Lin Psang Dain
J Opt Soc Am A Opt Image Sci Vis. 2020 Sep 1;37(9):1435-1441. doi: 10.1364/JOSAA.399620.
Our group recently showed that the Seidel primary ray aberration coefficients of an axis-symmetrical system can be accurately determined using the third-order Taylor series expansion of a skew ray R¯ on an image plane. This finding inspires us to determine the third-order derivative matrix of R¯ with respect to the vector X¯ of the source ray, i.e., R¯m3/X¯03, under reflection/refraction at a flat boundary. Finite difference methods using the second-order derivative matrix, R¯m2/X¯02, require multiple rays to compute R¯m3/X¯03 and suffer from cumulative rounding and truncation errors. By contrast, the present method is based on differential geometry. Thus, it provides a greater inherent accuracy and requires the tracing of just one ray. The proposed method facilitates the analytical investigation of the primary aberrations of an axis-symmetrical system and can be easily extended to determine the higher-order derivative matrices required to explore higher-order ray aberration coefficients.
我们的团队最近表明，轴对称系统的赛德尔初级像差系数可以通过在像平面上对斜光线(\overline{R})进行三阶泰勒级数展开来精确确定。这一发现促使我们确定在平面边界反射/折射时，斜光线(\overline{R})相对于源光线向量(\overline{X})的三阶导数矩阵，即(\frac{\partial^{3}\overline{R}}{\partial\overline{X}^{3}})。使用二阶导数矩阵(\frac{\partial^{2}\overline{R}}{\partial\overline{X}^{2}})的有限差分法需要多条光线来计算(\frac{\partial^{3}\overline{R}}{\partial\overline{X}^{3}})，并且存在累积舍入和截断误差。相比之下，本方法基于微分几何。因此，它具有更高的固有精度，只需要追踪一条光线。所提出的方法便于对轴对称系统的初级像差进行分析研究，并且可以很容易地扩展以确定探索高阶光线像差系数所需的高阶导数矩阵。
