## 3 Optical Error Analysis
### Simulation Experiment Results
#### 4.2.5 Reflectivity Variation Error of FPI  
Based on the analysis in Section 3.1.4, changes in reflectance \(R\) cause a change in \(a_{m}\), which is wavelength-dependent, and the sensitivity of \(a_{m}\) to changes in \(R\) varies for different orders of \(m\). Using \(B_{0}=1\) as the input spectrum in Figure 13, the effect of changes in \(R\) is analyzed based on \(I_{2}\). When \(R\) is fixed, \(I_{2}(\Delta)\) can be expressed as \(\sum_{m}\delta(\Delta\pm m\Delta_{FPI})\). When \(R\) changes, the difference between \([a_{m}+a_{m}^{\prime}\widetilde{R}_{m}(\Delta)]\)\(\ast\)\(\delta(\Delta\pm m\Delta_{FPI})\) and \(a_{m}\delta(\Delta\pm m\Delta_{FPI})\) allows us to evaluate the effect of reflectance variation on the interference pattern. Assuming an average reflectance of 40%, four smooth curves  
Figure 20: (**a**) Super-resolution spectra obtained using MJJ-HI with divergence angles of \(0^{\circ}\), \(1^{\circ}\), and \(2^{\circ}\); (**b**) local magnification at wavenumber \(1.5\times 10^{4}\) cm\({}^{-1}\). Here, \(B_{0}\) represents the input spectrum, and \(B_{sup-\theta}\) denotes the super-resolution spectra obtained under different divergence angles using MJJ-HI. with different complexities are randomly generated to represent the variations in reflectance. The curves are scaled proportionally so that their maximum-minimum values are set to PV = 2%, 6%, and 10%, which simulate film layer reflectance tolerances of \(\pm\)1%, \(\pm\)3%, and \(\pm\)5%. The simulation results for \([a_{m}+a^{\prime}_{m}\widetilde{\widetilde{K}}_{m}(\Delta)]*\delta(\Delta\pm m \Delta_{PPI})\) for \(m=0\) to 3 are shown in Figure 21. It can be observed that these \(\delta\) functions remain consistent across different PV values. It can be concluded that changes in reflectance do not affect the linear relationship between \(I_{1}\) and \(I_{2}\), and the inversion algorithm remains applicable. For \(m=0\), 2, 3, slight differences appear at the side lobes of the \(\delta\) functions as the PV value changes, whereas the \(\delta\) function for \(m=1\) remains mostly unchanged. This is because \(R_{0}=40\%\) is close to the zero point \(a_{1^{\prime}}\) of 41.3% in Equation (18), making \(a_{1}\) less sensitive to changes in reflectance. Using \(B_{0}=1\) as the input spectrum, the simulation results of the MJJ-HI super-resolved spectra for different PV values of the FPI reflectance are shown in Figure 22. It can be observed that the spectrum has become a curve due to the variation in \(R\), and its shape is consistent with the four curves. For the case of a reflectance PV of 10%, the maximum fluctuation in the super-resolved spectrum for the four curves is approximately 0.024 to 0.025, which accounts for 2.4% to 2.5% of \(B_{0}\), indicating that the impact on the spectrum is relatively small. Based on the above analysis, the primary errors in the FPI include OPD offset, non-parallelism, surface irregularity, and reflectivity variations. Among these, OPD offset affects the positions of frequency-shifted components but does not disrupt the system's linear relationship. This error can be calibrated by analyzing interference peaks in the joint interferogram. Non-parallelism and surface irregularity, which are errors of the same type, cause position-dependent optical path differences (OPDs) across the clear aperture. Their impact on the interferogram manifests as a convolution of each frequency-shifted component with the error distribution function, thereby degrading the linear relationship. When these errors are small, their effects can be neglected; however, for larger errors, the error distribution function must be incorporated into the system matrix by replacing the corresponding coefficients with vectorized representations, effectively simulating the convolution's impact algorithmically. Reflectivity variations have a negligible effect on the linear relationship and can be ignored, as reflectivity tolerances within \(\pm\)5% remain acceptable for the MJJ-HI operation.