Novel perturbation method for judging the stability of the equilibrium solution in the presence of passive harmonic cavities

When using the harmonic cavity to stretch the beam in electron storage rings, we expect the beam can be lengthened effectively and also stably. Semianalytical approaches can be used to obtain the equilibrium distribution but generally failed in the prediction of its stability. In this work, a novel perturbation method is proposed to check whether the calculated equilibrium solution can survive from the l = 1 mode perturbation. For the case of uniform beam filling, we derive a simple formula that can determine the threshold of l = 1 mode instability under specific parameters, which is well verified with tracking simulation for the Hefei Advanced Light Facility storage ring.