# Solutions of the Helmholtz equation given by solutions of the eikonal equation

## TL;DR

This paper establishes a specific relationship between the refractive index and solutions of the eikonal equation that produce exact solutions to the Helmholtz equation, enhancing understanding of wave propagation in variable media.

## Contribution

It derives the form of the refractive index that ensures solutions of the eikonal equation lead to exact Helmholtz solutions, providing a new analytical link between these equations.

## Key findings

- Identifies the refractive index form for exact Helmholtz solutions
- Connects eikonal solutions directly to Helmholtz solutions
- Provides a method to generate exact wave solutions in variable media

## Abstract

We find the form of the refractive index such that a solution, $S$, of the eikonal equation yields an exact solution, $\exp ({\rm i} k_{0} S)$, of the corresponding Helmholtz equation.