# Dynamic Measuring Method of Laser Beam Incident Angle for Laser Doppler Vibrometer
## 1 Introduction  
Doppler vibrometer technology was proposed in 1964 along with the fluid velocity measurement [1] and was gradually commercialized as a typical laser Doppler vibrometer (LDV). This technique measures the Doppler frequency shift caused by a light beam reflected by a vibrating object [2]. The Doppler frequency shift is proportional to the surface velocity of the target. It is not affected by the distance between the LDV and the target [3] (the distance will affect the received power of the reflected beam), so it is a convenient and non-contact method to measure the vibration velocity and displacement. At present, LDVs are widely used in various fields, such as the monitoring of bridge structures [4, 5], the oscillation amplitude of high-speed hard drives [6], defects in structural parts [7], fruit maturity [8], and angular displacement measurement [9], and have become adequate substitutes for traditional contact sensors. In these cases, the LDV measurements were not used directly for quantitative analysis; instead, features implicit in the measured signals, such as frequency, were explored. When the LDV values are used as an intermediate quantity of the final test result, it is necessary to calibrate the direction of the LDV laser beam. This is mainly because the direction of the laser beam determines the displacement/velocity component measured by the LDV. From the perspective of the measurement principle, LDV picks up the vibration response of a single point on the target surface in the direction of the laser beam (not necessarily equal to the vibration direction) [10]. If the laser beam is aligned with the vibration direction, the measurement is accurate; otherwise, the measurement willbe affected. Therefore, to accurately calculate the actual vibration response of the measured point, it is necessary to measure the incident angle of the laser beam accurately. However, affected by the precision error of the LDV itself, the processing error of the fixed structural parts, and the assembly error of the two, the direction of the laser beam may be different from the expected one, and the result will inevitably affect the subsequent measurement accuracy based on LDV values [1]. An excellent application of LDVs in the transportation industry is the Laser High-Speed Deflectometer [11], commonly referred to as the Traffic Speed Deflectometer (TSD) [12; 13; 14] or Laser Dynamic Deflectometer (LDD) [15; 16]. Multiple LDVs are installed on a stiff beam inside the trailer compartment of this detection instrument, which can efficiently measure the deformation velocity of the road surface under the rolling of the vehicle's weight in real-time [13; 17; 18] and back-calculate the deflection value based on the Euler-Bernoulli beam model. The practical results show that this measuring method is greatly affected by the incident angle of the LDV laser beam. [PERSON] et al. [19] pointed out that a deviation of 0.005\({}^{\circ}\) in the incident angle would cause a 25% error in the velocity measurement. Therefore, it is crucial to calibrate the incident angle of the LDV laser beam. As a successful case in the field of road detection, the manufacturers and users of laser high-speed deflectometers have proposed various ways to obtain the incident angle of LDV, but the inability to trace the source limits the timely promotion and application of this advanced technology. To obtain the accurate incident angle of the laser beam, the British Transport Research Laboratory, Denmark's Green Wood company, and [PERSON] et al. proposed the absolute calibration method [20], the ballast method [21], and the offset method [19], respectively. Although the above techniques can theoretically obtain the accurate incident angle of the LDV laser beam, it needs to embed the accelerometer, find the ideal long-distance rigid road surface, load and unload the counterweight, or lengthen the trailer body and other harsh conditions, and the implementation process is complicated. In addition, [PERSON] et al.