  The effects of quadratic order terms in the dispersion matrix near a mode
conversion are considered. It is shown that including the corrections due to
these quadratic terms gives a better matching between the local solution in the
mode conversion region, and the far-field WKB solutions for the incoming and
outgoing waves. This matching is demonstrated by comparison of the asymptotic
solution with a numerical solution for a simple one-dimensional conversion.
This procedure for obtaining the corrections due to quadratic order terms can
be extended to arbitrary order and, in principle, an outline for performing
such an extension is given.
