Temporal phase-shift shearography is generally only applicable to static measurements. Despite the fast temporal phase-shift algorithms that have been developed, the phase map quality and measurement range are limited and not suitable for many applications. Therefore, spatial phase-shift digital shearography (SPS-DS), in which phase-shift interferograms are obtained in different spatial positions at the same time to realize dynamic shearography measurements, was proposed. This section will describe spatial phase-shift shearography by introducing multichannel spatial phase-shift shearography and carrier-frequency spatial phase-shift shearography, respectively.

#### 3.2.1. Multichannel Spatial Phase-Shift Shearography

Multichannel spatial phase-shift shearography can be divided into the multidetector method and the single-detector method. Among them, the multidetector method generally uses three or more CCD cameras to simultaneously acquire a speckle interferogram with a known fixed phase difference [

28]. Although this method can perform dynamic measurements, the system is complicated, has a large error, and the setting of the phase shift between multiple cameras is difficult to implement, making the method unsuitable for practical applications. A method of obtaining spatial phase shift using only a single CCD camera was later proposed [

29]. One method of doing this is to divide a single CCD into at least three imaging regions, then introduce a fixed phase difference between each region. This method requires the use of a large number of wave plates, gratings, and polarizers, which is costly and difficult to adjust. Another method uses a single pixel as a channel to introduce a phase difference between pixels covered by the same speckle, resulting in a spatial phase shift [

30].

Figure 5 is a schematic of the method. The interpixel intensity is calculated to determine the phase value, which in turn obtains the phase value of the entire image.

By setting the reference beam at an appropriate angle

$\alpha $, the optical path difference (OPD) between two adjacent pixels can be created as follows:

where

$S$ is the pixel size. If the laser wavelength is

$\lambda $, the phase shift can be calculated by the following formula:

Thus, for a given wavelength and pixel size, the phase difference can be determined by setting the appropriate reference beam angle. Assuming that the optical path difference by adjusting the angle $\alpha $ is $\lambda /4$ and the phase difference between pixels is $\pi /2$, then the phase difference can be calculated using Equations (8) and (9). After loading, the same process can be used to calculate the phase distribution under loading conditions.

The adjacent pixel phase-shift method can obtain the phase distribution from a single speckle pattern and is suitable for dynamic measurement. However, in practical applications, this method has reduced spatial resolution due to its need for at least three pixel points to determine the phase distribution of one point. Another problem with this method is that these three or more pixels should have the same values for A and B as shown in Equations (3) and (4). Although it can be partly realized by adjusting the aperture of the camera, it is not highly accurate, resulting in a noisy phase map. Moreover, only one dimension can be measured in one speckle pattern. Therefore, a carrier frequency method was proposed to enhance the quality of the phase map and the ability to extract information from multiple dimensions in a single image.

#### 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.

After passing through the shearing device, the laser is divided into a sheared part and an unsheared part, which can be expressed as follows:

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:

The intensity recorded by the CCD is as follows:

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:

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).

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:

where

$\mathrm{Im}$ and

$\mathrm{Re}$ 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. Spatial phase-shift systems that introduce carrier frequencies with other carriers, such as a double-aperture mask and a Michelson interferometer, have been developed to solve these problems. In 2006, Bhaduri et al. proposed a digital shearography system using a double-aperture mask as a carrier to introduce the carrier frequency [

31,

32]. Compared to the system based on the Mach–Zehnder interferometer, the double-aperture-based system provides better phase map quality, comparable to that of the TPS-DS system. However, similar to the Mach–Zehnder system, the measurement size of the double-aperture-based system is also small due to the use of a collimated light source and dual apertures. The stability of the entire measurement system is also reduced due to its complex optical path, which also limits its application in the laboratory and industry.

Both the Mach–Zehnder-based SPS-DS system and double-aperture SPS-DS system have the disadvantage of limited measurement area. A Michelson-based SPS-DS system also exists, which can be used to obtain a larger measurement area by embedding the 4f system [

33,

34]. The SPS-DS system with a 4f system is shown in

Figure 8a. The imaging range for this system is no longer affected by the beam splitting prism but only depends on the lens and the camera.

Figure 8b,c show the field of view of the conventional shearography device and the 4f system. At the same measuring distance, the measuring area of the conventional device only covers the framed portion of the 4f system measurement result.

The Michelson-based system has a simpler setup than the Mach–Zehnder and double-aperture systems, so it is relatively insensitive to external disturbances. Additionally, it is easy to change the shearing amount and shearing direction, making it the most widely used shearing device. However, the Michelson-based system typically requires a large shearing amount to enable the two side spectra to be separated from the center spectrum, while the Mach–Zehnder-based setup can easily be performed with a small or large shearing amount. In order to solve this problem, a special slit aperture can be used instead of the usual circular aperture [

35]. Under the same shearing amount and aperture size, the slit aperture provides a larger spectral region containing the phase information. Experiments have shown that under the same experimental conditions, using the slit aperture requires a lower shearing amount, and the side spectral area after Fourier transform is much larger than that obtained using the circular aperture. This reduces the shearing amount required for the Michelson-based system and improves the quality of the phase map.

Figure 9 shows the results of an application of the Michelson-based SPS-DS system for NDT of delaminations/disbonds of a composite plate. All images were captured during dynamic loading. The settings of the camera parameters, such as exposure time, shutter mode, and resolution, should be noted. For dynamic measurements, a short exposure time and global shutter should be chosen for a clear image.

Based on the above introduction, the two methods are summarized in

Table 2.

In recent years, some new phase extraction algorithms and optimizations have been proposed [

36,

37,

38,

39]. In 2014, Albertazzi et al. proposed an alternative approach for retrieving phase information from a sequence of images with unknown phase shifts [

40]. The approach combines data from several good modulation points on the image to form an N-dimensional Lissajous ellipsis. An overall phase-shift value is determined from the fitted ellipsis for each interferogram, then the phase-shifted values are used to determine the phase value for each image pixel. In 2015, Xie et al. provided a review of recent developments of spatial phase-shift digital shearography and specifically introduced the problem of improving the phase map quality of spatial phase-shift digital shearography [

41]. In 2018, Aranchuk et al. developed a pulsed digital shearography system that utilizes the spatial phase-shifting technique, which allows for instantaneous phase measurements [

42]. In 2018, Kirkove et al. applied a method that combines the time-averaging and phase-shifting techniques, which improved the contrast and resolution compared to traditional time averaging [

43] and has proven to be particularly useful in vibration testing performed under industrial conditions.