A Combined Alignment Technique for Mid-spatial Frequency Error Suppression in Ion Beam Figuring of Aspheric Mirrors

The deterministic correction of mid-spatial-frequency (MSF) errors on complex aspheric surfaces is highly sensitive to coordinate consistency between the target surface error and the actual material-removal position. This study investigates how lateral deviations along the complete “removal function–ion-beam machine–workpiece–metrology data” coordinate chain affect the MSF correction capability of ion beam figuring (IBF), and proposes a full-chain composite coordinate alignment strategy for small-beam IBF correction of MSF errors. A quantitative relationship among the finite Gaussian-beam response, alignment-induced spatial phase mismatch, and residual surface error is established through two-dimensional frequency-domain decomposition and convolution analysis. The workflow uses the four-point method to calibrate the actual removal function and its center, combines a Faraday cup with a touch-trigger probe to transfer the physical datum, and uses in-situ ion-beam fiducials to correct the coordinate mapping between the metrology data and the workpiece. Comparative experiments on two ULE annular aspheric mirrors with similar initial surface errors show that, after three iterations, the conventional workflow reduced the surface error from 12.972 to 7.173 nm RMS, with an overall convergence ratio of 1.81, whereas the composite alignment workflow reduced it from 13.166 to 2.736 nm RMS, with an overall convergence ratio of 4.81. PSD analysis shows relatively stable error attenuation in the 0.1–0.4 mm⁻¹ band for the composite alignment group, and the model-predicted effective cutoff frequency agrees reasonably with the experimental result. The strategy helps preserve deterministic convergence of small-beam IBF and provides a quantitative basis for coordinate management and error-propagation assessment in IBF of complex aspheric surfaces.