# Super-Resolution Multicomponent Joint-Interferometric Fabry-Perot-Based Technique
## 3 Spectral Super-Resolution Inversion for MJI-HI  
Because the interferogram \(I(\Delta)\), obtained by adding the FPI, is a linear combination of multiple displacement components of the original interferogram \(I_{0}(\Delta)\), the latter cannot be calculated directly. Therefore, it is necessary to obtain multiple \(I(\Delta)\) values to establish a linear system of equations for \(I(\Delta)\). For this, based on Figure 2, a beam-splitter BS2 was added after the Michelson interferometer. As shown in Figure 5, one path of light behind BS2 is imaged on Detector 1 through lens L2, whereas the other is imaged on Detector 2 through the FPI and lens L3. It is worth noting that the effects produced by the reverse-transmitted beam of the FPI must be taken into consideration. The reverse-transmitted beam may go through the reflection of BS2, back into the Michelson interferometer, and then reflect back to BS2, which in turn affects Detector 1 and Detector 2. Therefore, a wedge substrate with a tiny angle is required in the FPI design to mitigate the problem of the reverse beam [21; 23; 24]. Figure 4: (**a**) The spectrum after FPI modulation; (**b**) the frequency spectrum of the spectrum after FPI modulation; the red curve is the m = 1 order displacement component; the green curve is the m = 2 order displacement component, and other higher order displacement components are omitted from the figure. Moving M2 changes the OPD for scanning, and Detectors 1 and 2 acquire a set of interferograms. The interferogram acquired on Detector 1 is similar to that of a classical interferometric spectral imager, \(I_{1}(\Delta)=I_{0}(\Delta)\), where \(\Delta\in[-L,L]\). Let the FPI interval \(d\) and refractive index of medium \(n\) be fixed to satisfy \(2 nd=2\ L\). The interferogram obtained for Detector 2 is then obtained, according to Equation (7) [28; 29; 30], by  
\[I_{2}(\Delta)=a_{0}I_{0}(\Delta)+\frac{1}{2}\sum_{m=1}^{\infty}a_{m}I_{0}( \Delta\pm 2 mL), \tag{8}\]  
where \(\Delta\in[-L,L]\). Because the spectral energy is mainly concentrated in the low-frequency region and coefficient \(a_{m}\) decreases with increasing \(m\), the displacement component has little energy within \(\Delta\in[-L,\ L]\). The information of the interferogram \(I_{0}(\Delta)\) outside \(\Delta\in[-2\ L,\ 2\ L]\) and the higher displacement components for \(m\geq 2\) are neglected. Thus, Equation (8) simplifies to  
\[I_{2}(\Delta)=a_{0}I_{0}(\Delta)+\frac{1}{2}a_{1}I_{0}(\Delta+2L)+\frac{1}{2} a_{1}I_{0}(\Delta-2L), \tag{9}\]  
where \(\Delta\in[-L,\ L]\). Because the interferograms \(I_{0}(\Delta)\), \(I_{1}(\Delta)\), and \(I_{2}(\Delta)\) are all even functions, and \(\Delta\) is discrete, Equation (9) can be expressed in the form of a linear system of equations [28; 29; 30]:  
\[ \begin{bmatrix}I_{1}(\Delta)\\ I_{2}(\Delta)\end{bmatrix} = \begin{bmatrix}1&0\\ a_{0}&\frac{1}{2}a_{1}\end{bmatrix}\begin{bmatrix}I_{0}(\Delta)\\ I_{0}(2L-\Delta)\end{bmatrix} , \tag{10}\]  
where \(\Delta\in[0,\ L]\). Since it is an even function, \(I_{0}(\Delta)\), \(\Delta\in[-2\ L,\ 2\ L]\) can be solved. This achieves a twofold increase in the spectral resolution. For the off-axis image point, the field-of-view angle is \(\omega=\theta\) when using the FPI with a vacuum interval. In this case, \(\Delta\) in Equation (10) should be changed to \(\Delta\)cos\(\theta\) and \(L\) to \(L\)cos\(\theta\). The proposed spectral super-resolution method can improve the maximal OPD of the off-axis image point from \(L\)cos\(\theta\) to \(2L\)cos\(\theta\). Therefore, the spectral resolution can be improved, regardless of whether it is the central image point of the field of view. Interestingly, the interferogram inversion process is theoretically the same for the whole field of view. Compared to the resampling method, our proposed method has greater practical applicability. The FPI does not need to satisfy \(2 nd=2\ L\), and an FPI with a different interval can be added to the optical path corresponding to Detector 1.