Theoretical spectral analysis of FEL radiation from multi-harmonic undulators

Abstract

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.

Об Излучении Гармоник В Рентгеновских Лазерах На Свободных Электронах С Изменяемым Параметром Дипольности Ондуляторов

“…Отметим, что в соответствующих выражениях (4) для плоского ондулятора присутствуют четные гармоники излучения за счет бетатронного вклада ∼ y 0 и за счет вклада угловых эффектов ∼ γθ. Вклад непериодических компонент поля в генерацию четных гармоник ОИ учитывается аналогично угловому вкладу в (4), где в качестве угла γθ используется значение угла изгиба γθ H , индуцированного непериодическим полем H d [51][52][53][54][55]:…”

Theoretical spectral analysis of FEL radiation from multi-harmonic undulators. Corrigendum

“…The author would like to correct a misprint in formula 3of Zhukovsky (2020) and communicate the change of the numerical values in equations (23), (24) and in Fig. 9.…”

“…was erroneously written in equation 3of Zhukovsky (2020) as a cosine instead of an exponential; formula (3) should correctly read as follows,…”

“…The undulator radiation (UR) harmonic powers in the undulator with N = 30 periods, k = 2.216 and period u = 2.3 cm, and the beam with energy spread e = 0.1%, are shown here in the revised versions of Figures 9(a) and 9(b), which replace the respective figures in the original paper. The ratio between the harmonic intensity of the undulator and the magnetic field (21) [see Zhukovsky (2020)] has somewhat changed, but, for the studied undulator field (22) in Zhukovsky (2020),…”

“…The changes of the numerical values in (26), (27), (29), (30) are minor and we omit them for brevity. The harmonic intensities of the UR spectrum, obtained with formula (3) for the undulator in Zhukovsky (2020), are now revised and reported in the revised Figures 9(a) and 9(b) for x-and ypolarizations, respectively. We consider [see also Zhukovsky (2020)] an undulator with period u = 2.3 cm, k = 2.216, magnetic field given by (22), electron beam relativistic parameter = 12.72, emittances " x = 1.5 mm mrad and " y = 0.35 mm mrad, Twiss parameters x = 43.66 cm and y = 28.75 cm, beam sections x = 809 mm and y = 317 mm, divergences x = 4.5 mrad and y = 1.6 mrad and electron energy spread e = 0.1%.…”