Extremum Problem of Optical Path in Imaging by a Single Spherical Surface Under Non-Paraxial Conditions

Fermat's principle indicates the characteristic that the actual optical path always takes an extremum. Based on this, it is possible to further determine the specific type of extremum. Single-spherical-surface imaging, as a typical example of curved-surface imaging, has a constant actual optical path in paraxial ideal imaging. However, for the non-paraxial case, the type of extremum of the actual optical path has not been specifically discussed in the literature. This paper uses the mathematical method of finding higher-order derivatives to focus on discussing the type of extremum of the actual optical path corresponding to a given image point after reflection or refraction of light rays emitted from a given object point on the principal axis under non-paraxial conditions, deepening the understanding of non-paraxial imaging and Fermat's principle.