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Coupled Landau Damping

Linear coupling between the transverse planes together with the tune separation Q Q h v is important for Landau damping of coherent instabilities. Often beam stability is found to be more critical in one of the transverse planes. Overall stability can then be improved by tuning the coupling and/or the tune split, to transfer a certain amount of Landau damping from the stable to the unstable plane. The transverse tune spreads, the skew gradient (due to skew quadrupoles or solenoids) and the tune separation are four major parameters that can be used to damp transverse coherent instabilities. A stability criterion for both coasting and bunched beams (with long and short-range interactions) is derived using these parameters, considering the smooth approximation and Lorentz spectra for the transverse tune spreads (an “elliptical” tune distribution is also discussed at the end of the paper and a simple example is treated). These assumptions should be sufficient to expose the essential physics of the topic. To obtain quantitative results concerning the sharing of Landau damping between the transverse planes, we discuss the case of a coasting beam without horizontal tune spread in a circular vacuum pipe of finite resistivity (the general case is then deduced).