## 3 Optical Error Analysis
### Simulation Experiment Results  
#### 4.2.1 Broadening of the FPI Response  
The non-parallelism and non-flatness errors of the FPI both lead to a decrease in multi-beam interference intensity and a broadening of the response function. Using \(B_{0}=1\) as the input spectrum, a simulation analysis of the interferogram \(I_{2}\) was conducted under different degrees of non-parallelism and non-flatness. According to Equation (17), at this point, \(I_{2}(\Delta)\) can be expressed as the sum of the response functions \(a_{0}\delta(\Delta)+a_{m}\delta(\Delta\pm m\Delta_{FPI})*Defect_{m}(\Delta)\), and the results of these response functions are shown in Figure 14. The simulations examined cases where \(FWHM_{pun}\) and \(FWHM_{dat}\) were set to 0, \(\lambda_{\min}/8(0.0625~{}\upmu\mathrm{m})\), \(\lambda_{\min}/4(0.125~{}\upmu\mathrm{m})\), and \(\lambda_{\min}/2(0.25~{}\upmu\mathrm{m})\), respectively. When the optical defects of the FPI are zero, its interferogram is consistent with Equation (6), where the response function of each frequency-shifted component, \(a_{m}\delta(\Delta\pm m\Delta_{FPI})\), is a \(\delta\) function with energy decreasing stepwise. However, when non-parallelism and non-flatness exist, its interferogram matches Equation (17), and the DC component remains unaffected, while the response functions of higher-order frequency-shifted components undergo convolution with \(g_{m}\) and \(p_{m}\), becoming \(a_{m}\delta(\Delta\pm m\Delta_{FPI})*Defect_{m}(\Delta)\). As the order \(m\) of the frequency shift increases, the broadening of the frequency-shifted components increases progressively by a factor of \(m\), and their amplitude decreases stepwise to \(1/m\). Larger optical defects lead to lower interference intensity. The simulation results are consistent with the analysis presented in Section 3. When the broadening caused by FPI defects is relatively small, its impact can be neglected. Considering both optical defects comprehensively, the full width at half maximum (\(FWHM_{D}\)) of \(Defect_{1}\) in Equation (17) is \(\lambda_{\min}/4(0.125~{}\upmu\mathrm{m})\), \(\lambda_{\min}/2(0.25~{}\upmu\mathrm{m})\), and \(\lambda_{\min}\) (\(0.5~{}\upmu\mathrm{m}\)), respectively, with the broadening caused by non-parallelism and non-flatness each accounting for 50%. The interferogram reconstruction is performed using both the improved algorithm from Section 3.1.3 and the unimproved algorithm, and the results are shown in Figure 15. The red curve in the figure represents \(I_{sup\text{-}\text{imp}}\), while the blue curve represents \(I_{sup}\). From the results, it can be observed that when \(FWHM_{D}\) is small, the results of the improved and unimproved algorithms are relatively close. As \(FWHM_{D}\) increases, the error of \(I_{sup}\) also increases, whereas the results of \(I_{sup\text{-}\text{imp}}\) remain more stable and accurate. By applying the IFT to these interferograms, the super-resolution spectra are obtained, and their errors are calculated based on the following equation:  
\[Error(dB)=10\mathrm{lg}\Bigg{[}\frac{\sum[B^{\prime}(v)-B(v)]^{2}}{\sum[B(v) ]^{2}}\Bigg{]}. \tag{36}\]  
When \(FWHM_{D}\) is \(\lambda_{\min}/4\), \(\lambda_{\min}/2\), and \(\lambda_{\min}\), the errors of the improved algorithm are \(-18.70\) dB, \(-18.74\) dB, and \(-18.38\) dB, respectively, while the errors of the unimproved algorithm are \(-18.60\) dB, \(-18.04\) dB, and \(-15.55\) dB. The error of the unimproved algorithm increases significantly as non-parallelism increases, whereas the improved algorithm maintains a smaller and nearly constant error. Therefore, the improved algorithm is feasible and effective. Figure 14: (**a1–a4**). Broadening of the displacement components of orders \(m=0\), 1, 2, and 3 caused by different non-parallelism; (**b1–b4**). broadening of the displacement components of orders \(m=0\), 1, 2, and 3 caused by different non-flatness.