Mapping Distortion Correction in Null Testing of Sector-Shaped Off-Axis Aspheric Segments for Large Segmented Telescope

Segmented primary mirror technology overcomes the manufacturing limitations of large-aperture monolithic mirrors by assembling the primary mirror from multiple off-axis aspheric sub-mirrors. However, null testing of such off-axis aspheric surfaces with a compensator introduces mapping distortion, which prevents the interferometric data from directly guiding high-precision deterministic figuring. This paper proposes a mapping distortion correction method for sector-shaped off-axis aspheric sub-mirrors tested with a null lens. The method exploits the fact that one-dimensional radial distortion symmetry of the null lens about its distortion center. First, the distortion center coordinates are determined from the radial boundary data of the distorted interferogram of the sector-shaped mirror. Then, an error function is constructed based on the deviation of the measured outer-to-inner radius ratio of the sector-shaped mirror from its nominal value, and the radial distortion coefficients are iteratively solved using a radial distortion model refined by a sparse set of fiducial marks. Finally, the mapping distortion is corrected through an inverse mapping with solved distortion center coordinates and radial distortion coefficients. The proposed method was validated on a sector-shaped sub-mirror with outer and inner radii of 357.5 mm and 75.5 mm, reducing the outer-to-inner radius ratio error from 0.3833 to 0.0094, with a maximum fiducial coordinate deviation of 0.91 mm. Based on the distortion-corrected map, a single deterministic figuring run reduced the surface figure RMS from 0.034λ to 0.025λ (λ = 632.8 nm), demonstrating the effectiveness of the method.