Validation of the paraxial beam-tracing method in critical cases

Paraxial Wentzel-Kramers-Brillouin (WKB) (beam-tracing) solutions of a scalar wave equation are compared to the exact solutions for the two important cases of reflection from a plasma with linearly increasing density and of oblique propagation in the presence of an absorbing half plane. In these situations, the beam-tracing technique is close to its limits of validity, yet these conditions are often met in practical applications. More specifically, the first case is relevant to reflectometry diagnostics, whereas the second case models the absorption of electron-cyclotron beams obliquely launched onto the resonance layer, as envisaged, for instance, for the ITER upper launcher [M. A. Henderson , Nucl. Fusion 48, 054013 (2008)]. In both cases, the beam-tracing approach is found to reproduce well the exact behavior of the wave for experimentally relevant parameters, confirming the robustness of WKB-based techniques close to or even beyond their range of applicability. For the latter case, moreover, the analysis presented here allows an evaluation of the reflection of part of the beam by the steep gradients of the absorption coefficient at the resonance layer, our results yielding an upper bound on the amount of reflected power.