Radio-frequency sheath voltages and slow wave electric field spatial structure

We investigate theoretically how sheath radio-frequency (RF) oscillations relate to the spatial structure of the RF parallel electric field emitted by Ion Cyclotron (IC) wave launchers, using a simple model of Slow Wave (SW) evanescence coupled with Direct Current (DC) plasma biasing via sheath boundary conditions in a plasma-filled 2-dimensional (parallel, radial) rectangle. Within a "wide sheaths" asymptotic regime, valid for large-amplitude near RF fields, our model becomes partly linear: the sheath oscillating voltage at open field line boundaries is a linear combination of elementary contributions by every source point of the radiated RF field map. These individual contributions are all the more intense as the SW emission point is toroidally nearer to the sheath walls. A limit formula is given for a source infinitely close to the sheaths. The decay of sheath RF voltages with the sheath/source parallel distance is quantified as a function of two characteristic SW evanescence lengths. Decay lengths are smaller than antenna parallel extensions. The sheath RF voltages at an IC antenna side limiter are therefore mainly sensitive to SW emission near this limiter, as recent observations suggest. Toroidal proximity effects could also explain why sheath oscillations persist with antisymmetric strap toroidal phasing, despite the parallel anti-symmetry of the radiated field map. They could also justify current attempts at reducing the RF fields induced near antenna boxes to attenuate sheath oscillations in their vicinity.