  We investigate the analytic continuation of wave equations into the complex
position plane. For the particular case of electromagnetic waves we provide a
physical meaning for such an analytic continuation in terms of a family of
closely related inhomogeneous media. For bounded permittivity profiles we find
the phenomenon of reflection can be related to branch cuts in the wave that
originate from poles of the permittivity at complex positions. Demanding that
these branch cuts disappear, we derive a large family of inhomogeneous media
that are reflectionless for a single angle of incidence. Extending this
property to all angles of incidence leads us to a generalized form of the
Poschl Teller potentials. We conclude by analyzing our findings within the
phase integral (WKB) method.
