## 3 Optical Error Analysis
### Simulation Experiment Results
#### 4.2.4 Collimation Error of FTIS  
According to the analysis in Section 3.2.1, the effect caused by the divergence angle is that the target interferogram \(I_{0}\) becomes \(I_{0\text{-}dive}(\Delta)=\int I_{0}(\Delta u)du/(1-\cos\theta_{0})\). Therefore, when \(\Delta\) is small, \(I_{0\text{-}dive}(\Delta)\) will approach \(I_{0}\), and when \(\Delta\) is large, \(\Delta u\) will cause a larger shift. Using the cosine function from Figure 13(c1,c2) as the input spectrum, the interferogram \(I_{2}(\Delta)\) will have \(\delta\) functions at \(\Delta=m\Delta_{FPI}\), \(\Delta=m\Delta_{FPI}+k_{0}\), and \(\Delta=m\Delta_{FPI}-k_{0}\). Simulations were conducted with maximum divergence angles \(\theta_{0}=0^{\circ}\).0.5\({}^{\circ}\),1.0\({}^{\circ}\),1.5\({}^{\circ}\), and 2.0\({}^{\circ}\), with the results shown in Figure 19. As shown in Figure 19(b1-b4), the interferograms of \(I_{2}(m\Delta_{FPI})\) are almost identical for different divergence angles, indicating that the effect on the center of each frequency-shifted component can be ignored. As shown in Figure 19(c1-c4,d1-d4), in the interferogram of \(I_{2}(m\Delta_{FPI}\pm k_{0})\), the \(\delta\) functions broaden, and the width increases with the divergence angle. Additionally, the \(\delta\) functions also shift, with the shift increasing as the divergence angle increases. As the order \(m\) increases, the broadening width and central shift remain unchanged. Therefore, the divergence angle affects \(I_{0\text{-}dive}(\Delta)\), but \(I_{2}\) and \(I_{1}\) still maintain a linear relationship, and the inversion algorithm does not introduce additional errors.  
Figure 18: **(a1)** Interferogram reconstruction result under combined errors when \(\varphi_{FTIS}-\varphi_{FPI}=0^{\circ}\), with (**a2–a4**) showing local magnifications; **(b1)** interferogram reconstruction result under combined errors when \(\varphi_{FTIS}-\varphi_{FPI}=30^{\circ}\), with (**b2–b4**) showing local magnifications; **(c1)** interferogram reconstruction result under combined errors when \(\varphi_{FTIS}-\varphi_{FPI}=60^{\circ}\), with (**c2–c4**) showing local magnifications; **(d1)** interferogram reconstruction result under combined errors when \(\varphi_{FTIS}-\varphi_{FPI}=90^{\circ}\), with (**d2–d4**) showing local magnifications; here, \(I_{0}\) represents the ideal interferogram, and \(I_{sup}\) denotes the MJI-HI reconstructed interferogram under joint errors.  
Using the Gaussian function from Figure 13(c1,c2) as the input spectrum, the effect of the divergence angle on the super-resolved spectrum is analyzed. Simulations were conducted with the maximum divergence angles of \(\theta_{0}=0^{\circ}\), \(1.0^{\circ}\), and \(2.0^{\circ}\), with the results shown in Figure 20. The simulation results show that the divergence angle has little effect on the Gaussian function with FWHM = 5000 cm\({}^{-1}\), which represents low-frequency information. However, for the Gaussian function with FWHM = 2.5 cm\({}^{-1}\), which represents high-frequency information, the divergence angle causes a shift in the center wavenumber and an increase in width. This is consistent with the conclusion from Equation (20), where the divergence angle reduces the coherent intensity, leading to a decrease in spectral resolution, and also produces a wavenumber drift factor (\(1-\cos\theta_{0}\))/2.