Electrostatics of gapped and finite surface electrodes

We present approximate methods for calculating the three-dimensional electric potentials of finite surface electrodes including gaps between electrodes, and estimate the effects of finite electrode thickness and an underlying dielectric substrate. As an example, we optimize a radio-frequency surface-electrode ring ion trap, and find that each of these factors reduces the trapping secular frequencies by less than 5% in realistic situations. This small magnitude validates the usual assumption of neglecting the influences of gaps between electrodes and the finite electrode extent.