## 2 Theoretical Analysis
### Principle of Double-Exposure Method
Specifically, it records holograms of the object before and after deformation on the same plate, which are later reconstructed under identical optical conditions. Observing through the hologram reveals interference fringes generated by the superposition of object waves from the two states. These fringes represent contour lines of equal displacement along the measurement sensitivity direction. The complex amplitudes of the reference wave and object waves (before and after deformation) are denoted as \(U_{R}(x\),\(y\)), \(U_{o}(x\),\(y)\), and \(U^{\prime}_{o}(x,y)\), respectively. \[U_{o}(x,y)=a_{o}(x,y)\exp[j\phi_{o}(x,y)] \tag{13}\]  
\[U^{\prime}_{o}(x,y)=a_{o}(x,y)\exp[j\phi(x,y)] \tag{14}\]  
\[U_{R}(x,y)=a_{R}(x,y)\exp[j\phi_{R}(x,y)] \tag{15}\]  
For small deformations, the amplitude of the object wave remains unchanged, while its phase undergoes a measurable shift. **First exposure**: The recorded complex amplitude and intensity on the hologram are  
\[A_{1}(x,y)=U_{o}(x,y)+U_{R}(x,y) \tag{16}\]  
\[I_{1}(x,y)=A_{1}(x,y)\cdot A_{1}^{*}(x,y)=\left[U_{o}(x,y)+U_{R}(x,y)\right] \cdot\left[U_{o}(x,y)+U_{R}(x,y)\right]^{*} \tag{17}\]  
**Second exposure**: The complex amplitude and intensity distribution obtained on the same hologram are  
\[A_{2}(x,y)=U_{r}^{o}(x,y)+U_{R}(x,y) \tag{18}\]  
\[I_{2}(x,y)=A_{2}(x,y)\cdot A_{2}^{*}(x,y)=\big{[}U_{o}^{t}(x,y)+U_{R}(x,y)\big{]} \cdot\big{[}U_{o}^{t}(x,y)+U_{R}(x,y)\big{]}^{*} \tag{19}\]  
Assuming equal exposure time \(t_{0}\), the total energy recorded on the hologram is  
\[B(x,y)=[I_{1}(x,y)+I_{2}(x,y)]\cdot t_{0} \tag{20}\]  
The amplitude transmittance of the hologram is  
\[T(x,y)=\alpha-\beta\cdot B(x,y)=\alpha-\beta\cdot[I_{1}(x,y)+I_{2}(x,y)]\cdot t _{0} \tag{21}\]  
For the holographic reconstruction of the above equation, the complex amplitude of the diffracted light wave passing through the hologram is  
\[E(x,y)=T(x,y)\cdot U_{R}(x,y)=\alpha U_{R}(x,y)-\beta\cdot[I_{1}(x,y)+I_{2}(x, y)]\cdot t_{0}\cdot U_{R}(x,y) \tag{22}\]  
Simplifying this expression yields three terms:  
\[E(x,y)=\Big{[}\alpha-2\beta t_{0}(a_{o}^{2}+a_{R}^{2})\Big{]}U_{R}-\beta t_{0 }a_{R}^{2}a_{o}\left\{ \begin{array}{l}\exp[j\phi_{o}(x,y)]+\exp[j\phi(x,y)] -\\ \beta t_{0}a_{R}^{2}a_{o}\left\{\begin{array}{l}\exp[-j(\phi_{o}(x,y)-2\beta t _{R}(x,y))]\\ +\exp[-j(\phi(x,y)]-2\beta t_{R}(x,y))\end{array}\right\}\end{array} \right\} \tag{23}\]  
The first term of the equation represents the transmitted wave, propagating along the original reference beam direction. The second term of the equation represents the first-order diffraction wave, which is a coherent superposition of the object waves before and after deformation, and is critical for interferometric analysis. The third term of the equation represents the conjugate object wave, propagating in the opposite direction. The first-order diffraction term (denoted as \(T_{1}(x,y)\)) is expressed as  
\[T_{1}(x,y)=-\beta t_{0}a_{R}^{2}a_{o}\{\exp[j\phi_{o}(x,y)]+\exp[j\phi(x,y)] \}=-\beta t_{0}a_{R}^{2}[U_{o}(x,y)+U_{o}^{t}(x,y)] \tag{24}\]  
The corresponding intensity is  
\[I(x,y)=I_{0}\{1+\cos[\Delta\phi(x,y)]\}=2I_{0}\cos^{2}\frac{\Delta\phi(x,y)}{2} \tag{25}\]  
where \(\Delta\phi(x,y)=\phi(x,y)-\phi_{o}(x,y)\) represents the phase difference caused by deformation. In the equation, \(I_{0}=2\beta^{2}t_{0}^{2}a_{R}^{4}a_{o}^{2}\). \[ \begin{array}{l}I(x,y)=4\beta^{2}t_{0}^{2}a_{R}^{4}a_{o}^{2}\cos^{2}\frac{ \Delta\phi(x,y)}{2}\\ =4\beta^{2}t_{0}^{2}a_{R}^{4}a_{o}^{2}\cos(\Delta\phi(x,y))+1\end{array}  \tag{26}\]  
Phase decoupling occurs via the arcosine operation, as shown below:  
\[\Delta\phi(x,y)=\frac{I(x,y)}{4\beta^{2}t_{0}^{2}a_{R}^{4}a_{o}^{2}}-1 \tag{27}\]  
Here, \(\Delta\phi(x,y)\) is defined only for angles between 0 and \(\pi\) in Equation (27). According to Equation (25), the intensity function of double exposure is a light field modulated by a cosine function.