# Polarization Radiometric Calibration in Laboratory for a Channeled Spectropolarimeter
## 3 Method of Polarization Radiometric Calibration
### Classical Radiometric Calibration
 \tag{17}\]  
By filtering out the desired channels and performing inverse Fourier transformations independently, the radiometric calibration coefficient \(A^{k}(\sigma,\theta)\) and diattenuation of fore-optics \(D_{fore}(\sigma,\theta)\) can be described as  
\[D_{fore}(\sigma,\theta)=\frac{2\mathcal{F}^{-1}\left\{G_{1}{}^{\theta}\right\} }{a_{2}b_{2}c_{2}\mathcal{F}^{-1}\left\{G_{0}{}^{\theta}\right\}\exp[-i\varphi _{2}(\sigma)]-2a_{2}b_{1}c_{1}\mathcal{F}^{-1}\left\{G_{1}{}^{\theta}\right\}}, \tag{18}\]  
\[A^{k}(\sigma,\theta)=2\mathcal{F}^{-1}\left\{G_{0}{}^{\theta}\right\}-2\frac{b_ {1}c_{1}}{b_{2}c_{2}}2\mathcal{F}^{-1}\left\{G_{1}{}^{\theta}\right\}\exp[i \varphi_{2}(\sigma)], \tag{19}\]  
where \(a_{i}\), \(b_{i}\), and \(c_{i}\) can be computed through \(\varepsilon_{1}\) and \(\varepsilon_{2}\) calibrated in Section 3.2, and \(\exp[i\varphi_{2}(\sigma)]\) as computed in Section 3.2. For the traditional imaging spectropolarimeter, the influence of polarization properties of optical system on radiometric calibration is not considered. \(A^{k^{\prime}}(\sigma,\theta)\) is directly taken as the radiometric calibration coefficient, so that there is a deviation between the spectral radiance obtained by calibration and the spectral radiance at the entrance pupil. Calibration of Retardation \(\exp\{-i[\varphi_{1}(\sigma,\theta)+\varphi_{2}(\sigma)]\}\) and Retardance of Fore-Optics \(\delta_{fore}(\sigma,\theta)\)  
The variation of the phase retardation of R\({}_{1}\) with the field of view will significantly influence the measurement accuracy of the fieldable CDISP [16], so \(\varphi_{1}(\sigma,\theta)\) cannot be regarded as a constant across the entire FOV. In addition, because the phase retardations of R\({}_{1}\) is always in the form of \(\exp\{-i[\varphi_{1}(\sigma,\theta)+\varphi_{2}(\sigma)]\}\) in the calibration and reconstruction of Stokes parameters, we just need to calibrate \(\exp\{-i[\varphi_{1}(\sigma,\theta)+\varphi_{2}(\sigma)]\}\). To calibrate \(\exp\{-i[\varphi_{1}(\sigma,\theta)+\varphi_{2}(\sigma)]\}\) with the change of field angle, a 0\({}^{\circ}\) linearly polarized beam is entered into the system and filled the entire entrance pupil. For a 0\({}^{\circ}\) linearly polarized beam, \(S_{0}(\sigma)=S_{1}(\sigma)\), \(S_{2}(\sigma)=0\), and \(S_{3}(\sigma)=0\). 0 and the 4 channels in Equation (11) then become  
\[C^{\theta}_{0,0^{\circ}}=\mathcal{F}\{X_{0}(\sigma,\theta)+a_{2}b_{1}c_{1}X_{ 1}(\sigma,\theta)\}, \tag{20}\]  
\[C^{\theta}_{4,0^{\circ}}=\mathcal{F}\{\frac{1}{4}a_{1}b_{2}(1-c_{1})\,X_{1}( \sigma,\theta)\exp\{-i[\varphi_{1}(\sigma,\theta)+\varphi_{2}(\sigma)]\}\}, \tag{21}\]  
where \(C^{\theta}_{0,0^{\circ}}\) and \(C^{\theta}_{4,0^{\circ}}\) denote the channels obtained when the 0\({}^{\circ}\) linearly polarized beam is entered into entire entrance pupil of the instrument. By filtering out the desired channels and performing inverse Fourier transformations independently, \(\exp\{-i[\varphi_{1}(\sigma,\theta)+\varphi_{2}(\sigma)]\}\) can be described as  
\[\exp\{-i[\varphi_{1}(\sigma,\theta)+\varphi_{2}(\sigma)]\}=\frac{4(1+a_{2}b_{ 1}c_{1})}{a_{1}b_{2}(1-c_{1})}\cdot\frac{\mathcal{F}^{-1}\{C^{\theta}_{0,0^{ \circ}}\}}{\mathcal{F}^{-1}\{C^{\theta}_{4,0^{\circ}}\}}. \tag{22}\]  
For the fieldable CDISP, the polarization properties caused by the fore-optics is more serious with the increase of the field angle. The retardance of fore-optics, is different with the change of field angle. To calibrate \(\delta_{fore}(\sigma,\theta)\), a 45\({}^{\circ}\) linearly polarized beam is entered into the system and filled the entire entrance pupil. For a 45\({}^{\circ}\) linearly polarized beam, \(S_{0}(\sigma)=S_{2}(\sigma)\), \(S_{1}(\sigma)=0\), and \(S_{3}(\sigma)=0\).