Analytic model for electrostatic fields in surface-electrode ion traps

The design of surface-electrode ion traps is studied analytically. The classical motion of a single ion in such a trap is described by coupled Mathieu equations. An analytic boundary-value solution for the Laplace equation in three dimensions with a piecewise-constant Dirichlet boundary condition is described. This solution is used to model the electrostatic potential field generated by a series of electrodes lying in a single plane. The model is applied to the problem of designing surface-electrode ion traps, including calculating important trap design parameters such as the center position of the trapped ion, trap depth, and stability parameters. The model is used to determine optimized dimensions for trap electrodes in various cases.