## 3 Optical Error Analysis
### Simulation Experiment Results
#### 4.2.3 The Combined Error of FPI and FTIS  
According to the analysis in Section 3.2.4, when both the FTIS tilt error and the FPI non-parallelism error are present, their combined effect on the interferogram can be equivalently considered as each \(m\)-th order frequency-shifted component experiencing a tilt error with an angle \(\beta_{m}\). The magnitude of \(\beta_{m}\) depends on the tilt angles of FTIS (\(\beta_{FTIS}\)) and FPI (\(\beta_{FPI}\)), as well as the angle \(\varphi_{FTIS}-\varphi_{FPI}\) between their respective tilt directions. Assuming that the FTIS has a tilt angle of \(\beta_{FTIS}=1^{\prime\prime}\) and the FPI has a tilt angle of \(\beta_{FPI}\)= \(1^{\prime\prime}\), with tilt direction angles of \(\varphi_{FTIS}-\varphi_{FPI}=0^{\circ}\), \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\), we simulate the interferograms \(I_{2}(0)\), \(I_{2}(\pm\Delta_{FPI})\), \(I_{2}(\pm\Delta_{FPI})\), and \(I_{2}(\pm\Delta_{FPI})\) under these tilt error conditions using an input spectrum of \(B(\
u)=1\). Based on Equation (35), under these simulation conditions, the interferograms are given by the following: \(I_{2\text{-}comb}(0)=a_{0}\otimes(\Delta)*\text{g}_{0\text{-}comb}(\Delta)\), \(I_{2\text{-}comb}(m\Delta_{FPI})=a_{m}\otimes(\Delta-m\Delta_{FPI})*\text{g}_ {m\text{-}comb}(\Delta)\), and \(I_{2\text{-}comb}(-m\Delta_{FPI})=a_{m}\otimes(\Delta+m\Delta_{FPI})*\text{g}_ {m\text{-}comb}(\Delta)\), which represent the response functions of each frequency-shifted component. The simulation results are shown in Figure 17, where \(I_{2}\) (\(\Delta\)) represents the ideal interferogram without errors, and \(I_{2\text{-}comb}(m\Delta_{FPI})\) and \(I_{2\text{-}comb}(-m\Delta_{FPI})\) are shifted to the same central position for comparison. It can be observed that as \(m\) increases, the convolution-broadening effect of \(\text{g}_{m\text{-}comb}(\Delta)\) and \(\text{g}_{m\text{-}comb}(\Delta)\) on the response functions becomes more pronounced, indicating that \(\beta_{m}\) increases with \(m\). For the \(m=0\) order, since \(\beta_{0}=\beta_{FTIS}\), the broadening caused by \(\text{g}_{0\text{-}comb}\) (\(\Delta\)) remains the same regardless of \(\varphi_{FTIS}-\varphi_{FPI}\). As shown in Figure 17(d1-d4,e1-e4), for \(m=1,2,3\), the difference between \(I_{2\text{-}comb}(m\Delta_{FPI})\) and \(I_{2\text{-}comb}(-m\Delta_{FPI})\) is the largest when \(\varphi_{FTIS}-\varphi_{FPI}=0^{\circ}\), and as the angle approaches \(90^{\circ}\), \(I_{2\text{-}comb}(m\Delta_{FPI})\) and \(I_{2\text{-}comb}(-m\Delta_{FPI})\) gradually converge. These simulation results are consistent with the analysis presented in Section 3.2.4. Figure 16: **(a1)** Interferogram reconstruction results when the _FWHM\({}_{tilt}\)_ of \(g_{FTIS}\) is \(\lambda_{\text{min}}/4\); **(a2–a5)** are its local enlargements; **(b1)** interferogram reconstruction results when the _FWHM\({}_{tilt}\)_ of \(g_{FTIS}\) is \(\lambda_{\text{min}}/2\); **(b2–b5)** are its local enlargements. Here, \(I_{0}\) represents the ideal interferogram, \(I_{sup}\) denotes the MJI-HI reconstructed interferogram under mirror tilt errors, and \(I_{0\text{-}tilt}\) is the interferogram obtained using FTIS with the same mirror tilt errors. Using the Hitran spectral data from Figure 13(a1,a2) as the input spectrum, the MJI-HI interferogram calculation results under the combined effects of FPI non-parallelism errors and FTIS tilt errors were obtained. We assume the FPI non-parallelism error satisfies \(FWHM_{pm}\)= \(\lambda_{\min}\)/4 (0.125 \(\mu\)m), as defined in Equation (13), and the FTIS tilt error satisfies \(FWHM_{tilt}\) = \(\lambda_{\min}\)/4 (0.125 \(\mu\)m).