The design of an electron gun capable of producing beam emittance one order of magnitude lower than current technology would reduce considerably the cost and size of a free electron laser emitting at 0.1nm. Field emitter arrays (FEAs) including a gate and a focusing layer are an attractive technology for such high brightness sources. Electrons are extracted from micrometric tips thanks to voltage pulses between gate and tips. The focusing layer should then reduce the initial divergence of each emitted beamlets. This FEA will be inserted in a high gradient diode configuration coupled with a radiofrequency structure. In the diode part very high electric field pulses (several hundreds of MV/m) will limit the degradation of emittance due to space charge effect. This first acceleration will be obtained with high voltage pulses (typically a megavolt in a few hundred of nanoseconds) synchronized with the low voltage pulses applied to the FEA (typically one hundred of volts in one nanosecond at frequency below kilohertz). This diode part will then be followed by an RF accelerating structure in order to bring the electrons to relativistic energies.