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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.

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.