An algorithm for solving the problem of local shape errors of optical surfaces correcting by a small-sized ion beam, based on the matrix representation is presented. The algorithm involves searching for points that rise relative to the average height on the surface, where exposure to an ion beam of a given shape will lead to a decrease in the RMS deviation of the surface shape from the calculated one. It is shown that the new approach makes it possible to significantly expand the range of spatial frequencies, in which the height of relief can be reduced using the given ion beam size, and provides better results than the method of minimizing the functional of convolution of the ion beam and the surface map. Moreover, the new approach does not lead to appearance of shape errors (concentric structure) defined by the ion beam size and scanning step. An experiment for minimizing of local surface shape errors based on an etching map calculated using new approach was done. Matrix algorithm allowed in one procedure to reduce surface shape errors for a concave spherical optical element made of fused silica with a diameter D = 100 mm and a radius of curvature R = −137.5 mm in the RMS by more than 3 times. The initial RMS was 4.5 nm, RMS after treatment is 1.36 nm.

提出了一种基于矩阵表示的小尺寸离子束校正光学表面局部形状误差的算法。该算法涉及搜索相对于表面平均高度上升的点，在该点上暴露于给定形状的离子束将导致表面形状的RMS偏差与所计算出的减小。结果表明，这种新方法可以大大扩展空间频率的范围，其中使用给定的离子束尺寸可以减小浮雕的高度，并且与使波峰的卷积函数最小化的方法相比，可以提供更好的结果。离子束和表面图。此外，新方法不会导致出现由离子束大小和扫描步骤所定义的形状误差（同心结构）。基于使用新方法计算出的蚀刻图，进行了使局部表面形状误差最小化的实验。矩阵算法允许通过一种程序来减少由直径为2毫米的熔融石英制成的凹球形光学元件的表面形状误差D  = 100毫米，曲率半径R = -137.5毫米（均方根值）超过3倍。初始RMS为4.5 nm，处理后的RMS为1.36 nm。