In recent years, echo-enabled harmonic generation (EEHG) demonstrated that it is capable to upshift the seed frequency in an FEL by almost two orders of magnitude. In this presentation, I will talk about the EEHG concept in FELs and show its connection to the echo effect known in other areas of physics (such as spin echo, plasma echo, echo in accelerators, etc.). The physics of the EEHG will be discussed as well is its major limitations.

Quantum diffusion is caused by the recoil effect that a particle experiences when it emits a photon [1]. Quantum diffusion due to the synchrotron radiation in high-energy electron and positron circular accelerators defines the main parameters of the beam: its energy spread and hence the bunch length, as well as the horizontal emittance. It is calculated as a single particle effect assuming incoherent radiation. This assumption is not valid in FELs where the radiation is coherent. In this work, we develop theory of the quantum diffusion in coherent radiation and show that it leads to the energy diffusion of the particles that is correlated between the different positions in the bunch.

Accurate measurements of the x-ray pulse duration produced by x-ray free-electron lasers (XFELs) typically rely on longitudinal electron beam phase space diagnostics, e.g. in a transverse deflecting cavity or TCAV, or from measurements of spectral correlations. All of the known spectral methods share the weakness that they will underestimate the pulse length in the case that the FEL spectrum is broadened due to the electron beam having an energy chirp. We present a statistical analysis of FEL radiation in the presence of a linear electron beam energy chirp which extends previous results by including an accurate description of the FEL gain process. In doing so, we show that with measurements of the spectral intensity correlations and the average spectrum, one can reconstruct the x-ray pulse length, e-beam chirp, and spectrometer resolution. Our approach is validated by comparison with 1D FEL simulations.