Self-injection by trapping of plasma electrons oscillating in rising density gradient at the vacuum-plasma interface

We model the trapping of plasma $e^-$ within the density structures excited by a propagating energy source ($\beta_{S}\simeq1$) in a rising plasma density gradient. Rising density gradient leads to spatially contiguous coupled up-chirped plasmons ($d{\omega^2_{pe}(x)}/{dx}>0$). Therefore phase mixing between plasmons can lead to trapping until the plasmon field is high enough such that $e^-$ trajectories returning towards a longer wavelength see a trapping potential. Rising plasma density gradients are ubiquitous for confining the plasma within sources at the vacuum-plasma interfaces. Therefore trapping of plasma-$e^-$ in a rising ramp is important for acceleration diagnostics and to understand the energy dissipation from the excited plasmon train \cite{LTE-2013}. Down-ramp in density \cite{density-transition-2001} has been used for plasma-$e^-$ trapping within the first bucket behind the driver. Here, in rising density gradient the trapping does not occur in the first plasmon bucket but in subsequent plasmon buckets behind the driver. Trapping reduces the Hamiltonian of each bucket where $e^-$ are trapped, so it is a wakefield-decay probe. Preliminary computational results for beam and laser-driven wakefield are shown.