Evaluation and prediction model of surface slope error for an off-axis aspheric mirror based on ultra-precision envelope grinding

Off-axis aspheric mirrors are critical components in focusing optical systems, where surface slope errors can significantly degrade imaging resolution, necessitating precise control. Envelope grinding is a common method for machining brittle-material optical mirrors. To investigate slope error formation mechanisms, this study developed a surface profile prediction algorithm based on discretized grid modeling and iterative cross-sectional layering. Slope errors were evaluated by analyzing sagittal height deviations between the simulated surface and ideal profiles. Grinding experiments were conducted to analyze the spatial distribution patterns of slope errors under three envelope grinding path strategies including constant step, constant arc-length, and constant scallop-height. Experimental results revealed that subtle differences in residual regions induced distinct spatial error distributions. Constant step grinding concentrated errors near the central region, constant arc-length grinding distributed errors bilaterally adjacent to the center, and constant scallop-height grinding localized errors at the peripheral edges. The algorithm achieved high prediction accuracy with experimental observations. Under identical parameters, constant step grinding achieved the smallest slope error with RMS value of 25.3 arcsec. These findings provide valuable insights for grinding path optimization and slope error mitigation in off-axis aspheric mirror manufacturing.