A solution to the linear wave conversion problem in weakly inhomogeneous plasmas

The wave equation for electromagnetic and electrostatic waves propagating in a weakly inhomogeneous plasma is studied near a conversion layer (any layer where the WKB approximation breaks down). The wave equation is approximated by a Laplace equation, [P(i d/dx)−xQ(i d/dx)]E(x)=0, where P and Q are polynomials of arbitrary order. The general solution is given in terms of contour integrals. Simple transformations on sets of contours allows connection of the WKBJ asymptotic solutions on both sides of the conversion layer. The mathematical technique is general (matrix operations) and is well suited to numerical computation. The method is illustrated by solving the fourth-order problems met in ion-cyclotron fast-wave and lower-hybrid wave conversion.