Discussion of the paraxial approximation and numerical analysis of spherical aberration

Textbooks intended to undergraduate students discuss image formation by mirrors and lens based on paraxial approximation. This conventional approach has ambiguities and lacks which induce students to conceptual mistakes. This paper points out those ambiguities and demonstrates that a basic proposition of paraxial approximation has a very limited scope. Furthermore, as a didactical tool to overcome conceptual difficulties, figures are presented depicting ray trajectories and spherical aberrations, computed precisely in terms of the focusing systematic error, for object points at an infinite distance from the optic system, in different cases of mirrors and lenses. Results show that a ) the aberration for each ray depends mainly on the angular distance of its incidence point, b ) the aberration behavior is the same in mirrors and lens, c ) in concave and convex mirrors, for the same angular distance, focusing errors are the same, d ) the aberration varies in lens according to their light-facing side.本教材面向本科生，基于近轴近似（paraxial approximation）讲解反射镜与透镜的成像原理。该传统方法存在歧义且存在不足，易导致学生产生概念误区。本文指出了这些歧义之处，并证明近轴近似的基本命题适用范围极为有限。此外，作为克服概念难点的教学工具，本文给出了多组图形，用于描绘不同反射镜与透镜场景下、物点距光学系统无穷远时的光线轨迹与球差（spherical aberrations），这些图形基于聚焦系统误差精确计算得到。研究结果表明：a）单条光线的像差主要取决于其入射点的角距离；b）反射镜与透镜的像差变化规律一致；c）对于凹面镜与凸面镜，在相同角距离下，其聚焦误差完全相同；d）透镜的像差会随其光线入射侧的不同而发生变化。