Lecture 5. Self-amplified spontaneous
emission. FLASH and the European
XFEL in Hamburg
X-Ray Free Electron Lasers
Igor Zagorodnov
Deutsches Elektronen Synchrotron
TU Darmstadt, Fachbereich 18
2. June 2014
Contents
Motivation
Shot noise in electron beam
Current modulation from shot noise
FEL start up from shot noise
Statistical properties of SASE radiation
FEL facilities
Motivation
Electrons produce
spontaneous undulators radiation
How to obtain a useful external
field?
SASE
Motivation
Low-energy undulator test line (LEUTL), USA
530 nm
Motivation
Shot-noise in electron beam
Fluctuations of the electron beam current density serve as
the input signal in the SAS EFEL
Laser pulse
Spectrum
~
ω ρω
∆
ω ω
∆
[ . ]
t a u
( )
P t
P
( )
ω
Shot-noise in electron beam
The electron beam current (at the undulator entrance)
consists from electrons randomly arriving at time t
k
1
( )
(
)
N
k
k
I t
e
δ
t
t
=
=
∑
−
The electron beam averaged over an ensemble of bunches
( )
( )
I t
≡
eNF t
The electron beam profile function can be, for example,
[0, ]
1
( )
( )
r
T
F t
t
T
χ
=
2
2
2
1
( )
2
T
t
g
T
F t
e
σ
πσ
−
=
[0, ]
1,
0
,
( )
T
t
T
t
χ
=
≤ ≤
Shot-noise in electron beam
In frequency domain
1
1
( )
( )
(
)
k
N
N
i t
i t
i t
k
k
k
I
ω
I t e
ω
d
ω
e
e
ω
δ
t
t d
ω
e
e
ω
∞
∞
=
=
−∞
−∞
=
∫
=
∫
∑
−
=
∑
It follows from central limit theorem that the real and
imaginary parts are normally distributed
The probability density distribution of spectral power
( )
2
2
2
1
,
Re ( ), or
Im ( )
2
x
x
x
p x
e
σ
x
I
ω
x
I
ω
πσ
−
=
=
=
( )
1
2
,
,
( )
x
p x
e
λ
λ
x
x
I
ω
λ
−
=
=
=
Shot-noise in electron beam
First-order correlation function
(
)
'
*
2
1
1
'
'
2
2
1
( )
( ')
k
n
k
k
n
N
N
i t
i
t
k
n
N
N
i
t
i t
i
t
k
k n
I
I
e
e
e
e
e
e
e
ω
ω
ω ω
ω
ω
ω
ω
−
= =
−
−
=
≠
=
=
=
+
∑ ∑
∑
∑
1
1
( )
( )
(
)
k
N
i t
i t
i t
k
k
F
F t e
d
t
t
e
d
e
N
ω
ω
ω
ω
∞
ω
∞
δ
ω
=
−∞
−∞
=
∫
=
∫
∑
−
=
*
2
2
*
( )
( ')
(
')
(
1) ( )
( ')
I
ω
I
ω
=
e NF
ω ω
−
+
e N N
−
F
ω
F
ω
Shot-noise in electron beam
First-order correlation function
*
2
2
*
( )
( ')
(
')
(
1) ( )
( ')
I
ω
I
ω
=
e NF
ω ω
−
+
e N N
−
F
ω
F
ω
2
2
2
( )
T
g
F
e
ω σ
ω
=
−
(
)
(
)
(
)
sin 0.5
( )
sinc 0.5
0.5
r
T
F
T
T
ω
ω
ω
ω
=
=
*
2
( )
( ')
(
')
I
ω
I
ω
≈
e NF
ω ω
−
I
( )
ω
2
≈
e N
2
*
( )
( ')
1, for
T
1
NF
ω
F
ω
<<
ωσ
>>
Current modulation from shot-noise
We consider a rectangular averaged current
[0, ]
1
( )
( )
r
T
F t
t
T
χ
=
[0, ]
1,
0
,
( )
0,
otherwise
T
t
T
t
χ
=
≤ ≤
( )
r
( )
I t
=
eNF t
(
)
( )
sinc 0.5
r
F
ω
=
ω
T
(
)
( )
r
( )
sinc 0.5
I
ω
=
eNF
ω
=
eN
ω
T
Current modulation from shot-noise
2
2
0
0
0
2
0
1
1 1
( )
( )
( )
1 1
( )
T
r
S
d
I t
dt
I
d
T
T
F
d
T
ω ω
ω
ω
π
ω
ω
π
∞
∞
∞
=
=
=
=
∫
∫
∫
∫
Spectral power density of averaged current
Parseval's theorem
( )
( )
2
2
(
)
~ sinc
0.5
0,
for
1
I
S
T
T
T
ω
ω
ω
ω
π
=
≈
>>
Current modulation from shot-noise
( )
2
2
0
( )
shot
I
e N
eI
S
T
T
ω
ω
π
π
π
≡
=
=
We are interested in an averaged spectral power density of
shot noise, which by analogy can be written as
The amplification takes place in bandwidth
∆ω
and we can
replace the power of the current in this bandwidth by power
of the equivalent current with fluctuations at
ω
at amplitude
1
1
1,shot
0
0
(
)
(
)
shot
rms
b
b
S
I
e
j
j
A
A
I
ω ω
ω
ω
π
∆
∆
≡
=
=
ɶ
2
2
( )
I
ω
≈
e N
2
1
1
(
)
(
)
rms
shot
I
ω
=
S
ω ω
∆
FEL start up from shot-noise
0
1
[
]
( )
4
x
z
r
cK JJ
d
E z
j
dz
µ
γ
= −
ɶ
ɶ
2
,
1, 2,...
n
u n
d
k
n
N
dz
ψ
=
η
=
High-gain FEL model with space-charge
2 2
2
(
)
[
]
(
)
2
n
i
n
z
n
x
e
r
r
e
d
eK JJ
eE
E e
dz
m c
m c
ψ
η
ψ
γ
γ
== −
ℜ
ɶ
−
(
) (
)
(
)
0
0
1
(
)
sgn
N
z
z
n
n
m
n
m
m
j
E
N
ψ
π
ψ ψ
ψ ψ
ωε
=
= −
∑
−
−
−
1
0
1
2
m
N
i
z
z
m
j
j
e
N
ψ
−
=
=
∑
ɶ
FEL start up from shot-noise
2
3
2
ˆ
2
ˆ
0
x
x
x
x
E
E
E
i
η
η
iE
′′′
′′
′
+
+
−
=
Γ
Γ
Γ
ɶ
ɶ
ɶ
ɶ
3
( )
1
( , )
( )
j
z
x
j
j
E
η
z
c
η
e
α η
=
=
∑
ɶ
0
0
0
2
0
γ γ
ω ω
η
γ
ω
−
−
=
≈
1
1
2
3
2
2
2
2
3
1
2
3
(0)
1
1
1
(0)
(0)
x
x
x
E
c
c
E
c
E
α
α
α
α
α
α
′
=
′′
ɶ
ɶ
ɶ
1
1
2
3
(0)
(0)
(0)
x
x
x
E
c
c
E
c
E
−
′
=
′′
A
ɶ
ɶ
ɶ
ˆ
η
η
ρ
=
FEL start up from shot-noise
0
1
[
]
(0)
(0)
4
x
z
r
cK JJ
E
µ
j
γ
′
= −
ɶ
ɶ
(0)
0
[
]
1
(0)
4
x
z
r
cK JJ
E
µ
j
γ
′′
= −
′
ɶ
ɶ
1
0
0
1
1
2
2
2
n
n
N
N
i
i
z
z
n
u z
n
n
n
j
ij
e
k ij
e
N
N
ψ
ψ
ψ
η
−
−
=
=
′
= −
∑
′
= −
∑
ɶ
1
0
1
2
n
N
i
z
z
n
j
j
e
N
ψ
−
=
=
∑
ɶ
d
n
2
k
u n
,
n
1, 2,...
N
dz
ψ
=
η
=
1
(0)
2
1
(0)
z
u
z
j
′
= −
i k
η
j
ɶ
ɶ
(0)
,
1, 2,...
n
n
N
η
≡
η
=
0
1
[
]
(0)
2
(0)
4
x
u
z
r
cK JJ
E
i k
η
µ
j
γ
′′
=
ɶ
ɶ
FEL start up from shot-noise
1
1
1
0
2
1
3
(0)
0
[
]
(0)
1
(0)
4
2
(0)
x
x
z
r
u
x
E
c
cK JJ
c
E
j
i k
c
E
µ
γ
η
−
−
′
=
=
−
′′
A
A
ɶ
ɶ
ɶ
ɶ
1
1
1
2
3
(0)
(0)
0
0
(0)
x
in
x
x
E
E
c
c
E
c
E
−
−
′
=
=
′′
A
A
ɶ
ɶ
ɶ
Start up from current modulation
FEL start up from shot-noise
On resonance energy
0
r
r
γ γ
η
γ
−
=
≡
3
0
x
x
E
iE
′′′
−
=
Γ
ɶ
ɶ
z
x
E
ɶ
=
Ae
α
α
3
= Γ
i
3
(
)
1
i
3
2
α
= +
Γ
Γ
Im
α
Re
α
(
)
2
i
3
2
α
= −
Γ
3
i
α
= − Γ
1
α
2
α
3
1
j
z
x
j
j
E
c e
α
=
=
∑
ɶ
*
1
1
2
3
2
2
2
1
2
3
1
1
1
1
1
3
3
α
α
α
α
α
α
−
=
=
*
A
A
FEL start up from shot-noise
*
1
1
1
0
0
*
2
1
1
2
*
3
3
0
[
]
1
[
]
1
(0)
(0)
4
3
4
0
z
z
r
r
c
cK JJ
cK JJ
c
j
j
c
α
µ
µ
α
γ
γ
α
−
=
−
= −
A
ɶ
ɶ
1
1
2
3
1
0
1
3
0
1
in
in
E
c
E
c
c
−
=
=
A
Start up from current modulation
Start up from seed field
0
0
,
1,
0
0
[
]
[
]
(0)
4
4
in shot
z shot
r
r
cK JJ
cK JJ
e
E
j
j
I
µ
µ
ω
γ
γ
π
∆
=
=
Γ
ɶ
Γ
Statistical properties of SASE radiation
Interference
Coherence
Coherence is a property of waves that enables interference.
Temporal coherence is the measure of correlation between
the wave and itself delayed. it characterizes how well a wave
can interfere with itself at a different time. The delay over
Statistical properties of SASE radiation
Coherence time
1
1
1
~
~
coh
τ
ω ω ρ
∆
The time-averaged intensity (blue) detected at the output of
an interferometer plotted as a function of delay. The
Statistical properties of SASE radiation
c
coh
I
N
l
ce
=
Typical length of one
spike
coh
l
1
1
:
b
L
M
l
τ
T
=
=
Coherence length
1
~
coh
c
c
l
τ
c
ρω
=
Number of cooperative
electrons
[
µ
m]
s
Laserpuls
[GW]
P
Number of spikes
(longitudinal modes)
6
M
=
M
=
2.6
Statistical properties of SASE radiation
1
~ 2
λ
ρλ
∆
Spikes in spectrum
V. Ayvazyan et al, Eur. Phys.Journ. D 20, 149 (2002)
Spectrum
long bunch (~100fs)
short bunch (~40fs)
( )
Statistical properties of SASE radiation
Statistical properties of SASE radiation
Fluctuations of SASE pulse energy (linear regime)
1
( )
(
)
M
M
Mu
M
M
u
p
u
e
M
−
−
=
Γ
rad
rad
U
u
U
=
1
0
( )
z
t
z
e dt
t
∞
− −
Γ
=
∫
Statistical properties of SASE radiation
Fluctuations of SASE pulse energy (after
saturation, 13 nm, FLASH)
0
10
20
30
40
0
0.5
1
1.5
2
Statistical properties of SASE radiation
b
P
W
ρ
g
z
L
3 3
ln
sat
c
g
L
N
L
=
+
SASE with
c
N
electrons on
coherence length
electrons
radiation
Statistical properties of SASE radiation
Longitudinal profile with large statical fluctuations
FEL facilities
FEL facilities
FEL facilities
FEL facilities
FEL facilities
TESLA Test Facility II ( 2002-2006)
From 2003 on, TTF1 was expanded
to TTF2, an FEL user facility for the
spectral range of soft x-rays, including
a new tunnel and a new experimental
hall (in the foreground). In April 2006,
the facility was renamed FLASH: FEL
in Hamburg
FEL facilities
FEL facilities
FEL facilities
Phase space linearization
rollover compression vs. linearized compression
~ 1.5 kA
~2.5 kA
Q=1 nC
Q=0.5 nC
Phase space linearization
In accelerator modules the energy of the electrons is
increased from 5 MeV (gun) to 1200 MeV (undulator).
Phase space linearization
In compressors the peak current I is increased from 1.5-50 A
(gun) to 2500 A (undulator).
Phase space linearization
FLASH
FEL radiation parameters
Wavelength Range
4.1 - 45 nm
Average Single Pulse Energy
10 - 400 µJ
Pulse Duration (FWHM)
50 - 200 fs
Peak Power (from av.)
1 - 3 GW
Average Power (5000 pulses/sec)
400 mW
Spectral Width (FWHM)
0.7 - 2 %
FEL facilities
FEL facilities
FEL facilities
FLASH 2
Photon Beam
HHG
SASE
Wavelength range
(fundamental)
10 - 40 nm
4 - 80 nm
Average single
pulse energy
1 – 50
µ
J
1 – 500
µ
J
Pulse duration
(FWHM)
<15 fs
10 – 200 fs
Peak power (from
av.)
1 – 5 GW
1 – 5 GW
Spectral width
(FWHM)
0.1 – 1 %
0.5 – 1.5 %
Peak
Brilliance*10 - 40
nm
10
28
- 10
31
10
28
- 10
31
FEL facilities
Intensity
distrubution
for
λ
= 0.14 nm
radiation power ~ GW
G.Gutt et al, PRL,
E= 3.5-14 GeV
FEL facilities
LCLS
P. Emma et al, Nature
Photon. 4, 641(2010)
radiation power ~ GW
Pulse length ~30 fs
G.Gutt et al, PRL,
108, 024801 (2012)
λ
=1.4
FEL facilities
European XFEL
-
kürzeste
Wellenlänge
-
größte
Brillanz
FEL facilities
FEL facilities
FEL facilities
FEL facilities
FEL facilities
FEL facilities
FEL facilities
European XFEL
Parameter
Value
SASE 1
SASE 2
SASE 3
photon energy [keV]
12.4 - 4.0
12.4 - 3.1
3.1 - 0.2
wavelength [nm]
0.1 - 0.31
0.1 - 0.4
0.4 - 6.4
peak power [GW]
24
22
100 - 135
average power [W]
72
66
300 - 800
photon beam size (FWHM) [µm]
110
110
65 - 95
photon beam divergence (FWHM) [µrad]
0.8
0.8
3 - 27
bandwidth (FWHM) [%]
0.09
0.08
0.28 - 0.73
coherence time [fs]
0.3
0.3
0.3 - 1.9
pulse duration (FWHM) [fs]
100
100
100
average brillance [x10^25, photons/(s
Linac Coherent
Light Source
(LCLS)
Spring-8 Angstrom
Compact Laser
(SACLA)
European
XFEL
Standort
USA
Japan
Deutschland
Start der
Inbetriebnahme
2009
2011
2015
Beschleuniger –
Technologie
normalleitend
normalleitend
supraleitend
Anzahl der
Lichtblitze pro
Sekunde
120
60
27 000
Minimale
Wellenlänge
0.15 nm
0.1 nm
0.05 nm
FEL facilities
Outlook
self-“seeding“
high harmonics of laser light
Methods for improving of coherence
Outlook
“Table-Top-FEL”
M.Fuchs et al, Nature
H.-P. Schlenvoigt et al, Nature
Physics 4, 130 (2008)
λ
=740 nm
λ
=17 nm
spontaneous undulator radiation
with a laser plasma accelerator
