Universal Description of Spherical Aberration Free Lenses Composed of Positive or Negative Index Media

TL;DR

This paper presents a universal mathematical description for spherical aberration free lenses that applies to both positive and negative refractive index media, expanding the theory of aplanatic optical surfaces.

Contribution

It introduces a unified surface function applicable to lenses with positive or negative indices, revealing new symmetries and extending classical lens formulas.

Findings

The surface function covers all conic sections for negative index media.

Classical lens formulas remain valid for negative indices in the thin lens limit.

The theory reveals asymmetries between positive and negative index lens profiles.

Abstract

We find that the function that describes the surface of spherical aberration free lenses can be used for both positive and negative refractive index media. With the inclusion of negative index, this function assumes the form of all the conic sections and expands the theory of aplanatic optical surfaces. There are two different symmetry centers with respect to the index that create an asymmetric relationship between positive and negative index lens profiles. In the thin lens limit the familiar formulas for image position and magnification hold for any index.