Characterization of the 1st Matthieu stability regime in auto-ponderomotive Paul-trap-like electron guides

We demonstrate a detailed study of the 1st Mathieu stability region for an electrostatic auto-ponderomotive electron guide. Analog to a radio-frequency-driven linear quadrupole mass filter, we find a roughly triangular stability region spanning the dimensionless $a$-$q$ parameters in Mathieu space. In our experiments, we achieve stable guiding for $a$ $\leq$ 0.16 and 0.03 $\leq$ $q$ $\leq$ 0.77. According to simulations, the pseudopotential trap depth was $\SI{0.44}{\eV}$, sufficient to confine guided electrons inside the trap radius of $\SI[separate-uncertainty]{290 +- 2}{\micro\metre}$. Additionally, we detect an instability inside the guiding regime which is due to parametric nonlinear excitation and matches well with simulation results. Our detailed study substantially improves the understanding of auto-ponderomotive guiding for charged particles. Most importantly, it shows that these electron guides indeed behave very similar to radio-frequency Paul trap-like quadrupole mass filters and, thus might enable electron-optical elements such as electron beam splitters and electron resonators.