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.

给出了波动性辐射（UR）谐波产生的分析说明，其中考虑了有限的电子束大小，发射率，离轴束偏差和电子能量散布以及恒定磁分量和场谐波的影响。为广义贝塞尔和艾里函数获得的精确解析表达式描述了考虑到上述因素的两频波荡器中的谱线轮廓和UR强度。所获得的分析公式可用于区分每个场分量，波状起伏和光束参数对自由电子激光器（FEL）的谐波辐射的贡献。考虑到有限的光束尺寸及其离轴偏差，分析了磁场对谐波辐射的影响。现象学模型用于FEL建模；在直线加速器相干光源（LCLS）和低能波荡器测试线（LEUTL）实验中，研究了谐波的产生，包括偶数次谐波。分析地证明，在LCLS实验中，X射线FEL中在λ= 0.75 nm波长处强的第二FEL谐波是由于电子轨迹在1.6 m的增益长度上偏离轴心15μm所致。与光束大小相当；在LEUTL实验中，在λ= 192 nm波长处的强第二FEL谐波可以归因于电子束本身的大横截面。建模结果与测量结果完全一致。发达的形式主义使得可以分析投射和建造的FEL及其辐射。它有助于最大程度地减少损耗并纠正磁场；