The Risley prism scanning system is a useful supplement to traditional rotating frame and mirror scanning systems. It features a compact structure, low optical loss, excellent dynamic performance, and a large scanning field of view, and has broad application prospects in lidars, laser communication, and laser guidance. In the practical applications of this system, it is important to select the scanning trajectory reasonably, which will directly affect the scanning efficiency of the system and the acquisition probability of the target. When the parameters and relative positions of the Risley prism are determined, the rotation velocity ratio of the Risley prism is variable and controllable to obtain the scanning trajectories of different shapes. We aim to study the relationship between the velocity ratio with the number of scanning points and petals, then summarize the internal rules of the velocity ratio and scanning trajectory, and evaluate the scanning time and coverage rate of the scanning trajectory under different velocity ratios. Therefore, our study has a guiding significance for selecting the scanning trajectory that meets the scanning efficiency requirements.

Firstly, the forward problem of the Risley prism is solved by the non-axial ray tracing algorithm, and the scanning trajectories under different velocity ratios can be obtained. Secondly, the number of scanning points is calculated according to the rotation velocity of the Risley prism and sampling interval, and the number of scanning petals is calculated according to the number of minimum points of the distance curve between scanning points and coordinate origin. Then, the velocity ratio is classified according to its absolute value and fractional part, and the formula for calculating the number of scanning petals by the velocity ratio is established. The scanning trajectory rules of the 2-element Risley prism are analyzed, and the scanning time and coverage rate under different velocity ratios are evaluated. Finally, the scanning trajectory of the 3-element Risley prism is regarded as the superposition and cancellation of the scanning trajectory of the 2-element Risley prism, and the scanning time and coverage rate can be evaluated according to the scanning trajectory rules of the 2-element Risley prism. Additionally, the condition for the 3-element Risley prism to obtain a regular symmetry scanning trajectory without a large scanning blind zone is proposed by analyzing the velocity ratio.

The scanning trajectory of 2-element Risley prism has the following rules (Table 1 and Fig. 4). When M is positive, the scanning trajectory is inner petal, and the trajectory is outer petal under negative M. When M is an integer, the scanning time under different velocity ratios is the same, and when M is a decimal, the scanning time under each type of velocity ratio is the same if the number of decimal places is the same. Under the different numbers of decimal places, the larger number of decimal places leads to longer scanning time. Therefore, the number of decimal places should not be too large. For each type of velocity ratio, when M is of the same sign, the larger |M| brings a larger coverage rate. The scanning trajectory of 3-element Risley prism has the following rules (Table 4 and Fig. 10): only when the scanning petals of 2-element Risley prism are doubled (1-2 times) with the velocity ratio of M₁ and M₂, and the scanning points are also doubled (1-2 times), the scanning trajectory of 3-element Risley prism is regular symmetry and has no large scanning blind zone. When M₁ and M₂ are both positive, the scanning trajectory is inner petal. When M₁ and M₂ are both negative or different signs, the scanning trajectory is the outer petal. Additionally, the scanning time and coverage rate of the 3-element Risley prism can be evaluated according to the scanning trajectory rules of the 2-element Risley prism.

As the scanning trajectory of the Risley prism determines the scanning efficiency of the system and the acquisition probability of the target, it is important to study the method of selecting the scanning trajectory by analyzing the velocity ratio. Based on the non-axial ray tracing algorithm, the forward problem of the Risley prism scanning system is solved. Then the petal-shaped scanning trajectories under different velocity ratios are obtained, and the number of scanning points and scanning petals are calculated, which is then adopted to summarize the rules between the scanning trajectory and velocity ratio. The scanning time and coverage rate of the scanning trajectory under different velocity ratios are evaluated. Meanwhile, the condition for the 3-element Risley prism to obtain a regular symmetry scanning trajectory without a large scanning blind zone is proposed. The obtained rules and conclusions can be employed to reasonably determine the velocity ratio in the practical applications of the Risley prism scanning system to select the scanning trajectory that meets the scanning efficiency requirements. However, the scanning trajectory of the Risley prism is sensitive to the velocity ratio, and there will be deviations between the actual and set velocity ratios in the rotation control. Therefore, the influence of such deviations on the scanning trajectory can be further explored.