Steady-state solutions of split beams in electron storage rings

Recently, a novel operation method for synchrotron light sources with transversely split beams has been explored to fulfill the rising demand for flexible and high-throughput X-ray sources required in such diverse fields as time-resolved X-ray spectroscopy, molecular chemistry in organic cells, high-resolution medical imaging, quantum materials science or sustainable energy research. Within that novel operation mode, additional stable regions are produced in the horizontal phase space by operating an electron storage ring on a resonance that is driven by the nonlinear sextupole or octupole magnets. In the longitudinal phase space, a similar split can be produced by introducing an oscillation of the synchrotron phase via a modulation of the phase of the radiofrequency resonator. Strong radiation damping in electron storage rings, however, has to be overcome before additional regions in phase space can become populated by particles and form stable islands. This damping mechanism changes the dynamics of the system and causes diffusion between the different islands in phase space, raising the question what kind of equilibrium state exists in the asymptotic temporal limit. In this paper, a finite-differences approximation in rotating action-angle coordinates is used to solve the Vlasov–Fokker–Planck equation and to study the obtained equilibrium states for the longitudinal as well as the transverse case. The number of solution vectors and the magnitude of the corresponding singular values of the matrix of the underlying finite-differences equation are used as abstract indicators to define the required parameter set that provides stable additional beamlets. As a consequence, the beamlets have a stability that is close to that of the main beam in terms of diffusion caused by the radiation damping and quantum excitation.