A general theory of beam breakup instabilities

The authors re-examine the beam breakup instabilities (BBU) in linear accelerators with the use of a simple continuum model. The standard mode coupling approach in microwave engineering is adopted to treat the cumulative, regenerative and convective BBU in a unified manner. In addition to being able to recover all major classical solutions as limiting cases, the simplified treatment, in several instances, provides a rather different perspective on the role of the focusing magnetic fields, spreads in betatron frequencies and spreads in the breakup mode frequencies, etc., than has been given in the past. Dimensionless parameters are constructed which provide an immediate assessment of the relative importance of these effects, as well as the anticipated BBU growth when these various factors are incorporated. A highly accurate numerical scheme has been devised which directly integrates the governing equations. Excellent agreement between the numerical results and the analytic theory is observed. The numerical studies include other effects such as stagger tuning and nonlinear focusing which defy analytical treatment