The phenomenological modeling of nonlinear harmonic generation in single-pass free electron laser (FEL) experiments was performed. The gradual saturation of the harmonic power and its evolution around the saturated regime was modeled. The model includes all major losses for each harmonic with account for the fact that the undulator radiation (UR) and the electron-wave interaction at higher harmonic wavelengths are more sensitive to the beam energy spread, divergence, and other factors. The results of the phenomenological modeling are compared with the experimental data of the corresponding FELs and with the numerical simulation. The proposed description of nonlinear harmonic generation appears to be in good agreement with the FEL experiments SALCA, LEUTL, LCLS, and others under various conditions. The results also match their numerical simulations; in contrast to the latter, our phenomenological model allows rapid, still, and accurate modeling of the FEL harmonic power and bunching evolution with a PC or even an engineering calculator. The model permits studying FEL radiation from undulators with arbitrary magnetic field configuration. Using the analytical FEL model, we demonstrate that, in contrast to the helical undulator, in whose spectrum the fundamental and the second harmonics prevail, the FEL radiation from the elliptical undulator with the third field harmonic has a distinct fifth UR harmonic as well as the fundamental one, whereas other harmonics can be ignored even close to the saturation length.

中文翻译：

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自由电子激光器中接近饱和区的非线性谐波产生的解析描述

进行了单次自由电子激光（FEL）实验中非线性谐波产生的现象学建模。对谐波功率的逐渐饱和及其在饱和状态下的演化进行了建模。该模型包括每个谐波的所有主要损耗，这是因为在较高谐波波长下，波动辐射（UR）和电子波相互作用对电子束能量的传播，发散和其他因素更加敏感。现象学建模的结果与相应FEL的实验数据以及数值模拟进行了比较。所提出的非线性谐波产生的描述似乎与各种条件下的FAL实验SALCA，LEUTL，LCLS等相吻合。结果也与他们的数值模拟相符。与后者相反，我们的现象学模型允许使用PC甚至工程计算器对FEL谐波功率进行快速，静态和准确的建模，并进行成束演化。该模型允许研究来自具有任意磁场配置的波荡器的FEL辐射。使用分析型FEL模型，我们证明，与螺旋波状起伏器不同，在其频谱中以基波和二次谐波为主，椭圆形起伏器的FEL辐射具有三次场谐波，并且具有明显的五阶UR谐波。基本谐波，而其他谐波甚至可以接近饱和长度而忽略不计。使用PC甚至是工程计算器对FEL谐波功率进行精确建模，并进行成束演化。该模型允许研究来自具有任意磁场配置的波荡器的FEL辐射。使用解析FEL模型，我们证明，与螺旋波状起伏器不同，在其频谱中以基波和二次谐波为主，椭圆形起伏器的FEL辐射具有三次场谐波，并且具有明显的五阶UR谐波。基本谐波，而其他谐波甚至可以接近饱和长度而忽略不计。使用PC甚至是工程计算器对FEL谐波功率进行精确建模，并进行成束演化。该模型允许研究来自具有任意磁场配置的波荡器的FEL辐射。使用分析型FEL模型，我们证明，与螺旋波状起伏器不同，在其频谱中以基波和二次谐波为主，椭圆形起伏器的FEL辐射具有三次场谐波，并且具有明显的五阶UR谐波。基本谐波，而其他谐波甚至可以接近饱和长度而忽略不计。