A novel method of gating electron emission from field-emitter cathodes for radio frequency (RF) electrode guns and a novel cathode that provides a focused electron beam without the need for magnetic fields or a curved cathode surface are provided. The phase and strength of a predefined harmonic field, such as the 3rd harmonic field, are adjusted relative to a fundamental field to cause a field emission cathode to emit electrons at predefined times for the generation of high-brightness electron beams. The emission time is gated responsive to the combined harmonic and fundamental fields and the response of the FE cathode to the combined fields. A planar focusing cathode includes a selected dielectric material, such as a ceramic material, to provide an electron beam emission surface. Metal surfaces are provided both radially around and behind the dielectric material to shape the electric fields that accelerate and guide the beam from the cathode surface.