Multi-dimensional error figuring model for ion beams in X-ray mirrors.

X-ray mirrors are core optical elements in synchrotron light sources, which are typically figured by ion beams. However, traditional ion beam figuring (IBF) methods cannot effectively correct both the height and 2D slope errors. Thus, what we believe to be a novel multi-dimensional error figuring model (MEFM) is proposed to integrate the height and 2D slope errors into a linear system of equations for simultaneous multi-dimensional error convergence. The weight values are determined based on specific manufacturing errors. The simulation results show that MEFM can achieve lower 2D slope and height errors than a single height or 2D slope errors in the figuring method. The power spectral density (PSD) curves before and after simulated figuring demonstrate that the low- and mid-frequency errors are corrected simultaneously. Figuring experiments on a 270 mm × 15 mm X-ray reflector showed that the height and 2D slope errors effectively converged. The results of the PSD curves indicate that MEFM is more effective than the traditional height figuring model in correcting mid-frequency errors, which further proves the validity of the solution model. This study provides an algorithmic reference for the IBF of ultra-high precision X-ray reflectors, thereby promoting the development of physics, medicine, materials, and other fields.