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Free-Electron Laser
A) Motivation and Introduction
C) Experimental Realization / Challenges
B) Theoretical Approach
v1
coherent radiation
micro-bunching
why SASE?
self amplifying spontaneous emission (SASE)
amplifier and oscillator
free electron ↔ wave interaction
why FELs?
need for short wavelengths
A) Motivation and Introduction
need for short wavelengths
state of the art:structure of biological macromolecule
needs ≈1015 samples
crystallized � not in life environment
reconstructed from diffractionpattern of protein crystal:
LYSOZYME MW=19,806the crystal lattice imposes restrictions on molecular motion
images courtesy Janos Hajdu, slide from Jörg Rossbach
resolution does not depend on sample qualityneeds very high radiation power @ λ ≈1Åcan see dynamics if pulse length < 100 fs
need for short wavelengths - 2
simulated image
we need a radiation source with
SINGLEMACROMOLECULE
• very high peak and average power• wavelengths down to atomic scale λ ~ 1Å• spatially coherent• monochromatic• fast tunability in wavelength & timing• sub-picosecond pulse length
courtesy Janos Hajdu
principle of a quantum laser
why FELs?
problem & solution:active medium → free electron – EM wave interaction
free electron ↔ wave Interaction
free electron in uniform motion + electric field lines
trajectory
undulator
(before undulator)
courtesy T. Shintakehttp://www.shintakelab.com/en/enEducationalSoft.htm
free electron is source of waves
cm∝uλÅ→lλ(in principle)
after undulator:waves ahead of particle
there are backward waves
uλλ 2≈
micro-bunching
longitudinal motion to 1st order is trivial, but
micro-bunching is a 2nd order effect → coupled theory of particle motion and
wave generation
transverse bunch structure is much largerthan longitudinal sub-structure→ 1d theory with plane waves
nm 1.0
nm 00005
amplifier and oscillator
amplifier:
in principle FEL
wave
electrons
oscillator:
amplified noise:
instability, driven by noise, growth until amplifier saturates
amplitude
frequency
amplitude
frequency
noise …
amplitude
frequency
saturation
self amplifying spontaneous emission (SASE)
bunch of electronswith increasing micro-
modulation
log(P/P0)
exponential gain
saturation
uniform random distribution of particles at entrance
incoherent emission of EM waves (noise, wide bandwidth)
amplification (→ resonant wavelength, micro-bunching)
saturation, full micro modulation, coherent radiation
why SASE?
oscillator needs resonator
alternative: seed laser + harmonic generation + amplifier
but there are no mirrors for wavelengths < 100 nm
seed laser
seedλmodulation ofelectron beam
micro bunching ofa particular harmonic
amplifier seedλλ nl =
bunch of electronswith increasing micro-
modulation
log(P/P0)
exponential gain
saturation
coherent radiation
no micro-bunching(only shot noise)
P0 ∝ N
saturation:full micro-bunching
P ∝ N2
with N = particles per λl
3ϕ1ϕ 2ϕ1ϕ 2ϕ 3ϕ
particle dynamics: undulator motion
about particles
independent parameter: z
particle dynamics: interaction with EM wave
longitudinal equation of motion
phase space and pendulum
FEL low gain theory
micro bunching
electrodynamics (1D)
FEL high gain theory (1D)
continuous phase space: Vlasov equation
FEL third order equation
B) Theoretical Approach
particle dynamics: undulator motion
field of planar undulator zkB uyu sin0eB −=
Bvv ×−= eme &γequation of motion
uλ uuk λπ2=
approach:
txx uωsinˆ≈
tztvz uω2cosˆ−≈ uu v λπω 2=with
uue k
KB
kmv
ex
βγγ== 02
ˆ with undulator parameter 10 ∝=ueckm
eBK
+−≈
−=2
122
2
2
2Kc
cK
vvvγβγ
uk
Kz
2
2
8ˆ
γ=
example: FLASH
mm 27=uλ
T 47.00 =B
2.1cmT
934.00 ≈≈ uBK λ
µm 6.2ˆ ==uk
Kx
γ
GeV 1≈W 1957≈→ γ
nm 2.08
ˆ2
2
==uk
Kz
γ
about particles
z
x
“reference particle”: only interaction with undulator fieldconstant energy, averaged velocity, …index “0” uλ
0γ L≈0z2
0000 2
−=
γK
vvv200 1 γ−= cv txx uωsinˆ00 ≈
ordinary particles:in interaction with undulator, external waves, self fields, …slowly variation of energy, velocity (compared to λu)index “ν” or skipped
z
x
wave( )t,rE lλ
νγ νv νv L
resonancecondition
new independent parameter: z
( ) zkxzx usinˆ00 =
( ) zkv
z
v
zzt u2cos
ˆ
0
0
00 +=
0γ 0v 0v
( ) ( ) zkv
z
zv
dztzt u
z
i 2cosˆ
0
0
0
, +⌡
⌠+≈ν
νν
( )zνγ ( )zvν ( )zvν approach: nearly constant on one period λu
slippage effects
( ) zkxxzx ui sinˆ0, +≈ νν
initial condition
energy parameter:0
0
γγγη ν
ν−=
“reference particle”:
ordinary particles:
( ) ( )ztzT 0+= ν
( )zxx i 0, += ν
particle dynamics: interaction with EM wave
lλwavelength of light
z
x
wave( )t,rE
( ) rrE dtedW ⋅−= ,
resonance condition (permanent energy transfer): ( ) →=− ul kvck 01
it is a condition for the energy of the reference particle or the wavelength λl
+=2
12 2
0
Kul γ
λλ
plane wave with kl=2π/λx polarization
( )( ){ } zkeK
zctzkEedz
dWul cos cos 0 γν
ν ⋅−−=
( ) ( )( ){ } zkzkzzcTkzvckKeE
dz
dWuull cos 2cosˆ 1 cos 00
0 ⋅+−−−= νν
γ
dx/dz
zkv
z
v
zTt u2cos
ˆ
0
0
0
++= νν
particle dynamics: interaction with EM wave
cos2
0ν
ν ψγKeE
dz
dW −=
averaged vs. undulator period ?=dz
dWν ( ) constzT ≈ν
estimation without longitudinal oscillation ( ) ψψ cos2
1cos2cosˆcos 0 =++ zkzkzzk uuu
( ) zcTkl ννψ = ponderomotive phase
with longitudinal oscillation: replace K by
+−
+=
2
2
12
2
0 2424ˆ
K
KJ
K
KJKK
(modified undulator parameter)
example: FLASH
mm 27=uλ
T 47.00 =B
2.1cmT
934.00 ≈≈ uBK λ
µm 6.2ˆ ==uk
Kx
γ
GeV 1≈W 1957≈→ γ
nm 2.08
ˆ2
2
==uk
Kz
γ
nm 62
12
2
2≈
+= Ku
l γλλ
+−
+=
2
2
12
2
0 2424ˆ
K
KJ
K
KJKK
06.1ˆ =K
νν ψ
γη
cos2
ˆ20
20
cm
KeE
dz
d
e
−≈
νν ηψ ukdz
d2≈
particle equations
longitudinal equation of motion
( ) ( ) ( ) ( ) ( )ztck
zztzTzt
l00 +=+= ν
ννψ
new longitudinal parameters
( ) ( )1
0
−=W
zWz ν
νη
ponderomotive phase
relative energy deviation
( )ν
ν
νν ηψu
lll k
v
ck
v
ck
z
zdTck
dz
d2
0
≈−==with and follow20
20 γcmW e=
equations are analog to mathematical pendulum
φ lm
two types of solution:
trajectories in phase space
15 particles with different initial conditions
“oscillation”“rotation”separatrix φ
v
0=t
0>t(end of undulator)
( ) 00 =νη
η
2πψ +separatrix
2πψ +
η
( ) 00 >νη
( ) 00 γγν >
or
phase space and pendulum
FEL low gain theory
neglect change of field amplitude
indirect gain calculation
( ) ( )energy field initial
outinenergy field initial
energy particle of lossenergy field initialenergy field of gain ΣΣ −=== WW
G
( ) ( )outin ΣΣ = WW
0=G
( ) ( )outin ΣΣ > WW
0>G
( ) 00 =νη
η
2πψ +( ) 00 γγν >
or
2πψ +
η
( ) 00 >νη
=uN periods of undulator
FEL low gain theory
uN
η
G
neglect change of field amplitude
micro-bunching
Iz(ψ)
Iz,0
Fourier analysis→ amplitude of micro modulation
( )∑ −∝ νψiI expˆ
(fundamental mode)
electrodynamics (1D)
Maxwell equations↓
wave equation↓
1D wave equation
xx Jt
Etcz ∂
∂=
∂∂−
∂∂
02
2
22
2 1 µ ( ) ( )( ) ( )tzJzv
zvtzJ z
z
xx ,, =
( ) ( ) ( )( ){ } L+− ctzikzItzJ lz expˆRe~, 1
( ) ( ) ( )( ){ } L+− ctzikzEtzE lxx expˆRe~,
( ) 10
0 ˆ4
ˆˆ I
KczE
dz
dx γ
µ−=
approach with slowly varying amplitude
zkkx uu cosˆ0
FEL high gain theory (1D)
( ){ }νν ψ
γη
iEcm
Ke
dz
dx
e
expˆRe2
ˆ20
2−≈
νν ηψ ukdz
d2≈
particle equations
( )∑ −∝ νψiI expˆ
10
0 ˆ4
ˆˆ I
KcE
dz
dx γ
µ−≈
micro modulation
FEL high gain theory (1D)
numerical solution(Mathcad)
3rd order equation
numerical
P(z)/P(0)
phase space
modulation
FEL high gain theory (1D)
continuous phase space, Vlasov equation
10 −= γγη
ψ
many point particles → continuous density distributionνν ηψ , ( )zF ,,ηψ
continuous phase space, Vlasov equation
( )zF ,,ηψ
( ) ( )zFz ,,,, ηψηψ
ηψ
′′
=J
( ) 0,, =∂∂+⋅∇
z
FzηψJ
( ) ( )0=
∂∂+
∂∂′+
∂′∂+
∂′∂
z
FFFF
ηη
ηη
ψψ
( )zgz
,ψηη =∂∂=′
( )zfz
,ηψψ =∂∂=′
0=∂∂+
∂∂′+
∂∂′
z
FFF
ηη
ψψ 0=
dz
dF
phase space density:
continuity equation:
with “current density”
therefore
with particle equations
Vlasov equation: or
FEL 3rd order equation
perturbation approach: ( ) ( ) ( ) ( ){ }ψηηηηψ izFFzF exp,ˆRe,, 1off0 +−≈
0ˆˆ
4ˆ
4ˆ
32off
22
2
off3
3
=Γ−−+ xx
ux
ux Ei
dz
Edk
dz
Edik
dz
Ed ηη
with energy offset0
0off γ
γγηη ν
ν−
==
gain parameter 330
20
ˆ
4
1
γµ Kk
A
I
cm
e u
e
=Γ beam current I
beam cross-section A
solution ( ) zAzAzAzEx 332211 expexpexpˆ ααα ++=
FEL 3rd order equation
no energy offset: 0γγν =0off =η or
( ) zAzAzAzEx 332211 expexpexpˆ ααα ++=
Γ+=2
31
iα
Γ−=2
32
iα
Γ−= i3α
positive real part → exponential growth !
3rd order equation
numerical
( )( )
2
0ˆ
ˆ
x
x
E
zE
power gain length: ( ) ( ) 3 exp2exp 121 Γ∝→ zzAzP α
1/Lg
C) Experimental Realization / Challenges
Linac Coherent Light Source - LCLS
scales
challenges
rf gun
European X-FEL
bunch compression
Linac Coherent Light Source- LCLS
SLAC mid-April 2009 – first lasing at 1.5 Å
injector
nC 25.0≈q
linac & bunch compression
m10 3∝L m 100 ∝L
undulator
GeV 6.13=E
kA 3=I
cm 3≈uλ
Å 5.1=lλ
m 3.3≈gL
saturation> 10 GW
exponential gain
incoherent
410≈=ec
IN lλ
particles per λl
scales
total length m10 3∝L
undulator length m 100 ∝uL
undulator period m10 2−∝uλ
cooperation length m10 8−∝lL
bunch length m 10 5−∝bL
power gain length m 10 .. 1≈gL
Rayleigh length RL (scale of widening of photon beam)
saturation length uggs LLLL <≈ 20 ..10
photon wavelength m10 10−∝lλ 2γλu∝
transverse oscillation m10ˆ 6−∝x (undulator trajectory)
bunch width Rlw Lλσ ∝wavem10 5bunch −∝wσ width of photon beam
overlap of particle beamwith photon beam
challenges
+=
21
2
2
2
Kul γ
λλ
312
2
34
3
1
=
IKe
mcL ru
g
σλγµ ● high peak current >~ kA
● λl → Å
● Energy → 10 .. 20 GeV
● gain length Lg <~ 10 m
● transverse beam size σr ∝ 10 µm
● energy spread
● overlap electron-photon beam
longitudinal: compression
(undulator parameter K ∝ 1)
glr Lλσ ∝2
transverse: generate low emittance beampreservation of emittance
diagnostic and steering
acceleration
undulator alignment
qc
L
Ibrr
22 σσ =volume
bunch charge
space charge forces:
22
1
rsq
qE
σγ∝
rf gun
nC 1.0∝q MeV 5∝E A 5∝I
typical parameters of FLASH & European XFEL:
10∝γlongitudinal compression 1 → 0.001 needed ! (5 kA)
magnetic bunch compression
magnetic compression: path length depends on energy
γ >>1 → velocity differences are too small for effective compression
accelleration “off crest” →head particle with less energy than tail
t
Ezz
r
B
r
B
r
B
r
B
FLASH:
gun accellarating module ~ 10 m 1st bunch compressor
beam dynamics with space charge and CSR effects
4 magnet chicane
2nd
com
pres
sor,
mor
e m
odul
esun
dula
tor
magnetic bunch compression - 2
LCLS (14 GeV, 0.15 nm)
FLASH (1.2 GeV, 4 nm)
European XFEL (0.1 nm)
BCs at 130 MeV, 500 MeV and 2 GeV
1.2 GeV450 MeV130 MeV
European XFEL
total length 3.4 kmaccelerator length 1.7 kmenergy 17.5 MeVminimal wavelength 1 Åbunches / second 30000bunch charge 0.01 .. 1 nCtypical peak current 5 kA
European X-FEL - 2
superconducting cavity, 1.3 GHz Eacc → 40 MeV/m23.5 MeV/m are needed
FLASH tunnel: cryo module undulator
European X-FEL - 3
beamlines
multi bunch operation