Relativistic theory of the collective atomic recoil laser

We propose a new method for producing coherent, tunable radiation at short wavelength using the collective Compton backscattering of a counter-propagating radiation field from a relativistic two-level particle beam (generically called atoms). Under proper conditions one can have exponential growth of the co-propagating, back-scattered radiation at the Doppler up-shifted frequency at the expense of the recoil energy of the particle beam. This growth arises from self-bunching of the particle beam which generates the coherent back scattering of the incident field. We show that the gain and efficiency of the system, for a given density, scale as the inverse cube root of the beam energy instead of the inverse of the energy as in the free electron laser (FEL). Examples of an ion beam and of electric spins are discussed.