Abstract: High efficiency amplification of radio frequencies to very high power levels including: establishing a cylindrical cloud of electrons; establishing an electrical field surrounding and coaxial with the electron cloud to bias the electrons to remain in the cloud; establishing a rotating electrical field that surrounds and is coaxial with the steady field, the circular path of the rotating field being one wavelength long, whereby the peak of one phase of the rotating field is used to accelerate electrons in a beam through the bias field in synchronism with the peak of the rotating field so that there is a beam of electrons continuously extracted from the cloud and rotating with the peak; establishing a steady electrical field that surrounds and is coaxial with the rotating field for high-energy radial acceleration of the rotating beam of electrons; and resonating the rotating beam of electrons within a space surrounding the second field, the space being selected to have a phase velocity equal to that of the rota