# Super-Resolution Multicomponent Joint-Interferometric Fabry-Perot-Based Technique
## 4 Results and Discussion
### Evaluation and Analysis  
To evaluate the restoration accuracy, the error of the spectrum was quantified using the following formula [18]:  
\[Error(\text{dB})=10\text{lg}\left[\frac{\sum[B^{\prime}(\
u)-B(\
u)]^{2}}{\sum[B (\
u)]^{2}}\right] \tag{11}\]  
As calculated by Equation (11), when the input spectrum is spectrum data, the spectral error of \(B_{classic}(\
u)\) is \(-27.80\) dB, whereas that of \(B^{\prime}(\
u)\) is \(-33.42\) dB. The reduction in this part of the error is mainly due to the improvement in detail. The reflectivity \(R\) determines the coefficient \(a_{m}\), which in turn determines the linear equation system of Equation (10). Therefore, \(R\) has a significant influence on the algorithm. The \(T_{FPI}\) at different \(R\)s is shown in Figure 12a. When \(R\) is determined, \(\{a_{m}\}\) is a proportional sequence, and its relationship to \(R\) is shown in Figure 12b. With the increase in \(R\), the error caused by Equation (9) for model simplification becomes more significant. Figure 12c shows the errors under different reflectance values calculated using Equation (11). In fact, the most critical step in the design of super-resolution systems is to establish a linear relationship between \(I_{1}(\Delta)\) and \(I_{2}(\Delta)\), \(\Delta\in[-L,L]\) with respect to \(I_{0}(\Delta)\), \(\Delta\in[-2\ L,2\ L]\), similar to that presented in Equation (10):  
\[\left[ \begin{array}{c}I_{1}\\ I_{2}\end{array} \right]=A\cdot I_{0}, \tag{12}\]  
where the inverse matrix \(A\) is a large, sparse matrix, and \(A\) must have a high numerical stability. We use the 2-Norm condition number \(\kappa\) to evaluate the numerical stability of \(A\). The reflectance \(R\) of the FPI determines the coefficients of the inversion matrix \(A\), and the interval \(d\) determines its structure. Therefore, \(R\) and \(d\) together determine the stability of \(A\). We calculated \(\kappa\) for different values of \(R\) and \(d\). The calculation results show that when \(2 nd=2\ L\) or \(L\), \(\kappa\) is small; that is, it has a high numerical stability, as shown in Figure 13. Under this condition, an \(R\) that is too large or too small will increase \(\kappa\). Nonetheless, \(\kappa\) can be maintained at 60% in the range of 20% to 80%, which is excellent numerical stability. Compared to the resampling method, our proposed technique is more relaxed in terms of the reflectance requirement of the FPI. This advantage makes the technique more suitable for the enhancement of the spectral resolution for large-aperture or wide-spectral-range interferometric spectrometers with lower FPI preparation costs. Figure 12: (**a**) The \(T_{\text{FPI}}\) at different RS; (**b**) the relationship between the reflectivity \(R\) and the coefficient \(a_{m}\); (**c**) the curve of Error vs. the reflectivity \(R\). For a static FPI, the condition of \(2 nd=2\ L\) (or \(L\)) is relatively easy to implement, and it has a higher stability without changing the interval \(d\), as compared to the resampling method. If this condition cannot be satisfied, it is necessary to ensure that the interferogram is recoverable by sacrificing a part of the spectral resolution. For example, when \(nd=0.9\ L\), \(2 nd=2\ \times(0.9L)\) is satisfied, so \(I_{0}(\Delta)\), \(\Delta\in[-1.8L,\ 1.8L]\) can be obtained based on the inversion of \(I_{1}(\Delta)\), \(I_{2}(\Delta)\), \(\Delta\in[-0.9L,\ 0.9L]\), where the spectral resolution is enhanced by 1.8, instead of 2, times. Or, when \(nd=0.4\ L\), \(2 nd=0.8\ L\) is satisfied, and \(I_{0}(\Delta)\), \(\Delta\in[-1.6L,\ 1.6L]\) can be obtained based on the inversion of \(I_{1}(\Delta)\), \(I_{2}(\Delta)\), \(\Delta\in[-0.8L,\ 0.8L]\), and the spectral resolution is enhanced by 1.6 times. Therefore, interval \(d\) determines the ability of the FPI to improve the spectral resolution. Based on this approach, \(d\) has a wide range of values and provides greater flexibility in the design of the FPI.