## 2 Theoretical Analysis
### Principle of Double-Exposure Method  
Figure 2 depicts the schematic diagram of the double-exposure method. The double-exposure method involves two sequential recordings. First, the optical field \(\widetilde{O}\) of the object in its initial state is recorded by interfering with it using reference beam \(\widetilde{R}\), forming hologram \(I\):  
\[I=\left|\widetilde{R}+\widetilde{O}\right|^{2}=\left|\widetilde{R}\right|^{2}+ \left|\widetilde{O}\right|^{2}+\widetilde{R}^{*}\widetilde{O}+\widetilde{R} \widetilde{O}^{*} \tag{9}\]  
Figure 2: (**a**) First exposure; (**b**) second exposure; (**c**) reconstructed optical fields in double-exposure holographic interferometry. Subsequently, after the object undergoes deformation, the modified optical field \(\widetilde{\mathcal{O}}^{\prime}\) (with identical exposure parameters) is recorded by using the same reference beam \(\widetilde{R}\), generating hologram \(I^{\prime}\):  
\[I^{\prime}=\left|\widetilde{R}+\widetilde{\mathcal{O}}^{\prime}\right|^{2}= \left|\widetilde{R}\right|^{2}+\left|\widetilde{\mathcal{O}}^{\prime}\right|^{ 2}+\widetilde{R}^{*}\widetilde{\mathcal{O}}^{\prime}+\widetilde{R}\widetilde{ \mathcal{O}}^{\prime*} \tag{10}\]  
By analyzing and counting the interference fringes, the wavefront variation can be quantified, thereby determining changes in the measured physical quantities. Unlike the single-exposure method, the holographic plate is exposed twice, resulting in a composite hologram \(I_{H}\) expressed as  
\[I_{H}=I+I^{\prime}=2\left|\widetilde{R}\right|^{2}+\left|\widetilde{\mathcal{O }}^{\prime}\right|^{2}+\left|\widetilde{\mathcal{O}}\right|^{2}+\widetilde{R}^ {*}\widetilde{\mathcal{O}}+\widetilde{R}\widetilde{\mathcal{O}}^{*}+ \widetilde{R}\widetilde{\mathcal{O}}^{\prime}+\widetilde{R}\widetilde{ \mathcal{O}}^{\prime*} \tag{11}\]  
During reconstruction, the hologram is illuminated with the original reference beam \(\widetilde{R}\), producing the following optical fields:  
\[ \begin{array}{l}\widetilde{U}=\widetilde{R}I_{H}=\widetilde{R}(2\left| \widetilde{R}\right|^{2}+\left|\widetilde{\mathcal{O}}^{\prime}\right|^{2}+ \left|\widetilde{\mathcal{O}}\right|^{2}+\widetilde{R}^{*}\widetilde{\mathcal{O }}+\widetilde{R}\widetilde{\mathcal{O}}^{*}+\widetilde{R}^{*}\widetilde{ \mathcal{O}}^{\prime}+\widetilde{R}\widetilde{\mathcal{O}}^{\prime*})\\ =\widetilde{R}(2\left|\widetilde{R}\right|^{2})+\widetilde{R}(\left|\widetilde {\mathcal{O}}^{\prime}\right|^{2}+\left|\widetilde{\mathcal{O}}\right|^{2})+ \widetilde{R}\widetilde{R}^{*}(\widetilde{\mathcal{O}}+\widetilde{\mathcal{O }}^{\prime})+\widetilde{R}\widetilde{R}(\widetilde{\mathcal{O}}^{*}+ \widetilde{\mathcal{O}}^{\prime*})\\ =\widetilde{U}_{1}+\widetilde{U}_{2}+\widetilde{U}_{3}+\widetilde{U}_{4}\end{array}  \tag{12}\]  
In the equation, \(\widetilde{U}_{1}=\widetilde{R}(2\left|\widetilde{R}\right|^{2})\), \(\widetilde{U}_{2}=\widetilde{R}(\left|\widetilde{\mathcal{O}}^{\prime}\right| ^{2}+\left|\widetilde{\mathcal{O}}\right|^{2})\), \(\widetilde{U}_{3}=\widetilde{R}\widetilde{R}^{*}(\widetilde{\mathcal{O}}+ \widetilde{\mathcal{O}}^{\prime})\) and \(\widetilde{U}_{4}=\widetilde{R}\widetilde{R}(\widetilde{\mathcal{O}}^{*}+ \widetilde{\mathcal{O}}^{\prime*})\). As shown in Figure 2c, for \(\widetilde{U}_{3}=\widetilde{R}\widetilde{R}^{*}(\widetilde{\mathcal{O}}+ \widetilde{\mathcal{O}}^{\prime})\), the propagation direction deviates from the conjugate path of the original object beam. By extending the deviated beam backward, a virtual image with visible interference fringes can be observed. For \(\widetilde{U}_{4}=\widetilde{R}\widetilde{R}(\widetilde{\mathcal{O}}^{*}+ \widetilde{\mathcal{O}}^{\prime*})\), the propagation direction deviates from the conjugate path of the original object beam. The intersection of these deviated beams forms a real image with measurable interference fringes. Double-exposure holographic interferometry is the most widely used quantitative method for analyzing surface deformations or displacements [17].