  We investigate theoretically how sheath radio-frequency (RF) oscillations
relate to the spatial structure of the near RF parallel electric field E//
emitted by Ion Cyclotron (IC) wave launchers. We use a simple model of Slow
Wave (SW) evanescence coupled with Direct Current (DC) plasma biasing via
sheath boundary conditions in a 3D parallelepiped filled with homogeneous cold
magnetized plasma. Within a "wide sheaths" asymptotic regime, valid for
large-amplitude near RF fields, the RF part of this simple RF+DC model becomes
linear: the sheath oscillating voltage VRF at open field line boundaries can be
re-expressed as a linear combination of individual contributions by every
emitting point in the input field map. SW evanescence makes individual
contributions all the larger as the wave emission point is located closer to
the sheath walls. The decay of |VRF| with the emission point/sheath poloidal
distance involves the transverse SW evanescence length and the radial
protrusion depth of lateral boundaries. The decay of |VRF| with the
emitter/sheath parallel distance is quantified as a function of the parallel SW
evanescence length and the parallel connection length of open magnetic field
lines. For realistic geometries and target SOL plasmas, poloidal decay occurs
over a few centimeters. Typical parallel decay lengths for |VRF| are found
smaller than IC antenna parallel extension. Oscillating sheath voltages at IC
antenna side limiters are therefore mainly sensitive to E// emission by active
or passive conducting elements near these limiters, as suggested by recent
experimental observations. Parallel 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 finally
justify current attempts at reducing the RF fields induced near antenna boxes
to attenuate sheath oscillations in their vicinity.
