Lecture 4. High-gain FELs
X-Ray Free Electron Lasers
Igor Zagorodnov
Deutsches Elektronen Synchrotron
TU Darmstadt, Fachbereich 1819. May 2014
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 2
Contents
Review
Exponential growth and saturation
Bunching factor and space charge field
Linearized model
FEL parameters
FEL bandwidth
3D effects
Outlook
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 3
Review
undulator radiation
low gain FEL
high gain FEL
electrons
Interaction
EM field
EM field
EM field
Model
FEL radiation
electrons
electrons
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 4
Review
What is shown? (Exercise 7)
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 5
Review
( ) sin( )wu
ux t t
k
2
8( ) sin(2 )w
uz uk
z t v t t
2
22
14
zK
v c
u z u z uv k ck
Electron motion
wK
trajectory
z 0
e u
eBK
m ck
2u uk
2e
e
W
m c
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 6
Review
Exercise 7Exercise 4
, 0,( )
e , , 02
i
iz
RR
2
8
, ,2
( ) sin(2 )
sin( ) ,
w
u
w
u u z
z z uk
wz u
u
ik k
z k
i k elsek
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 7
Review
zv c
the electron should be slower by one wavelength
- the electron is slower than the light
Resonance condition
2
2122
u K
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 8
Review
020
[ ]cos
2 e
eK JJ Ed
dt m c
2 ud
k cdt
Low gain FEL model0 constE
( )uk k z t
0
0
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 9
Review
data from FLASH
W. Ackermann et al, Nature Photonics 1, 336 (2007)
rad ~ elP N 2rad ~ elP N
[μJ]E
[ ]z m [nm]The amplification is very high
Microbunching
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 10
Review
01
[ ]( )
4x zr
d cK JJE z j
dz
2 , 1,2,...n u nd
k n Ndz
2 2
[ ]( )
2ni
n xe r
d eK JJE e
dz m c
High-gain FEL model
2
1 010
1 2m
Nii
z z zm
j j e d j eN
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 11
Exponential growth and saturation
(0) 0.1MV/mxE
0 5 10 15 20 250
2
4
-3 -2 -1 0 1 2 3-5
0
5
-3 -2 -1 0 1 2 30
10
20
0mz
r
[kA]I
[GW]P
[m]z
Exercise 6
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 12
Exponential growth and saturation
0 5 10 15 20 250
2
4
-3 -2 -1 0 1 2 3-5
0
5
-3 -2 -1 0 1 2 30
10
20
saturation: beam fully modulated
20mz [GW]P
[m]z
r
[kA]I
Exercise 6
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 13
Exponential growth and saturation
0 5 10 15 20 250
2
4
-3 -2 -1 0 1 2 3-5
0
5
-3 -2 -1 0 1 2 30
10
20
21mz [GW]P
[m]z
r
[kA]I
Exercise 6
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 14
Exponential growth and saturation
0 5 10 15 20 250
2
4
-3 -2 -1 0 1 2 3-5
0
5
-3 -2 -1 0 1 2 30
10
20
22mz [GW]P
[m]z
r
[kA]I
Exercise 6
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 15
Exponential growth and saturation
0 5 10 15 20 250
2
4
-3 -2 -1 0 1 2 3-5
0
5
-3 -2 -1 0 1 2 30
10
20
23mz [GW]P
[m]z
r
[kA]I
Exercise 6
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 16
0 5 10 15 20 25
10-5
100[GW]P
[m]z
Exponential growth and saturation
linear
saturationExercise 6
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 17
Bunching factor and space charge field
( ) inn
n
f x f e
0
1( )
Tinx
nf f x e dxT
0 0( ) ( ) ( )x f x dx f x
*( ) ( )f x f x *n nf f
( ) ( )f x f x
Fourier series of real periodic function
Dirac delta-function
01
( )2
inn
n
ff x f e
*0 01 1
( )2
in in in innn n n
n n n
ff x f e f f e f e f e
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 18
Bunching factor and space charge field
01
( ) inz z zn
n
j j j e
2
1 0 10
1( ) 2i
z z zj j e d j b
2
00
1( )
2z zj j d
0
1
2( ) ( )
N
z z nn
j jN
11
1n
Ni
n
b eN
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 19
Bunching factor and space charge field
0 E
01
( ) inz z zn
n
E E E e
01
( ) inn
n
e
0
inin n
zne
E ez
0
( ) nu znin k k E
0
nznE i
kn
0
znzn
jE i
n
0
1
2m
Nin
zn zm
j j eN
( )uk k z t
uk k
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 20
Bunching factor and space charge field
01 1
0
0 01 1 1
0
0 1 1
0
0 1
( )
2 1Im Im
sin2
sgn
m
in inz z zn zn
n n
Nininzn z
n n m
Nmz
m n
Nz
m mm
E E E e E e
j je e
n N n
nj
N n
j
N
1
sgn( ) sin( )
2 n
x x nx
n
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 21
Bunching factor and space charge field
x x z zdW
e ev E ev Edt
v E
Revision of energy equation
2 2
[ ]( )
2ni
n xe r
d eK JJE e
dz m c
2 2 2
( )[ ]( )
2nin z n
xe r r e
d eEeK JJE e
dz m c m c
0
0 1
( ) sgnN
zz n n m n m
m
jE
N
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 22
Bunching factor and space charge field
01
[ ]( )
4x zr
cK JJdE z j
dz
2 , 1,2,...n u nd
k n Ndz
High-gain FEL model with space-charge
2 2 2
( )[ ]( )
2nin z n
xe r r e
d eEeK JJE e
dz m c m c
0
0 1
( ) sgnN
zz n n m n m
m
jE
N
1 01
2m
Ni
z zm
j j eN
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 23
Linearized model
0 1( , , ) ( ) Re ( , ) iF z F F z e
1 0F F
0( , ) ( , , )z F z d
1 0 1( ) ( , )F z d
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 24
Linearized model
0dF F F d F d
dz z dz dz
Vlasov-Maxwell equations
00 1
[ ]( ) ( , )
4x zr
cK JJdE z j F z d
dz
2 2 2
( )[ ]( )
2
i zx
e r r e
eEd eK JJE e
dt m c m c
2 u
dk
dz
0 ( ) ( )F - monoenergetic beam
According to the Liouville Theorem the phasespace volume occupied by an ensemble of particles is conserved along theparticle trajectory.
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 25
FEL parameters
2 23 2
ˆˆ ˆ2 0x x xp x
E E Ei iE
12 2 3
0 03
[ ]
4u
r e
eK JJ k j
m c
Gain parameter Detuning
22
2ˆ pp
Space charge parameter
ˆ
2 0 02 up
r e
k j e
m
2 uk
Pierce parameter
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 26
FEL parameters
ˆ , 1, 2,...ˆ n nd
n Ndz
Normalized high-gain FEL model with space-charge
2ˆ ˆ ˆ ˆ( ) ( )ˆ
ninx p z n
dE e E
dz
1
1ˆ ( ) sgnN
z n n m n mm
EN
11
2ˆ mN
iz
m
j eN
2 2
02
[ ]e rm c
EeK JJ
0
ˆ xx
EE
E
1ˆ ˆ( )ˆ x zdE z j
dz
z z
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 27
FEL parameters
Power gain length
0r
r
30x
xE
iE
zxE Ae 3 3i
1 3 2i
Im
Re
2 3 2i 3 i
12
31 21 2 3
zz zxE C e C e C e
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 28
FEL parameters
01
22 ( ) 3
0~
2g
z
Lx zES e e e
Z
31 21 2 3
zz zxE C e C e C e 1
1z
xE C ez
Power gain length
01
3gL
Field gain length
02
3FgL
1( )~ z
xE e
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 29
FEL parameters
Power amplification
(0)x inE E
( , ) cosseedx inE z t E kz t
(0) 0xE
(0) 0xE 31 2
3zz zin
xE
E e e e
beamA
P Sdxdy 0
9g
z
LinPP e S E H
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 30
FEL parameters
0
9g
z
LinPP e
“lethargy regime”
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 31
FEL parameters
?satP
( ) 1
9g
z
L
in
P ze
P
?satz
Saturation
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 32
0 5 10 15 20 25
2468
x 109 field amplitude vs undulator length
z[m]
|Ex|[V
/m]
-3 -2 -1 0 1 2 3
-4-202
x 10-3 phase space
psi[rad]
eta
-3 -2 -1 0 1 2 30
5000
current
psi[rad]
I[A
]
0FgL
Exponential growth and saturation
Exercise 6( ) ?x
dE z
dz
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 33
FEL parameters
Saturation power
11
1n
Ni
n
b eN
2
1 0 10
1( ) 2i
z z zj j e d j b
1 1b
ˆ ( ) 2ˆ xdE z
dz
0
ˆ xx
EE
E
z z
0( ) 2xdE z E
dz
1ˆ ˆ( )ˆ x zdE z j
dz
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 34
FEL parameters
Saturation power
, 0 04
( )3
Fx sat x g
dE E z L E
dz
2, 2 0
00 0 0
1 1 8 4
2 3 3x sat
sat beam beam
E IP A E P
Z Z j
0( ) 2xdE z E
dz
2 0beam
IP mc
e
0 0 0 0( ) ( ) ( )f x f x f x x x
2 2
02
[ ]e rm c
EeK JJ
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 35
FEL parameters
4
3sat beamP P
0
( ) 1
9g
z
LP ze
P
020 usat gz L
Saturation
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 36
FEL Bandwidth
0
0
2
2122
u K
2020
122
uk K
k
0
0
02
0
Detuning for resonance frequency
Detuning for resonance energy
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 37
FEL Bandwidth
and are equivalent and present a “detuning” in the resonance condition
2
2
11
2 22u
u
k Kk
k
0 2
02
00
11 (2)
2 22( )u
u
k Kk O
k
0 2
02
00
11 (2)
2 2 22u
u
k k K kk O
k k
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 38
FEL Bandwidth
0 0
0 02
23 2
ˆ ˆ2 0x x xx
E E Ei iE
ˆ z
xE Ae
3 2 2ˆ ˆ ˆ ˆ ˆ2 0i i
2ˆ ˆ ˆi i
zxE Ae ˆ
z z
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 39
FEL Bandwidth
3Re 0
2
1 20
1Re ( ) 1
2 9gL
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 40
FEL Bandwidth
2
1 20
1Re ( ) 1
2 9gL
2 2
2 20 0 09 2( , )
( , ) ~ g g g
zz zL L Lin
in
P z PG z e e e e
P
202 9
2gL
z
0 0
0 02
00 02 3 2 gL
z 0 02
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 41
FEL Bandwidth
-2 -1 0 1 20
20
40
60
80
100
120Fig. 2.5. from SSY
02
(0)x
x
E
E
0~
11gz L
amplification at resonance energy small positive detuning increases amplification
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 42
3D Effects
Diffraction
Diffraction reduces the field amplitude on the axis
012 ( ) 2
4xKd
ik E z i jdz
Δ
PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 43
Outlook
Electrons produce spontaneous undulators radiation
How to obtain a useful external field?
SASE
Self-Amplified Spontaneous Emission (SASE)
A. Kondratenko, E. Saldin, Part. Accelerators 10, 207 (1980)
R.Bonifacio et al, Opt. Comm.50, 373 (1984)