# Polarization Radiometric Calibration in Laboratory for a Channeled Spectropolarimeter
## 3 Method of Polarization Radiometric Calibration
### Classical Radiometric Calibration  
According to the classical radiometric calibration method [19], the dark current response \(C(\sigma,\theta)\) and modulated radiometric calibration coefficient \(A^{k^{\prime}}(\sigma,\theta)\) can be calibrated by using the lights generated by integrating sphere. The integrating sphere provides unpolarized polychromatic lights that \(S_{0}(\sigma)\
eq 0\),\(S_{1}(\sigma)=0\),\(S_{2}(\sigma)=0\), and \(S_{3}(\sigma)=0\). Substituting these equations into Equation (6), we know that, due to the existence of polarization properties, the calibrated \(A^{k^{\prime}}(\sigma,\theta)\) is not \(A^{k}(\sigma,\theta)\) as we needed, but the result of its coupling with the polarization parameters. \(A^{k^{\prime}}(\sigma,\theta)\) is given by \[\begin{array}{l}A^{k\prime}(\sigma,\theta)=A^{k}(\sigma,\theta)\cdot\left\{1+a_{2 }b_{1}c_{1}D_{fore}(\sigma,\theta)\right.\\ \left. \begin{array}{l}+\frac{1}{2}a_{2}b_{2}c_{2}D_{fore}(\sigma,\theta) \left[\exp[-i\varphi_{2}(\sigma)]+\exp[i\varphi_{2}(\sigma)]\right]\\ -\frac{1}{4}a_{1}b_{2}(1+c_{1})D_{fore}(\sigma,\theta)\exp[-i[\varphi_{1}( \sigma,\theta)-\varphi_{2}(\sigma)]]\\ -\frac{1}{4}a_{1}b_{2}(1+c_{1})D_{fore}(\sigma,\theta)\exp[i[\varphi_{1}( \sigma,\theta)-\varphi_{2}(\sigma)]]\\ +\frac{1}{4}a_{1}b_{1}c_{2}D_{fore}(\sigma,\theta)\exp[-i[\varphi_{1}( \sigma,\theta)]+\exp[i\varphi_{1}(\sigma,\theta)]]\\ +\frac{1}{4}a_{1}b_{2}(1-c_{1})D_{fore}(\sigma,\theta)\exp[-i[\varphi_{1}( \sigma,\theta)+\varphi_{2}(\sigma)]]\\ +\frac{1}{4}a_{1}b_{2}(1-c_{1})D_{fore}(\sigma,\theta)\exp[i[\varphi_{1}( \sigma,\theta)+\varphi_{2}(\sigma)]]\end{array}  \tag{16}\]  
Based on Equation (16), the radiometric calibration coefficient is coupled with the polarization properties in the process of radiometric calibration for fieldable CDISP. It is necessary to decouple them to achieve accurate polarization radiometric calibration. Before decoupling them, the basic polarization parameters need to be calibrated. Self-Correction of Alignment Errors \(\varepsilon_{1}\),\(\varepsilon_{2}\) and Retardation \(\exp[i\varphi_{2}(\sigma)]\)  
The alignment errors of two high-order retarders do not change with the field of view. In addition, the variation of the phase retardation of R\({}_{2}\) with the field of view is too small, so that it can be ignored [16], so we can use the value of \(\varphi_{2}(\sigma)\) at the central fields of view (FOV) as the values of the entire FOV. At central FOV, the beam light is incident vertically to the interface of each lens of fore-optics, so its polarization properties can be negligible, \(D_{fore}(\sigma,0)\approx 0\) and \(\delta_{fore}(\sigma,0)\approx 0\). Substituting these into the Equation (16), we obtain that \(A^{k}(\sigma,0)=A^{k^{\prime}}(\sigma,0)\). Therefore, the modulated radiometric calibration coefficient of central FOV can be used to acquire the modulation spectrum radiance after passing the PSIM module. Based on the calibrated modulation spectrum radiance, the above polarization parameters, \(\varepsilon_{1}\),\(\varepsilon_{2}\), and \(\exp[i\varphi_{2}(\sigma)]\), can be obtained by using the self-correction method [17]. Decoupling of Radiation Calibration Coefficient \(A^{k}(\sigma,\theta)\) and Diattenuation of Fore-Optics \(D_{fore}(\sigma,\theta)\)  
In the process of radiometric calibration, calibration coefficient \(A^{k}(\sigma,\theta)\) and polarization property \(D_{fore}(\sigma,\theta)\) are modulated to different frequencies by two high-order retarders. The Fourier transformation of \(A^{k^{\prime}}(\sigma,\theta)\) gives the autocorrelation function, which can be expressed as  
\[ \begin{array}{l}G^{\theta}(L)\ \ =G^{\theta}_{0}(L)+G^{\theta}_{1}(L-L_{2})+G^{ \theta}_{-1}(-L-L_{2})\\ +G^{\theta}_{2}[L-(L_{1}-L_{2})]+G^{\theta}_{-2}[-L-(L_{1}-L_{2})]\\ +G^{\theta}_{3}(L-L_{1})+G^{\theta}_{-3}(-L-L_{1})\\ +G^{\theta}_{4}[L-(L_{1}+L_{2})]+G^{\theta}_{-4}[-L-(L_{1}+L_{2})].\end{array}