Let us first consider a simple cylinder of constant radius 
mm.
The Dirichlet boundary condition (perfectly
conducting wall) is applied at 
and Neumann conditions are
assumed at both ends, z=0 and 
. A cold annular electron beam with
a pitch
angle 
and energy 
kV
is injected at z=0 into the cylinder, immersed in an uniform magnetic
field 
. This geometry can be considered as the
simplest model for the beam tunnel connecting the gun region to the
gyrotron resonator. The simulation starts from noise and, for a given
beam current I, the tunnel
length 
is chosen sufficient for saturation to take place before
the beam enters the resonator. A quasi-steady state is reached
after about three electron transit times along
the cylinder. Typical longitudinal profiles of spreads at this
state are illustrated in Fig. 3, showing the convective
nature of the instability. By performing a time Fourier analysis of the
potential along the axis, 
, after the quasi steady-state has been
reached, it can be seen that the instabilities occur, indeed, at
frequencies close to the relativistic electron cyclotron frequency
(see Fig. 4).
The Fourier transform in z of
this signal, displayed in Fig. 5, clearly shows that, in addition
to the Doppler shifted electron cyclotron branch
, the low
frequency Langmuir wave
has also been excited.
In Fig. 6, the saturated velocity and energy spreads
are displayed versus the beam density. For each value of the
beam density, the system length is chosen sufficiently
large to reach saturation. The results from the 1D
simulation described above are also reported in figure. 6, showing
that the 2D spreads are only slightly larger than the 1D ones: the
finite Doppler shift 
does not seems to
be an important effect for the electrostatic instability
with uniform B. In most gyrotrons, the
beam density 
is usually smaller than
since high current beams are designed to also have large
cross-sections.
As a consequence, the maximum perpendicular velocity spread
due to the electrostatic electron cyclotron instability is expected
to be around 10%. However, the spreads in 
can be as large as
1% which exceeds by at least an order of magnitude what can be
expected from DC space charge effects alone. As shown in
Ref. [1], such values of 
could significantly
reduce the gyrotron interaction efficiency. In order to assert that
the instability can occur in the beam tunnel, and hence degrade the beam
quality before reaching the gyrotron resonator, it is important to
consider the effects of non-uniform magnetic fields in the beam tunnel.
In Ref.[4], the effects of the magnetic field gradient have been
considered using a linear magnetic taper. It was shown that
the nonlinear saturated spreads are not affected even though the
linear amplification rates 
can change (see Fig. 5 of
[4]). This change is due to the increase (decrease) of the
density and 
in a positive (negative) magnetic field taper,
consistent with the linear growth rate given in Eq. (8).
In the following, we will use the real magnetic field found in two existing gyrotrons: the first (gyrotron I) is a quasi-optical gyrotron operating at 90 GHz[12] and the second (gyrotron II) is a cylindrical cavity gyrotron operating at 118 GHz[13]. Experimental measurements have shown that the quasi-optical gyrotron efficiency saturates at a maximum value of 15% instead of the predicted 22%, while in the second device, the measured efficiencies agree well with the theory prediction. In the simulation, we consider only the beam tunnel extending from the end of the acceleration region to the center of the resonator, as shown in Fig. 7. The magnetic field for both cases is computed from the gyrotron coils. The phase space distribution of the injected beam is specified using the equilibrium obtained from the electron gun design code Daphne[14].
The steady state frequency spectra of the electrostatic potential 
at
r=0, in the case of gyrotron I,
are plotted at different axial positions in Fig. 8,
revealing that electrostatic instabilities are developed at
frequencies close to the local electron cyclotron frequency
. As in the uniform B case, a weak excitation at
low frequencies could also be observed.
The evolution of the computed energy spreads, displayed in
Fig. 7, shows
that the maximum spreads in gyrotron I are much larger than those
in gyrotron II. In addition, the instability developed in gyrotron II
is not yet saturated, due to the shorter beam tunnel.
The main cause of the larger spreads found in gyrotron I is, however, the larger
beam density reached in the maximum magnetic compression region
as can be inferred from Fig.9. Notice that
the correlation between the maximum spread and
the normalized beam density 
(with 
calculated using the maximum B field) is
practically the same as that found in the constant magnetic field case,
shown in Fig. 6. We thus get the simple result that
gyrotron II has smaller energy spreads because the beam was designed
to have a small density. Referring to the calculations
of the efficiency versus the energy spread shown in Fig. 12 of [1],
the efficiency in gyrotron I is expected to decrease to 18% from its
mono-energetic beam prediction of 22%; while in gyrotron II, the
effects of the spreadsinduced by the electrostatic instability
should remain negligible on the efficiency.
The space charge depression induced by an opening in the quasi-optical
gyrotron is simulated, using the geometry illustrated in
Fig. 10. In order to single out the effect of the beam
depression, the external magnetic field is assumed uniform. The
simulated domain is 20 cm long with an opening of 12 cm located at
its center. A 
and V=70 kV cold annular beam is injected at the left
side. The steady state velocity and energy spreads calculated at
the right boundary are shown versus the beam current in
Fig. 11. By comparing these spreads with those obtained
without beam depression (see e.g. Fig. 6), it is clear that the
space charge depression has no effect on the electrostatic instability.
Finally, the presence of a background plasma is considered by adding a population of non-drifting cold electrons in the simulation with a specified density. These cold electrons could roughly model an imperfect vacuum in the gyrotron or the secondary electrons which can be emitted by the gyrotron walls. We have observed no noticeable effects on the instability, except when the background electron density largely exceeds the beam density. In that case, a strong two-stream instability occurs, drastically changing the phase space distribution of the beam electrons.