# Super-Resolution Multicomponent Joint-Interferometric Fabry-Perot-Based Technique
## 4 Results and Discussion
### Simulation Results  
The ideal interferogram \(I_{0}(\Delta)\) is obtained by directly Fourier transforming the input spectrum \(B(v)\), and the super-resolution interferogram \(I^{\prime}{}_{0}(\Delta)\) is obtained by MJI-HI. Figure 7 shows a comparison diagram of \(I^{\prime}{}_{0}(\Delta)\) and \(I_{0}(\Delta)\), where the latter exhibits an interference peak at \(\Delta\) = \(\pm\)5 cm when the input spectrum is a cosine function. Interference peak information is lost when using a classical interferometric imaging spectrometer. In contrast, \(I^{\prime}{}_{0}(\Delta)\) recovers the interference peak at \(\Delta\) = \(\pm\)5 cm, as shown in Figure 7a. An enlarged view of this part in Figure 7b clearly indicates that the result of the calculation of \(I^{\prime}{}_{0}(\Delta)\) has a high level of accuracy. Since the input spectral range is 1.0-2.0 \(\upmu\)m, \(I_{0}(\Delta)\) is here a curve similar to the Sinc function. The inverse Fourier transform of \(I^{\prime}{}_{0}(\Delta)\) yields MJI super-resolution spectrum \(B^{\prime}(v)\). The [PERSON] sampling theorem is satisfied by adding zeros at both ends of the interferogram. Figure 6: Input spectrum: (**a**) cosine function; (**b**) Gaussian function; and (**c**) spectral data. Figure 7: (**a**) The interferogram when the input spectrum is a cosine function; (**b**) the partial magnification comparison of this interferogram. To ensure that the improvement in spectral resolution does not originate from the sampling density, we set the same amount of data as \(I^{\prime}{}_{0}(\Delta)\), \(\Delta\in[-2\ L,\,2\ L]\) by adding zeros at both ends of \(I_{0}(\Delta)\ \Delta\in[-L,\,L]\). Then, we set the same sampling density for \(B^{\prime}(\
u)\) and \(B_{classic}(\
u)\). Figure 8 shows the comparison between \(B^{\prime}(\
u)\) with the input spectrum \(B(\
u)\) and the low-resolution spectrum \(B_{classic}(\
u)\), obtained from a classical interference imaging spectrometer with the same maximal OPD. It can be seen that \(B^{\prime}(\
u)\) can restore the cosine waveform well, whereas \(B_{classic}(\
u)\) completely loses the high-frequency information of the spectrum. This proves that MJI using a FPI can superimpose high-frequency spectral information into low-frequency spectral information, and high-frequency information can be obtained using the inversion algorithm. When the input spectrum is a Gaussian function, as shown in Figure 9a, the spectral resolution of \(B^{\prime}(\
u)\) is higher, and the value is more accurate than that of \(B_{classic}(\
u)\). Through Gaussian fitting, the FWHM of \(B^{\prime}(\
u)\) and \(B_{classic}(\
u)\) can be obtained as 0.101 cm\({}^{-1}\) and 0.197 cm\({}^{-1}\), respectively, as shown in Figure 9b, which also proves that the spectral resolution is improved by a factor of two. This extremely narrow Gaussian function often appears in the form of an absorption peak of a certain gaseous substance in the spectrum. Thus, information, such as the concentration of this gaseous substance, could thus be obtained by the inversion and calculation of the size of the absorption peak. In this application, the resolution and accuracy of spectral absorption peak measurements have a crucial impact on the accuracy of the inversion results. When the input spectrum is chosen as spectral data, the result \(I^{\prime}{}_{0}(\Delta)\), compared to \(I_{0}(\Delta)\), is as shown in Figure 10a. In Figure 10b-d, \(I^{\prime}{}_{0}(\Delta)\) and \(I_{0}(\Delta)\) are amplified and compared at the low-, middle-, and high-frequency parts of \(\Delta\in[L,2\ L]\), respectively. It can be seen that the main, overall waveforms of the two interferograms are relatively consistent. Figure 8: The result of the super-resolution spectrum when the input spectrum is a cosine function.