## 3 Optical Error Analysis
### Optical Errors of FTIS
#### 3.2.4 The Combined Error of FPI Non-Parallelism and FTIS Mirror Tilt Error
The relationships between \(\beta_{m}\), \(\varphi_{m}\), and the FTIS/FPI parameters are illustrated in Figure 12. In polar coordinates, the FTIS tilt error is represented as the vector \(\overrightarrow{x}_{FTIR}\)= (\(\beta_{FTIR}\),\(\varphi_{FTIR}\)), while the non-parallelism error of the FPI's \(m\)-th order  
Figure 11: (**a**) In FTIS, the mirror tilt error causes the wavefront (blue surface) to deviate from the ideal wavefront (yellow surface), tilting in the \(\varphi_{FTIR}\) direction with a tilt angle of \(2\beta_{FTIR}\); in FPI, the non-parallelism error causes the wavefront (blue surface) to deviate from the ideal wavefront (yellow surface), tilting in the \(\varphi_{FPI}\) direction with a tilt angle of \(2\beta_{FPI}\), where the Z-axis represents the optical axis. (**b**) The relative tilt direction along the optical axis between the FTIS mirror tilt error and the FPI non-parallelism error. frequency-shifted component is expressed as \(\overrightarrow{x}_{FPI}=(m\beta_{FPI},\varphi_{FPI})\). The equivalent tilt angle is then derived through vector summation in polar coordinates:  
\[\left\{ \begin{array}{l}(\beta_{m},\varphi_{m})=\overrightarrow{x}_{FTIS}- \overrightarrow{x}_{FPI}\\ (\beta_{-m},\varphi_{-m})=\overrightarrow{x}_{FTIS}+\overrightarrow{x}_{FPI} \end{array} \right.. \tag{33}\]  
Only when the tilt direction of FTIS is perpendicular to that of FPI, i.e., \(\mid\varphi_{FTIS}\textasciitifteq}_{FPI}\mid=\pi/2\), the tilt angles \(\beta_{m}\) and \(\beta_{-m}\) are equal. Substituting Equation (32) into Equation (31) gives the following:  
\[\begin{array}{lcl}I_{2-comb}(v,\Delta)&=&a_{0}B(v)\frac{2I_{1}(2\pi v2r_{0} \beta_{FTIS})}{2\pi v2r_{0}\beta_{FTIS}}\cos(2\pi v\Delta)\\ &+&\sum\limits_{m=1}^{\infty}a_{m}B(v  
Figure 13(a1) shows the atmospheric spectrum calculated based on the HITRAN database, which is used to simulate actual target spectral detection [35]. Figure 13(a2) presents the corresponding interferograms \(I_{1}\) and \(I_{2}\), where, due to the effect of the FPI, \(I_{2}\) exhibits multiple interference peaks at \(\Delta=m\Delta_{FPI}\). The input spectrum in Figure 13(b1) is obtained by subtracting two Gaussian functions with FWHM of 5000 and 2.5 cm\({}^{-1}\), where the Gaussian function at 5000 cm\({}^{-1}\) represents the low-frequency information of the spectrum, and the Gaussian function at 2.5 cm\({}^{-1}\) represents the high-frequency information. Figure 13(b2) shows the corresponding interferogram. Figure 13(c1) represents an input spectrum modeled as a cosine function with a periodic frequency of \(k_{0}=1.8\) mm: \(1+\cos(2\pi k_{0}\
u)/2\); thus, under ideal conditions, the interferogram \(I_{1}\) can be expressed as \(I_{1}=\delta(\Delta)+\delta(\Delta-k_{0})/4+\delta(\Delta+k_{0})/4\), while \(I_{2}\) contains delta functions at \(\Delta=m\Delta_{FPI}\) and \(\Delta=m\Delta_{FPI}\pm k_{0}\). Thus, using a cosine function as the input spectrum allows the analysis of optical errors affecting different frequency-shifted components in \(I_{2}\) at both the central position \(\Delta=m\Delta_{FPI}\) and the shifted positions \(\Delta=m\Delta_{FPI}\pm k_{0}\). Figure 13(d1) represents an input spectrum of a constant value of 1, and the corresponding interferogram \(I_{2}\) exhibits delta functions at \(\Delta=m\Delta_{FPI}\), enabling the analysis of how optical errors affect the response functions of different frequency-shifted components. By using different input spectra, the effects of various errors on the response functions of the frequency-shifted components are simulated and analyzed, along with the changes in the final super-resolution interferograms or spectra obtained using MJI-HI.