Ray and Wave Aberrations Revisited: A Huygens-Like Construction yields Exact Relations

TL;DR

This paper derives exact analytical equations linking wave and ray aberrations in optical systems, quantifies the classical approximation error, and provides conditions for its smallness, with implications for high numerical aperture systems.

Contribution

It introduces precise conditions and exact relations for wave and ray aberrations, improving understanding of the classical approximation's limitations.

Findings

Exact equations for wavefront and ray aberrations derived

Error bounds for classical approximation established

Numerical simulations show larger errors for high NA and high-frequency OPD functions

Abstract

The optical aberrations of a system can be described in terms of the wave aberrations, defined as the departure from the ideal spherical wavefront; or the ray aberrations, which are in turn the deviations from the paraxial ray intersection measured in the image plane. The classical connection between the two descriptions is an approximation, the error of which has, so far, not been quantified analytically. We derive exact analytical equations for computing the wavefront surface, the aberrated ray directions, and the transverse ray aberrations in terms of the wave aberrations (OPD) and the reference sphere. We introduce precise conditions for a function to be an OPD function, show that every such function has an associated wavefront, and study the error arising from the classical approximation. We establish strict conditions for the error to be small. We illustrate our results with numerical simulations. Our results show that large numerical apertures and high-frequency OPD functions yield larger approximation errors.