Free-Electron Laser
SASE free-electron laser: resonance wavelength, Pierce parameter ρ with the beam size, 3D gain length from Ming Xie's fit, saturation power and length, and the power along the undulator.
Electron beam and planar undulator
Resonance wavelength λ₁
823.9 nm
γ = 196.7; [JJ] = 0.8514
Pierce parameter ρ
0.004131
beam size σ_x = √(βε_n/γ) = 159.4 µm
Power gain length L_G
0.5730 m
1D cold beam L_G0 = 0.3337 m
Saturation length
12.28 m
L_G ln(9P_sat/P_n), noise P_n = 0.09998 W
Saturation power
22.53 MW
1.6ρ(L_G0/L_G)² of P_beam = 10.05 GW
Power at the undulator end
2.253e7 W
saturated
Xie parameters η_d, η_ε, η_γ
0.86, 0.13, 0.14
diffraction, emittance, energy spread; keep ≪ 1
λ₁ = λ_u(1 + K²/2)/(2γ²); ρ = [(1/16)(I/I_A)K²[JJ]²/(γ³σ_x²k_u²)]^(1/3) with I_A = 4πε₀mc³/e = 17.045 kA and [JJ] = J₀(ξ) − J₁(ξ), ξ = K²/(4 + 2K²) (Huang–Kim). L_G0 = λ_u/(4π√3 ρ); L_G = L_G0(1 + Λ) from Ming Xie's fit in η_d = L_G0/(2k₁σ_x²), η_ε = (L_G0/β)(4πε/λ₁), η_γ = 4π(L_G0/λ_u)σ_δ, within about 10 % of numerical solutions.
SASE from shot noise: P(z) = (P_n/9) exp(z/L_G) up to P_sat, with the 1D noise P_n = ρ²cγmc²/λ₁. The plot stops growth at P_sat; the real power overshoots and oscillates there.
Power along the undulator
- P(z), exponential gain to P_sat
- Undulator end
Gain length against undulator K
- L_G with emittance, diffraction, energy spread (Xie)
- L_G0, 1D cold beam
- Input K
Resonance wavelength against electron energy
- λ₁
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Planar undulator. ρ from Huang–Kim with the rms beam size √(βε_n/γ); the gain length includes diffraction, emittance and energy spread through Ming Xie's fit (within about 10 % of numerical solutions). Start-up from the 1D shot-noise power and Xie's empirical saturation power 1.6ρ(L_G0/L_G)²P_beam; the plot caps the exponential growth at P_sat. Checked against Xie's LCLS design table and the LCLS ρ = 4.5e-4. The fit is extrapolated (flagged) once the energy spread σ_δ exceeds ρ; no saturation length when 9P_sat ≤ P_n. No helical undulators, taper or harmonics.
- Huang Z., Kim K.-J. (2007). Review of x-ray free-electron laser theory. Phys. Rev. ST Accel. Beams 10, 034801. doi:10.1103/PhysRevSTAB.10.034801
- Xie M. (1995). Design optimization for an X-ray free electron laser driven by SLAC linac. Proc. 1995 Particle Accelerator Conference, 183–185. accelconf.web.cern.ch