Single-mode coherent synchrotron radiation instability

The microwave instability driven by the coherent synchrotron radiation (CSR) has been previously studied [S. Heifets and G. V. Stupakov, Phys. Rev. ST Accel. Beams 5, 054402 (2002)] neglecting effect of the shielding caused by the finite beam pipe aperture. In practice, the unstable mode can be close to the shielding threshold where the spectrum of the radiation in a toroidal beam pipe is discrete. In this paper, the CSR instability is studied in the case when it is driven by a single synchronous mode. A system of equations for the beam-wave interaction is derived and its similarity to the 1D free-electron laser theory is demonstrated. In the linear regime, the growth rate of the instability is obtained and a transition to the case of continuous spectrum is discussed. The nonlinear evolution of the single-mode instability, both with and without synchrotron damping and quantum diffusion, is also studied.