# Inverse Solution Error Analysis and Correction of Beam Steering System Based on Risley Prisms
## 1 Introduction  
Risley prisms are a widely used beam-steering mechanisms, with their compact structure, high precision, low moment of inertia, and low cost [1, 2], they are often used as beam-pointing devices and beam-scanning devices in laser communication [3], large-FOV imaging [4], photoelectric tracking [5], lidar [6], laser scanning [7], and other fields. Risley prisms consist of a pair of optical wedges that rotate around a common axis independently, which enables high-resolution arbitrary beam-pointing over a wide range. The features and advantages of Risley prisms make them suitable as a coarse or fine pointing mechanism for laser communication acquisition, tracking, and pointing (ATP) systems, especially for inter-satellite and airborne laser communication. Figure 1 shows the optical layout of the laser communication system based on the Risley prism's coarse tracking mechanism. The accuracy of the beam-steering mechanism of the ATP system is one of the key factors affecting the system's performance. The pointing accuracy of the Risley prism is mainly determined by two factors: the accuracy of the solution method and the impact of the error on the solution. The former refers to the precision of solving the mathematical relationship between the rotation angle of prisms \(\theta_{1},\theta_{2}\) and the spatial orientation of the outgoing beam \(\Phi,\Theta\). The definition of the prism rotation angle \(\theta_{1},\theta_{2}\) and the deflection angle and azimuth of the outgoing beam \(\Phi,\Theta\) are shown in Figure 2. The trapezoid formed by the dotted line in the figure is the principal section of the prism, and the red arrows represent the light rays. The solution consists of two aspects: solving the direction of the emitted light according to the prism's rotation angles, called the forward solution, and solving the prism's rotation angles with a known direction of the emitted beam, called the inverse solution. [PERSON] used the vector form of Snell's law to perform nonparaxial ray tracing and gave an exact forward analytical solution for the first time [8]. For the inverse solution, the two-step method [9], table look-up method, third-order approximate expression solution [10], damped least-squares iterative solution [11], and forward iterative optimization solution [12] have been proposed. The systematic errors of Risley prisms also impact the accuracy of the forward and inverse solution. The systematic error refers to the deviations between the actual and theoretical parameters of the Risley prisms system, which usually has a greater impact than the solution errors. For the impact of systematic errors on the performance and forward solution accuracy, a lot of work has been carried out. [PERSON] investigated the optical distortions of scan patterns caused by the component errors and alignment errors of prisms [13]. [PERSON] et al. analyzed the influence of component errors, prism orientation errors, and assembly errors on pointing accuracy. The allowances of the error sources for a given pointing accuracy were evaluated [14]. Figure 1: Optical layout of a laser communication system with a Risley prism coarse tracking mechanism. Figure 2: Schematic diagram illustrating the coordinate systems for Risley prisms. The incident ray, rotation axis of prisms, and \(Z\)-axis are collinear. There are relatively few studies focusing on compensating for the impacts of system errors. [PERSON] et al. proposed an error correction method based on the paraxial approximation model [15]. Under the paraxial approximation model, an error vector was added to compensate for the system error. The method model is simple, easy to implement, and considers the influence of the thickness and spacing of the prism on the outgoing beam. However, the solutions obtained by the paraxial approximation method will deviate from the exact solution, especially when the deflection angle is large [16]. So, it is not suitable for large-angle deflection applications. [PERSON] et al. proposed an error parameter identification method [17].