Objective Internal temperature drift, mechanical structure deformation, thermal effects of optical components, and other factors lead to misalignment of the emitted laser beam, causing drift of the spot on the target surface (Fig.1) and affecting laser applications. In order to suppress this phenomenon, it is necessary to establish a beam pointing control system. Currently, there are primarily two approaches of reflective approach using fast steering mirrors and transmission approach using rotating prisms. Control system with single fast steering mirror can correct the angle deviation of beam and the deflection angle is easy to calculate. However, it cannot address the coupling misalignment between the position and angle of the beam. Control system with dual fast steering mirror can simultaneously correct the position and angle deviation of the beam. However, it introduces coupling issues and makes it difficult to find suitable deflection angles. Currently, there have been numerous theoretical analyses on the reverse angle problem of prism rotation in the transmission approach, while the coupling theory of the dual fast steering mirrors remains unclear, and the linear fitting of mirror deflection angle and spot position offset lacks theoretical support. Therefore, it is necessary to explore in depth the inherent relationship between beam pointing and the deflection angles of dual fast steering mirrors and find a rational method to calculate the deflection angles.

Methods The dual fast steering mirror beam pointing control system consists of two branched optical paths (Fig.2). Using ray tracing, a theoretical model of this system has been established (Tab.1). This model reflects the mathematical relationship between the beams and the deflection angles of the dual fast steering mirrors. Based on this model, a simulation environment can be created. By analyzing this theoretical model, it is observed that the complex coupling relationship between the deflection angles of the fast steering mirrors and the spot position deviation can be approximated as a linear relationship under small deflection angles. A fast calculation method for the deflection angles of the dual fast steering mirrors is proposed. Firstly, a data set of spot position deviations and deflection angles is collected using an iterative convergence strategy (Fig.5). Then, a shallow neural network is trained based on the historical data accumulated from iterative convergence strategy (Fig.8). Finally, the trained neural network is used to quickly determine the output deflection angles of the dual fast steering mirrors based on the input detector's spot position deviation. The data collection and neural network training processes of the iterative convergence strategy can be performed offline, without increasing the time required for beam pointing control.

Results and Discussions  The experimental results in the simulation environment demonstrate that the proposed iterative convergence strategy effectively solves the deflection angles for beam pointing control (Fig.6-7), with an average iteration step of 9.09. The fast calculation method based on shallow neural networks establishes a direct mapping between spot position deviation and deflection angles, and the result can be obtained after a single computation. The experimental results in the simulation environment show that compared to the beam state before control, the position deviation in the X and Z directions is reduced by 99.32% and 99.46% respectively, the angle deviation is reduced by 99.07% and 98.98% with the average comprehensive deviation being reduced by 99.16% (Tab.2). This method effectively suppresses the original beam misalignment. The shallow neural network only requires one-step solving process, eliminating the need for multiple iterations and greatly improving the calculation speed.

Conclusions After the derivation and analysis of the theoretical model of the dual fast steering mirror beam pointing control system, the coupling phenomenon is explained, and it is demonstrated that there exists an approximate linear relationship between the deflection angle of the fast steering mirrors and the spot displacement under small angle conditions. The simulation experimental results show that with the proposed fast calculation method the deflection angles of dual fast steering mirrors for beam pointing control can be solved quickly and effectively. In engineering applications, a simulation model can be constructed based on the actual optical path parameters and form a simulation dataset. Subsequently, iterative convergence control can be performed in the actual optical path to form a real dataset. Integrated learning, transfer learning, and model fusion can be performed based on both datasets to reduce the requirement for a large training dataset for shallow neural networks.