On the Generation of Even Harmonics in Planar and Elliptical Undulators of Free Electron Lasers

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

“…Just as for the spontaneous UR, the spectral characteristics of the FEL radiation are therefore largely determined by the Bessel coefficients f n;x,y , where all main effects are accounted for, while the corrections (13) in Equation ( 12) are influent if the respective losses are non negligible. The harmonic powers obtained with the above formula have proved to be in good agreement with the known documented data of all major FELs [31][32][33][34][35][36][37][38].…”

“…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.…”

“…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]:…”

“…However, for large emittances, a proper account should be done with ζ just as we use convolution ( 8) instead of ( 6) in this case for the spontaneous UR intensity. The details of the phenomenological model can be found in our preceding publications (see, for example, [32][33][34][35][36][37][38][39]).…”