# Polarization Radiometric Calibration in Laboratory for a Channeled Spectropolarimeter
## 1 Introduction
### Establishment of Polarization Radiometric Calibration Model  
According to the principle of classical radiometric calibration [19; 20], the relationship between the digital number (DN) of CCD detector and the spectral radiance at entrance pupil in the actual operation of the instrument can be given by  
\[\begin{array}{ll}DN(\sigma,\theta)&=A(\sigma,\theta)\cdot&M_{spec}(\sigma, \theta)\cdot M_{imag}(\sigma,\theta)\cdot M_{PSIM}(\sigma,\theta)\cdot M_{ fore}(\sigma,\theta)\cdot S(\sigma)+C(\sigma,\theta)\\ &=A^{k}(\sigma,\theta)\cdot&B(\sigma,\theta)+C(\sigma,\theta)\\ &=A^{k}(\sigma,\theta)\cdot&\{X  
where \(DN(\sigma,\theta)\) is the original DN of the CCD detector, \(A(\sigma,\theta)\) represents the detector pixel response, and \(C(\sigma,\theta)\) is the DN of CCD dark current. For a dispersive spectropolarimeter, \((\sigma,\theta)\) represents the positions of different pixels on the CCD detector. And  
\[ \begin{array}{ll}A^{k}(\sigma,\theta)=\frac{1}{2}A_{spec}(\sigma,\theta)A_{ image}(\sigma,\theta)A_{fore}(\sigma,\theta)A(\sigma,\theta)\\ \times\Big{[}1+D_{spec}(\sigma,\theta)+D_{imag}(\sigma,\theta)+D_{spec}( \sigma,\theta)D_{imag}(\sigma,\theta)\Big{]},\end{array}  \tag{7}\]  
is a function of the transmittance and diattenuation of sub-modules. Their values are different at different pixel positions of the CCD detector, that is, they change with the wavenumber and field of view. Therefore, we simply need to determine \(A^{k}(\sigma,\theta)\) with no need for determining the transmittance and diattenuation of sub-modules independently in the calibration. \(A^{k}(\sigma,\theta)\) is called radiometric calibration coefficient. Moreover \(S(\sigma)=\left[S_{0}(\sigma),S_{1}(\sigma),S_{2}(\sigma),S_{3}(\sigma) \right]^{T}\) is the spectral radiance at entrance pupil of the instrument. The spectral radiance at the entrance pupil, contains spectral, intensity and polarization information of the detected object, is modulated by the PSIM module into modulation spectrum. The modulation spectrum radiance could be obtained by radiometric calibration of the DN of CCD detector, which can be expressed as  
\[ \begin{array}{ll}B(\sigma,\theta)=&X_{0}(\sigma,\theta)+b_{1}c_{1}X_{12}( \sigma,\theta)\\ &+\frac{1}{2}b_{2}c_{2}X_{12}(\sigma,\theta)\{\exp[-i\varphi_{2}(\sigma, \theta)]+\exp[i\varphi_{2}(\sigma,\theta)]\}\\ &-\frac{1}{4}b_{2}(1+c_{1})X_{123}^{*}(\sigma,\theta)\exp[-i[\varphi_{1}( \sigma,\theta)-\varphi_{2}(\sigma,\theta)]\}\\ &-\frac{1}{4}b_{2}(1+c_{1})X_{123}(\sigma,\theta)\exp[i[\varphi_{1}(\sigma, \theta)-\varphi_{2}(\sigma,\theta)]\}\\ &+\frac{1}{2}b_{1}c_{2}X_{123}^{*}(\sigma,\theta)\exp[-i\varphi_{1}(\sigma, \theta)]\\ &+\frac{1}{2}b_{1}c_{2}X_{123}(\sigma,\theta)\exp[i\varphi_{1}(\sigma,\theta) ]\\ &+\frac{1}{4}b_{2}(1-c_{1})X_{123}^{*}(\sigma,\theta)\exp[-i[\varphi_{1}( \sigma,\theta)+\varphi_{2}(\sigma,\theta)]]\\ &+\frac{1}{4}b_{2}(1-c_{1})X_{123}(\sigma,\theta)\exp[i[\varphi_{1}(\sigma, \theta)+\varphi_{2}(\sigma,\theta)]],\end{array}  \tag{8}\]  
where  
\[\left\{ \begin{array}{ll}a_{1}=\sin 2\varepsilon_{1}\\ a_{2}=\cos 2\varepsilon_{1}\end{array} \right.,\left\{ \begin{array}{ll}b_{1}=\sin 2 \varepsilon_{2}\\ b_{2}=\cos 2\varepsilon_{2}\end{array} \right.,\left\{ \begin{array}{ll}c_{1}= \sin 2(\varepsilon_{2}-\varepsilon_{1})\\ c_{2}=\cos 2(\varepsilon_{2}-\varepsilon_{1})\end{array} \right., \tag{9}\]  
\[\left\{ \begin{array}{ll}X_{0}(\sigma,\theta)=S_{0}(\sigma)+D_{fore}(\sigma, \theta)S_{1}(\sigma)\\ X_{1}(\sigma,\theta)=D_{fore}(\sigma,\theta)S_{0}(\sigma)+S_{1}(\sigma)\\ X_{2}(\sigma,\theta)=S_{2}(\sigma)+\delta_{fore}(\sigma,\theta)S_{3}(\sigma) \\ X_{3}(\sigma,\theta)=S_{3}(\sigma)-\delta_{fore}(\sigma,\theta)S_{2}(\sigma) \\ X_{12}(\sigma,\theta)=a_{2}X_{1}(\sigma,\theta)+a_{1}X_{2}(\sigma,\theta)\\ X_{123}(\sigma,\theta)=a_{1}X_{1}(\sigma,\theta)-a_{2}X_{2}(\sigma,\theta)+iX _{3}(\sigma,\theta)\end{array} \right., \tag{10}\]  
and the superscript \(*\) denotes the complex conjugate.