为利用有限元法和面形检测结果反演出光学元件的面形, 对面形检测结果进行分解, 并对旋转平均法面形检测原理进行分析, 讨论采用忽略光学元件自身面形的理想几何模型对其旋转非对称项面形误差进行有限元计算的理论可行性.在此基础上提出了基于有限元法反演光学元件面形的反演模型.以三点球支撑6 inch平面镜为例, 建立接触有限元模型计算旋转非对称项面形误差, 对比了数值法和N步旋转平均法所获得的镜面旋转非对称项面形误差, 结果显示, 二者的旋转非对称项面形均方根值为分别为2.944 nm和2.762 nm, 两种方法获得的面形相减结果分别为二者的6.31%和6.73%.最后对比了面形反演的面形结果与N步旋转平均法所获得的面形检测结果, 结果显示, 二者的面形均方根值分别为3.535 nm 和3.351 nm, 两种方法获得的面形相减结果分别为二者的11.67%和11.06%.证明提出的反演模型准确可靠.

In order to inversed compute the optical element surface profile using the finite element method and the surface shape interference test results, the composition of interference test results was decomposed, and the testing principles of the rotational averaging method were analysised. The feasibility of using simplified geometrical model to caculate the deformation of an optical element with arbitrary given surface shape was discussed. Surface profile inversion model was proposed based on finite element method. A finite element analysis model was then presented based on surface-surface contact modeling. A plane mirror supported by three rigid spheres as the simulation and experimental model, the rotationally asymmetrical surface deviation was quantitative analysis using the numerical method presented in this paper and the N-step rotational averaging method. The obtained results show that the root mean square of rotationally asymmetric surface is 2.933 nm obtained through numerical method and the root mean square of rotationally asymmetric surface is 2.75 nm obtained through N-step rotational averaging method. Subtracted result of the two rotationally saymmetric surface show that the root mean square of the subtraction are 6.31% of the numerical method result and 6.73% of the N-step rotational averaging method result. Finally, comparing the optical element surface profile computed by proposed inversion model with the surface profile tested by N-step rotational averaging method. Results show that the root mean square of optical element surface profile are 3.535 nm and 3.351 nm respectively. Subtracted result of the two surface profile show that the root mean square of the subtraction are 11.67% of the inversiong result and 11.06% of the result tested by N-step rotational averaging method .The proposed inversion model is accurate and reliable.