# Dynamic Measuring Method of Laser Beam Incident Angle for Laser Doppler Vibrometer
## 3 Static Measuring Method of Incident Angle
### Static Measuring Device and Results  
The experimental setup for static measurement is shown in Figure 4. A high-precision air-floating marble platform provides horizontal support for the entire practical device. A coordinate system _o-xyz_ was established on the marble plane, and the marble plane was on the _oxy_ plane. A piece of coordinate paper (120 mm \(\times\) 170 mm) was pasted on this plane. One LDV (Model: Polytec OFV 505) was mounted on a sliding table perpendicular to the marble table and moved in the z-direction. The incident angle of the LDV laser beam was adjusted so that the included angle \(\beta\) with the \(z\)-axis was about 2\({}^{\circ}\). According to Equation (3), the measuring process of angle \(\beta\) was transformed into the accurate measurement of two length parameters: \(d\) and \(l\). Their operation steps and calculation methods are as follows:  
Step 1. Adjust the focal length of the LDV to minimize the spot of the laser beam on the coordinate paper, and record the positions of the laser spot and the LDV, i.e., A and C;  
Step 2. Move the sliding table in the \(z\) direction to change the distance between the coordinate paper and the LDV;  
Figure 4: Static measuring experimental device, (**a**) Main view; (**b**) LDV at position C; (**c**) LDV at position C\({}^{\prime}\); (**d**) High-precision digital display handle; (**e**) Coordinate paper and image recording equipment. Figure 3: Schematic diagram of the principle of angle static measuring method. (**a**) View \(1\), (**b**) View \(2\). Step 3. Adjust the focal length of the LDV again, and record the positions of the laser spot and the LDV, i.e., A\({}^{\prime}\) and C\({}^{\prime}\);  
Step 4. Calculated distance \(d\) and \(l\), respectively, and then calculate angle \(\beta\) according to Equation (3). On the one hand, the actual displacement \(l\) of the LDV moving along the \(z\)-axis is calculated according to Equation (4). Parameters \(z_{1}\) and \(z_{2}\) represent the corresponding coordinates of the LDV at positions C and C\({}^{\prime}\), respectively. They are provided to the digital display handle by the steel tape raster gauge (position accuracy: 0.001 mm) installed on the side of the marble platform, as shown in Figure 4d. \[l=z_{2}-z_{1} \tag{4}\]  
On the other hand, this paper introduces the image recognition method to measure the distance \(d\) accurately, that is, to locate points A and A\({}^{\prime}\). The coordinate paper containing the laser spot A (or A\({}^{\prime}\)) was photographed using an iPhone 12 (system version: 14.5); see Figure 4e. The camera system was fixed on the horizontal plane, and the coordinate paper had no relative movement. This process can ensure that the view fields of multiple images are consistent, thereby reducing image differences. Subsequently, pixel positions (coordinates) of the laser spots, A and A\({}^{\prime}\), on the two images were identified, and their centers of gravity were, respectively, recorded as (\(x_{1}\), \(y_{1}\)) and (\(x_{2}\), \(y_{2}\)), as shown in Figure 5. The pixel distance \(d_{\rm pix}\) between points A and A\({}^{\prime}\) was calculated using Equation (5). \[d_{\rm pix}=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}} \tag{5}\]  
However, the pixel distance of the image is not equal to the actual physical distance. To improve the accuracy of the measuring results, a physical distance of 10 mm on the coordinate paper was selected for pixel calibration to determine the pixel value \(d\)p corresponding to each millimeter distance on the coordinate paper. Assuming that the total number of pixels corresponding to 10 mm is \(P\), \(d\)p pixels in the image correspond to a physical distance of 1 mm on the coordinate paper, \(d\)p = \(P\)/10. In this calibration process, the mean value of 5 measurements were selected as the actual value of \(P\). Finally, in these images, the pixel point corresponding to 10 mm was \(P\) = 202.5, and \(d\)p = 20.25.