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FEL I : Introduction to FELs
Atoosa Meseck
Synchrotron Radiation
Phenomenologically:
A consequence of the finite
value of the velocity of light.
Electrical field lines
Poynting
Vector S
charge
cT
A
long. Electric field
component
B
magnetic field
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Synchrotron Radiation: Electrical Field Component
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Retarded Potentials!
1873  J.C. Maxwell formulated his unifying electromagnetic theory.
1887  H. Hertz succeeded to generate, emit and receive again
electromagnetic waves.
1898 Lienard
1900 Wiechert
Independently, derived the expressions for
“retarded” potentials of point charges.
Lienard-Wiechert Potentials relate the scalar and vector
potential of electromagnetic fields at the observation point to
the location of the emitting charges and currents at the time of
emission.
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Retarded Potentials
Lienard-Wiechert Potentials
q
 (t )
 
40 r 1  n  

A(t )
1


ret

v
 
2
4c  0 r 1  n  
q


ret
And the electromagnetic fields:
 1 
A
0
C t


B   A
Lorentz gauge

E   

A
t
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Moving Charge
Coulomb regime
updates the direction of the field toward the
instantaneous position of the charge




q
n
1
E (t ) 

 3 2 2 

40  1  n    r 
ret

radiation regime
 


q  n  n    

3

4c 0  1  n  



magnetic field
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


1  
B (t )  n  E
c


1

r
 ret

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Radiation
Poynting Vector
Observer time
retarded time
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Radiation in a Solid Angle
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Observed for the First Time
April 24, 1947
visible radiation was Observed for the first
time at the 70 MeV Synchrotron built at
General Electric.
Since then, this radiation is called
Synchrotron Radiation.
The theory was developed by
Ivanenko, Pomeranchuk 1944,
and Schwinger 1946.
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Radiation Power
Bending magnet
 = Bending radius
E = Electron beam energy
 = Lorentz factor
Electron beam
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Wiggler and Undulator
Oscillation frequency:  w
ku   c
lu
K > 1 ; Qw > 1/ 
qw
s
Trajectory
qw
1


lu  e  B
K
2  me  c

Undulator
parameter
Brilliance ~
2N
K <1 ; Qw < 1/

Res. Wavelength:
lw
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2


K
2
2
 1 
    q0 
2 
2

2 
lu
Brilliance ~ N2
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Peak Brilliance
Synchrotron Radiation Sources
Year
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FEL- Interaction
Interchange between electron beam and radiation field:
W


e  


 
EL ds
  


e   v  EL dt

lu
N
Vx
S
N
W
0
W  0

v

v


EL


EL
S
Resonance condition
for FEL:
s
s lL
2

K 

 1 
2 
2 
2 
lu
s
s
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FEL: Equations of Motion
d
 ( s)
2
N  ku
r
ds
d
  ( s )
ku  KL  K
 ( s)
2  r
ds

r
d

F ( N   )  sin (  ( s) )
 ds
 ( s)

0

  r

KL
e  EL  0
k u  me  c
2
Pendulum:
d2
ds
2
 
 ( s)   L
2
 sin (  ( s ) )

0

0
Frequency:
L
N  k u  KL  K
r
2

F (N   )

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
Separatrix
0

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
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Amplification ( Low Gain )
• The amplification of radiation intensity depends on the electron density.
• For small electron densities the amplification per turn is small by the Undulator.
• For  > 0 a net intensity amplification is expected.
0
0


0
0
Separatrix
Separatrix

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0


0
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
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Low Gain
0

0
Separatrix

0
Madey -Theorem:

The amplification (Gain) is proportional
to the negative derivative of the
“resonance-curve” of the spontaneous
undulator spectrum.
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Low Gain FEL
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Low-Gain FEL Example : ALICE
FEL 2011
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Mirrors for FELs
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High Gain - SASE FEL
• Extremely high electron densities lead to a permanent amplification of the
radiation intensity.
• The electrons are bundled into packages: micro-bunching
• The electrons radiate coherently.
0
Separatrix
Separatrix

 
0
0
 
Separatrix
0

0
Separatrix

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0
Separatrix
0
0

 

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Separatrix

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High Gain - SASE FEL
• Extremely high electron densities lead to a permanent amplification of the
radiation intensity.
• The electrons are bundled into packages: Micro-bunching
• The electrons radiate coherently.
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SASE FEL: Efficiency Parameter
A fundamental scaling parameter for
a SASE FEL is the dimensionless
Pierce parameter:
 K 2 f b2 lu2 


2
 32l p 
1/ 3
Once the FEL interaction has started,
the radiation intensity starts to grow
exponentially along the undulator.
The e-folding length of the radiation
power called the gain length is given
by
lg 
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lu
4 3 
Radiation wavelength:
lu
2
l  2 1  K 
2
Radiation Power:
Pout    Pbeam    ( I  E )
The fact that the repulsive space
charge force inside the bunch
counteracts the formation of
microbunches needs to be taken into
account::
lp 
2
2I

3
I A 2 
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SASE FEL: Slippage , bandwidth, cooperation
The radiation propagates faster than
the electrons. It “slips” by λ per
undulator period; thus electrons
communicate with the ones in front,
only if their separation is less than the
total slippage:
The high gain FEL cuts and amplifies
only a narrow frequency band of the
initial
spectrum.
bandwidth
The high
gain FEL The
cuts typical
and amplifies
only
of
the amplified
spectrum
ofinitial
the
a narrow
frequency
band ofis
the
order
of : The typical bandwidth of the
spectrum.
amplified spectrum is of the order of 2ρ.
l
S  Nu l
l
A single shot spectrum of a radiation
pulse having the duration T contains
spikes with a typical width of 1/T. The
number of spikes in the spectrum and
thus in the pulse profile is about:
2 cT 2 
l
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 2
Each spike (wavepacket) has a
length of λ / 4πρ. Thus, the
cooperation length (slippage in one
gain length) is defined as:
lc 
lg
lu
l
l
4 3
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SASE-FEL Example : FLASH
http://photon_science.desy.de/facilities/flash/the_f
ree_electron_laser/how_it_works/sase_self_amplif
ied_spontaneous_emission/index_eng.html
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Spectral Properties of the SASE -FELs
Power [GW]
40
30
20
10
1.0
The statistic behavior of the FEL
100 200 300
400 500
radiation
is caused by the statistic
100 200 300 400 500 600
Time [fs] fluctuations in the 0density
distribution of the electron bunches.
Spectrum [a.u.]
0.8
0.6
0.4
0.2
0.0
10.2
10.3
10.4
10.5
Wavelength [nm]
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Benefits of Seeding
HGHG
SASE
Power [GW]
40
3
2
1
Spectral power [a.u.]
100
200 300 400
Time [fs]
10
200
300
400
500
Time [fs]
1.0
0.8
0.6
0.4
0.2
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100
1.0
10.2
30
500
Spectrum [a.u.]
Power [GW]
4
10.3
10.4
Wavelength [nm]
10.5
0.8
0.6
0.4
0.2
0.0
10.2
10.3
10.4
10.5
Wavelength [nm]
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High Gain Harmonic Generation (HGHG)*
Kurzer Laserpuls (Ti:Sa)
intensiver hochbrillanter Elektronenstrahl
Dispersive Strecke
Radiator

Modulator
-


*Developed by L.-H. Yu et al.,BNL
-n
n
Phys. Rev. A44/8 (1991) 5178
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Cascaded HGHG-FEL
1.Stage
Laser
ls
lr1= ls/n1
lr2= ls/n1 /n2
1.0
4
0.8
3
Power [GW]
Spectral power [a.u.]
Final Amplifier
2.Stage
0.6
0.4
0.2
10.2
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10.3
10.4
Wavelength [nm]
10.5
lf= lr2
2
1
100
200 300 400
Time [fs]
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500
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Seeding : Radiation
1. Energy modulation
2. Coherent emission
(fundamental or harmonics)
3. Amplification
Dispersive section
Expected Power:
Microbunching
depends on the
energy modulation.
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2
n
Pcoh. ~ b
I2 K2
2 2
Radiator
For given period length, the
desired Wavelength
determines this ratio via
resonance condition.
Electron beam brightness is
determined by the injector and
accelerator.
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Seeding: Electron Beam
1. Energy modulation
2. Coherent emission
(fundamental or harmonics)
3. Amplification
Radiator
Dispersive section
necessary
Modulation:
  n  
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Energy modulation by
the seed:
 ~
K

Lmod
Pseed
ls
Total energy deviation at
the radiator entrance:
  ,tot
 2
  
2
2
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Linking Bunch, Seed and Undulator
8.5
Gain length [m]
7.5
lu = 2.5 cm
K=1
E = 2.3 GeV
I = 2000 A
6.5
5.5
4.5
3.5
2.5
1.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Energy spread [MeV]
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Fresh Bunch Technique
The seeded bunch part is no longer
suitable for a further seeding process .
Use a long bunch and shift the
interaction region for each stage.
1st Stage
Electron bunch
2nd Stage
Final Amplifier
seed
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HGHG-FEL Example : FERMI
NATURE PHOTONICS | VOL 7 |
NOVEMBER 2013 | p913-918
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Modulator Cascade – ECHO Scheme
Laser1
Modulator
Laser2
1
Modulator
2
Radiator 1
Electron beam
Chicane
Chicane
Modulator 1
+ Chicane 1
Modulator 2
+ Chicane 2
G. Stupakov
FEL 2009
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EEHG Example : SDUV-FEL
NATURE PHOTONICS | VOL 6 |
JUNE 2012 | p 360-363
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Figure 3 | Spectra for FEL radiation. a,
Experimental results (red line, HGHG;
blue line, EEHG; green line, intermediate
state between HGHG and EEHG). b,
Simulation results (red line, HGHG; blue
line, EEHG; green line, intermediate state
between HGHG and EEHG).
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Signal to Noise Ratio and Seed Power
For successful seeding we have to ensure that
• the phase correlation and pulse length are conserved!
• the shot-noise effects are suppressed!
 Ps 
1
   2
 Pn out n
*
 Ps 
 
 Pn in
• Limits the total harmonic number
• High seed power required
* E. Saldin et a., Opt. Comm. 202 (2002) 169
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Self-Seeding Example: LCLS
NATURE PHOTONICS |
VOL 6 | OCTOBER 2012 | p
693- 698
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Seeding with HHG-Sources , Example: sFLASH
Electron beam
LASER
HHG process
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Future Application of LASERs
Laser-Compton source
Design of a clean, highbrightness light source is
presented for extreme
ultraviolet/soft
x-ray
(EUV/SXR) lithography
research
and
mask
inspection.
Design of high brightness laser-Compton source for extreme ultraviolet
and soft x-ray wavelengths
Kazuyuki Sakaue ; Akira Endo ; Masakazu Washio
J. Micro/Nanolith. MEMS MOEMS. 11(2), 021124 (May 03, 2012).
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Future Application of LASERs in FELs?
The wavelength in the electron rest frame
In the lab frame the wavelength, neglecting recoil,
of the scattered radiation parallel to the z-axis:
The Lorentz invariant gain
per wavelength or per cycle
of interaction.
The
primary
effect
of
the
ponderomotive
impulse
which
occurs when the pump laser pulse
starts to overlap with the electron
bunch is to decrease the axial
velocity of the bunch. This effect has
been included using:
Nearly copropagating sheared laser pulse FEL undulator for soft xrays , J. E. Lawler et al., J. Phys. D: Appl. Phys. 46 (2013) 325501 (11pp)
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Future Application of LASERs in FELs?
c02
c 

 2
 0  0   n
2
1
c02
n  n  i n   
     i  
     i  
      < 0  n < 0
As a not-laser-physicist I ask myself:
• How does the developing microbunching
change the speed of the light inside the
laser-cavity?
• Do we expect some kind of equillibrium
after some time?
• Do we need to adjust the the cavity length
fast, slow, at all?
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Permittivity (ω):
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Damping term
41
Exciting Future Application of LASERs
All-optical Compton gamma-ray source
K. Ta Phuoc et al.
“Here, we present a simple and compact
scheme for a Compton source based on
the combination of a laser-plasma
accelerator and a plasma mirror. This
approach is used to produce a broadband
spectrum of X-rays extending up to
hundreds of keV and with a 10,000-fold
increase in brightness over Compton X-ray
sources
based
on
conventional
accelerators”
Thank you for your attention!
NATURE PHOTONICS | VOL 6 | MAY 2012 | p 308-311
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