Analytical Account for Off-Axis Effects in X-Ray Radiation of Harmonics of Free-Electron Lasers

Analytical description of generation of undulator radiation (UR) harmonics is given with account for the effects of finite electron beam size, emittance, off-axis beam deviation, and electron energy spread as well as of constant magnetic components and field harmonics. Exact analytical expressions obtained for the generalized Bessel and Airy functions describe the spectrum line profiles and the UR intensities in a two-frequency undulator with account for the above-enumerated factors. The obtained analytical formulas can be used to distinguish contributions of each field component and undulator and beam parameters to harmonic radiation of free-electron lasers (FELs). The effect of the field on harmonic radiation is analyzed with account for the finite beam size and its off-axis deviation. A phenomenological model is employed for FEL modeling; with its help, generation of the harmonics, including even ones, is studied in the Linac Coherent Light Sourсe (LCLS) and Low Energy Undulator Test Line (LEUTL) experiments. It is demonstrated analytically that in the LCLS experiment, the strong second FEL harmonic in the x-ray FEL at the wavelengths λ = 0.75 nm is due to the off-axis deviation of electron trajectories by 15 μm on a 1.6 m gain length, which is comparable to the beam size; the strong second FEL harmonic in the LEUTL experiment at the wavelength λ = 192 nm can be attributed to the large cross section of the electron beam itself. The results of modeling are in complete agreement with measurements. The developed formalism allows the projected and built FELs and their radiation to be analyzed; it helps losses to be minimized and magnetic fields to be corrected; it also shows physical background and reasons for each harmonic radiated power in the FEL under study.