Radiation from Elliptical Undulators with Magnetic Field Harmonics

Numerical study of the second harmonic generation in FELs

“…They have a double lobe structure, as was recorded by Ratner et al (2011). Our formalism can also be applied to other magnetic field configurations, for example for asymmetric elliptic undulators (Kalitenko & Zhukovskii, 2020) and two-frequency undulators (Zhukovsky & Kalitenko, 2019).…”

Beam Asymmetry, Divergence and Energy Spread Effects on the Radiation from Planar Undulators

“…The analytical tool for the study of the UR is based on the formalism of the generalized Bessel functions. In the most general case it involves very complex integrals to account for field harmonics and the resulting expressions can be rather cumbersome when multiple field harmonics in both directions are considered for the radiation off the undulator axis (see for details, for example, [28][29][30][31][32][33][34][35][36][37][38][39]). In most cases, however, single period planar undulators are used with weak field harmonic content; it has minor effect on the radiation.…”

“…is the detuning parameter from the resonance ω n = 2πc/λ n ; the Bessel coefficients f n;x,y for the xand y-polarizations depend on the parameters of the beam and the undulator. Omitting the field harmonics effect, which is a stand-alone subject addressed in [23,25,26,31], et al, the Bessel coefficients f n;x,y in (2) reduce for a planar undulator to the following rather simple form:…”

“…Omitting the detailed description of the phenomenological model for FEL harmonics, based on the early versions in [46,47] and presented in its current form in recent publications (see [23][24][25][26][28][29][30][31][32][33][34][35][36][37][38][39]), we briefly outline that in a FEL the interaction of the radiation with electrons in the undulator causes qualitative change of the power growth along undulators. The quadratic growth at the beginning of a FEL, where the spontaneous radiation dominates, changes to the exponential growth P(z) ∝ P 0 e z/L g0 of the power from bunched electrons; the initial power is P 0 , and it can originate from a seeding laser in an amplifier FEL or from coherent fluctuations of the initial noise of a bunch in a SASE FEL; in the latter case it reads as follows: P n,noise ≈ 1.6ρ 2 n e4πcP beam /(I 0 λ n ) [48].…”

“…where the effects of the energy spread σ ε and the emittance ε x,y are phenomenologically described by the coefficients, and verified for FELs in the wide radiation range from infrared to hard X-rays [25][26][27][28][29][30][31][32][33][34][35][36][37]:…”