Analytic solution for low-frequency rf sheaths in pulsed discharges.

The equations governing the evolution of rf-driven sheaths are solved analytically in the regime where the rf frequency is small compared to both the ionic plasma frequency and the ion transit time in the sheaths. Poincaré's map of first return is used to gain geometric insight into the dynamics of the circuit-sheath system. The requirements of minimizing wall bombardment while maximizing the efficiency of the coupling to the substrate sheath are shown to lead to an optimum value for the blocking capacitance in asymmetric discharges. This optimum value is also favorable for rapid relaxation to the steady state in pulsed discharges. The analytic solution is applied to the problem of negative-ion extraction in afterglow plasmas.