Theoretical Analysis of Radiation Properties of X-Ray Free-Electron Lasers

The Influence of Increased Electron Energy Spread on the Radiation of the Second Harmonic in Free Electron Lasers

“…Angle θ is specific for FEL radiation; the spontaneous UR does not involve photon-electron interactions. It was demonstrated in [47][48][49][50][51][52] that accounting for the effective angle θ of electron-photon interaction, analytical formulae can describe the FEL harmonic powers in agreement with their measured values for all major FELs in the range from visible to X-rays. Moreover, the FEL harmonic powers calculated accounting for the electron-photon interaction angle θ in [47][48][49][50][51][52] were closer to the experimental data than early numerical simulations for FELs were in the years 2000-2009; the results for FEL harmonics in [47][48][49][50][51][52] compare with the numerical simulations for FELs in the last decade.…”

“…∝ ω n γ ; for relativistic beams γ>>1, thus, we get ω β << ω n and conclude that betatron harmonics p are tightly packed within a UR harmonic n. More precisely, the split and contribution of each betatron harmonic within a radiation line were computed for major FELs in [47,48,52]. The number of betatron harmonics p within one radiation line varies from one to two up to one to two dozens.…”

“…However, the second-order harmonic response of the matter must be separated also from the second harmonic of the FEL. The second harmonic in FEL Symmetry 2023, 15, 2180 2 of 18 radiation appears due to multiple reasons, mainly due to off-axis effects in undulators (see for example, [45][46][47][48][49][50][51][52][53][54][55]); it is reflected by the examined sample and mixed with the detected signal of the SHG. Such contamination shadows the SHG and makes unclear the detected response; its interpretation and understanding may be erroneous in this case.…”

“…A phenomenological value of an effective angle can be specified [50,51], in which the electrons view the UR in the beam section on the FEL gain length. It was demonstrated in [47][48][49][50][51][52] for all major FELs, operating in the range from visible to X-rays, that the contribution of the above-specified angular term to the even harmonic FEL power well exceeds the contributions from the betatron oscillations in the beams, especially for X-ray FELs, where the relativistic factor is high, γ~10 3 . The deflection angle of a beam in an FEL is usually smaller than the effective angle of a photon-electron interaction in an FEL; however, they may compare with each other in some cases, for example, for LCLS FEL.…”

“…Usually, an account for emittance alone yields the calculated power of the second FEL harmonic one order of magnitude less than the measured value. In order to get realistic prediction and description of even harmonic powers in FELs [47][48][49][50][51][52], we must consider that stimulated radiation occurs due to electron bunching, which in turn, is a result of the interaction of radiation with the electrons of the macro bunch. Each act of electron-photon interaction happens at a random angle, specific for the electron-photon pair.…”