  We propose and analyze a novel regenerative amplifier free electron laser
(FEL) to produce fully coherent x-ray pulses. The method makes use of
narrow-bandwidth Bragg crystals to form an x-ray feedback loop around a
relatively short undulator. Self-amplified spontaneous emission (SASE) from the
leading electron bunch in a bunch train is spectrally filtered by the Bragg
reflectors and is brought back to the beginning of the undulator to interact
repeatedly with subsequent bunches in the bunch train. The FEL interaction with
these short bunches not only amplifies the radiation intensity but also
broadens its spectrum, allowing for effective transmission of the x-rays
outside the crystal bandwidth. The spectral brightness of these x-ray pulses is
about two to three orders of magnitude higher than that from a single-pass SASE
FEL.
