  We propose an unconventional electron gun structure in which the emitter is
located on a concave cathode surface with a non-uniform electric field. Such a
design violates the intuition that an emitter should place close to a uniform
electric field to reduce the velocity spread. The commonly employed design
guide based on the adiabatic condition predicts a huge velocity spread of 24%,
but the simulation using EGUN code and verified with CST particle studio shows
a very low spread of 2.8%. Examining the magnetic moment and the kinetic energy
of the beam reveals that the electrons experience a relatively long
acceleration process due to the much weak electric field. That's why the
non-adiabatic effect matters. In addition to the cyclotron compression and the
E$\times$B drift, the "resonant" polarization drift plays a crucial role in
reducing the overall velocity spread.Simulations show a decent beam quality
with the pitch factor of 1.5 and the transverse velocity spread of 2.8% over a
wide range of the magnetic field (7.4-8.0 T) and the beam voltage (12-22 kV)
with a high structural tolerance. The promising results with the wide working
range enable the development of continuous frequency-tunable gyrotrons.
