# Digital Shearography for NDT: Phase Measurement Technique and Recent Developments
## 2 Brief Review of Digital Shearography  
### Formation of Fringes in Digital Shearography  
Digital shearography is a laser-based optical interferometry technique that requires a shearing device to be placed in front of the camera to introduce image shearing, thereby allowing light reflected from two different object points to interfere at one point in the image plane. Different shearing devices, such as an optical wedge or biprism [19], a Mach-Zehnder interferometer [20], and a modified Michelson interferometer [21], have been used for digital shearography measurements. The modified Michelson interferometer is the most common shearing device because of its simple structure and ease in changing the shearing amount and shearing direction. This section introduces the basic principle of digital shearography with the modified Michelson interferometer as the shearing device. Figure 1 shows the classic shearographic setup using a modified Michelson interferometer to create shearing amount and make interference at point P. The shearing can be created by tilting mirror 1 by a very small angle. Then, light waves from points P1 and P2 on the surface of the object can be transmitted to a point P on the sensor plane. These light waves interfere with each other and produce a speckle pattern, or speckle interferogram, which is registered by a CCD camera and saved in the computer [22]. The total light field \(U_{t}\) at the point P on the image plane can be expressed by the following equation:  
\[U_{t}=U_{1}+U_{2}=a_{1}e^{i\theta_{1}}+a_{2}e^{i\theta_{2}} \tag{1}\]  
where \(\theta_{1}\) and \(\theta_{2}\) represent the random phase angle of points P1 and P2, respectively, and \(a_{1}\) and \(a_{2}\) are the light amplitudes. Thus, the intensity at point P can be expressed as follows:  
\[ \begin{array}{ll}I&=U_{t}U_{t}{}^{*}=(a_{1}e^{i\theta_{1}}+a_{2}e^{i\theta_{2}} )(a_{1}e^{-i\theta_{1}}+a_{2}e^{-i\theta_{2}})\\ &=({a_{1}}^{2}+{a_{2}}^{2})+a_{1}a_{2}[e^{i(\theta_{1}-\theta_{2})}+e^{-i( \theta_{1}-\theta_{2})}]\\ &=({a_{1}}^{2}+{a_{2}}^{2})+2a_{1}a_{2}\cos(\theta_{1}-\theta_{2})\\ &=A+B\cos\phi\end{array}  \tag{2}\]  
where \(U_{t}^{*}\) is the complex conjugate of \(U_{t}\), \(\cos(\theta_{1}-\theta_{2})=[e^{i(\theta_{1}-\theta_{2})}+e^{-i(\theta_{1}- \theta_{2})}]/2\) (Euler equation), \(A={a_{1}}^{2}+{a_{2}}^{2}\), \(B=2a_{1}a_{2}\), and \(\phi=\theta_{1}-\theta_{2}\) is the phase difference between the light waves from the two points. After the object is loaded, the phase difference between the two points becomes \(\phi^{\prime}=\phi+\Delta\), where \(\Delta\) is the relative phase change, resulting from a relative deformation between points P1 and P2, between the light waves from two points before and after loading. The intensity of the speckle pattern after loading then becomes the following:  
\[I^{\prime}=A+B\cos\phi^{\prime}=A+B\cos(\phi+\Delta) \tag{3}\]  
A fringe pattern, or shearogram, which is shown in Figure 1, can be obtained by direct subtraction between the intensity images before and after loading. Because the intensity value must be positive, the absolute value will be displayed, represented as follows:  
\[|I_{s}|=\left|I^{\prime}-I\right|=B|\cos(\phi+\Delta)-\cos(\phi)| \tag{4}\]  
where \(\Delta=2n\pi(n=1,2,3\ldots)\). Dark fringes appear on the shearogram when \(|I_{s}|=0\). The shearogram is therefore displayed as a highly visible fringe pattern. Due to the relative phase difference between two adjacent fringes being equal to \(2\pi\), the phase at each fringe can be determined by counting the shearogram fringes. Then, the derivative of the deformation can be determined based on the relationship between the relative phase change and the relative deformation between points P1 and P2. For digital shearography NDT, the angle between the illumination direction of the laser and the observation direction of the CCD camera is usually small or close to zero, and the shearing amount \(\delta w\) is also small.