Transcript UW Spring 2008 Accelerator Physics

UW Spring 2008
Accelerator Physics
Topic IX
Wigglers, Undulators, and FELs
Joseph Bisognano
Engineering Physics &
Synchrotron Radiation Center
University of Wisconsin
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Bending Magnet Radiation
CERN School 1998
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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
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Insertion Devices
CERN School 1998
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Light Source
CERN School 1998
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Ideal ID Field Pattern
(infinite pole tips in x)
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Magnetic scalar potential
 ( s, z )  f ( z ) cos( 2
s
u
)  f ( z ) cos( ku s)
Maxwell  Laplace 
 ( s, z )  A sinh( ku z ) cos( ku s )
B0
Bz 
cosh( ku z ) cos( ku s)
g
cosh(  u )
~
Bz  B cosh( ku z ) cos( ku s)
~
Bs   B sinh( ku z ) sin( ku s ) by divB  0
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 IDs
Neglecting vertical motion, we have
e
Bz ( s )
x   s
m
e
Bz ( s )
s   x
m
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First Order Solution
Assume
x  s, s  c; x  x' c

~
eB
x"  
cos( ku s ); with   1
mc
~
Since there is a Bs, one
u eB
x' ( s) 
sin(ku s ) can get a vertical force;
2mc
i.e., focusing
2 ~
u eB
x( s)  2
cos( ku s )
4 mc
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Basic Parameters
Maximum Angle
~
1 u eB K
K

 
 2mc  natural angle
Undulator K  1  interference
Wiggler K  1
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Second Order
Energy Conservation says that if x is moving it’s at the
expense of longitudinal energy
1
 *    1  2 [1  K 2 / 2]
2
Kc
x (t )  
sin( u t )

cK 2
s(t )   * c  2 cos( 2u t )
4
K
x(t )  
cos(u t )
ku 
K2
s(t )   * ct 
sin( 2u t )
2
8ku 
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In Beam Frame
Beam frame coordinates t
and frequency
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Lorentz Transforms and Radiating
In beam frame, electron sees shortened
wiggler with period  /
Since it' s still going almost c, it oscillated with
a frequency   c/ or  beam   ku  c
Now, we need to go back into the lab frame
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Photon Frequency in Lab
Expect 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
 beam
 lab 
 (1   cos  )
ku  c
 lab 
(1   cos  )
 lab  2 2 ku c(1  K 2 / 2   2 2 )
K and  give tuning knobs
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Undulator Spectrum
Since 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 School
Higher harmonics add to reach of an undulator
Require care in phase errors of undulator periodic fields
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Cern School
Fundamental power/total power=1/(1+K2/2)1/2
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R Walker
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Spontaneous Emission
Spontaneou s radiation (planar undulator)
0  2 2 (2c / w ) /(1  K 2 / 2)
eBww
K
Note that for higher frequency, you
2
2mc
need higher energy or shorter
undulator period

1

Shorter undulator period implies

Nw
 
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
1
Nw
smaller gap
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R Walker, CERN School
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Brightness/Brilliance
Brightness 
Flux / PhaseSpaceArea
Flux
B
(2 ) 2  x x  z z
x    
2
x
2
R
x   x2   R2
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L / 2
R 
4
2
 R 
L
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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
Spontaneous radiation (helical wiggler)
 0  2 (2c / w ) /(1  K )
eBw w
K
2
2mc

1

 Nw
2
2
1 1
 
 Nw
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FEL Dynamics I
Consider

Bw  Bw ( xˆ cos k w s  yˆ sin k w s )


  0s  
1 K 2
0  1 
2 2

K
   ( xˆ cos k w s  yˆ sin k w s )

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FEL Dynamics II
Suppose circularly polarized radiation

E  E 0 ( xˆ sin( ks  t   0 )  yˆ cos( ks  t   0 )


B  sˆ  E
e  
 
 E
mc
eE0 K
 
sin 
mc
  (k w  k ) s  t   0 ; with s   0 c
k 1 K 2
   w (1 
)
2
k w 2
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FEL Dynamics III
We' ll get net energy exchange if   0
 0   w [2 2 /(1  K 2 )]   w /(1   0 )
Electron moves through one period in time
w /  0 c
photon, in w / c
w 1   0 
w /  0 c  w / c 
  rad period
c 0
c
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Another Pendulum Equation
Let

(1  K 2 )
2 w
 r

r
r 
Then
2
  
sin 
2 w
  2 w
2 
2 weE0 K
2
mc r
   2 sin   0
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η
φ
Gain only when energy of beam doesn’t quite match “ideal”
energy
CERN School
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If wiggler is two long, process reverses, unless wiggler is
“tapered”
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FEL parameters
electron energy loss
•G 
energy in field
G  ( )mc N /(
2
2 E0 2
8
V)
G  4(4N w ) f (4N w  )

3
4e ( N / V ) K
 
2
2
16 m w
2
3
2
Need high beam density
Bucket height is  1/NW , need small energy spread
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SYLee Text
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SYLee Text
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Looks like derivative of undulator
power spectrum: fluctuationdissipation or Madey’s theorem
SYLee Text
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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:
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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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LCLS Project Overview
LCLS layout
Ma x Cor nac chia,
SLAC
J. J. Bisognano
BESAC, Feb. 26-27, 2001
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Spectrum From a SASE FEL
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A SASE FEL amplifies random electron density modulations
/ (%)
t (fs)
Graves
The SASE radiation is powerful, but noisy!
Solution: Impose a strong coherent modulation with an
external laser source
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Bill Graves
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e-
Brookhaven Laser Seeding
Demonstration
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outpu
t
266 nm
Laser
800 nm
Modulator
Buncher
High Gain Harmonic
Generation (HGHG)
Radiator
HGHG
•Suppressed SASE noise
•Amplified coherent signal
SASE x105
•Narrowed bandwidth
•Shifted wavelength
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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
Stage 1 output at
Stage 2 output at
50 seeds 2nd stage
250 seeds 3rd stage output at 5N0
…Nth stage
Input
seed 0
1st stage
2nd stage
stage
W. Graves, MIT
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…Nth
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Key facility elements
Bunch
Bunch
compresso
compresso
r
r
Photoinjector
SRF linac
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Ebeam
switch
SRF linac
Seed laser
Photocathode
laser
W. Graves, MIT
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Undulators
UW Spring 2008
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High Harmonic Generation (HHG) Seeding
UW Spring 2008
Courtesy of
B. Sheehy
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1240 eV FEL seeded by 100 kWUWat
100
eV
Spring 2008
Fiber link synchronization
HHG laser
seeds at 100 eV
Mod 1
100 eV
3m
Rad 1
Rad 2
Rad 3
Rad 4
100 eV
200 eV
600 eV
1200 eV
1.5 m
2m
3m
22 m
2.5 GeV
ebeam
Spent
ebeam is
dumped
1.2 GW @
1200 eV
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.
Graves, MIT
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Performance Goals
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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
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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
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Femtochemistry
Allows 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.
Time Resolved Imaging and Coherent Scattering
Taking 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 highspeed high-density magnetic storage devices.
Bunch compressors
SRF
Electron
Injectors
Biological Systems
Complex 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.
Superconducting
Electron Accelerator
Resonant Inelastic X-ray
Scattering
This is a powerful
technique for studies of
low energy electronic and
magnetic excitations in
materials.
RF Separation
FEL
Undulatorss
Ultrahigh Resolution Spectroscopy
Photoemission spectroscopy is the
tool 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.
Monochromators
Jacobs and Moore, SRC
J. J. Bisognano
Exotic Materials, Clusters and
Nanostructures
The 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.
Experimental
Areas
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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
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