Starting conditions for relativistic backward wave oscillators at low currents

This paper contains a theoretical treatment of the start oscillation condition for backward wave oscillators with rippled wall slow wave structures, driven by relativistic, low current, annular electron beams. In the low current, or Compton, limit of device operation, this condition amounts to setting a minimum value on the product of the beam current and the cube of the structure length; above the minimum value, the device will spontaneously break into oscillations, even in the absence of an input signal. Also computed is the small signal growth rate of the oscillations, which is shown to be less than the single mode growth rate for an infinitely long structure. Approximate closed form expressions are found in the limit of shallow structure ripples, while numerical results are presented for deeper wall ripples.