Based on the rotational Doppler effect (RDE), the vortex beam carrying orbital angular momentum (OAM) can be sensitive to the rotation movement along the normal direction, so the vortex beam is widely used for measuring the rotational speed. At present, the commonly used beam in RDE research is the Laguerre-Gaussian (LG) beam. However, most studies focus on the relationship between the azimuthal index in the vortex beam and OAM, frequency shift, and other physical quantities. Few researchers have paid attention to the radial index. Compared with Bessel beams, the LG beam with a high radial index shows better self-healing. Under equal conditions, the LG beam with a high radial index shows a certain degree of nondiffracting characteristics, and the beam quality is better during propagation. Although the influence of the non-zero radial index on RDE has been explored in previous literature, it is only limited to the case where the vortex beam coincides with the rotational center of the object. In practical noncooperative measurement, it is more common that the LG beam does not coincide with the object's rotating axis. Therefore, this research mainly explores the influence of the LG beam's radial index on the rotational Doppler signal under the misaligned condition. It is found that different from the alignment condition, although the radial index of the LG beam can enhance the amplitude of the rotational Doppler signal under the condition of lateral misalignment, which is helpful for rotational speed extraction, the relationship is not linear. With the increase in the radial index, the amplitude of the frequency signal increases first and then decreases. There is an optimal radial index, which maximizes the signal amplitude. This discovery expands the practical application range of RDE and helps to select the appropriate radial index, so as to improve the detection range under the condition of lateral misalignment. Therefore, it is of great significance for reference.

In the study, LG beams with different radial indexes are used as the detection light source. First, the physical meaning of the radial index is described, and the influence of the radial index on the intensity distribution of the LG beam is analyzed. Combined with the small scattering model, we have theoretically analyzed the influence of radial index on the rotational Doppler signal under the misalignment condition and the reason. Second, a proof-of-concept experiment is designed for rotational speed detection by using different radial indexes. The experiment shows that the radial index affects the accuracy of target speed measurement and the amplitude of the frequency signal when the LG beam axis is not coaxial with the rotation axis of a spinning object. The experimental results are in good agreement with the theory. Third, different conditions are set to verify the generality and correctness of the rule through repeated experiments.

By comparing the experimental results (Fig. 4), it is found that the average amplitude of the rotational Doppler signal first increases and then decreases with the increase in the radial index. When the radial index is 1 or 2, the effective average frequency amplitude is maximum. This is because, in the beginning, the increase in the radial index makes the number of concentric vortices increase. In addition, the beam intensity gradually increases, and the effective reflected signal and the frequency amplitude rise. Then, with the increase in the radial index, the energy of the central optical ring is concentrated, but the radius of the innermost ring gradually decreases. On the basis of the small scattering model, the radius decreases obviously, the rotational Doppler spectrum gradually broadens, and the average frequency amplitude decreases. After that, we change the topological charge of the vortex beam and do several repeated experiments. The measurement results (Fig. 6) have shown the same rule and proved the generality and correctness of the rule.

This study analyzes the influence of the radial index in the LG beam on the rotational Doppler signal under the condition that the object rotation axis is misaligned with the vortex beam axis. The experimental results show that with the increase in the radial index, the average signal frequency amplitude increases first and then decreases. Therefore, there is an optimal radial index to maximize the signal amplitude. Under the condition that the surface material of the rotating object is tinfoil, the average frequency signal is maximum when the radial index is 1 or 2. It is worth noting that when the influence of the radial index is considered, the material and shape distribution on the surface of the spinning object cannot be ignored. The optimal radial index for different materials has its particularity, which needs to be specifically considered when the rotational speed of the object based on RDE is measured. This research provides a reference for improving the rotational speed measurement ability in practical engineering, expands the practical application range of RDE, and contributes to selecting the appropriate radial index to increase the detection range of the spinning object. Therefore, it is of great reference significance for the engineering of rotational speed detection of spinning objects.