Quadratic corrections to the metaplectic formulation of resonant mode   conversion
  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. 
A. S. Richardson, E. R. Tracy