Since the Hubble space telescope (HST) was successfully launched in 1990, wide field of view (FOV) and high-resolution sky surveys have become a hotspot in the development of space telescopes under the guidance of the research on dark matter, dark energy, and gravity theory. In large space telescopes with wide FOV, such as the James Webb space telescope (JWST) and the ultraviolet/optical/infrared surveyor (LUVOIR), fast steering mirrors (FSMs) are usually employed in optical systems as the execution mechanism of high-accuracy image stabilization due to their small inertia moment, high positioning accuracy, high bandwidth, and fast response speed. Finally, the vibration and pointing errors reduced by the attitude controlling system of the satellite platform are compensated for additionally to adjust the transient location of the image on the focal plane. However, due to the FSMs located near the exit pupil and in a convergent optical path, the tip-tilt process will also cause the inclination between the optical focal plane and the detector plane, resulting in projection distortion effects. As a result, the motion of the image points caused by the tip-tilt of the FSM at different FOV angles will lose the synchronization, while an additional distortion has been involved, which means decreasing image stabilization precision. Therefore, we theoretically analyze the processing mechanism and influencing factors of the projection distortion effects to determine mitigating methods during the telescope design and further improve the image stabilization accuracy in precise image stabilization.

Our analyses are based on the imaging principle of geometric optics. The image point displacement at different FOVs caused by the tip-tilt of FSM is analytically modeled by adopting the ray-tracing method, and the effects of shift, rotation, and projection distortion are demonstrated respectively. Meanwhile, all these parameters involved in the analytical model are determined to be the same coordinate system (focal plane coordinate system) by employing the homogeneous coordinate transformation method. Then the image motion calculation results of this model are compared to those given by CODE V by taking the parameters of the China space station telescope (CSST) as an example. Finally, the projection distortion effects are separated from the image motion caused by the tip-tilt of FSMs, and possible factors that may influence the projection distortion effects are analyzed, including the FOV of the optical system, motion range of the FSM, location of the FSM, and incident angle on the focal plane.

The projection distortion effects are caused by different directions of the principal rays incident on FSMs in different FOVs, and this is reflected in the analytical model for image motion. The analytical model for image motion is demonstrated to be accurate enough in most telescope conditions, which only shows errors less than 0.01 μm compared to the results given by CODE V (Table 1). Thanks to this model, the FOVs (Fig. 4), motion range of the FSM (Fig. 5), and incident angle on the focal plane (Fig. 8) are identified as the major factors in the projection distortion effects, and the projection distortion effects increase with the rising factors, while the locations of the FSM have few relations, including the distance from the center of rotation of the FSM to the center of the FOV of the focal plane (Fig. 6), and the initial pitch and azimuth angles of FSMs (Fig. 7). Since the FOVs are determined according to the astronomical observation requirements, it is impossible to reduce the projection distortion effects by decreasing the FOV size. However, we can reduce the projection distortion effects by reducing the motion range of the FSM and incident angle on the focal plane.

We analyze the projection distortion effects caused by the tip-tilt of the FSM. The calculation accuracy of the built analytical model for the image motion can meet the practical application needs. The analytical model for the image motion and the analysis results for the influencing factors of projection distortion effects will provide valuable references for designing image stabilization systems of space telescopes. The most effective ways to reduce the projection distortion effects caused by the FSM are lowering the tip-tilt range of the FSM by improving the accuracy of the first-stage pointing control and reducing the incident angles on the focal plane.