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We model the evolution of radiation harmonics in a single-pass free-electron laser using a phenomenological model, in which the main losses are taken into account separately for each of the harmonics. The modeling results are compared with the corresponding FEL experiments and the simulations, which we performed using the PERSEO code. A wide wavelength range (0.15–500 nm) is considered. The phenomenological description based on the use of several basic FEL parameters, such as the electron beam current, average energy, energy spread, and emittance of electrons, yields the results that correspond well to the LEUTL, SPARC, and LCLS experiments in various conditions. The evolution of the radiation power, bunching, and electron energy spread in the Spring 8 FEL at the follow-up stage is also considered. The necessity to improve electron beam parameters for generation of harmonics in this FEL is shown. The phenomenological model allows assessing the operation of the available and future FELs fast. Using it, one can model FELs with nearly any undulator with allowance for higher harmonics of their magnetic fields. Moreover, this model makes it possible to describe easily various effects in designed FELs, e.g., those with dephasing of electrons and photons, filtering of harmonics, and other features.

The paper presents a theoretical study of the harmonic power evolution in a single-pass free-electron laser (FEL) and a comparison with some of the experimental results. The phenomenological approach which considers the main FEL parameters (current density, the Lorentz factor, electron energy spread, and the beam geometry) provides a description of any undulator. With due regard to the real laser beam parameters, the effect from high-order harmonic radiation and the beam deviation from the undulator axis on the FEL irradiation is investigated. The Bessel coefficients are calculated for the undulator in the presence of the second periodic field. The phenomenological approach provides the free-electron laser simulation experiments at Sorgente Pulsata Auto-amplificata di Radiazione Coerente (SPARC), SPring-8 Angstrom Compact Free-Electron Laser (SACLA) and Linac Coherent Light Source (LCLS). These studies include the high-harmonic generation, a comparison of the emission power evolution with the experimentally obtained data on the harmonic generation including even and odd high-order harmonics, gain and saturation lengths and the respective power. A possible effect from the third harmonic of the planar undulator magnetic field is studied in relation to the FEL irradiation. It is shown that this effect is insignificant even at the magnetic-field harmonic amplitude equaling ~1/10 of the amplitude of the ideal magnetic field in the periodic undulator. It is found that the second-harmonic generation in FEL lasing is rather weak, that is in good agreement with the measurement results as well as the behavior of the high-order harmonics.

We present a theoretical study and a computer simulation of characteristics of the undulator radiation in single-pass free-electron lasers (FELs). Using a phenomenological model describing the dynamics of the radiated power in FELs with allowance for the basic loss, we study generation of harmonics in the X-ray range in a FEL with a two-frequency undulator. We study the possibility to achieve a hundredfold increase in the radiation intensity of the nth harmonic in a FEL, in which the electron-phase shift by kπ/n with respect to photons occurs between undulator sections, where k = 2, 4, . . . . The advantages of using a two-frequency undulator in a single-pass FEL and the possibility of generating the high-power X-ray radiation by the FEL at the harmonic wavelengths 2.3–3.3 nm in the linear regime are demonstrated. The FEL is compared with the two-frequency undulator and the conventional plane undulator. Additionally, generation of radiation having a power of tens of megawatts is studied at the wavelength λ ≈ 3.27 nm in a multistage FEL with a length of 40 m, an off-the-shelf excimer ultraviolet seed laser, which operates at a wavelength of 157 nm, and an electron beam having an energy of about 0.6 GeV.