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 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.