  Exact solutions are obtained for all the modes of wave propagation along an
anisotropic cylindrical waveguide. Closed-form expressions for the energy flow
on the waveguide are also derived. For extremely anisotropic waveguide where
the transverse permittivity is negative while the longitudinal permittivity is
positive, only transverse magnetic (TM) and hybrid modes will propagate on the
waveguide. At any given frequency the waveguide supports an infinite number of
eigenmodes. Among the TM modes, at most only one mode is forward wave. The rest
of them are backward waves which can have very large effective index. At a
critical radius, the waveguide supports degenerate forward- and backward-wave
modes with zero group velocity. These waveguides can be used as phase shifters
and filters, and as optical buffers to slow down and trap light.
