Synchrotron Radiation in Periodic Magnetic Fields of FEL Undulators—Theoretical Analysis for Experiments
Abstract
:1. Introduction
2. Spontaneous UR intensity and Spectrum Distortions
3. Analysis of the Harmonic Generation in Some FEL Experiments
3.1. SACLA FEL Experiment
3.2. POHANG FEL X-ray Experiments
4. Conclusions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Phenomenological Model of Harmonic Power Evolution in High-Gain FELs
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Beam parameters: relativistic factor γ = 1526, beam power PE = 234 GW, current I0 = 300 A, current density J = 2.9 × 1010 A/m2, beam section ∑ = 1.03×10−8 m2, emittances μm, βx = 6 m, βy = 4 m, beam size ≈40 μm, divergence ≈ 8 μrad, ~40 μrad, γθ ≈ 0.06, energy spread (per slice) σe = 1.6 × 10−3 | |||||
Undulator parameters: k = 2.1, λu = 1.8 cm, N = 259, section length 4.66 m | |||||
Calculated FEL properties: saturated length Ls = 13 m, gain length Lgain = 1.1 m, radiation beam size ≅ 36 μm | |||||
Harmonic number | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 |
Bessel coefficient fn | 0.79 | 0.09 | 0.32 | 0.09 | 0.18 |
Pierce parameter | 0.0015 | 0.0003 | 0.0008 | 0.0003 | 0.0006 |
Harmonic wavelength λn, nm | 12.4 | 6.2 | 4.1 | 3.1 | 2.5 |
Saturated power PF,n,W | 1.9 × 108 | — | 6 × 105 | — | 3 × 104 |
Beam parameters: relativistic factor γ = 15264, beam power PE = 78 TW, current I0 = 10 kA, current density J = 3.04×1012 A/m2, beam section ∑ = 3.29×10−9 m2, emittances μm, βx,y = 20m, beam size ≈ 22μm, divergence ≈ 1.1 μrad, ~ 9 μrad, γθ ≈ 0.14, energy spread σe = 0.926 × 10−3 | |||||
Undulator parameters: k = 2.1, λu = 1.8 cm, N = 277, section length 4.66 m | |||||
Calculated FEL properties: saturated length Ls = 38 m, gain length Lgain = 2.6 m, radiation beam size ≅ 11 μm | |||||
Harmonic number | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 |
Bessel coefficient fn | 0.79 | 0.19 | 0.27 | 0.19 | 0.11 |
Pierce parameter | 0.00075 | 0.0003 | 0.00037 | 0.0003 | 0.0002 |
Harmonic wavelength λn, nm | 12.4 | 6.2 | 4.1 | 3.1 | 2.5 |
Saturated power PF,n,W | 1.9 × 1010 | 9 × 106 | 5 × 107 | 5 × 106 | 1.6 × 105 |
Beam parameters: γ = 5870, beam power PE = 6.60 TW, current I0 = 2.2 kA, current density J = 1.246 × 1011 A/m2, beam section ∑ = 1.766 × 10−8 m2, emittances = 0.55 μm, β = 30 m, beam size = 53 μm, divergence ≈ 1.8 μrad, ≈ 15 μrad, energy spread σe = 0.5 × 10−3 | |||||
Undulator parameters: k = 2, λu = 3.5 cm, section length 5 m | |||||
Calculated FEL properties: saturated length Ls = 31 m, gain length Lgain = 2.0 m, radiation beam size ≈0.29 mm | |||||
Harmonic number | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 |
Bessel coefficient fn | 0.80 | 0.13 | 0.32 | 0.13 | 0.16 |
Pierce parameter | 0.0010 | 0.0003 | 0.0005 | 0.0003 | 0.0003 |
Harmonic wavelength λn, nm | 1.52 | 0.76 | 0.51 | 0.38 | 0.30 |
Saturated power PF,n,W | 8.2 × 109 | 3.2 × 106 | 5.4 × 107 | 1.6 × 106 | 2.0 × 106 |
Beam parameters: γ = 15,660, beam power PE = 20.0 TW, current I0 = 2,5 kA, current density J = 3.16 × 1011 A/m2, beam section ∑ = 7.91 × 10−9 m2, emittances μm, β ≈ 36 m, beam size = 35 μm, divergence ≈ 1 μrad, ≈4.5 μrad, energy spread σe = 0.18 × 10−3 | |||||
Undulator parameters: k = 1.87, λu = 2.571 cm, section length 5 m | |||||
Calculated FEL properties: saturated length Ls ~ 55 m, gain length Lgain = 3.4 m, radiation beam size ≈15 μm | |||||
Harmonic number | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 |
Bessel coefficient fn | 0.82 | 0.09 | 0.31 | 0.09 | 0.16 |
Pierce parameter | 0.0004 | 0.00009 | 0.0002 | 0.00009 | 0.00014 |
Harmonic wavelength λn, nm | 0.144 | 0.072 | 0.048 | 0.036 | 0.029 |
Saturated power PF,n,W | 1.0 × 1010 | 2.0 × 106 | 1.0 × 108 | 1.0 × 106 | 6.0 × 106 |
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Zhukovsky, K. Synchrotron Radiation in Periodic Magnetic Fields of FEL Undulators—Theoretical Analysis for Experiments. Symmetry 2020, 12, 1258. https://doi.org/10.3390/sym12081258
Zhukovsky K. Synchrotron Radiation in Periodic Magnetic Fields of FEL Undulators—Theoretical Analysis for Experiments. Symmetry. 2020; 12(8):1258. https://doi.org/10.3390/sym12081258
Chicago/Turabian StyleZhukovsky, Konstantin. 2020. "Synchrotron Radiation in Periodic Magnetic Fields of FEL Undulators—Theoretical Analysis for Experiments" Symmetry 12, no. 8: 1258. https://doi.org/10.3390/sym12081258