Braat Joseph J M, Janssen Augustus J E M
J Opt Soc Am A Opt Image Sci Vis. 2013 Jun 1;30(6):1213-22. doi: 10.1364/JOSAA.30.001213.
Various authors have presented the aberration function of an optical system as a power series expansion with respect to the ray coordinates in the exit pupil and the coordinates of the intersection point with the image field of the optical system. In practical applications, for reasons of efficiency and accuracy, an expansion with the aid of orthogonal polynomials is preferred for which, since the 1980s, orthogonal Zernike polynomials have become the reference. In the literature, some conversion schemes of power series coefficients to coefficients for the corresponding Zernike polynomial expansion have been given. In this paper we present an analytic solution for the conversion problem from a power series expansion in three or four dimensions to a double Zernike polynomial expansion. The solution pertains to a general optical system with four independent pupil and field coordinates and to a system with rotational symmetry in which case three independent coordinate combinations have to be considered. The conversion of the coefficients is analytically in closed form and the result is independent of a specific sampling scheme or sampling density as this is the case for the commonly used least squares fitting techniques. Computation schemes are given that allow the evaluation of coefficients of arbitrarily high order in pupil and field coordinates.
多位作者将光学系统的像差函数表示为关于出射光瞳中的光线坐标以及与光学系统像场交点坐标的幂级数展开。在实际应用中，出于效率和准确性的考虑，借助正交多项式进行展开更为可取，自20世纪80年代以来，正交泽尼克多项式已成为这方面的参考。在文献中，已经给出了一些将幂级数系数转换为相应泽尼克多项式展开系数的方案。在本文中，我们给出了一个从三维或四维幂级数展开到双泽尼克多项式展开的转换问题的解析解。该解适用于具有四个独立光瞳和视场坐标的一般光学系统，以及具有旋转对称性的系统，在这种情况下必须考虑三个独立的坐标组合。系数的转换以解析形式给出，结果与特定的采样方案或采样密度无关，而常用的最小二乘拟合技术则依赖于这些。文中给出了计算方案，可用于计算光瞳和视场坐标中任意高阶的系数。
