Determination of Mathieu Stability Plot for Planar Trap With Circular Trapping Region

Planar toroidal ion traps with circular trapping regions are investigated using a toroidal multipole representation of the electrostatic potential. The multipole expansion coefficients are calculated and employed to model the electric potential in the vicinity of the trapping region. The accuracy of the analytical representation is validated through comparison with potentials computed using the boundary element method (BEM), showing excellent agreement near the trapping circle. Ion trajectories are numerically simulated for various combinations of RF and DC voltages to identify stable and unstable operating regimes. These regimes are subsequently mapped onto the Mathieu parameter space to construct Mathieu stability diagrams. The resulting stability regions reveal localized instability zones within the broader stable regions, highlighting the influence of nonlinear electric field components on ion dynamics. The secular frequencies predicted using the Mathieu parameters are compared with those extracted directly from ion trajectory simulations using the harminv tool. Good agreement is obtained between the two approaches, with relative errors remaining below 3% over a wide range of operating conditions, thereby confirming the validity of the theoretical and numerical methods employed in this study.