# Numerical analysis of eikonal equation

## TL;DR

This paper explores the numerical analysis of the eikonal equation, derived from wave optics, and applies geometric methods to analyze optical devices like Maxwell and Luneburg lenses.

## Contribution

It demonstrates the transformation of the eikonal equation into an ODE system for optical device analysis using geometric methods.

## Key findings

- Successful transformation of the eikonal equation into an ODE system
- Application of geometric methods to Maxwell and Luneburg lenses
- Enhanced understanding of wave and geometric optics connection

## Abstract

The Maxwell equations have a fairly simple form. However, finding solutions of Maxwell's equations is an extremely difficult task. Therefore, various simplifying approaches are often used in optics. One such simplifying approach is to use the approximation of geometric optics. The approximation of geometric optics is constructed with the assumption that the wavelengths are small (short-wavelength approximation). The basis of geometric optics is the eikonal equation. The eikonal equation can be obtained from the wave equation (Helmholtz equation). Thus, the eikonal equation relates the wave and geometric optics. In fact, the eikonal equation is a quasi-classical approximation (the Wentzel-Kramers-Brillouin method) of wave optics. This paper shows the application of geometric methods of electrodynamics to the calculation of optical devices, such as Maxwell and Luneburg lenses. The eikonal equation, which was transformed to the ODE system by the method of characteristics, is considered. The resulting system is written for the case of Maxwell and Luneburg lenses.