizer axis angles \(\theta\) of each channel are \(0^{\circ}\), \(60^{\circ}\), and \(120^{\circ}\). Therefore, three equations can be created, which include the Stokes vector of the incent light [\(I\), \(Q\), \(U\)]. The Stokes vector of the incent light [\(I\), \(Q\), \(U\)] can be obtained by solving the Equations (5)-(7). The degree of polarization DoLP and the orient of polarization \(\sigma_{\rm orient}\) of the target can be calculated. To simplify the result, the terms are ignored that include the power exponent of the extinction ratio, as the extinction ratio is estimated to be 1/200. The simplified result is  
\[I=\frac{4\Gamma_{1}+4\varepsilon_{1}\cos 2\theta  
parameters can be obtained by analyzing the change of radiance intensity. This situation is shown schematically in Figure 6.  
When the solar diffuser is an ideal Lambertian source, the incent light is unpolarized light. The Stokes vector is \(\mathbf{S}^{\prime}_{in}=[I^{\prime}00]^{T}\), where \(I^{\prime}\) represents the radiance intensity of the unpolarized light. The polarizer axis angle of the linear analyzer changes from \(0^{\circ}\) to \(360^{\circ}\), which is represented by \(\theta^{\prime}\). The received light of each channel is \(\mathbf{S}^{\prime}_{out}=T_{i}\mathbf{M}_{\text{P}}(E_{i},\theta)\times \mathbf{M}_{\text{R}}(\theta_{1},\varepsilon_{1},\omega,\varphi)\times\mathbf{ S}^{\prime}_{in}\), which can be rewritten as  
\[ \begin{array}{ll}\mathbf{S}^{\prime}_{out}&=\frac{T_{\text{org}}}{4}\left[ \begin{array}{ccc}1+E_{i}&(1-E_{i})\cos 2\theta&(1-E_{i})\sin 2\theta^{ \prime}\\ (1-E_{i})\cos 2\theta^{\prime}&(1-E_{i})\cos^{2}2\theta^{\prime}+2\sqrt{E_{i}}\sin ^{2}2\theta^{\prime}&(1-\sqrt{E_{i}})^{2}\cos 2\theta^{\prime}\sin 2\theta^{ \prime}\\ (1-E_{i})\sin 2\theta^{\prime}&(1-\sqrt{E_{i}})^{2}\cos 2\theta^{\prime}\sin 2 \theta^{\prime}&(1-\sqrt{E_{i}})^{2}\sin 2\theta^{\prime}+2\sqrt{E_{i}} \end{array}\right]\times\\ &\left[\begin{array}{ccc}1&\varepsilon_{1}\cos 2\theta_{1}&\varepsilon_{1}\sin 2 \theta_{1}\\ \varepsilon_{1}\cos 2\theta_{1}&\cos^{2}2\theta_{1}+\sqrt{1-\varepsilon_{1}^{2}}\sin ^{2}2\theta_{1}&(1-\sqrt{1-\varepsilon_{1}^{2}})\cos 2\theta_{1}\sin 2\theta_{1}\\ \varepsilon_{1}\sin 2\theta_{1}&(1-\sqrt{1-\varepsilon_{1}^{2}})\cos 2\theta_{1} \sin 2\theta_{1}&\sin^{2}2\theta_{1}+\sqrt{1-\varepsilon_{1}^{2}}\cos^{2}2 \theta_{1}\end{array}\right]\left[\begin{array}{c}I^{\prime}\\ 0\\ 0\end{array}\right].\end{array}  \tag{13}\]  
The received radiance intensity is  
\[I^{\prime}_{out}=\frac{T_{\text{sys}}I^{\prime}}{4}[1+E_{i}+(1-E_{i}) \varepsilon_{1}\cos 2(\theta^{\prime}-\theta_{1})]. \tag{14}\]  
We can see from Equation (13) that the change of the received radiance intensity is an approximate cosine curve. The relationship between the response of the detector and the polarizer axis angle \(\theta^{\prime}\) is as Equation (14), when the polarimetric camera has been taken radiometric calibration. The polarization parameters, the diattenuation \(\varepsilon_{1}\), its angle of linear polarization \(\theta_{1}\) and the extinction ratio \(E_{i}\), can be obtained, by taking the least-squares fitting to the response data. The proposed method of on-orbit polarization calibration not only takes full advantage of the on-board solar diffuser, but also doesn't need an extra complex polarizer. What it needs is only a micro precision rotation.