  Theories of monochromatic high-frequency electromagnetic fields have been
designed by Felsen, Kravtsov, Ludwig and others with a view to portraying
features that are ignored by geometrical optics. These theories have recourse
to eikonals that encode information on both phase and amplitude -- in other
words, are complex-valued. The following mathematical principle is ultimately
behind the scenes: any geometric optical eikonal, which conventional rays
engender in some light region, can be consistently continued in the shadow
region beyond the relevant caustic, provided an alternative eikonal, endowed
with a non-zero imaginary part, comes on stage. In the present paper we explore
such a principle in dimension $2.$ We investigate a partial differential system
that governs the real and the imaginary parts of complex-valued two-dimensional
eikonals, and an initial value problem germane to it. In physical terms, the
problem in hand amounts to detecting waves that rise beside, but on the dark
side of, a given caustic. In mathematical terms, such a problem shows two main
peculiarities: on the one hand, degeneracy near the initial curve; on the other
hand, ill-posedness in the sense of Hadamard. We benefit from using a number of
technical devices: hodograph transforms, artificial viscosity, and a suitable
discretization. Approximate differentiation and a parody of the
quasi-reversibility method are also involved. We offer an algorithm that
restrains instability and produces effective approximate solutions.
