为了研究主动支撑条件对超薄镜面形误差的校正能力，以一个直径0.5 m的超薄镜为例进行了面形校正的仿真分析及实验验证。分析了致动器作用力与超薄镜面形的关系，引入了一些需校正的面形误差，如初级球差、慧差、像散及重力变形等，确定了致动器作用力的优化目标，用求解非线性约束问题的优化算法——序列二次规划法计算了校正面形误差所需的致动器作用力，得到了超薄镜面形残余误差。仿真分析表明，对于归一化系数为1的初始球差、慧差、像散以及它们的叠加，用本文提供的致动器排布方式可以将面形误差校正到RMSλ/24以内，且对初级像散的校正能力最强，慧差和球差次之；竖直放置时的重力变形加上3种低阶像差的叠加也可被校正到RMSλ/24。在得到主动支撑的0.5 m实验镜的初始面形结果后，重新计算了优化力和面形误差，结果表明,计算结果和实际装调结果基本一致，RMS约为λ/7。计算分析了超薄镜面形未能达到预期目标的原因，提出了适当增加致动器和提高超薄镜初始面形精度的改进方案，并最终使超薄镜面形达到RMSλ/20的要求。

In order to study correcting capabilities for figure errors of ultra-thin mirrors with active supports, a 0.5 m demonstration mirror is designed and tested.Firstly, the relevant relations between actuator forces and figure forms are analyzed, and some lower-order figure errors, such as primary spherical aberration, coma, astigmatism and distortion due to gravity, are introduced.Then, the optimization objective is set up, and the Sequential Quadratic Programming (SQP) method for nonlinear constraint problem is applied to calculating optimum actuator forces.The residual figure errors of the ultra-thin mirror are given.Simulation analysis shows that by using the actuator arrangement proposed in this paper,the normalized primary spherical aberration, coma, astigmatism as well as their summation can be corrected to be less than λ/24 rms,and the correcting capability for astigmatism is the strongest and for spherical aberration and coma are less; moreover,the deformation due to gravity combined with the summation of above three lower-order aberrations can also be corrected to be less than λ/24 rms.After the actual surface errors from the interferometry based on the 0.5 m demonstration mirror are given, the optimum actuator forces and form errors are solved.The results show that the surface quality is about λ/7 rms ,which is in accord with the test data.The causes that the figure error can not meet the desired objective are analyzed,then an improved method is proposed.By the method, the figure error of the 0.5 m ultra-thin mirror with active supports satisfies the requirements equal to or less than λ/20 rms.