# Polarization Radiometric Calibration in Laboratory for a Channeled Spectropolarimeter
## 1 Introduction
### Principle of the Fieldable Channeled Dispersive Imaging Spectropolarimetry  
The optical schematic of the fieldable CDISP is depicted in Figure 1. In order to facilitate the establishment of polarization radiometric transmission model for optical system, we advanced a method modeling to divide into four sub-modules: fore-optics, PSIM module, imaging optics, and dispersive imaging spectrometer. Polarization information from a detected object is collected and collimated by the fore-optics, and then modulated by PSIM module. After passing though the imaging optics, the modulation spectrum is finally received by a dispersive imaging spectrometer. The fore-optics is used to meet the requirement that all incident angles on R\({}_{1}\) and R\({}_{2}\) are limited to less than about 5\({}^{\circ}\) in the PSIM module [3], and to reduce the size of polarization elements and the processing difficulty. The PSIM module consists of two high-order retarders, R\({}_{1}\) and R\({}_{2}\), with thicknesses d\({}_{1}\) and d\({}_{2}\), respectively, and a linear analyzer, A. The fast axis of R\({}_{1}\) is aligned with the transmission axis of A, and R\({}_{2}\) is oriented with its fast axis at 45\({}^{\circ}\) to the transmission axis of A in an ideal condition. According to Mueller matrix theory and radiation transmission theory, if the incident Stokes vector is \(S_{in}=\left[S_{0},S_{1},S_{2},S_{3}\right]^{T}\) (where T denotes the transpose operation), the Stokes vector of the target light passing through the PSIM module can be given by  
\[S_{out}(\sigma)=M_{PSIM}(\sigma)\cdot S_{in}(\sigma)=M_{A}(0^{\circ})\cdot M_ {R_{2}}[45^{\circ},\varphi_{2}(\sigma)]\cdot M_{R_{1}}[0^{\circ},\varphi_{1}( \sigma)]\cdot S_{in}(\sigma), \tag{1}\]  
where \(M_{R_{2}}\), and \(M_{R_{1}}\) denote the Mueller matrices of A, R\({}_{2}\), and R\({}_{1}\), respectively [18], and \(\varphi_{2}(\sigma)\) and \(\varphi_{1}(\sigma)\) are the ideal phase retardations of R\({}_{2}\) and R\({}_{1}\), respectively. However, for a fieldable CDISP, \(\varphi_{2}(\sigma)\) and \(\varphi_{1}(\sigma)\) are related to the incident angle of the ray to the high-order retarders and will change with the increase of the field angle. In addition, alignment errors, \(\varepsilon_{1}\) and \(\varepsilon_{2}\) of R\({}_{1}\) and R\({}_{2}\), are inevitable for the manufacturing and alignment tolerances, which have significant influences on the reconstructed Stokes parameters. The Mueller matrix of the PSIM module becomes  
\[M^{\prime}{}_{PSIM}(\sigma,\theta)=M^{\prime}{}_{P}(0^{\circ},\sigma,\theta) \cdot M^{\prime}{}_{R_{2}}[45^{\circ}+\varepsilon_{2},\varphi_{2}(\sigma, \theta),\theta]\cdot M^{\prime}{}_{R_{1}}[\varepsilon_{1},\varphi_{1}(\sigma, \theta),\theta]. \tag{2}\]  
The polarization properties of optical system include diattenuation, retardance and depolarization. Thereinto, depolarization is intrinsically associated with scattering and a loss of coherence in the polarization state. A small amount of depolarization is associated with the scattered light from all optical components [18]. Therefore, only diattenuation and retardance were studied as the polarization properties of CDISP. Figure 1: Optical schematic of the fieldable CDISP. The fore-optics and imaging optics can be regarded as refractors. According to the Mueller matrix of each lens interface of optics and its series properties [18], the equivalent Mueller matrices of the fore-optics and imaging optics at the field of view \(\theta\) can be expressed as  
\[M_{fore}(\sigma,\theta)=A_{fore}(\sigma,\theta)\cdot\left[ \begin{array}{cccc} 1&D_{fore}(\sigma,\theta)&0&0\\ D_{fore}(\sigma,\theta)&1&0&0\\ 0&0&1&\delta_{fore}(\sigma,\theta)\\ 0&0&-\delta_{fore}(\sigma,\theta)&1\end{array} \right], \tag{3}\]  
\[M_{imag}(\sigma,\theta)=A_{imag}(\sigma,\theta)\cdot\left[ \begin{array}{cccc} 1&D_{imag}(\sigma,\theta)&0&0\\ D_{imag}(\sigma,\theta)&1&0&0\\ 0&0&1&\delta_{imag}(\sigma,\theta)\\ 0&0&-\delta_{imag}(\sigma,\theta)1\end{array}