Abstract: The generation of very high power pulses of coherent electromagnetic radiation that are continuously tunable in frequency is accomplished by means of a free electron laser in which a hollow relativistic electron beam is projected along the longitudinal axis of an evacuated drift tube. A first magnetic field expands the electron beam into an annular peripheral interaction region of the drift tube where the beam interacts with a second periodic radial magnetic field. Frequency is varied by changing the electron velocity of the electron beam or by changing the periodicity of the radial magnetic field. The device can be made to operate as an oscillator by the inclusion of resonant cavity defining mirrors within the interaction region, or as an amplifier by injecting a coherent radiation signal into the interaction region. Both oscillator and amplifier functions can be incorporated into a single device.