Echo-enabled harmonic generation free electron laser

In this paper, we systematically study the echo-enabled harmonic generation (EEHG) free electron laser (FEL). The EEHG FEL uses two modulators in combination with two dispersion sections that allow one to generate in the beam a high harmonic density modulation starting with a relatively small initial energy modulation of the beam. After presenting an analytical theory of the phenomenon, we address several practically important issues, such as the effect of incoherent synchrotron radiation in the dispersion sections, and the beam transverse size effect in the modulator. Using a representative realistic set of beam parameters, we show how the EEHG scheme enhances the FEL performance and allows one to generate a fully (both longitudinally and transversely) coherent radiation. As an example, we demonstrate that 5 nm coherent soft x rays with GW peak power can be generated directly from the 240 nm seeding laser using the proposed EEHG scheme.