# Effect of Wheel Path in Raster Grinding on Surface Accuracy of an Off-Axis Parabolic Mirror
###### Abstract  
Off-axis parabolic mirrors have extensive applications in X-ray optics, with the precision of their curvature directly impacting grazing-incidence focusing performance. Notably, the off-axis parabolic surface has non-rotating and non-symmetrical characteristics. Ultra-precision raster grinding utilizing a diamond wheel is a common method. Crucially, establishing an optimal wheel path stands as the key to ensuring surface accuracy during off-axis paraboloid grinding. In this study, according to the double curvature property of an off-axis parabolic surface, two different wheel paths were compared: one tracing the meridian direction (parabolic generatrix) and the other following the arc vector direction (arc). The results showed that the wheel path in raster grinding stepping along the arc vector direction can obtain a smaller scallop height and higher surface accuracy. The surface accuracy of one step along the arc vector direction is 9.6 \(\upmu\)m, and that of the other step along the meridian direction is 32.6 \(\upmu\)m. A model of the scallop height was established based on the relative relationship between adjacent wheel paths, and the error is within 5%. According to the correlation between scallop height and shape error, we conducted an analysis of the spatial distribution of shape errors under varying wheel paths. The wheel path that steps along the arc vector is more suitable for raster grinding of the off-axis paraboloid. The above study can provide theoretical guidance for the wheel path planning of off-axis parabolic mirrors with high surface accuracy. X-ray optics; off-axis parabolic; raster grinding; scallop height; surface accuracy +
Footnote †: journal: Applied  
**Clinical:** [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Effect of Wheel Path in Raster Grinding on Surface Accuracy of an Off-Axis Parabolic Mirror. via a cutting experiment. [PERSON] et al. [16] used the freeform generation technique of a slow tool servo with a diamond grinding wheel (STS-DGW) previously proposed by us to manufacture free-form optics made of hard and brittle materials, established a theoretical model of the scallop height during biconical free-form grinding, and studied the effects of curvature changes on scallop height. [PERSON] et al. [17] carried out grinding of a monocrystal silicon rotary aspheric surface in parallel and cross paths, and analyzed the influence of grinding wheel machining attitude and different wear states of the grinding wheel on the surface quality and surface shape accuracy of the machined surface in two modes. [PERSON] et al. [18] studied the ultra-precision grinding of non-rotating asymmetric biconical free-form optics with a raster grinding path. Surface roughness down to Ra 8.4 nm was successfully achieved on the monocrystalline silicon biconical free-form optics surface via ultra-precision grinding with a vertical raster grinding path. To improve grinding accuracy and processing efficiency, many scholars have optimized and analyzed the wheel path based on the principle of scallop height generation. In the ultra-precision and high-efficiency grinding of large-diameter aspheric lenses, [PERSON] et al. [19] optimized and adjusted the raster step pitch under the premise of controlling the surface waviness after grinding, which improved the material removal efficiency in complex surface grinding. [PERSON] et al. [20] studied the wheel-path generation method for the multi-axis NC machining of sculptured surfaces, proposed a time-minimal wheel-path, and discussed local and global wheel-path generation, eliminating the influence of scallop height on machining accuracy. [PERSON] et al. [21] proposed a nonuniform rational basis spline (NURBS) interpolation method based on grinding wheel path planning. The characteristic parameters of the NURBS machining program were used to calculate the accuracy of curve interpolation path planning, and the grinding wheel path control points were optimized to effectively improve the grinding accuracy and efficiency. [PERSON] et al.