The Gaussian formula and spherical aberration of the static and moving curved mirrors from Fermat's principle

TL;DR

This paper investigates the Gaussian formula and spherical aberrations of static and relativistic curved mirrors using Fermat's principle, revealing how relativistic effects like Lorentz contraction influence optical properties and aberrations.

Contribution

It introduces a comparative analysis of static and relativistic curved mirrors, demonstrating the impact of relativistic effects on focal lengths and aberrations based on Fermat's principle.

Findings

Relativistic mirrors' focal lengths obey Lorentz contraction.

Spherical aberration relations are affected by relativistic speeds.

Limits for paraxial approximation and minimum system speed are identified.

Abstract

The Gaussian formula and spherical aberrations of the static and relativistic curved mirrors are analyzed using the optical path length (OPL) and Fermat's principle. The geometrical figures generated by the rotation of conic sections about their symmetry axes are considered for the shapes of the mirrors. By comparing the results in static and relativistic cases, it is shown that the focal lengths and the spherical aberration relations of the relativistic mirrors obey the Lorentz contraction. Further analysis of the spherical aberrations for both static and relativistic cases have resulted in the information about the limits for the paraxial approximation, as well as for the minimum speed of the systems to reduce the spherical aberrations.