Saturation of a quasi-mode parametric instability in an inhomogeneous plasma

Quasi-mode parametric instabilities (PI) can result in significant power loss in RF-plasma interactions. However, theoretical difficulties still remain in the analysis of their saturation mechanisms within the framework of the Wenzel-Kramers-Brillouin (WKB) approximation, mainly due to the uncertainty of the impact of an inhomogeneous plasma. In this article, we present a systematic theoretical model under the WKB framework to analyze the saturation process of quasi-mode PIs, using the spatial gradient of the dielectric function to replace the linear dispersion relation to evaluate the effects of plasma inhomogeneity. The amplification factor of a quasi-mode PI in an inhomogeneous plasma with a finite pump profile is calculated using such a method to analyze the dominant mechanism of saturation.