Research on Stability of Removal Function in Figuring Process of Mandrel of X-Ray-Focusing Mirror with Variable Curvature

Over the past 30 years, researchers have developed X-ray-focusing telescopes by employing the principle of total reflection in thin metal films. The Wolter-I focusing mirror with variable-curvature surfaces demands high precision. However, there has been limited investigation into the removal mechanisms for variable-curvature X-ray mandrels, which are crucial for achieving the desired surface roughness and form accuracy, especially in reducing mid-spatial frequency (MSF) errors. It is essential to incorporate flexible control in deterministic small-tool polishing to improve the tool’s adaptability to curvature variations and achieve stable, Gaussian-like tool influence functions (TIFs). In this paper, we introduce a curvature-adaptive prediction model for compliance figuring, based on the Preston hypothesis, using a compliant shaping tool with high slurry absorption and retention capabilities. This model predicts the compliance figuring process of variable-curvature symmetrical mandrels for X-ray grazing incidence mirrors by utilizing planar tool influence functions. Initially, a variable-curvature pressure model was developed to account for the parabolic and hyperbolic optical surfaces’ curvature characteristics. By introducing time-varying removal functions for material removal, the model establishes a variable-curvature factor function, which correlates actual downward pressure with parameters such as contact radius and contact angle, thus linking the variable-curvature surface with a planar reference. Subsequently, through analysis of the residence time distribution across different TIF models, hierarchical filtering, and PSD distribution, real-time correction of the TIFs was achieved to enable customized variable-curvature polishing. Furthermore, by applying a time-varying deconvolution algorithm, multiple rounds of flexible polishing iterations were conducted on the mandrels of a rotationally symmetric variable-curvature optical component, and the experimental results demonstrate a significant improvement in form accuracy, surface quality, and the optical performance of the mirror.