## 2 Theory of Interferometric Spectral Imaging Based on Liquid Crystal Delayers
The response times for decay (\(\tau_{\mathrm{d}}\)) and rise (\(\tau_{\mathrm{r}}\)) can be described by the following equation [26]:  
\[\tau_{\mathrm{d}}=\frac{\gamma_{1}L^{2}}{4\mathrm{K}_{33}\left(\frac{\pi}{2}- \frac{\theta_{\mathrm{p}}}{\theta_{\mathrm{m}}}\right)^{2}} \tag{2}\]  
\[\tau_{\mathrm{r}}=\frac{\gamma_{1}}{\left|\varepsilon_{0}\left|\Delta\varepsilon \right|\mathrm{E}^{2}-\frac{4\mathrm{K}_{33}}{\mathrm{L}^{2}}\left(\frac{\pi} {2}-\frac{\theta_{\mathrm{p}}}{\theta_{\mathrm{m}}}\right)^{2}\right|} \tag{3}\]  
where \(\gamma_{1}\) is the LC rotational viscosity, L is the LC cell thickness, K\({}_{33}\) represents the elastic constant, \(\varepsilon_{0}\Delta\varepsilon\mathrm{E}^{2}\) is the electric field energy density, \(\Delta\varepsilon\) is the LC dielectric anisotropy, \(\theta_{\mathrm{p}}\) is the pretilt angle, and \(\theta_{\mathrm{m}}\) is the maximum tilt angle in the center of the LC cell. In addition to the character of the liquid itself, the temperature and the pretilt angle of the liquid crystal will have an influence on the response time. K\({}_{33}\) and \(\gamma_{1}\) decrease with an increase in temperature [27]. However, when the temperature changes, \(\gamma_{1}\) has a greater influence than K\({}_{33}\). Therefore, the relaxation time decreases with increasing temperature. In addition, the pretilt angle of the LC will have an effect on the response time. In most cases, the maximum tilt angle is far larger than the pretilt angle. So, the response time will increase with the growth of the pretilt angle. It is worth noting that the bias voltage (V) has effects on the LC response time, since \(\theta_{\mathrm{m}}\) decreases when the voltage becomes smaller. The pretilt angle effect becomes more pronounced when the voltage (V) gets close to the threshold voltage (V\({}_{\mathrm{th}}\)). In the V\({}_{\mathrm{th}}\)\(<\) V \(<\) 4V\({}_{\mathrm{th}}\), \(\theta_{\mathrm{m}}\) increases significantly when the applied voltage increases and eventually approaches 90\({}^{\circ}\)[26]. When the voltage applied to the LC exceeds the V\({}_{\mathrm{th}}\), the amount of delay begins to decrease as the voltage rises. Between the V\({}_{\mathrm{th}}\) and the 4V\({}_{\mathrm{th}}\), the phase delay falls faster, and above the 4V\({}_{\mathrm{th}}\), the phase delay changes slowly and closes to the minimum. The bias voltage V affects the minimum delay acquired by the LC, because of the angle of LC molecules. When light with a wavelength \(\lambda\) is normally incident on the surface of the LC cell, the phase difference \(\Delta\varphi\) between the two polarized light beams can be expressed as \(\Delta\varphi=2\pi\Delta nL/\lambda\). The LC cell was located between two parallel polarizers, and the polarization axis was oriented at 45\({}^{\circ}\) with respect to the LC optical axis. The schematic in Figure 1a shows the delay between the dual beams emitted from the LC. Figure 1b illustrates the relationship between the molecular deformation of the LC and the dual-beam delay during the relaxation time. From left to right in the figure, the delay between the dual beams corresponds to the time when the LC returns from the high-energy state to the initial stable orientation distribution state. The dual-beam delay changes rapidly with the relaxation time of the LC. The relationship between the transmitted light intensity and the incident light intensity is given by Equation (4) [28]:  
\[I=I_{0}\Big{(}1-\sin^{2}(\Delta\varphi/2)\Big{)} \tag{4}\]  
From the equation above, the light intensity reflects the phase change. Based on the above principles, the key to interferometric spectroscopy is to obtain the optical path difference and interference information at various instances in time. The relaxation time of thick LCs can be used to obtain a large optical path difference in a short time. Therefore, the variation in interference intensity with different optical path differences can be obtained in a short time by using the electro-controlled birefringence and relaxation properties of the LC.