Suppression of the standing wave effect in high frequency capacitive discharges using a shaped electrode and a dielectric lens: Self-consistent approach

The standing wave effect causes nonuniform plasma excitation in high frequencies capacitive discharges when the electrode size is not considerably smaller than the excitation wavelength. A shaped electrode was proposed by Sansonnens and Schmitt [Appl. Phys. Lett. 82, 182 (2003)] to suppress this unwanted effect. The shape of the electrode was calculated in the vacuum approximation (no plasma was present between the electrodes), and was found to be Gaussian. The authors postulated that the presence of plasma would not significantly modify the solution. However, it was shown [Chabert et al., Phys. Plasmas 11, 1775 (2004)] using a self-consistent nonlinear transmission line model that the presence of plasma significantly shortens the wavelength for a system composed of two parallel plate electrodes. It was therefore legitimate to expect the optimized shape of the electrode and lens to be different when a plasma is present. Here it is shown that to suppress the standing wave effect the current flowing in the electrodes must be proportional to the discharge radius. This condition is independent of the medium present between the plates and indeed requires a Gaussian electrode.