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.

中文翻译：

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FELs中二次谐波产生的数值研究

提出了对电子加速器振荡对自由电子激光器（FEL）中二次谐波产生影响的数值研究。得出了有效耦合强度因子的解析表达式，可以清楚地区分亚谐波和辐射的每个极化的所有贡献。提出了一个三维时间相关的数字FEL代码，该代码考虑了主要FEL效应以及每个电子对二次谐波的贡献。同样，X-和Y分析了二次谐波的极化。在软X射线条件下，在高级光子源（APS）低能波荡器测试线（LEUTL）和直线加速器相干光源（LCLS）的实验中检测到了二次谐波。本文中介绍的方法对于XFEL的全面研究和诊断很有用。在本文中，对LCLS和浦项加速器实验室X射线自由电子激光（PAL-XFEL）实验进行了建模。仿真结果与实验数据吻合良好。