  We show that transparent dielectrics with strong optical anisotropy support a
new class of electromagnetic waves that combine the properties of propagating
and evanescent fields. These "ghost waves" are created in tangent bifurcations
that "annihilate" pairs of positive- and negative-index modes, and represent
the optical analogue of the "ghost orbits" in the quantum theory of
non-integrable dynamical systems. Similarly to the regular evanescent fields,
ghost waves support high transverse wavenumbers, but in addition to the
exponential decay show oscillatory behavior in the direction of propagation.
Ghost waves can be resonantly coupled to the incident evanescent waves, which
then grow exponentially through the anisotropic media - as in the case of
negative index materials.As ghost waves are supported by transparent dielectric
media, they are free from the "curse" of material loss that is inherent to
conventional negative index composites.
