  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.
