A theoretical study of the spontaneous and stimulated undulator radiation (UR) from electrons in undulators with multiple periods in both transversal directions is presented. Exact expressions are derived for the UR intensities in terms of the generalized Bessel and Airy functions, accounting for undulator field harmonics of arbitrary strength and for real parameters of the beams and installations. Theoretical results are verified with numerical and experimental data for SWISS‐XFEL, PAL‐XFEL, LEUTL, LCLS etc. The spectrum, UR line shape and width, and the harmonic evolution along the undulators are analyzed and compared with the available data for these experiments. Moreover, the effect of the field harmonics is elucidated. It is demonstrated that the third field harmonic can cause distinct odd UR harmonics. The asymmetric undulator field configuration is identified, which allows intense radiation of these harmonics. The power evolution in a free‐electron laser (FEL) with such an undulator is studied by means of an analytical FEL model. The latter is enhanced by a true description of the gradual power saturation of harmonics. A FEL with elliptic undulator and electron–photon phase‐shifting is proposed and modeled. It is shown that the resulting harmonic power from the phase‐shifted elliptic undulator can be significantly higher than from a planar undulator with the same phase‐shifting.

中文翻译：

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多谐波波荡器的FEL辐射的理论光谱分析

提出了在两个横向方向上具有多个周期的波状起伏器中电子自发和受激波状起伏辐射（UR）的理论研究。根据广义贝塞尔函数和艾里函数推导了UR强度的精确表达式，其中考虑了任意强度的波动场谐波以及梁和装置的实际参数。理论结果已通过SWISS‐XFEL，PAL‐XFEL，LEUTL，LCLS等的数值和实验数据验证。分析频谱，UR线的形状和宽度以及沿波荡器的谐波演变，并将其与这些实验的可用数据进行比较。此外，阐明了磁场谐波的影响。结果表明，三次磁场谐波会引起明显的奇数UR谐波。识别出不对称的波荡器场配置，该配置允许强烈辐射这些谐波。通过分析FEL模型研究了带有这种波荡器的自由电子激光器（FEL）中的功率演化。通过真实描述谐波的渐进功率饱和，可以增强后者。提出并建模了具有椭圆波荡器和电子-光子相移的FEL。