Analytical formalism for the description of free electron laser (FEL) spectrum is proposed with account for the electron beam size, divergences, beam energy spread, the off-axis deflection and the periodic harmonic and non-periodic constant magnetic field components in undulators. The exact analytical expressions for the spontaneous undulator radiation (UR) are obtained in terms of generalized Bessel and Airy functions, which describe the spectrum line shape and intensities. The obtained analytical formulae distinguish the effects of each installation and beam parameter on the FEL harmonic radiation. The enhanced phenomenological model is employed for the FEL modeling; it describes the harmonic powers evolution along the undulators and gradual saturation around the saturation length with account for the saturated powers oscillations. With this model, we study the generation of FEL harmonics, including even, in several instances of FEL experiments at LCLS, LEUTL and PAL-XFEL. We analytically demonstrate the even harmonics reasons in the LCLS and LEUTL FEL experiments. The spectrum line split due to the beam size is demonstrated analytically and compared with experimentally measured spectral characteristics. The PAL-XFEL experiment is analyzed and compared with the LCLS experiment; the possibility of high harmonic radiation in it is explored. The modeling results fully agree with the measured values in the studied FELs. The analytical formalism shows physical background for each harmonic in every FEL. It allows the analysis of the radiation and beam alignment in projected, built and working FELs and it can help minimizing losses and correcting magnetic fields as well as projecting experiments with FELs.

提出了描述自由电子激光 (FEL) 光谱的分析形式，其中考虑了电子束尺寸、发散度、束能量扩展、离轴偏转以及波荡器中的周期性谐波和非周期性恒定磁场分量。自发波荡器辐射 (UR) 的精确解析表达式是根据广义 Bessel 和 Airy 函数获得的，这些函数描述了谱线形状和强度。获得的解析公式区分了每个安装和光束参数对 FEL 谐波辐射的影响。FEL建模采用增强的现象学模型；它描述了沿波荡器的谐波功率演变和围绕饱和长度的逐渐饱和，并考虑了饱和功率振荡。使用这个模型，我们研究了 FEL 谐波的产生，包括偶数，在 LCLS、LEUTL 和 PAL-XFEL 的几个 FEL 实验实例中。我们分析证明了 LCLS 和 LEUTL FEL 实验中的偶次谐波原因。由于光束尺寸而导致的谱线分裂被分析证明并与实验测量的光谱特性进行比较。PAL-XFEL实验与LCLS实验分析比较；探讨了其中存在高次谐波辐射的可能性。建模结果与所研究的 FEL 中的测量值完全一致。分析形式显示了每个 FEL 中每个谐波的物理背景。它允许分析投影中的辐射和光束对准，