accelerator physics topic ix wigglers, undulators, and fels
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Accelerator Physics Topic IX Wigglers, Undulators, and FELs. Joseph Bisognano Engineering Physics & Synchrotron Radiation Center University of Wisconsin. Bending Magnet Radiation. CERN School 1998. Wiggler or Undulator (Insertion Devices). CERN School 1998. More flux or higher brightness - PowerPoint PPT PresentationTRANSCRIPT
J. J. Bisognano
Topic Nine: Wiggler to FELs
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UW Spring 2008
Accelerator Physics
Accelerator PhysicsTopic IX
Wigglers, Undulators, and FELs
Joseph Bisognano
Engineering Physics &
Synchrotron Radiation Center
University of Wisconsin
J. J. Bisognano
Topic Nine: Wiggler to FELs
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Accelerator Physics
Bending Magnet Radiation
CERN School 1998
J. J. Bisognano
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Wiggler or Undulator (Insertion Devices)
CERN School 1998
More flux or higher brightness
Wigglers: high field, broad spectrum
Undulators: low field, interference peaked spectrum
J. J. Bisognano
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Insertion Devices
CERN School 1998
J. J. Bisognano
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Light Source
CERN School 1998
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Ideal ID Field Pattern(infinite pole tips in x)
0divBby )sin()sinh(~
)cos()cosh(~
)cos()cosh()cosh(
)cos()sinh(),(
)cos()()2cos()(),(
potentialscalar Magnetic
0
skzkBB
skzkBB
skzkB
B
skzkAzs
LaplaceMaxwell
skzfzfzs
uus
uuz
uugz
uu
us
u
u
Gap and period go hand in hand
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Gap Dependence of Magnetic Field
CERN School 1998
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Equation of Motion of Electrons in IDsNeglecting vertical motion, we have
)(
)(
sBme
xs
sBme
sx
z
z
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First Order Solution
)cos(4
~)(
)sin(2
~)('
1);cos(~
"
';,
2
2
skcm
Besx
skcm
Besx
withskcmBe
x
cxxcssx
Assume
uu
uu
u
Since there is a Bs, one can get a vertical force; i.e., focusing
J. J. Bisognano
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Accelerator Physics
Basic Parameters
1
1
2
~1
KWiggler
ceinterferenKUndulator
anglenaturalKK
mcBe
AngleMaximum
u
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Accelerator Physics
Second OrderEnergy Conservation says that if x is moving it’s at the expense of longitudinal energy
)2sin(8
*)(
)cos()(
)2cos(4
*)(
)sin()(
]2/1[2
11*
2
2
2
2
22
tk
Kctts
tk
Ktx
tcK
cts
tKc
tx
K
uu
uu
u
u
J. J. Bisognano
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In Beam Frame
Beam frame coordinates t and frequency
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Lorentz Transforms and Radiating
frame lab the into back go to need weNow
ck or c/frequency a
withoscillated it c, almost going stillsit' Since
/period withwiggler
shortened seeselectron frame, beam In
ubeam
,
J. J. Bisognano
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Photon Frequency in LabExpect a “blue” shift since waves get pushed together as beam is moving toward observer
Use fact that energy of photon is hf, momentum is hf/c
knobstuninggiveandK
Kck
ck
ulab
ulab
beamlab
)2/1(2
)cos1(
)cos1(
2222
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Undulator SpectrumSince train is of finite length (N cycles), there is a width to spectrum, but it is very narrow, order 1/N
If one includes that motion is really not perfectly sinusoidal (remember the figure 8 and energy modulation) but that it does repeat in time, there is harmonic generation
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Cern SchoolHigher harmonics add to reach of an undulator
Require care in phase errors of undulator periodic fields
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Fundamental power/total power=1/(1+K2/2)1/2Cern School
R Walker
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J. J. Bisognano
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Spontaneous Emission
w
w
ww
w
N
N
mc
eBK
Kc
11
12
)2/1/()/2(2
undulator)(planar radiation sSpontaneou
2
220
Note that for higher frequency, you need higher energy or shorter undulator period
Shorter undulator period implies smaller gap
J. J. Bisognano
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R Walker, CERN School
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Brightness/Brilliance
L
L
FluxB
AreaPhaseSpaceFlux
Brightness
RRxx
RRxx
zzxx
2
4
2/
)2(
/
22
22
2
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Physics of FELs• An electron beam moving on a linear trajectory
will have no net energy coupling to a co-moving E&M wave, just “jiggled”
• In a wiggler (really undulator), an electron beam develops a transverse oscillation, as we’ve just seen
• If the oscillation stays in phase with the fields, there can be a net exchange of beam energy to the wave; i.e., the electron beam acts to amplify the electromagnetic wave
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Oscillators and SASEs• If one puts beam/wiggler into optical resonantor, there is a
feedback loop that generates an oscillator and a laser
• If the wiggler is long enough, the energy modulation of the electron beam can generate “microbunches” which can radiate coherently, generation self-amplified spontaneous emission (SASE) from the Schottky noise on the beam, lasing without mirrors from a beam instability
• Or one can “seed” the beam with an energy modulation induced by an external laser
• Sources are tunable (beam energy or wiggler field) and coherent
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Basic FEL Configuration•
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Jlab FEL
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Spontaneous Emission
w
w
ww
w0
N
N
mceB
K
Kc
wiggler)(helical radiation sSpontaneou
11
12
)1/()/2(2
2
22
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FEL Dynamics I
)sinˆcosˆ(
21
1
)sinˆcosˆ(
2
2
0
0
skyskxK
K
s
skyskxBB
Consider
ww
wwww
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FEL Dynamics II
)2
11(
;)(
sin
Emc
e
Eˆ
)cos(ˆ)sin(ˆ(EE
radiation polarized circularly
2
2
00
0
000
K
k
k
cswithtskk
mc
KeE
sB
tksytksx
Suppose
ww
w
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FEL Dynamics III
period radcc
cc
c in photon
c
time in period one through moves Electron
K
0 if exchangeenergy net get llWe'
www
w
w
ww0
0
00
0
0
22
1//
/,
/
)1/()]1/(2[
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Another Pendulum Equation
0sin
2
2
sin2
)1(2
2
202
2
2
r
w
w
w
r
r
wr
mc
KeE
Then
K
Let
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CERN School
Gain only when energy of beam doesn’t quite match “ideal” energy
If wiggler is two long, process reverses, unless wiggler is “tapered”
η
φ
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FEL parameters
•
eadenergy spr smallneed1/N is height Bucket
mKVNe
NfNG
VNmcG
field inenergy lossenergy electron
G
W
w
ww
E
,
16)/(4
)4()4(4
)/()(
22
223
3
822 2
0
Need high beam density
J. J. Bisognano
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SYLee Text
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SYLee Text
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SYLee Text
Looks like derivative of undulator power spectrum: fluctuation-dissipation or Madey’s theorem
J. J. Bisognano
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High Gain Regime
So far, we haven’t included how the increasing electromagnetic wave affects the continued electron motion
Also, there is a density variation developing
Also, at high enough frequencies there are no good mirrors to make an optical resonator
“High Gain” regime, really an instability saves the day, and points to X-ray lasers
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Basic Principle: Coherent Synchrotron Radiation
If we can get “microbunching” of electron beam, strong enhancement over incoherent
synchrotron radiation
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High Gain FEL to the Rescue: Basic Feedback Loop
• Electron beam responds to co-traveling electromagnetic wave in a wiggler/undulator
– Electrons radiate by stimulated emission in wiggler– Electrons move relative to each other: density variations at wavelength of
radiation
• Density variations radiate coherently in wiggler/undulator• Electromagnetic field is enhanced, with changes to both its amplitude and
phase• Electron move relative to each other in response to to co-traveling
electromagnetic other: density variations grow at wavelength of radiation
• Genuine instability with exponential growth of both the density variation and the electromagnetic radiation
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Further Details
• Can send beam through a dispersive compressor where the microbunching through energy variation is enhanced, “optical klystron”
• Generates higher harmonics
• Since Schottky (shot) noise is “noisy,” can instead seed with laser
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Zhirong Huang, SLAC
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Max Cornacchia,SLAC
LCLS Project Overview
BESAC, Feb. 26-27, 2001
LCLS layout
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(fs)
The SASE radiation is powerful, but noisy!
Solution: Impose a strong coherent modulation with an external laser source
A SASE FEL amplifies random electron density modulations
Spectrum From a SASE FEL
Graves
J. J. Bisognano
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Bill Graves
J. J. Bisognano
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Brookhaven Laser Seeding Demonstration
Buncher
e-
Laser
800 nm
Modulator
266 nm
output
Radiator
•Suppressed SASE noise
•Amplified coherent signal
•Narrowed bandwidth
•Shifted wavelength
High Gain Harmonic Generation (HGHG)
SASE x105
HGHG
L.H. Yu et al., Phys. Rev. Lett. 91, 74801 (2003).
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To Produce Transform-Limited Hard X-ray Pulses
Use “cascaded” High Gain Harmonic Generation methods
Input
seed 0
1st stage 2nd stage …Nth
stage
Stage 1 output at
50 seeds 2nd
stage
Stage 2 output at
250 seeds 3rd
stage
…Nth stage
output at 5N0
W. Graves, MIT
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Key facility elementsKey facility elements
Photoinjecto
r
SRF
linac
SRF
linac
Bunch
compress
orEbeam
switch
Undulators
Photocathod
e laser
Bunch
compress
or
Seed laser
W. Graves, MIT
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High Harmonic Generation (HHG) Seeding
Courtesy of B. Sheehy
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HHG laser seeds at 100 eV Mod 1
100 eV 100 eV 200 eV
Rad 1
Buncher magnets
1. Initial seed is 3 nJ at 100 eV.
2. Radiator 1 amplifies the seed laser.
3. Buncher magnets control the power in each succeeding section by changing the magnitude of harmonic bunching.
1240 eV FEL seeded by 100 kW at 100 eV
2.5 GeV ebeam
1.2 GW @ 1200 eV3 m 1.5 m 2 m 22 m
Spent ebeam is dumped
Fiber link synchronization
1200 eV600 eV
3 m
Rad 2 Rad 3 Rad 4
Graves, MIT
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•Transform-limited output – longitudinal and transverse
•Many beamlines operating simultaneously
•Complete tunability from 6 – 1200 eV
•Fully tunable polarization
•Peak power and brilliance much larger than current XUV sources
•Average flux and brilliance much larger than best synchrotrons and ERLs
•Synchronization of ~10 fs to user lasers
Performance GoalsPerformance Goals
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Three Standard Operating Modes
• Single-shot—Experiments that require the highest available peak brilliance/flux and cannot be cycled rapidly.
• kHz-class experiments—often driven by pump lasers and operate from 10-1000 Hz. Requires CW SC linac.
• MHz-class experiments—includes experiments which can cycle rapidly, where time constants of interest are less than a micro-second. Also includes experiments in the energy domain needing high energy resolution and high flux. Requires CW SC linac and gun.
• All available at the same time
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Breakthrough Science
SRF Electron Injectors Superconducting
Electron Accelerator
FEL Undulatorss
Experimental AreasJacobs and Moore, SRC
Monochromators
RF Separation
Time Resolved Imaging and Coherent ScatteringTaking advantage of the short duration and variable polarization of the FEL x-ray pulse, this technique is particularly suited to study magnetization dynamics. Examples include new materials for high-speed high-density magnetic storage devices.
Resonant Inelastic X-ray ScatteringThis is a powerful technique for studies of low energy electronic and magnetic excitations in materials.
FemtochemistryAllows chemists to follow the dynamics of chemical reactions over extremely short time scales. This may enable chemists to better control reactions to create new products.
Biological SystemsComplex biological processes (for example, photosynthesis, or the transport of information from the eye to the brain) can be studied in snap shot experiments utilizing the precise time pattern and tunability of the FEL.
Exotic Materials, Clusters and NanostructuresThe FEL can be used to the selectively fabricate of atomic clusters and other nanostructures (a billionth of a meter in scale) with specifically tailored medical or material properties.The FEL can also be used to characterize the properties of these new nanoscale materials.
Ultrahigh Resolution SpectroscopyPhotoemission spectroscopy is thetool of choice to study highly correlated systems such as high Tc superconductors, now done with energy resolution in the meV range. With a pulsed FEL source energy resolution of several 10 μeV should be possible.
Bunch compressors
J. J. Bisognano
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Homework Problems
• In the text, the vertical focusing in an undulator is derived from a hamiltonian. From a more newtonian approach in an expansion in 1/γ show that there is vertical focusing when the expansion is carried out to second order.
• Starting at equation 5.17 of Lee, fill in the details to get to equation 5.32.
• Lee 5.1.1