Generation of Coherent X-ray Harmonic Radiation in a Single-Pass Free-Electron Laser with Phase Shift of Electrons Relative to Photons

Evolution of bunching and harmonic power is theoretically studied for a single-pass free-electron laser (FEL) in which electrons are periodically phase-shifted relative to photons in intervals between undulators with harmonics of magnetic field. Analytical calculations are used to reveal the configuration of magnetic field of an asymmetric elliptical and planar undulators that make it possible to suppress the fundamental and amplify several higher harmonics. The evolution of power and bunching is analyzed using a phenomenological model of FEL that describes violation of bunching between the undulator stages and a relatively high sensitivity of electron-photon interaction at wavelengths of higher harmonics to energy spread, size, deflection of the beam from the axis, etc. Expressions for the Bessel coefficients of the planar and elliptical undulators with field harmonics are analytically studied. Such undulators can be used in phase-shifted FELs for an increase in the power of higher (X-ray) harmonics by a factor of up to 100 in comparison with conventional FELs.