## 2 Principles
### Principles of MJI-HI  
The schematic diagram of the MJI-HI system principle based on TSMFTIS and FPI is shown in Figure 2, where system L\({}_{0}\) collimates the object beam into parallel light entering FTIS to generate two coherent beams with a defined OPD. A beam splitter then splits the emergent light into two optical paths: one beam converges through imaging system L\({}_{1}\) onto Detector\({}_{1}\), while the other first traverses the FPI before being focused by imaging system L\({}_{2}\) onto Detector\({}_{2}\), with both paths utilizing identical imaging systems and detectors differing only in the FPI's presence. Assuming \(I_{0}\) denotes the ideal interferogram spanning an OPD range of \(-\infty\) to \(+\infty\), Detector\({}_{1}\) acquires the FTIS interferogram \(I_{1}\) constrained to an OPD range of \([-L\), \(L]\) (\(L\) being the maximum OPD of FTIS), whereas Detector\({}_{2}\) captures the joint FTIS-FPI interferogram \(I_{2}\) within the same OPD limits \([-L\), \(L]\). The maximum OPD of interferograms \(I_{1}\) and \(I_{2}\) is governed by the maximum OPD of FTIS. Subsequent algorithmic processing synthesizes a new interferogram with doubled OPD range through computational integration of \(I_{1}\) and \(I_{2}\), thereby enabling spectral super-resolution via this hybrid optical-computational methodology. The FTIS generates two coherent beams with a specific OPD, which propagate through L\({}_{1}\) and undergo two-beam interference upon imaging at Detector\({}_{1}\), where the resultant interferogram \(I_{1}\) exhibits identical characteristics to those produced by conventional FTIS configurations. \[I_{1}(\Delta)=\int R_{FTIS}T_{FTIS}B(v)[1+\cos(2\pi v\Delta)]\;dv, \tag{2}\]  
where \(R_{FTIS}\) and \(T_{FTIS}\) denote the reflectivity and transmissivity of the FTIS beam splitter, respectively, \(v\) is the wavenumber, and \(B(v)\) represents the source spectral radiance. The OPD generated by the [PERSON] interferometer is defined as \(\Delta=2\mathrm{xcos}\theta\), where \(x\) is the displacement of the moving mirror, and \(\theta\) is the field angle. Neglecting the DC component, the interferogram \(I_{1}\) can be expressed as follows:  
\[I_{1}(\Delta)=FT[B(v)]. \tag{3}\]  
The finite OPD inherent in FTIS imposes fundamental constraints on system performance. Let \(L\) denote the maximum OPD, thereby defining the truncated interferogram as \(I_{1}(\Delta)=I_{0}(\Delta)\) for \(\Delta\in(-L\), \(L)\), where \(I_{1}(\Delta)\) constitutes a windowed subset of \(I_{0}(\Delta)\) restricted to this OPD range. This truncation results in the loss of high-frequency interference components beyond \(|\Delta|>L\), which directly limits the spectral resolution of TMFTIS systems. Figure 2: Diagram of MJI-HI system structure. In the alternate optical path, the two coherent beams emitted from FITS propagate through the FPL. They are subsequently focused onto Detector\({}_{2}\) via imaging system L\({}_{2}\), simultaneously undergoing both two-beam and multiple-beam interference. The resultant composite interferogram arising from this joint interference mechanism can be expressed as follows:  
\[ \begin{array}{ll}I_{2}(\Delta)&=\int B(v)T_{FPI}(v)[1+\cos(2\pi v\Delta)]\, dv\\ &=\int B(v)[1+\cos(2\pi v\Delta)]\,\left[a_{0}+\sum\limits_{m=1}^{\infty}2a_{m} \cos(m\cdot 2\pi v\Delta_{FPI})\right]dv\end{array} . \tag{4}\]  
The FPI employs a fixed cavity spacing \(d\), maintaining a constant transmission function \(T_{FPI}(v)\) throughout the FITS scanning process. Consequently, the interferogram \(I_{2}\) remains solely dependent on the OPD parameter \(\Delta\). By suppressing the DC component, the interferogram can be formulated as follows:  
\[I_{2}(\Delta)=FT[B(v)T_{FPI}(v)]=I_{0}(\Delta)*FT[T_{FPI}(v)], \tag{5}\]  
where \(*\) denotes the convolution operator, and the joint interferogram fundamentally corresponds to the interferogram of \(B(v)T_{FPI}(v)\).