## 2 Theory of Interferometric Spectral Imaging Based on Liquid Crystal Delayers  
Interferometric imaging spectroscopy is an imaging technology that simultaneously obtains spatial and spectral information on the measured object through an interferometric imaging system. It uses the Fourier transform relationship between the interferogram and the restored spectrum and obtains the spectral information on the measured object using the Fourier integral transform of the interferogram. Based on the basic principles of interference spectrum imaging [22], Equation (1) shows that \(I(\Delta)\) and \(B(\sigma)\) are a pair of Fourier transform pairs, \(B(\sigma)\) is the spectrum of the polychromatic light, \(I(\Delta)\) is the two-beam interference intensity distribution, \(\Delta\) is the optical path difference delay between the two beams, and \(\sigma\) is the wavenumber. \[B(\sigma)=\int_{-\infty}^{\infty}I(\Delta)\cos(2\pi\sigma\Delta)d\Delta \tag{1}\]  
This principle can be used for spectral imaging to obtain the intensity distribution for different wavelengths. The acquisition of the interferometric intensity with different optical path differences is the most important part of interference spectral imaging. A LC cell is made of a nematic LC mixture (HTD028200-200) that is inserted between two quartz glass substrates coated with an indium tin oxide (ITO) conductive film. Polyimide (PI) is deposited over the ITO and rubbed to make the LC molecules essentially parallel to the substrates. The LC cell we used has a small pretilt angle (6\({}^{\circ}\)). An external electric or magnetic field can change the alignment of the LC molecules. According to the electro-optical Fredericks effect [23], the LC does not deform when the driving voltage is less than the threshold voltage. When the applied voltage is higher than the threshold voltage, the LC undergoes deformation to minimize the free energy. The Fredericks transformation studied in this paper was carried out under strong anchoring conditions. When a LC material with positive dielectric anisotropy (positive LC) is used, the LC molecules will change from the arrangement along the surface to the vertical plane under the action of an electric field along the normal direction of the LC surface. This effect changes the refractive index of the extraordinary axis of the LC, while the ordinary axis remains constant. This means that \(\Delta n=n_{\mathrm{e}}-n_{0}\) changes with the driving voltage, where \(n_{0}\) and \(n_{\mathrm{e}}\) are the ordinary and extraordinary refractive indices, respectively. However, the change in the direction of the LC molecules due to the minimum free energy takes some time, defined as the relaxation time of the LC molecules. For a thin LC, the relaxation time of the LC is approximately 10-200 ms [24; 25] (p. 244) (p. 184). The rising relaxation time of a LC is the time required for the LC molecules to reach a new equilibrium position from their initial alignment orientation under a loading external field. Correspondingly, the falling relaxation time is the time required for the LC molecules to return from their orientation position in the high-energy state to the initial stable state when the driving field is removed, owing to its elasticity. In LC interferometric spectroscopy imaging, the birefringence effect of the LC can be used to obtain the interference intensity at different delay times by adjusting the voltage. However, this method is time-consuming. Using the relaxation characteristics of the LC, different optical path differences can be obtained for different refractive index differences in the LC during the loading or unloading of the driving voltage.