One of the main prerequisites for efficient operation of high power microwave
sources, such as the gyrotron and the free-electron-laser, is the good
quality of the electron beams. For gyrotrons in particular, the
velocity spreads may affect the beam coupling to the electromagnetic
wave. The beam energy spread is important as well, since the energy
extraction efficiency depends strongly on the detuning parameter
where 
is the
non-relativistic electron cyclotron frequency, 
the
electron relativistic factor and 
the wave frequency.
In general, beam optics can only induce
velocity spreads, producing an almost mono-energetic beam, since the
energy spread induced by the DC space charge across the thin beam is
negligible. This is also the case when considering the effects of the
surface roughness of the gun emitter. On the other hand, beam instabilities
which can occur in the beam tunnel, can induce both velocity and
energy spreads, and thus could deteriorate the beam and its
interaction with the microwave in the resonator. On the basis of
single-mode calculations, a substantial reduction of the gyrotron
efficiency has been predicted in [1] due to an energy
spread 
as small as 
and less.
Beam diagnostics have been employed recently to measure the beam velocity distribution in gyrotrons. They are based on the retarding potential technique[2] and the electron cyclotron emission [1]. Both methods provide only the parallel velocity distribution and assume a mono-energetic beam in order to estimate the perpendicular velocity distribution.
In the present paper, numerical simulations using the Particle-In-Cell
(PIC) method are presented to study the electron beam instabilities
occurring close to the electron cyclotron frequency 
.
The principal
focus is the determination of the various spreads mentioned above,
induced by the instabilities in realistic conditions (geometry,
external magnetic profile). Both electrostatic and electromagnetic models
are considered. These types of simulations have already been considered
in 1D [3] and in 2D [4] in the context of
electrostatic noise amplification in gyrotron amplifiers.
In sec. II, the PIC simulation models are described in detail. The linear theory of the electrostatic electron cyclotron instability for a uniform plasma and perfectly aligned gyrocenter beam is briefly reviewed and compared to the 1D PIC simulations in sec. III. The 2D electrostatic and electromagnetic simulation results are presented in section IV and V respectively and finally the section VI contains some concluding remarks.