Synchrotron Radiation in Periodic Magnetic Fields of FEL Undulators—Theoretical Analysis for Experiments

Abstract

A theoretical study of the synchrotron radiation (SR) from electrons in periodic magnetic fields with non-periodic magnetic components is presented. It is applied to several free electron lasers (FELs) accounting for the real characteristics of their electron beams: finite sizes, energy spread, divergence etc. All the losses and off-axis effects are accounted analytically. Exact expressions for the harmonic radiation in multiperiodic magnetic fields with non-periodic components and off-axis effects are given in terms of the generalized Bessel and Airy-type functions. Their analytical forms clearly distinguish all contributions in each polarization of the undulator radiation (UR). The application to FELs is demonstrated with the help of the analytical model for FEL harmonic power evolution, which accounts for all major losses and has been verified with the results of well documented FEL experiments. The analysis of the off-axis effects for the odd and even harmonics is performed for SPRING8 Angstrom Compact free-electron LAser (SACLA) and Pohang Accelerator Laboratory (PAL-XFEL). The modelling describes theoretically the power levels of odd and even harmonics and the spectral line width and shape. The obtained theoretical results agree well with the available data for FEL experiments; where no data exist, we predict and explain the FEL radiation properties. The proposed theoretical approach is applicable to practically any FEL.

Theoretical Analysis of the Radiation Properties of Some Major X‐Ray Free Electron Lasers

“…The analytical expressions for the Bessel coefficients allow fast analysis of the spontaneous UR and stimulated UR in single-pass FELs; proper results were obtained, for example, in refs. [57][58][59][60][61][62] Omitting most cumbersome expressions for undulators with field harmonics, (see refs. [92,[57][58][59][60][61][62]), we give the most useful and simple analytical formulae for the Bessel coefficients for a common planar undulator:…”

“…[57][58][59][60][61][62] Omitting most cumbersome expressions for undulators with field harmonics, (see refs. [92,[57][58][59][60][61][62]), we give the most useful and simple analytical formulae for the Bessel coefficients for a common planar undulator:…”

“…The comparative contributions of different terms to the Bessel coefficients and the respective radiation intensity contributions were studied in refs. [57][58][59][60][61][62]. The results for the Bessel coefficients f n close to the axis of a common planar undulator are as follows: the smallest contribution is due to the betatron oscillations:f 𝛽 n,p ≈ 10 -2 , the angular terms give an order of magnitude larger contribution: f angular n ≈ 0.1, the odd harmonics Bessel coefficients are by far the largest: f n=1,3,5 ≈ 0.8, 0.3, 0.15, although these values strongly depend on the deflection parameter k, so that the values f n=1,3,5 ≈ 0.8, 0.3, 0.15 show the hierarchy of the values for n = 1,3,5.…”

“…For example, the difference for the pulse energy along the undulators in the FEL experiment SPARC, evaluated in various numerical codes, [56] was up to one order for magnitude for the first harmonic and even more for high harmonics. In our study we use the analytical approach, which combines the exact formulation for the UR harmonic Bessel factors, calculated analytically in terms of the generalized Bessel and Airy functions, and phenomenological approach to the FEL power and bunching modeling, verified with FEL experiments, [26][27][28][57][58][59][60][61][62] and numerical simulations. [63][64][65] In what follows we will analyze, compare and discuss the harmonic generation in major FELs constructed in the 21st century and operating from visible to X-ray range.…”

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]:…”