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.