Algorithm for Lens Calculations in the Geometrized Maxwell Theory

TL;DR

This paper presents an algorithm based on differential geometry for solving inverse problems in geometrical optics, specifically finding propagation paths of electromagnetic rays in various media using effective metrics and geodesic curves.

Contribution

It introduces a novel geometric algorithm for inverse lens calculations, applying differential geometry to determine media parameters from ray trajectories.

Findings

Successfully applied to Maxwell and Luneburg lenses

Demonstrated similarity between classical and geometric results

Provides a new approach for inverse optical problems

Abstract

Nowadays the geometric approach in optics is often used to find out media parameters based on propagation paths of the rays because in this case it is a direct problem. However inverse problem in the framework of geometrical optics is usually not given attention. The aim of this work is to demonstrate the work of the proposed the algorithm in the framework of geometrical approach to optics for solving the problem of finding the propagation path of the electromagnetic radiation depending on environmental parameters. The methods of differential geometry are used for effective metrics construction for isotropic and anisotropic media. For effective metric space ray trajectories are obtained in the form of geodesic curves. The introduced algorithm is applied to well-known objects - Maxwell and Luneburg lenses. The similarity of results obtained by classical and geometric approach is demonstrated.