694
M. Dohlus et al.
spectral component within the FEL bandwidth. The effective shot-noise power and
modulated current density are [7, 22]
P n = ργ m e c
2 σ ω / (2π) , ˜
j 1 (0) ≈
eI 0 σ ω /A b .
(11.84)
The computed power rise for typical parameters of the soft x-ray FEL FLASH
(see e.g. [7]) is shown in Fig. 11.16. Saturation is achieved at an undulator length
L u ≈ 20L g0 . The SASE bandwidth at saturation is σ s
ω /ω ≈ ρ. SASE radiation
exhibits shot-to-shot fluctuations in its output spectrum. The coherence length at
saturation is
L coh ≈
√
πc/σ
s
ω = λ l /
2
√
πρ
≈ 11
λ l
λ u
L g0 .
(11.85)
For a bunch length L b > L coh , the average number of spikes in the wavelength
spectra is M = L b /L coh (assuming full transverse coherence). M can be interpreted
as the number of coherent modes of the FEL pulse. In the exponential gain regime
the normalized radiation pulse energy u = U rad / rad fluctuates according to the
gamma distribution [14]
ρ M (u) =
M M u M−1
Γ (M)
e
−Mu , σ
2
u = 1/M.
(11.86)
Phase Space and Simulation of Microbunching
The FEL dynamics resembles the synchrotron oscillations of a proton in a synchrotron or storage ring. In the (ψ, η) phase space the particles rotate clockwise,
hence particles in the right half of an FEL bucket transfer energy to the light wave,
while those in the left half withdraw energy, see Fig. 11.18 and compare Fig. 11.14.
Equation (11.77) are well suited for modelling the microbunching. For z ≥ 12L g0
pronounced microbunches evolve in the right halves of the FEL buckets and increase
the light intensity, while beyond 18L g0 they move into the left halves and reduce it.
The FEL power oscillations in Fig. 11.16 are caused by this rotation in phase space.
Higher Harmonics
Close to saturation, the periodic sequence of narrow microbunches (see Fig. 11.18)
corresponds to a modulation current with rich harmonics contents. In a planar
undulator, odd higher harmonics will be amplified. The third (fifth) harmonic can
reach 1% (0.1%) of the fundamental power.
11.1.2.2 Three Dimensional Effects
The realistic description of high-gain FELs has to be based on a three-dimensional
(3D) theory, taking into account electron beam emittance and energy spread, and
optical diffraction. In idealized cases, e.g. a round beam with uniform longitudinal
charge density, an FEL eigenmode equation including all these effects can be
M. Dohlus et al.
spectral component within the FEL bandwidth. The effective shot-noise power and
modulated current density are [7, 22]
P n = ργ m e c
2 σ ω / (2π) , ˜
j 1 (0) ≈
eI 0 σ ω /A b .
(11.84)
The computed power rise for typical parameters of the soft x-ray FEL FLASH
(see e.g. [7]) is shown in Fig. 11.16. Saturation is achieved at an undulator length
L u ≈ 20L g0 . The SASE bandwidth at saturation is σ s
ω /ω ≈ ρ. SASE radiation
exhibits shot-to-shot fluctuations in its output spectrum. The coherence length at
saturation is
L coh ≈
√
πc/σ
s
ω = λ l /
2
√
πρ
≈ 11
λ l
λ u
L g0 .
(11.85)
For a bunch length L b > L coh , the average number of spikes in the wavelength
spectra is M = L b /L coh (assuming full transverse coherence). M can be interpreted
as the number of coherent modes of the FEL pulse. In the exponential gain regime
the normalized radiation pulse energy u = U rad / rad fluctuates according to the
gamma distribution [14]
ρ M (u) =
M M u M−1
Γ (M)
e
−Mu , σ
2
u = 1/M.
(11.86)
Phase Space and Simulation of Microbunching
The FEL dynamics resembles the synchrotron oscillations of a proton in a synchrotron or storage ring. In the (ψ, η) phase space the particles rotate clockwise,
hence particles in the right half of an FEL bucket transfer energy to the light wave,
while those in the left half withdraw energy, see Fig. 11.18 and compare Fig. 11.14.
Equation (11.77) are well suited for modelling the microbunching. For z ≥ 12L g0
pronounced microbunches evolve in the right halves of the FEL buckets and increase
the light intensity, while beyond 18L g0 they move into the left halves and reduce it.
The FEL power oscillations in Fig. 11.16 are caused by this rotation in phase space.
Higher Harmonics
Close to saturation, the periodic sequence of narrow microbunches (see Fig. 11.18)
corresponds to a modulation current with rich harmonics contents. In a planar
undulator, odd higher harmonics will be amplified. The third (fifth) harmonic can
reach 1% (0.1%) of the fundamental power.
11.1.2.2 Three Dimensional Effects
The realistic description of high-gain FELs has to be based on a three-dimensional
(3D) theory, taking into account electron beam emittance and energy spread, and
optical diffraction. In idealized cases, e.g. a round beam with uniform longitudinal
charge density, an FEL eigenmode equation including all these effects can be