Design and measurement considerations of hairpin resonator probes for determining electron number density in collisional plasmas

The hairpin resonator probe has been developed in recent years into a sophisticated diagnostic technique capable of measuring spatially resolved electron number densities in sub-Torr discharges. In this paper, we extend the use of this technique to discharges at pressures greater than 1 Torr. In this regime, the effects of electron-neutral collisions become significant and a suitable correction is applied in conjunction with the sheath correction. We also describe elements of hairpin design and coupling that need to be more carefully controlled in order to maximize the range of electron densities that can be detected at higher pressures. Finally, we discuss limitations to the transmission-line model used routinely to interpret hairpin data as they apply to measurements in a nonuniform plasma