# Micro Vibration Measurement with Microscopic Speckle Interferometry Based on Orthogonal Phase
## 4 Experiment
### Speckle Interference Experiment  
The measured object used in the experiment is the model 3325 silicon wheat produced by Goertek company, with the size being \(3\times 2\times 1.2\) mm\({}^{3}\) (\(L\times W\times H\)). After the light beam passes through the microscope objective lens, a circular spot with a diameter of 1.5 mm will be generated on its surface, as shown in Figure 6.  
Figure 5: The optical path structure diagram. Figure 6: (**a**) The measured object used for speckle interference; (**b**) The circular spot formed by the beam passing through the microscope objective; (**c**) The schematic diagram of magnified microscopic speckle. The speckle interference generated by the rough surface has the characteristics of random distribution, so the initial phase difference \(\varphi\) is randomly distributed in the range of (0-\(\pi\)). We need to find out the points that satisfies \(\varphi\) equal to \(\pi/2\). In our experiments, we found that when a \(6\times 6\) pixel window is used to search, we can get at least one pixel that satisfies the orthogonal condition among these 36 pixels. Combining the random distribution characteristics of speckle interference and cosine curve profile for analysis, it is reasonable to exist more than one pixel at the position of \(\varphi\) equal to \(\pi/2\) or very close to \(\pi/2\) in 36 pixels. Using the algorithm we have written in MATLAB, the light intensity curves of 36 points can be obtained. According to the distribution characteristics described above, the points that satisfy the orthogonal characteristic can be found out. If we cannot find any point that meets the condition, we need to further expand the search area. In order to make full use of the spatial resolution of the CCD, we flexibly choose different acquisition frame rates for different measured frequencies. Figure 7 shows images collected by the CCD at 50 frames/s. Figure 8 shows the light intensity curves obtained under the condition with the same amplitude (40 nm in the article) and different frequencies. The vibration frequencies of (a) and (b) are 0.5 Hz and 1 Hz, respectively. The acquisition frame rate of CCD is set to 50 frames/s, and the 200 frames shown in the figures correspond to 4 s. The vibration frequencies of (c) and (d) are 10 Hz and 20 Hz, respectively. At this time, the acquisition frame rate of CCD is set to 400 frames/s. In order to make the light intensity curves not too dense, we show the light intensity curves of 800 frames in 2 s.  
Then, we measure the vibrations with the same vibration frequency (10 Hz selected in the article) and different amplitudes, as shown in Figure 9. For the same reason mentioned above, we choose the light intensity curves of 800 frames in 2 s.  
Figure 7: The images captured by CCD. (**a**,**b**) are the independent imaging of the measured object and the reference object, respectively; (**c**) is one of the speckle interferograms formed by (**a**,**b**). Figure 8: The light intensity curves at different frequencies when the amplitude is 40 nm. (a) f = 0.5 Hz; (b) f = 1 Hz; (c) f = 10 Hz; (d) f = 20 Hz. Figure 9: The light intensity curves of different amplitudes when the frequency f is 10 Hz. (a) A = 10 nm; (b) A = 30 nm; (c) A = 50 nm; (d) A = 80 nm. In order to obtain the vibration amplitude, we need to calculate the values of I\({}_{1}\), I\({}_{2}\) and the peak value \(I_{peak}\). When frequency doubling phenomenon occurs, the maximum value I\({}_{\text{max}}\) or minimum value I\({}_{\text{min}}\) can be obtained. When the amplitude is too large, as shown in Figure 9d, we can read I\({}_{\text{max}}\) and I\({}_{\text{min}}\) simultaneously in one image. More commonly, I\({}_{\text{max}}\) and I\({}_{\text{min}}\) are read out, respectively, as shown in Figure 3b,h. Taking Figure 9d as an example, the average values of I\({}_{\text{max}}\) and I\({}_{\text{min}}\) are 160 and 12, and (I\({}_{\text{max}}+I_{\text{min}}\))/2 is 86.