Theory of coupled Landau damping

The influence of linear coupling between the transverse planes on Landau damping of coherent instabilities is assessed using two typical frequency distributions (Lorentzian, , and "elliptical", where ). A general stability criterion is derived in both cases that includes the coupling strength and the distance from the resonance It reveals the possibility of sharing the "stabilising" frequency spreads between the two planes. This can significantly improve the coherent beam stability, especially in cases where the situation is more critical in one plane. Another important observation is the fact that the influence of a large imaginary part in the beam-environment impedance, which normally requires a large frequency spread for Landau damping, can be compensated (at least in one plane) by a judicious choice of the coupling. The conjunction of these two features could explain why a machine like the CERN-PS can be stabilised by tuning close to a coupling resonance and can be used to determine optimum values for the tune split and the coupling strength.