1 X-Ray Sources at Large-Scale Facilities
33
phase and hence A tot = n e A. The intensity will be proportional to n
2
e (see Fig. 1.19).
This is the key to the ‘runaway’ amplification in SASE: the more the electrons bunch
and radiate in phase (that is, coherently), the stronger is the EM-field interaction
with them. This in turn further enhances the microbunching phenomenon associated
with coherent emission, thus increasing the degree of coherent emission in a positive
feedback loop.
This runaway instability causes the light intensity to grow exponentially along the
undulator until the process saturates [20]. The microbunches produced by the SASE
process each contain approximately n e = 10
9 electrons, resulting in a peak brilliance
for XFELs approximately one billion times larger than that from fourth-generation
synchrotron sources!
The quality of a high-gain XFEL is encapsulated in the dimensionless so-called
Pierce, or FEL, parameter, ρ FEL , given approximately by
ρ FEL =
λ
2
u r 0 n e K
2
32πγ 3
1/3
,
(1.42)
where λ u is the undulator period length, r 0 = 2.82 × 10
−15 m is the Thomson scattering length (also known as the classical electron radius), N e is the electron density
of the pulse bunch as it enters the undulator, K is the deviation parameter, and
γ = E/m e c
2 is the Lorentz factor. Equation (1.42) can be re-expressed in practical
units as
ρ FEL =
1.55 × 10
−5
E[GeV]
(λ u [mm])
2 N e [mm
−3
]K
2
1/3 .
(1.43)
Electron densities in the compressed electron bunch entering the undulator are
typically 10
4
µm
−3 (that is, 10
9 electrons in a volume of the order of 10
5
µm
3 ); the
undulator period λ u = 15–30 mm, and K -values of approximately 2–3 are normal.
The LINACs produce electron energies of the order of 10 GeV. Because of the weak
cube-root dependency of ρ FEL on all the design variables except E, its spread is fairly
narrow across all facilities designed or commissioned to date, and normally assumes
values of around 5 × 10
−4 .
ρ FEL determines three important properties of high-gain XFEL radiation. Firstly,
it describes the fraction of the electrons’ power converted at SASE saturation to
photon power, that is
ρ FEL =
P ph
P e
.
(1.44)
The larger is ρ FEL , the more efficiently electron energy is converted into photon
energy. ρ FEL also describes the relative spectral BW at saturation,
ρ FEL =
ω
ω
,
(1.45)
33
phase and hence A tot = n e A. The intensity will be proportional to n
2
e (see Fig. 1.19).
This is the key to the ‘runaway’ amplification in SASE: the more the electrons bunch
and radiate in phase (that is, coherently), the stronger is the EM-field interaction
with them. This in turn further enhances the microbunching phenomenon associated
with coherent emission, thus increasing the degree of coherent emission in a positive
feedback loop.
This runaway instability causes the light intensity to grow exponentially along the
undulator until the process saturates [20]. The microbunches produced by the SASE
process each contain approximately n e = 10
9 electrons, resulting in a peak brilliance
for XFELs approximately one billion times larger than that from fourth-generation
synchrotron sources!
The quality of a high-gain XFEL is encapsulated in the dimensionless so-called
Pierce, or FEL, parameter, ρ FEL , given approximately by
ρ FEL =
λ
2
u r 0 n e K
2
32πγ 3
1/3
,
(1.42)
where λ u is the undulator period length, r 0 = 2.82 × 10
−15 m is the Thomson scattering length (also known as the classical electron radius), N e is the electron density
of the pulse bunch as it enters the undulator, K is the deviation parameter, and
γ = E/m e c
2 is the Lorentz factor. Equation (1.42) can be re-expressed in practical
units as
ρ FEL =
1.55 × 10
−5
E[GeV]
(λ u [mm])
2 N e [mm
−3
]K
2
1/3 .
(1.43)
Electron densities in the compressed electron bunch entering the undulator are
typically 10
4
µm
−3 (that is, 10
9 electrons in a volume of the order of 10
5
µm
3 ); the
undulator period λ u = 15–30 mm, and K -values of approximately 2–3 are normal.
The LINACs produce electron energies of the order of 10 GeV. Because of the weak
cube-root dependency of ρ FEL on all the design variables except E, its spread is fairly
narrow across all facilities designed or commissioned to date, and normally assumes
values of around 5 × 10
−4 .
ρ FEL determines three important properties of high-gain XFEL radiation. Firstly,
it describes the fraction of the electrons’ power converted at SASE saturation to
photon power, that is
ρ FEL =
P ph
P e
.
(1.44)
The larger is ρ FEL , the more efficiently electron energy is converted into photon
energy. ρ FEL also describes the relative spectral BW at saturation,
ρ FEL =
ω
ω
,
(1.45)