# Super-Resolution Multicomponent Joint-Interferometric Fabry-Perot-Based Technique
## 2 Principle Analysis of MJI-HI Spectral Super-Resolution Based on a FPI
### Principle Analysis of MJI  
Previous studies have found that MJI has the possibility for compatible synergistic effects. First, the transmission spectrum of the FPI is a periodic function, which means that it can be used as a modulating wave for spectral information to improve the spectral resolution [18; 19]. In addition, the transmission function frequency is related to the OPD of the FPI, which can reach the same order of magnitude as the OPD of the dual-beam interferometer; thus, the frequency can meet the requirement [17; 26; 27].  
Available knowledge indicates that the choice of different modulation principle types for the dual-beam interferometer (i.e., temporal, spatial, and joint spatial-temporal modulations) has less influence on the joint-interferometric technique based on a FPI and dual-beam interferometry. Therefore, here we use a Michelson interferometer (Figure 2) as an example to illustrate the principle underlying this technique.  
First, the object beam is collimated by L1 into the double-beam interferometer. Then, the parallel light reflected by the fixed mirror and the moving mirror enters the FPI. The outgoing light then passes through L2 and converges at the imaging point. At this stage, the light at each image point exhibits double-beam interference and multi-beam interference. It is worth noting that the aperture size of the reflecting surfaces, on both sides of the FPI, must be fully calculated during the FPI design. The position and size of the collimated beam incident on the FPI after passing through the Michelson interferometer must be calculated based on the maximum field of view. Because the collimated beam is reflected several times inside the FPI, the apertures of both the FPI exit surface and rear optical system are larger than the incident surface. The incident and exit aperture sizes should be fully calculated during the FPI's design to ensure that multiple-beam interference can occur at the maximum field-of-view image point. The intensity of light at the image plane with wave number \(\
u\) is  
\[I_{D}(\Delta,\
u)=R_{MI}T_{MI}B(\
u)T_{FPI}(\
u)[1+\cos(2\pi\Delta\
u)], \tag{4}\]  
where \(R_{MI}\) and \(T_{MI}\) are the reflectance and transmittance of the Michelson interferometer beam splitter, respectively, and \(B(\
u)\) is the spectrum of the object. Removing the DC component, the total light intensity at the image plane after MJI-HI is:  
\[\begin{array}{l}I(\Delta)=\int_{0}^{\infty}B(\
u)T_{FPI}(\
u)\cos(2\pi \Delta\
u)d\
u\\ =I_{0}(\Delta)\otimes FT[T_{FPI}(\
u)],\end{array} \tag{5}\]  
Figure 2: MJI-HI optical system structure.  
where \(I_{0}(\Delta)\) is the Fourier transform of the object spectrum \(B(\
u)\), and \(I_{0}(\Delta)=FT\) [\(B(\
u)\)]. The obtained interferogram \(I(\Delta)\) is the convolution of the original interferogram \(I_{0}(\Delta)\) and the Fourier spectrum \(FT\) [\(T_{FF}(\
u)\)] of the FPI transmission spectrum \(T_{FPI}(\
u)\) in Equation (3). \(I_{0}(\Delta)\) shifts in frequency during the convolution.  
Structured light microscopy is a technique for the super-resolution of spatial domain information that enables a breakthrough in the optical diffraction limit [28; 29; 30]. Its illumination light is a high-spatial-frequency streak light, which is equivalent to generating a modulating wave in the frequency domain to modulate the sample spatial information. This allows high-frequency spatial information outside the cutoff frequency of the optical system to be mixed and encoded into the low-frequency-domain space within the cutoff frequency. The inversion algorithm can unmix the high-frequency information of the original frequency spectrum and expand the cutoff frequency. The maximal OPD of an interferometric imaging spectrometer is the cutoff frequency of the probe spectrum. With the modulation and unmixing approach of nonlinear structured light, it is also possible to apply modulated waves to the interferometric imaging spectral information. This enables the achievement of spectral mixing and then inversion to improve the spectral resolution.