为了实现超薄球面镜的非球面成形,提出了一种采用非球面梯度法求致动器排布初始解的方法。给出了该方法的理论依据、排布方法以及计算非球面度梯度的两种公式;以一大口径离轴超薄非球面镜为例,用非球面度梯度的两种计算公式分别求出了致动器排布初始解,完成了非球面成形的有限元分析,得到了满足面形精度RMS值为21.09 nm的最终解;最后,介绍了致动器排布的优化步骤,比较了非球面梯度法、正方形法和环形法的差异。结果显示,用非球面度梯度平均值公式求出的初始解与最终解最接近,符合非球面度梯度变化率与致动器面密度的关系,而通过优化还能进一步减少致动器数量和面形残差。在相同面形精度下,非球面梯度法排布的致动器个数约为正方形和环形排布的1/2或更少;在相同致动器个数下,非球面梯度法排布的面形残差RMS值约为正方形和环形排布的1/3或更小。结果表明,非球面梯度法更适合在非球面成形领域用于求解致动器排布初始解。

In order to shape aspheric surfaces for ultra-thin spherical mirrors, a new method for the initial solutions of actuator arrangements solved by asphericity gradient was presented.The theoretical analysis, arrangement method and two formulas for calculating the asphericity gradient were given. As shaping an off-axis large-aperture aspheric mirror for example, the initial solutions of actuator arrangements were obtained by two formulas above, respectively. By using Finish Element Method(FEM), the deformation analysis of shaping aspheric surfaces with form error of 21.09 nm was accomplished, and the final solutions were given. Moreorer, the optimizations of actuator arrangements were discussed. Finally, three kinds of methods for actuator arrangements were compared with each other. The results indicate that the initial solution based on average value of asphericity gradient is most similar to the final solution, which is in accord with the relation between the change rate of asphericity gradient and the surface density of actuator arrangements. The optimization can reduce the number of actuators and can improve the form error further. Under the same form error, the number of actuators by the aspheric gradient method is 1/2 or less than that by the square and circular methods; under the same actuator numbers, the RMS of form error by the aspheric gradient method is about 1/3 or less than that by the square and circular methods. These data reported here show that the aspheric gradient method is suitable to solve an initial solution of actuator arrangements in the field of shaping aspheric surfaces.