Transverse Mode Coupling Instability in the Presence of Linear Coupling

Linear coupling between the transverse planes can be helpful in raising the threshold for the transverse mode coupling instability, as has already been shown for the low intensity coherent modes of oscillation. The general determinant, which has to be solved in the presence of external non-linearities (due to octupoles), linear coupling (due to skew quadrupoles) and coupling between two consecutive modes, is given. In the absence of linear coupling, or neglecting mode coupling, the known results of the uncoupled theory are obtained, as expected. In the absence of frequency spreads, a fourth order equation has to be treated. The necessary condition for stability, which is a generalization of the one-dimensional stability criteria, has been found using a perturbation method, and verified numerically in the four following cases with the PS parameters: (i) a very short proton bunch interacting with the broadband impedance, (ii) a long proton bunch interacting with the resistive-wall impedance, (iii) a short positron bunch interacting with the broadband impedance, and (iv) a long proton bunch interacting with the 10 MHz RF cavities.