Lecture 1. Introduction.
Acceleration of charged particles
X-Ray Free Electron Lasers
Igor Zagorodnov
Deutsches Elektronen Synchrotron
TU Darmstadt, Fachbereich 18
20. April 2015
General information
Lecture: X-Ray Free Electron Lasers
Place: S2|17, room 114, Schloßgartenstraße 8, 64289 Darmstadt
Time: Monday, 11:40-13:20 (lecture), 13:30-15:10 (exercises)
1.
(07.04.14)
Introduction. Acceleration of charged particles
2.
(14.04.14)
Synchrotron radiation
3.
(05.05.14)
Low-gain FELs
4.
(12.05.14)
High-gain FELs
5.
(19.05.14)
Self-amplified spontaneous emission.
FLASH and the European XFEL in Hamburg
6.
(02.06.14)
Numerical modeling of FELs
7.
(23.06.14)
New FEL schemes and challenges
8.
(30.06.14)
Exam
General information
Lecture: X-Ray Free Electron Lasers
Literature
K. Wille, Physik der Teilchenbeschleuniger und
Synchrotron-strahlungsquellen, Teubner Verlag, 1996.
P. Schmüser, M. Dohlus, J. Rossbach, Ultraviolet and Soft X-Ray
Free-Electron Lasers, Springer, 2008.
E. L. Saldin, E. A. Schneidmiller, M. V. Yurkov, The Physics of Free
Electron Lasers, Springer, 1999.
Lecturer:
PD Dr. Igor Zagorodnov
Deutsches Elektronen Synchrotron (MPY)
Notkestraße. 85, 22607 Hamburg, Germany
phone: +49-40-8998-1802
Contents
Motivation. Free electron laser
Particle acceleration
Betatron. Weak focusing
Circular and linear accelerators
Strong focusing
RF Resonators
Bunch compressors
Motivation
Laser – a special light
monochromatic
(small bandwidth)
parallel
(tightly collimated)
coherent
(special phase relations)
The laser light allows to make
accurate interference images
gas
mirrors
energy pump
light
accelerator
undulator
bunch
Motivation
non quantized electron energy
the electron bunch is the energy
source und the lasing medium
Quantum Laser
Free electron laser (FEL)
Free electron laser
laser light
John Madey, Appl. Phys. 42,
1906 (1971)
„Light Amplification by
Stimulated Emission of
Radiation“
Motivation
no mirrors
under 100 nm
no long-term excited states for the population
inversion
Motivation
Why FEL?
Motivation
FEL as a source of X-rays
peak brilliance
[ph/(s mrad
2
mm
2
0.1% BW)]
Photon flux is the number of
photons per second within a
spectral bandwidth of 0.1%
photons
s 0.1 BW
Φ =
⋅
Brilliance
2
' '
4
xy
x y
B
π
Φ
=
Σ Σ
,
,
xy
σ σ
x e
y e
Σ =
' '
,
,
x y
σ
θ
ph
σ
θ
ph
Σ
=
Motivation
brilliant
extremely short pulses (~ fs)
ultra short wavelengths (atom
details resolution)
coherent
(holography at atom
level)
Motivation
H.Chapman et al,
Nature Physics,
2,839 (2006)
Experiment with FEL light
FEL puls
32 nm
Motivation
reconstructed
image
example structure
in 20 nm membran
1
µ
m
Experiment with FEL light
diffraction
image
H.Chapman et al,
Nature Physics,
2,839 (2006)
Motivation
data from FLASH
„High-Gain“ FEL
W. Ackermann et al, Nature Photonics
1, 336 (2007)
rad
~
el
P
N
P
rad
~
N
el
2
[
µ
J]
E
[ ]
z m
λ
[nm]
Exponential growth
Motivation
FLASH („
F
ree Electron
LAS
er in
H
amburg)
Motivation
FLASH („
F
ree Electron
LAS
er in
H
amburg)
Particle acceleration
short gain length
2
1
~
λ
γ
short radiation wavelength
2
( ) ~
gz
L
E z
e
5 4
2
5 6
1 2
~
1
g
L
O
I
I
γ
ε σ
ε
+
Requirements on the beam
high beam energy
high peak current
low emittance
low energy energy spread
[
µ
J]
E
[ ]
Particle acceleration
2
2
2
x
x
x
xx
ε
=
′
−
′
ε
n x
,
=
γε
x
- the normalized emittance is
conserved during acceleration
Emittance
x
z
p
dx
x
dz
p
′ =
=
- trajectory slope
Particle acceleration
Methods of particle acceleration
The energy of relativistic particle
2 4
2 2
0
E
=
m c
+
p c
with the relativistic momentum
0
p
=
γ
m v
(
)
0.5
2
1
γ
= −
β
−
/
v c
β
=
can be changed in EM field
2
2
1
1
L
E
d
q
d
qU
∆ =
∫
⋅
=
∫
⋅
= −
r
r
r
r
F
r
E
r
(
)
L
=
q
× +
F
v B E
-19
-19
1eV=1.602 10
×
C 1V=1.602 10
×
×
J
Cockroft-Walton
generator(1930)
Particle acceleration
Daresbury, ~20MeV
Acceleration in electrostatic field
Van de Graff
accelerator
The energy capability of this sort of devices is limited
by voltage breakdown, and for higher energies one is
Particle acceleration
No pure acceleration is obtained.
The electric field exists outside the plates. This field
decelerates the particle.
Time dependent electromagnetic field!
Maxwell‘s equations (1865)
The particles are sent repeatedly through the electrostatic
field.
Acceleration to higher energy?
t
∂
∇× = +
∂
H
J
D
t
∂
∇× = −
∂
E
B
ρ
∇ =
D
0
∇ =
B
Faraday‘s law
Coulomb‘s law
absence of free magnetic poles
generelized Ampere‘s law
Particle acceleration
Betatron
RF resonators
Acceleration to higher energy?
Faraday‘s law
d
d
t
∂
= −
∂
∫
E r
∫∫
B s
B
E
R
Betatron
main coils
corrector coils
yoke
vacuum chamber
beam
The magnetic field is changed in a way, that the particle circle orbit
remains constant.
The accelerating electric field appears according to the Faraday’s
law from the changing of the magnetic field.
B
E
R
Betatron
ϕ
R
d
= −
d
∫
E r
∫∫
B s
ɺ
0
0
E
ϕ
=
E
0
0
z
B
=
B
2
2
π
RE
ϕ
= −
π
R B
ɺ
av
2
1
av
z
B
B ds
R
π
=
∫∫
ɺ
ɺ
p
ɺ
ϕ
=
qE
ϕ
2
fug
mv
F
R
ϕ
=
.
L
z
F
=
qv B
ϕ
z
p
ɺ
ϕ
= −
qRB
av
B
=
ɺ
Constant orbit condition
2
av
R
p
ɺ
ϕ
= −
q
B
ɺ
Centrifugal force
Is equal to the Lorentz force
From Faraday’s law
From Newton’s law
=
p
ɺ
F
x
y
Betatron. Weak focusing
Betatron oscillations near the reference orbit
z
z
B
R
n
B
r
∂
= −
∂
- field index
0
< <
n
1
- orbit stability condition
Transverse oscillations are called betatron oscillations for all
accelerators.
Betatron. Weak focusing
2
2
0
(
)
1
1
fug
mv
mv
r
r
F
R
r
F
R
r
R
R
R
ϕ
ϕ
∆
∆
+ ∆ =
≈
−
=
−
+ ∆
0
fug
( )
L
( )
z
F
=
F
R
= −
F R
= −
qv B
ϕ
0
(
)
(
)
[
( )
z
]
(1
)
L
z
z
B
r
F R
r
qv B R
r
qv B R
r
F
n
r
R
ϕ
ϕ
∂
∆
+ ∆ =
+ ∆ ≈
+
∆ = −
−
∂
0
(
)
(
1)
rad
L
fug
r
F
R
r
F
F
F
n
R
∆
+ ∆ =
+
=
−
<
The radial force is pointed to the design orbit if
Radial stability
ϕ
R
+ ∆
r
z
B
R
n
= −
∂
0
0.2
0.4
0.6
0
0.5
1
1.5
2
field index
t[mks]
-1
0
1
-1
-0.5
0
0.5
1
orbit
x[m]
y
[m
]
0
0.2
0.4
0.6
0.9
1
1.1
1.2
relative radius
t[mks]
0
0.2
0.4
0.6
1
1.05
1.1
1.15
1.2
relative moment
Betatron. Weak focusing
Betatron. Weak focusing
0
r
z
z
B
B
z
r
=
∂
∂
=
∂
∂
0
(
)
(
)
r
z
z
r
B
B
z
F
z
qv B
z
qv
z
qv
z
F n
z
r
R
ϕ
ϕ
∂
ϕ
∂
∆
∆ = −
∆ ≈ −
∆ = −
∆ = −
∂
∂
0
n
>
0
(
)
(
1)
rad
r
F
R
r
F
n
R
∆
+ ∆ =
−
0
(
)
z
z
F
z
F n
R
∆
∆ = −
n
>
0
1
n
<
The vertical force is pointed to the design orbit if
The orbit is stable in all directions if
< <
Vertical stability
(
)
L
=
q
× +
F
v B E
t
µ
∂
∇× =
+
∂
B
J
D
Circular and linear accelerators
Circular accelerators: many runs
through small number of cavities.
Strong focusing
BESSY II, Berlin
PETRA III, Hamburg
S. Kahn, Free-electron
lasers. (a tutorial review)
Journal of Modern Optics
Strong focusing
multipolar expansion
equations of motion
transfer matrix (quadrupole)
RF Resonators
Maxwell equations in vacuum
0
∇ =
E
0
µ
∇× = −
E
H
ɺ
0
ε
∇× =
H
E
ɺ
From
∇×∇× = ∇∇ − ∇
2
F
F
F
follows wave equations
2
2
1
0
c
∇ −
E
E
ɺɺ
=
We separate the periodical time dependance und use the
representation (traveling wave)
0
∇ =
H
2
2
1
0
c
∇
H
−
H
ɺɺ
=
(
)
( , )
t
( )
e
i
ω
t k z
−
z⊥
=
E r
E r
x
⊥
=
r
x
y
=
r
Waveguides
RF Resonators
2
( )
k
c
( )
0
⊥
⊥
⊥
∆
E r
+
E r
=
∆
⊥
H r
( )
⊥
+
k
c
2
H r
( )
⊥
=
0
2
2
2
c
z
k
=
k
−
k
k
=
ω
/
c
For the space field distribution in transverse plane we obtain
The smallest wave number (cut frequency) k
c
Wave propagation in the waveguide is possible only if k>k
c
.
If k<k
c
then the solution exponentially decays along z.
k
2
2
z
c
k
=
k
+
k
2
2
1
c
ph
z
z
z
k
ck
v
c
c
k
k
k
ω
=
=
=
+
>
Phase velocity is larger than the
light velocity
Waveguides
c
RF Resonators
Unlike free space plane wave the waves in waveguides
have longitudinal components
(
)
0
sin
sin
,
0.
zi
t k z
z
z
m x
m y
E
E
e
a
b
H
ω
π
π
−
=
=
(
)
0
(
) cos(
)
,
0.
zi
t k z
z
m
c
z
E
E J
k r
m
e
H
ω
ϕ
−
=
=
TM waves
( ) 0cos
cos
,
0.
z i t k z zm x
m y
H
H
e
a
b
E
ω
π
π
−=
=
(
)
0
(
) cos(
)
,
0.
zi
t k z
z
m
c
H
H J
k r
m
e
E
ω
ϕ
−
=
=
TE waves
2 2 2,
1, 2,...;
1, 2,...
cm
n
k
m
n
a
b
π
π
=
+
=
=
,
0,1, 2,...;
1, 2,...
mn cx
k
m
n
a
=
=
=
J
m(
x
mn)
=
0
′ ′ =
Waveguides
RF Resonators
RF Resonators
The cylindrical waveguide were an ideal accelerator structure, if it were
possible to use E
z
component of TM wave. However the velocity of the
particle is always smaller than the wave phase velocity v
ph
.
waveguide with irises
(traveling waves)
RF resonators
(standing waves)
RF Resonators
Through tuning of phase velocity according to the particle velocity it is
possible to obtain, that the bunches synchronously with TM wave fly
and obtain the maximal acceleration.
Waveguide with irises (traveling wave)
z
k
k
waveguide
with irises
cylindrical
waveguide
phv
<
c
phv
=
c
2L
π
L
RF Resonators
Acceleration with standing and traveling waves
mode
RF Resonators
We separate only the periodic time dependence
and take the represantation (standing wave)
( , )
t
=
( )
e
i t
ω
E r
E r
( , )
t
=
( )
e
i t
ω
H r
H r
2
( )
k
( )
0
∆
E r
+
E r
=
2
( )
k
( )
0
∆
H r
+
H r
=
For the space field distribution we obtain
mode
RF Resonators
0
( )
0
z
E
=
E r
0
( )
0
H
ϕ
=
H r
2
2
2
E
z
E
z
k E
z
0
r r
r
∂
+
∂
+
=
∂
∂
( )
0
0
z
E
=
E J
kr
2.405
.
k
R
=
( )
0
1
E
H
J kr
c
ϕ
=
TM
010
-Welle
Pillbox
RF Resonators
The electron beam energy is
converted in RF energy.
Klzstron
Strahl
P
=
η
UI
klystron efficiency (45-65%)
η
−
Klystron
RF Resonators
The exact resonance frequency could be tuned. The resonator
is exited through an inductive chain. The waveguide from
RF Resonators
the concept of
wake fields
is used to describe the integrated
kick (caused by a source particle, seen by an observer
particle)
self field of cavity
(driven by bunches)
short range wakes describe interaction of particles in same
bunch long range wakes describe multi bunch interactions
important for FELs:
longitudinal single bunch wakes
change
the energy chirp and
interfere with bunch compression
Bunch compressors
0
s
δ
≡
(
2
3
)
1
0
0
56
566
5666
s
= + ∆ = −
s
s
s
R
δ
+
T
δ
+
U
δ
Bunch compressors
M. Dohlus et al.,Electron Bunch Length
Compression, ICFA Beam Dynamics
Newsletter, No. 38 (2005) p.15
Phase space linearization
In accelerator modules the energy of the electrons is
increased from 5 MeV (gun) to 1200 MeV (undulator).
1,1
1,1
cos(
1,1
)
E
eV
ks
ϕ
∆
=
+
FLASH
1,3
1,3
cos(3
1,3
)
E
eV
ks
ϕ
∆
=
+
2
2
1
2
cos(
)
E
eV
ks
ϕ
∆
=
+
3
3
cos(
2
3
)
E
eV
ks
ϕ
∆ =
+
Phase space linearization
In compressors the peak current I is increased from 1.5-50 A
(gun) to 2500 A (undulator).
(
2
3
)
56
566
5666
i
s
R
δ
T
δ
U
δ
∆ = −
+
+
FLASH
Phase space linearization
rollover compression vs. linearized compression
~ 1.5 kA
~2.5 kA
Q=1 nC
Q=0.5 nC
0 0.2 0.4 0.6 0.8 1 1.2
