## 3 Results  
The experiments in this study have been designed to verify the DSPI. A 532 nm laser (Changchun New Industries Optoelectronics Technology Co., Ltd., continuous wave, 200 mW) is used as the laser source. A high-resolution camera (Basler ace, acA1600-20 um, 4.4 um \(\times\) 4.4 um pixel size, 1626 pixels \(\times\) 1236 pixels) is used to record the speckle pattern images. The focal length of the imaging lens is 150 mm. The angle between each wave vector of observation direction and the z-axis is 4\({}^{\circ}\). The diameter of each aperture stop is 4 mm, and the distance of two aperture stops, through which light interferes, is 8 mm. This setting of aperture diameter and distance makes the different terms, shown in Figure 4, separates from each other. If the aperture is too large, or if the distance is too close, the different terms will alias. If the aperture is too small, the amount of light passing through will also be small, and the speckle noise will be large. By choosing an aperture size and distance appropriately, the different terms are evenly distributed to achieve better results. The object under test is a metal plate with a diameter of 130 mm, which can be precisely rotated in-plane by a micro-head, as illustrated in Figure 5. In-plane displacement is produced by the rigid-body rotation. A magnified portion of the recorded speckle pattern image is displayed in Figure 6, which reveals the two orthogonal spatial carrier fringes within the speckle. Figure 5: Object capable of rotating in-plane by a micro-head. Figure 6: Enlarged view of the speckle pattern obtained using DSPI. Figure 7 shows the spectrum distribution of the speckle pattern image after Fourier transformation, which is similar to that obtained theoretically as illustrated in Figure 4. The spectrum shown in Figure 7 is the logarithm of the original spectral intensity to make the figure clearer. The different terms, schematically presented in Figure 4, can be clearly separated in Figure 7. The two terms marked by a red circle and labeled with A and B are corresponding to \(U_{x1}\otimes U_{x2}^{*}\), and \(U_{y1}\otimes U_{y2}^{*}\), respectively. According to Equation (3) and Figure 4, the lower terms should be the conjugate of the upper ones. The negative displacement would be determined form the lower terms. For both interferograms before and after rotation, by selecting the terms A and B, shown in Figure 7, and applying a windowed inverse Fourier transformation to each term, phase differences \(\Delta_{x}\) and \(\Delta_{y}\), according to Equation (6), can be extracted. Only two interferograms are needed to evaluate the phase by using spatial carrier phase-shift technique. The phase map is smoothed by a low-pass filter algorithm as displayed in Figure 8(a1,a2). After phase unwrapping, the in-plane displacement in two directions, calculated using Equation (8), is displayed in Figure 8(b1,b2). Figure 8: The smoothed phase and in-plane displacement extracted by the DSPI: **(a1)** and **(a2)** are the phase differences \(\Delta_{x}\), and \(\Delta_{y}\), respectively; **(b1)** and **(b2)** are in-plane displacements in the x-axis direction and y-axis direction respectively; **(c1)** and **(c2)** are in-plane displacements errors in the x-axis direction, and y-axis direction, relative to the fitted value, respectively. Figure 7: Spectrum of the recorded interferogram obtained by Fourier transformation. Generally speaking, the measurement accuracy and sensitivity of DSPI are mainly determined by the phase map quality and number of times the images are smoothed. Speckle interference is usually able to reach a measuring sensitivity of about 2\(\pi\)/20 to 2\(\pi\)/10 (2\(\pi\) corresponds one fringe) and a measuring accuracy of about 2\(\pi\)/5 to 2\(\pi\)/2. A detailed discussion regarding this topic can be found in reference [33]. In this research, the in-plane displacement is produced by the rigid-body rotation, which follows certain rules demonstrated by [PERSON] [7].