从测量值计算非球面参数

Calculation of Aspheric Parameters from Measurement Values

为解决按测量值计算非球面光学系统成像质量的问题,提出了一种从三维坐标测量值提取标准非球面方程的参数的方法.首先在两个相互垂直的近似的中心剖面附近取值,并分别投影到这两个平面上;然后拟合曲线;再用这两条曲线在各自的平面内截取一组平行直线成一组线段,改变直线的斜率,使各线段中点共线,确定非球面的对称轴和顶点坐标;最后,平移、旋转坐标系后,将三维坐标降维,用多项式拟合成二维曲线,同时将标准形式方程中的无理项展为泰勒级数,对比二者的系数并推广,得到任意偶数阶标准非球面方程的参数.大量随机取值计算的统计结果表明,算法的误差是测量误差的1/2.

In order to solve the problem of calculating image quality of aspheric optical system from measurement values, a method of calculating aspheric parameters from 3D coordinates on aspheric surface was proposed. The data are collected near two approximate central sections, which are vertical to each other. These coordinates are projected on the two planes respectively and then utilized to fit out two curves. A group of parallel lines are intersected by the curves to obtain some line segments on its plane. The axis of symmetry and the top of the lines are determined by changing the slope of the lines until all mid-points are collinear. The coordinates are fitted to be a 2D curve after the coordinate system is translated and rotated to reduce the dimension. Meanwhile, the irrational item in the standard aspheric equation is expressed as Taylor series. The standard parameters of arbitrary even-order aspheric equation can be obtained by comparing the correspond coefficients and extending the results. Statistical figures from a great number of random data show that the errors of the algorithm are half of measuring errors.