## 2 Theoretical Analysis  
### Principle of Single-Exposure Method  
Figure 1 illustrates the schematic diagram of the single-exposure method. The procedure involves first recording the optical field \(\widetilde{O}\) of the object in its initial state. Subsequently, the hologram is illuminated with the original reference beam to reconstruct the optical field \(\widetilde{O}\), which then interferes with the modified optical field \(\widetilde{O}\) generated after the object undergoes changes. This interference enables the detection of variations in the object. The hologram is formed by the interference between reference beam \(\widetilde{R}\) and object beam \(\widetilde{O}\):  
\[I=\left|\widetilde{R}+\widetilde{O}\right|^{2}=\left|\widetilde{R}\right|^{2}+ \left|\widetilde{O}\right|^{2}+\widetilde{R}^{*}\widetilde{O}+\widetilde{R} \widetilde{O}^{*} \tag{1}\]  
During detection, only the modified object beam \(\widetilde{O}^{\prime}\) (after object alteration) illuminates the hologram \(I\), reconstructing the optical field:  
\[ \begin{array}{l}\widetilde{U}_{0}=\widetilde{O}^{\prime}I=\widetilde{O}^{ \prime}(\left|\widetilde{R}\right|^{2}+\left|\widetilde{O}\right|^{2}+ \widetilde{R}^{*}\widetilde{O}+\widetilde{R}\widetilde{O}^{*})=\widetilde{O}^{ \prime}(\left|\widetilde{R}\right|^{2}+\left|\widetilde{O}\right|^{2})+ \widetilde{R}^{*}\widetilde{O}^{\prime}\widetilde{O}+\widetilde{R}\widetilde{ O}^{\prime}\widetilde{O}^{*}\\ =\widetilde{U}_{0o}+\widetilde{U}_{0-1}+\widetilde{U}_{01}\end{array}  \tag{2}\]  
In the equation, \(\widetilde{U}_{0o}=\widetilde{O}^{\prime}(\left|\widetilde{R}\right|^{2}+ \left|\widetilde{O}\right|^{2})\), \(\widetilde{U}_{0-1}=\widetilde{R}^{*}\widetilde{O}^{\prime}\widetilde{O}\), \(\widetilde{U}_{01}=\widetilde{R}\widetilde{O}^{\prime}\widetilde{O}^{*}\). When the detection system illuminates hologram \(I\), solely with the recorded reference beam \(\widetilde{R}\), the reconstructed optical field becomes  
\[ \begin{array}{l}\widetilde{U}_{R}=\widetilde{R}I=\widetilde{R}(\left| \widetilde{R}\right|^{2}+\left|\widetilde{O}\right|^{2}+\widetilde{R}^{*} \widetilde{O}+\widetilde{R}\widetilde{O}^{*})=\widetilde{R}(\left|\widetilde{R} \right|^{2}+\left|\widetilde{O}\right|^{2})+\widetilde{O}+\widetilde{R} \widetilde{R}\widetilde{O}^{*}\\ =\widetilde{U}_{Ro}+\widetilde{U}_{R1}+\widetilde{U}_{R-1}\end{array}  \tag{3}\]  
In the equation, \(\widetilde{U}_{Ro}=\widetilde{R}(\left|\widetilde{R}\right|^{2}+\left| \widetilde{O}\right|^{2})\), \(\widetilde{U}_{R1}=\widetilde{O}\), \(\widetilde{U}_{R-1}=\widetilde{R}\widetilde{R}\widetilde{O}^{*}\). After single-exposure processing, the transmittance function of the hologram is expressed as  
\[T=\alpha-\beta t_{0}I(x,y) \tag{4}\]  
In Equation (4), T denotes the amplitude transmittance of the hologram, describing the intensity attenuation of light passing through the holographic plate. \(\alpha\) represents the initial transmittance of the unexposed plate, determined by the substrate material and independent of photosensitive reactions. \(\beta\) signifies the sensitivity coefficient of the recording medium, characterizing the change in transmittance per unit of exposure energy. t\({}_{0}\) indicates the exposure time. \(I(x,y)\) corresponds to the interference intensity between the object wave and reference wave. Upon re-illumination with the original reference beam, the complex amplitude of the diffracted field recorded through the hologram is  
\[A_{1}(x,y)=T(x,y)\big{[}U_{R}(x,y)+U_{o}^{\prime}(x,y)\big{]} \tag{5}\]  
In Equation (5), \(A_{1}(x,y)\) represents the complex amplitude of the diffracted optical field; \(T(x,y)\) is the single-exposure hologram transmittance function; \(U_{R}(x,y)\) denotes the complex amplitude distribution of the reconstructed optical field when the hologram  
Figure 1: (**a**) Recording process of single-exposure holographic interferometry; (**b**) reconstructed optical fields in single-exposure holographic interferometry.