Comparison of a Few Perturbation Methods to Evaluate Axial Secular Frequencies in a Nonlinear Planar Dual-Frequency Paul Trap

Modified homotopy and modified Lindstedt–Poincare perturbation methods have been applied to the dynamics of an ion in a dual-frequency planar Paul trap in the presence of hexapole, octopole, and decapole field aberrations. Approximate analytical expressions for axial secular frequency correct up to second order have been evaluated for the ion trapped inside the device. The motion of the ion in rapidly oscillating fields is transformed into a motion under the influence of an effective potential encapsulated by a nonlinear Duffing-like differential equation. The results obtained using perturbation methods have been matched with the exact axial secular frequency and are well in agreement for different voltage and frequency ratios for the dual-frequency Paul trap.