The particle motion through the fringing field of a toroidal condenser in a third-order approximation

The ion trajectories through the extended fringing field of a toroidal condenser are calculated including the effects of curved field boundaries. The optical fringing-field effects are expressed by analytical formulas including fringing-field integrals using the transfer matrix method. These formulas describe in an effective third-order approximation all effects in the plane of deflection as well as in the perpendicular surface to this plane. The obtained expressions are compared to precise numerical ray-tracing calculations through typical field distributions.