## 2 Principle of PSIFTIS and the Information Reconstruction Method  
### Principles of System Operation and Structural Parameters  
The principle of a static infrared snapshot polarization spectroscopy imaging system is shown in Figure 1. The system consists of a pre-system, a micro-lens array, two multi-stage micro-reflectors, a beam splitter, a relay optical system, a polarizer array, and a detector. The target light field is incident into the multiplex imaging interferometer system through the collimation system, and the incident light field is imaged by the micro-lens array in multiple channels. Two high and low rooftop-shaped multi-stage micro-mirrors are placed orthogonally, and the step height conforms to the principle of optical path difference sampling complementarity. The imaging unit is subjected to distributed phase modulation to form an interference image array. The polarizer array modulates the interference image unit with different polarization states, so as to obtain the polarization-modulated interference image on the detector that is shown in Figure 1 as well as the polarization-modulated spectral image by Fourier transform, and to realize the polarization image reconstruction by Stokes polarization calculation that is shown in Figure 1b.  
The system mainly works in the mid-infrared band; the spectral range is 3.7-4.8 \(\upmu\)m, divided into 4 polarization interference channels; each polarization state corresponding to the number of interference channels is 8 \(\times\) 8, the spectral resolution is 7.8 \(\upmu\)m\({}^{-1}\), and the modulation transfer function of each optical element is greater than 0.5@17 lp/mm.  
The core interferometric system of this system is composed of two orthogonal high- and low-stepped rooftop-shaped multi-stage micro-reflectors and a beam splitter, which contribute to the phase modulation of the imaging light field of the micro-lens array. The parameters of the rooftop-shaped multi-stage micro-reflector are as follows.  
It can be seen from Table 1 that the rooftop-shaped multi-stage micro-mirror has a total of 2 N = 16 steps, 8 on the left and 8 on the right, which satisfies the narrow-band sampling theorem. The sampling frequency F\({}_{\mathrm{s}}\) of the interferometer system is greater than 2 times the bandwidth BW of the emission spectral line of the detected target, and the height of the steps d meets the requirement of being less than or equal to 1/(4 BW). According to the spectral resolution \(\Delta_{\mathrm{v}}\) = 1/(2 N\({}^{2}\)d) = 7.8 cm\({}^{-1}\), required by the system design, the low height of the steps d\({}^{\prime}\) can be calculated to be 10 \(\upmu\)m, with the high height of the steps  
Figure 1: PSIFTIS principle: (**a**) polarization imaging, (**b**) polarization spectroscopy.  
Nd' being 80 um. The width of the steps matches the size of a single square lens of the micro-lens array (4 mm \(\times\) 4 mm), and the total length of the mirror matches the number of lenses of the micro-lens array (16 \(\times\) 16), which is made of silicon material, while the polarizer array is designed with a silicon substrate grid polarizer, so as to meet the working band of the system, which should be 3.7-4.8 um. The system adopts the model Gavin615A HgCdTe material medium wave-refrigerated infrared detector, with a working wavelength of 3.7-4.8 um, a pixel number of 640 \(\times\) 512, a pixel size of 15 um \(\times\) 15 um, an array size of 9.6 mm \(\times\) 7.68 mm, an F# of 4, a cold shutter aperture of 5.1 mm, and a rear focal length of 19.8 mm. The system structure built is shown in Figure 2.