  Radio frequency (RF) cavities are commonly used to accelerate charged
particle beams. The shape of the RF cavity determines the resonant
electromagnetic fields and frequencies, which need to satisfy a variety of
requirements for a stable and efficient acceleration of the beam. For example,
the accelerating frequency has to match a given target frequency, the shunt
impedance usually has to be maximized, and the interaction of higher order
modes with the beam minimized. In this paper we formulate such problems as
constrained multi-objective shape optimization problems, use a massively
parallel implementation of an evolutionary algorithm to find an approximation
of the Pareto front, and employ a penalty method to deal with the constraint on
the accelerating frequency. Considering vacuated axisymmetric RF cavities, we
parameterize and mesh their cross section and then solve time-harmonic
Maxwell's equations with perfectly electrically conducting boundary conditions
using a fast 2D Maxwell eigensolver. The specific problem we focus on is the
hypothetical problem of optimizing the shape of the main RF cavity of the
planned upgrade of the Swiss Synchrotron Light Source (SLS), called SLS-2. We
consider different objectives and geometry types and show the obtained results,
i.e. the computed Pareto front approximations and the RF cavity shapes with
desired properties. Finally, we compare these newfound cavity shapes with the
current cavity of SLS.
