Linear and nonlinear theory of single wave-beam interactions

A general theory of damped, beam-driven oscillations of a single mode is developed. This work represents a new approach to beam-cavity instabilities and extends previous work on beam-plasma instabilities while unifying both problems under a single theory. The fully non-linear model of the system is derived and then various instabilities are looked at by linearizing the equations. Of particular significance, it is noted that cavity-mode instabilities may begin to play a more important role in accelerator physics as high-quality cavities are put into use. The damping mechanisms for these instabilities are discussed and it is pointed out that the common name of Landau damping in plasma physics and accelerator physics refers to completely different phenomena. Numerical simulations show that different nonlinear bunch behavior can develop, depending on the amount of cavity damping. If the damping is large, the bunch breaks off and loses energy, staying coherent as it changes frequency. This bunch formation and frequency change in the highly-damped regime accounts for the phenomenon of overshoot: the increase of beam frequency spread beyond the Keil-Schnell stability threshold. The amount of wave damping in a synchrotron bounce time gives the scaling with accelerator parameters of the value of cavity quality factor that distinguishes these behaviors. The expression for this quality factor agrees within a few percent with simulation results. For the highly-damped case, an adiabatic theory yields an equation for the rate of change of the bunch frequency in terms of the cavity quality factor. This frequency change agrees with the computer simulations to within 3% for a pre-bunched beam and to within roughly a factor of 2 for an initially unbunched beam.