# Digital Shearography for NDT: Phase Measurement Technique and Recent Developments
## 3 Phase Measurement Technique
### Phase Measurement Technique: Spatial Phase-Shift Shearography
#### 3.2.2 Carrier-Frequency Spatial Phase-Shift Shearography  
In addition to the multichannel spatial phase-shift shearography method introduced above, another phase extraction method suitable for dynamic measurements is the carrier-frequency spatial phase-shift shearography. Compared with the multichannel spatial phase-shift method, the carrier-frequency spatial phase-shift method can provide higher spatial resolution. A typical carrier-frequency spatial phase-shift setup is based on the Mach-Zehnder interferometer [20]. In this case, the carrier frequency is introduced by tilting a mirror in the interferometer as shown in Figure 6.  
Figure 5: Schematic of single-detector spatial phase-shift technique. After passing through the shearing device, the laser is divided into a sheared part and an unsheared part, which can be expressed as follows:  
\[ \begin{array}{l}u_{1}(x,y)=|u_{1}(x,y)|\exp[i\phi(x,y)]\\ u_{2}(x,y)=|u_{2}(x+\Delta x,y+\Delta y)|\exp[i\phi(x+\Delta x,y+\Delta y)-2 \pi i\cdot f_{c}\cdot x]\end{array}  \tag{11}\]  
where \(u_{1}\) and \(u_{2}\) are the components of the sheared and unsheared part, respectively, and \(\Delta x\) and \(\Delta y\) represent the shearing distances in the \(x\) and \(y\) directions, respectively. The carrier frequency component \(f_{c}\) introduced by tilting angle \(\beta\) can be expressed as follows:  
\[f_{c}=\frac{\sin\beta}{\lambda} \tag{12}\]  
The intensity recorded by the CCD is as follows:  
\[ \begin{array}{ll}I&=(u_{1}+u_{2})(u_{1}^{*}+u_{2}^{*})\\ &=u_{1}u_{1}^{*}+u_{1}u_{2}^{*}+u_{2}u_{1}^{*}+u_{2}u_{2}^{*}\end{array}  \tag{13}\]  
where \(*\) represents the complex conjugate of \(u_{i}\). To extract the phase, a Fourier transform is performed on the formula to convert it from the spatial domain to the Fourier domain as follows:  
\[ \begin{array}{ll}F(I)&=U_{1}(f_{x},f_{y})\otimes U_{1}^{*}(f_{x},f_{y})\\ &+U_{2}(f_{x}+f_{c},f_{y})\otimes U_{2}^{*}(f_{x}+f_{c},f_{y})\\ &+U_{1}(f_{x},f_{y})\otimes U_{2}^{*}(f_{x}+f_{c},f_{y})\\ &+U_{2}(f_{x}+f_{c},f_{y})\otimes U_{1}^{*}(f_{x},f_{y})\end{array}  \tag{14}\]  
where \(\otimes\) denotes the convolution operation. Figure 7 is a schematic diagram of the Fourier spectrum of the image (with shearing in the x direction). Corresponding to the formula, the four items can be divided into three spectra in the frequency domain, where the central spectrum is a low-frequency term \(F(u_{1}u_{1}^{*}+u_{1}u_{2}^{*})\), containing information from the background. The high-frequency terms \(F(u_{2}u_{1}^{*})\) and \(F(u_{1}u_{2}^{*})\) on both sides contain the phase information of the interferogram, which are located at the positions \((f_{c},0)\) and \((-f_{c},0)\) on the spectrum, respectively, and the width is determined by the size of the aperture (AP). Figure 6: Spatial phase-shift digital shearography (SPS-DS) based on Mach–Zehnder interferometer. By appropriately selecting an illumination angle, the three spectrums on the Fourier domain can be separated. By appropriately selecting the windowed inverse Fourier transform (WIFT), the phase distribution can be calculated as follows:  
\[\phi+2\pi\cdot f_{\mathrm{c}}\cdot x=\arctan\frac{\mathrm{Im}[F(u_{1}u_{2}^{*})]} {\mathrm{Re}[F(u_{1}u_{2}^{*})]} \tag{15}\]  
where Im and Re denote the imaginary and real parts of the complex number, respectively. The relative phase difference due to deformation can be obtained after loading using a similar process. The carrier-frequency spatial phase-shift method can evaluate the phase map from a single pair of speckle images obtained before and after loading. The acquisition rate is no longer limited by the phase-shift steps but only by the camera function, thereby achieving dynamic measurements with high speed. However, there are still limitations, such as small measurement size, poor phase map quality, and the shearing and carrier-frequency both being controlled by tilting mirrors.