Explicit complex-valued solutions of the 2D eikonal equation

TL;DR

This paper develops a method to explicitly solve the complex 2D eikonal equation using complex analysis, providing insights into wave propagation, caustics, and shadow regions in optics.

Contribution

It introduces a novel approach to obtain explicit complex solutions of the 2D eikonal equation using holomorphic functions and pseudo-analytic function theory.

Findings

Explicit solutions depend on holomorphic functions and exponential factors.

Solutions reveal formation of caustics and shadow regions.

Applicable to both constant and non-constant refractive indices.

Abstract

We present a method to obtain explicit solutions of the complex eikonal equation in the plane. This equation arises in the approximation of Helmholtz equation by the WKBJ or EWT methods. We obtain the complex-valued solutions (called eikonals) as parameterizations in a complex variable. We consider both the cases of constant and non-constant index of refraction. In both cases, the relevant parameterizations depend on some holomorphic function. In the case of non-constant index of refraction, the parametrization also depends on some extra exponential complex-valued function and on a quasi-conformal homeomorphism. This is due to the use of the theory of pseudo-analytic functions and the related similarity principle. The parameterizations give information about the formation of caustics and the light and shadow regions for the relevant eikonals.