## 3 Deformation Calculation for Glass  
Figure 4 illustrates the relationship between surface deformation of a diffuse object and the resulting interference fringe field. For any point P on the surface of the diffuse object, if it displaces to point P\({}_{1}\) after deformation, the phase change \(\Delta\phi(P)\) observed at point Q in holographic interferometry arises from the displacement of P before and after deformation. Optical fields: \(\overrightarrow{s_{1}}\): optical field at point P before deformation; \(\overrightarrow{s_{2}}\): optical field at point P\({}_{1}\) after deformation. Figure 4: Relationship between surface deformation of a diffuse object and interference fringe field. Vectors: \(\overset{\rightarrow}{D}\): displacement vector from P to P\({}_{1}\). Illumination vectors: directed toward the object surface (\(\overset{\rightarrow}{\varsigma_{1}},\overset{\rightarrow}{\varsigma_{2}}\)). Observation vectors: directed toward the observer (\(\overset{\rightarrow}{\Omega_{1}}\) from P to Q\(\colon\overset{\rightarrow}{\Omega_{2}}\) from P\({}_{1}\) to Q). The phase difference \(\Delta\phi(P)\) denotes the phase change at observation point Q induced by the object wave before and after deformation. The phase difference \(\Delta\phi(P)\) caused by deformation is expressed as  
\[\Delta\phi(P)=\frac{2\pi}{\lambda}\Delta(P) \tag{37}\]  
The phase change calculation is  
\[\overset{\rightarrow}{D}=\overset{\rightarrow}{P_{1}}-\overset{\rightarrow} {P} \tag{38}\]  
Defining the following equations,  
\[\overset{\rightarrow}{s}=\frac{1}{2}\left(\overset{\rightarrow}{s_{1}}+ \overset{\rightarrow}{s_{2}}\right);\;\Delta\overset{\rightarrow}{s}=\frac{1} {2}\left(\overset{\rightarrow}{s_{1}}-\overset{\rightarrow}{s_{2}}\right)\]  
\[\overset{\rightarrow}{q}=\frac{1}{2}\left(\overset{\rightarrow}{q_{1}}+ \overset{\rightarrow}{q_{2}}\right);\;\Delta\overset{\rightarrow}{q}=\frac{1 }{2}\left(\overset{\rightarrow}{q_{1}}-\overset{\rightarrow}{q_{2}}\right)\]  
Optical path difference (\(\Delta(P)\) denotes the change in optical path length caused by deformation). \[ \begin{array}{l}\Delta(P)=\left(\overset{\rightarrow}{s}+\Delta\overset{ \rightarrow}{s}\right)\cdot\overset{\rightarrow}{SP}+\left(\overset{ \rightarrow}{q}+\Delta\overset{\rightarrow}{q}\right)\cdot\overset{ \rightarrow}{P}Q-\left(\overset{\rightarrow}{s}-\Delta\overset{ \rightarrow}{s}\right)\cdot\overset{\rightarrow}{SP_{1}}-\left(\overset{ \rightarrow}{q}-\Delta\overset{\rightarrow}{q}\right)\cdot\overset{ \rightarrow}{P_{1}}Q\\ =\overset{\rightarrow}{D}\cdot\overset{\rightarrow}{q}-\overset{ \rightarrow}{D}\cdot\overset{\rightarrow}{s}+\Delta\overset{ \rightarrow}{s}\cdot\left(\overset{\rightarrow}{SP}+\overset{\rightarrow}{ SP_{1}}\right)+\Delta\overset{\rightarrow}{q}\cdot\left(\overset{ \rightarrow}{PQ}+\overset{\rightarrow}{P_{1}}Q\right)\end{array}  \tag{39}\]  
\[\Delta\phi(P)=\frac{2\pi}{\lambda}\overset{\rightarrow}{D}(P)\left[\overset{ \rightarrow}{q}(P)-\overset{\rightarrow}{s}(P)\right]=\frac{2\pi}{\lambda} \overset{\rightarrow}{e}(P)\cdot\overset{\rightarrow}{D}(P) \tag{40}\]  
The displacement sensitivity vector \(\overset{\rightarrow}{e}(P)\) denotes the magnitude representing the sensitivity of the interferometer to the displacement direction; it determines the fringe shift per unit displacement. \[\overset{\rightarrow}{e}(P)=\overset{\rightarrow}{q}(P)-\overset{ \rightarrow}{s}(P) \tag{41}\]  
Then,  
\[\Delta\phi(P)=\frac{2\pi}{\lambda}\overset{\rightarrow}{e}(P)\cdot\overset{ \rightarrow}{D}(P) \tag{42}\]  
The displacement vector is  
\[\overset{\rightarrow}{D}(P)=\left[\overset{\rightarrow}{D_{x}}(P),\overset{ \rightarrow}{D_{y}}(P),\overset{\rightarrow}{D_{z}}(P)\right] \tag{43}\]  
Illumination vector \(\overset{\rightarrow}{s}(P)\) denotes the vector from the light source to the undeformed point P:  
\[\overset{\rightarrow}{s}(P)=\left\{ \begin{array}{l}\overset{\rightarrow}{ s}_{x}(P)\\ \overset{\rightarrow}{s}_{y}(P)\\ \overset{\rightarrow}{s}_{z}(P)\end{array}