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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.

We analyze undulator radiation in some multiperiodic magnetic fields. We consider undulators with planar, helical, and elliptical axisymmetric magnetic fields with higher field harmonics. For these undulators, we obtain exact analytic expressions for generalized Bessel functions and Bessel coefficients. Analytic results are compared with the results of numerical simulation. We analyze the effect of additional third field harmonic on radiation emitted by these undulators. For a helical undulator with an additional asymmetric third harmonic field, it is found that the fifth harmonic of undulator radiation prevails over the third harmonic. We perform 3D simulation of radiation from free-electron laser with such an undulator using the specially developed numerical program that accounts for the electron energy spread in the beam as well as betatron oscillations. We analyze two-frequency undulator with a harmonic elliptically polarized magnetic field and simulate radiation of a free-electron laser with such an undulator.

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.

A numerical study of the effect of betatron oscillations on the second harmonic generation in free-electron lasers (FELs) is presented. Analytical expressions for the effective coupling strength factors are derived that clearly distinguish all contributions in subharmonics and each polarization of the radiation. A three-dimensional time-dependent numerical FEL code that takes into account the main FEL effects and the individual contribution of each electron to the second harmonic generation is presented. Also, the X- and Y-polarizations of the second harmonic are analyzed. The second harmonic was detected in experiments at the Advanced Photon Source (APS) Low Energy Undulator Test Line (LEUTL) and Linac Coherent Light Source (LCLS) in the soft X-ray regime. The approach presented in the article can be useful for a comprehensive study and diagnostics of XFELs. In the paper, the LCLS and Pohang Accelerator Laboratory X-ray Free-Electron Laser (PAL-XFEL) experiments are modeled. The simulation results are in a good agreement with the experimental data.

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.

The harmonic power and bunching evolution in X-ray single-pass free-electron lasers (FELs) is modelled and the harmonic generation in a phase-shifted two-frequency FEL is explored. The advanced phenomenological FEL model, which is validated numerically and experimentally, is employed. The model accounts for major losses for each harmonic individually; it is compared with reported experimental data and with PERSEO numerical simulations, which are performed here for a variety of experiments. The latter cover the radiation wavelength range 0.15–300 nm. The phenomenological description is based on a few key FEL parameters: electron beam section, current, energy and its spread and divergence. The model is employed for modelling harmonic bunching and power evolution in a phase-shifted X-ray FEL with a two-frequency undulator, where lower harmonics with numbers less than nth are suppressed by the electron–photon phase shift of kπ/n, k = 2, 4, …, between the undulator sections. The benefits of the two-frequency phase-shifted FEL are highlighted. FEL-induced energy spread is shown to be three times lower than in a FEL without the phase-shift. The high-power harmonic and sub-harmonic radiation in such a FEL is demonstrated. In particular, powerful ∼14 GW X-ray radiation at λ5 = 0.15 nm from electrons with energy of 5.47 GeV and beam current ∼3.66 kA is possible in a two-frequency phase-shifted FEL at 30 m; this constitutes half of a FEL length where a common planar undulator radiates the same wavelength and power at the fundamental harmonic. Moreover, about a three times lower energy spread is induced by the dominant fifth harmonic, and the harmonic power can be thousands of times higher than in a common planar undulator FEL.