## 2 Theory of Interferometric Spectral Imaging Based on Liquid Crystal Delayers
Calibration of the Optical Path Difference at Different Times during the Relaxation of LC and Spectral Imaging Analysis  
As shown in Figure 2, a LC cascade was placed between two parallel linear polarizers. The polarization direction of the two polarizers was 45\({}^{\circ}\) relative to the optical axis of the thick LC. A single-frequency distributed-feedback (DFB) fiber laser at 1053 nm, or a halogen light, was used to pass light through a collimating lens, which was selected by a flip-flop. Figure 1: (**a**) Principle of measurement for the delay between the two beams provided by liquid crystal (LC). (**b**) Schematic of the relationship between the molecular deformation of the LC and the dual-beam delay during the relaxation time. For the 100 \(\upmu\)m LC cell, the impact of uniformity cannot be ignored. Nine measuring points at different LC positions are shown in Figure 5a. Figure 5b shows the optical path difference and error bars for each position limiting the vibration and temperature. From Figure 5b, the error is generally within 2% at different places during the same relaxation time. In addition, from the error bars at different places, it can be seen that the thickness of the right area of the LC is more uniform than that of the left area. Choosing a suitable beam size and position will reduce measurement errors. However, temperature changes are known to affect the birefringence and the response time of the LC. To reduce the impact of such errors, the system can be calibrated at different temperatures. The premise of interconversion between interferogram data and spectrogram data is that the sampling interval must satisfy the sampling theorem, and samples must be at equal intervals. The optical path difference interval corresponding to the sampling interval of the interferogram in the experiment is \(\Delta_{e}\leq 1/2\sigma_{\text{max}}\), where \(\sigma_{\text{max}}\) is the maximum wave number of the measured spectrum. Therefore, the calibration curve in Figure 4 was sampled with a constant optical path difference. Figure 6 shows the interferogram using the sampling theorem with 1053 nm laser light and wide-spectrum halogen light, respectively, with a constant step size. Figure 6a shows that the interferogram of the 1053 nm light is approximately equally spaced, which is typical of monochromatic light interference. Figure 6b shows typical broad-spectrum interference. Figure 4: Relationship between the optical path difference delay and the LC relaxation time obtained at 1053 nm. Figure 3: Interference light intensity variation with the relaxation time. t = 0 is the moment when the LC cell driving voltage \(V_{\text{m}}=5\) V is turned off. (**a**) The 1053 nm DFB fiber laser; (**b**) the halogen light. An effective interferogram was obtained after denoising, phase correction, and processing with an apodization function [22]. By applying the wavelet transform to the interferogram containing noise, an interferogram with reduced noise was obtained [31]. The wavelet transform has the advantages of Fourier analysis while being superior to it. Choosing a suitable number of decomposition layers and an adaptive threshold further enhances the denoising effect. Based on the principle of the single-side beyond-zero path difference, cascaded LC cells were used to obtain a single-sided asymmetric interferogram that passed through a 0 optical range difference, as shown in Figure 6b [32]. Using the interference maximum at 0 optical path difference, we obtained a double-sided symmetric interferogram. The interferogram in the experiment could only be collected at the position of the maximum optical range difference. Hence, to reduce the effect of side lobes caused by data truncation, the apodization function must be considered in the processing of the interferogram [33].