Explicit complex-valued solutions of the 2D eikonal equation

Abstract

We present a method to obtain explicit solutions of the complex eikonal equation in the plane. This equation arises in the approximation of the Helmholtz equation by the WKBJ or Evanescent Wave Tracking methods. We obtain the complex-valued solutions (called eikonals) as parametrizations in a complex variable. We consider both the cases of constant and non-constant index of refraction. In both cases, the relevant parametrizations depend on some holomorphic function. In the case of a 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 parametrizations give information about the formation of caustics and the light and shadow regions for the relevant eikonals.

Keywords:

• Complex eikonal equation
• geometrical optics
• explicit solutions
• pseudo-analytic functions

2010 Mathematics Subject Classifications:

• 78A05
• 35F20
• 35C05
• 35A22
• 35A30
• 30G20

Acknowledgments

The author wishes to thank Giulio Ciraolo for some useful discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The paper is partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) dell'Istituto Nazionale di Alta Matematica (INdAM).