Control of optical surface gradient error via directional matching of a non-rotationally symmetric removal function

Surface gradient RMS error (GRMS) is a key factor limiting the beam quality of high-power laser systems, yet conventional height-RMS-based deterministic figuring often yields inefficient GRMS convergence. This difficulty arises from the vectorial and frequency-weighted nature of gradient error. Accordingly, a gradient-error control method based on directional matching of a non-rotationally symmetric removal function is proposed. Principal component analysis is used to identify the dominant gradient direction, and the machining azimuth is selected to align the strongest removal direction with it. A coupled gradient-height objective is further introduced for dwell-time optimization. Controlled simulations on the same fused-silica surface achieve a GRMS convergence ratio of 8.02, compared with 3.01 for height-based figuring. In a two-workpiece engineering comparison, the corresponding experimental ratios are 2.51 and 1.43, respectively, providing experimental evidence for the practical feasibility of the proposed framework under realistic polishing conditions.