P. Elleaume, CAS, Trieste October 2 - 14, 2005 1/46
Introduction to Insertion DevicesPascal ELLEAUME
European Synchrotron Radiation FacilityGrenoble, France
P. Elleaume, CAS, Trieste October 2 - 14, 2005 2/46
What is an Insertion Device ?
- Insertion Devices are also called Undulators and Wigglers- Can be 1 to 20 m long, (typical 5 m ) with a small magnetic gap (5-15 mm)- Intense Source of Synchrotron Radiation in e- Storage Ring Sources- Control of damping times in Electron Colliders (LEP, CESR,…)
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Table of Content
• Beam Dynamics
• Radiation
• Technology
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Beam Dynamics
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Electron Trajectory in an Insertion Device
: , x z sAssume v v v c = − −
'
( ) ( ') '
( ) ( '') '' '
sxz
s s
z
v s e B s dsc mc
ex s B s ds dsmc
γ
γ
−∞
−∞ −∞
= −
= −
∫
∫ ∫and similar expression for ( ) and ( )zv s z s
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Electron Trajectory in a Planar Sinusoidal Undulator
00
( 0, sin(2 ), 0 )sConsider B Bλ
= π
0
2 22
0
0
0
cos(2 )
0
11 (1 cos (2 ))2
sin (2 )2
x
z
s
v K scvcv sKc
K sx
γ λ
γ λλ
γ λ
= π
=
= − + π
− ππ
0 00 0 0.0934 [ ] [ ]2
eBK B T mmmcλ λ= =
π
Example : ESRF , Energy= 6GeV , Undulator λ0 = 35 mm, B0 = 0.7 T
=> K = 2.3, Kγ= 200 µrad,
λ02π
Kγ= 1.1 µm !!
with
K is a fundamental parameter called : Deflection Parameter
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201 1
2zz ID
eBK ds LF mcγ
⎛ ⎞= = ⎜ ⎟
⎝ ⎠∫
00 0
00 0
0
cosh(2 ) cos(2 )
sinh(2 ) sin(2 )
x
z
s
Bz sB B
z sB B
π πλ λ
π πλ λ
=
=
= −
Undulator Field Satisfying Maxwell Equation
dvm ev Bdt
γ = ×
Lorentz Force Equation
2nd Order in 1 γ −
( )
2
2
2220 0
20
0
1 sinh(4 )2 4 z
d xds
eBd z z K zds mc
λ πγ π λ
=
⎛ ⎞= − −⎜ ⎟
⎝ ⎠
A vertical Field Undulator is Vertically Focusing !
201
2zeBKmcγ
⎛ ⎞= ⎜ ⎟
⎝ ⎠
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Interference with the beam dynamics in the ring lattice
• An Insertion Device is the first component of a photon beamline. Its field setting is fully controlled by the Users of the beamline . The beam dynamics in the whole ring may be altered if the field of an ID is changed => crosstalk of the source parameters in each beamline must be avoided .
• As far as the lattices are concerned, Insertion devices should ideally behave like drift space but the reality is somewhat different :
– Closed Orbit distortion (non zero field integrals generated by design and field errors) – Tune shift (induced by nominal field and by field errors)– Reduction of dynamic aperture (=>Lifetime reduction & reduced injection efficiency ) induced by
varying focusing properties inside the aperture for the beam => critical for modern sources operating in topping-up mode.
– Very high field IDs may change the damping time, emittance , energy spread …
• By combining field shimming and local steering corrections, most of the perturbations are able to be solved.
• The problem of the reduction of dynamic aperture is severest on low energy rings with many insertion devices.
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Radiation from IDs
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Electron bunchSynchrotron light
Electrons
Bending Magnet
1γ
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Critical Energy of Bending Magnet Radiation
140x10 12
120
100
80
60
40
20
0
Phot
ons/
s/.1
%/m
rad2
120100806040200Photon Energy [keV]
Computed for 6 GeV, I = 200 mA, B =1 tesla
Electric Field in the Time Domain Angular Flux in Frequency Domain
323 3
4 4chc heE B
mγ γ
π ρ π= =
600x10 3
500
400
300
200
100
0
Elec
tric
Fiel
d
140x10 -12120100806040200Distance
23ABt Bcρ γγ
≈ ≈
cE
2[ ] 0.665 [ ] [ ]cE keV E GeV B T=
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Undulator
0λ
e-
Source Points
1.5x10 6
1.0
0.5
0.0
-0.5
-1.0
-1.5
Elec
tric
Fiel
d
1.4x10 -91.21.00.80.60.40.20.0
Distance [m]
A
D
C
B A
B
C
D
Electric Field in Time DomainElectron trajectory
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Electric Field and Spectrum vs K
1.5x10 6
1.0
0.5
0.0
-0.5
-1.0
-1.5
Elec
tric
Fiel
d
1.4x10 -91.21.00.80.60.40.20.0
Distance [m]
K=1
K=2
Computed for 6 GeV, I=200 mA, 35 mm period
7x10 15
6
5
4
3
2
1
0Ph
oton
s/se
c/0.
1%/m
rad2
100806040200
Photon Energy [keV]
8x10 15
6
4
2
0Pho
tons
/sec
/0.1
%/m
rad2
100806040200
Photon Energy [keV]
2x10 6
1
0
-1
-2
Elec
tric
Fiel
d
2.5x10 -92.01.51.00.50.0
Distance [m]
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2x10 6
1
0
-1
-2
Elec
tric
Fiel
d
2.5x10 -92.01.51.00.50.0
Distance [m]
Fundamental Wavelength of the Radiation Field
λ
22 20
2
20
0 0 2
2
cos (1 )12 (1 (1 ))
2 2
(1 )2 2AC
c KtK
λθλ λ θ λ
γ
λ γ θγ
= − ≅ − − ≅ + +− +
0λ
e-
θ
0 cosλ θ
AC
0 02
21(1 (1 ))
2 2
ACs
tKv c
λ λ
γ
= =< > − +
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Wavelength of the Harmonics
02
0 0
, : , 1, 2,3,..:
:
: : 0.0934 [ ] [ ]
:
thn nE Wavelength Energy of the n harmonic
n Harmonic numberUndulator period
E mc Electron EnergyK Deflection Parameter B T mm
Angle between observer direction and e
λ
λ
γλ
θ
=
==
− beam
22 20
2 (1 )2 2nK
nλλ γ θγ
= + +
2
22 2
0
9.5 [ ][ ][ ](1 )
2
nn E GeVE keV
Kmmλ γ θ=
+ +
In an equivalent manner , the energy En of the harmonics are given by
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Undulator Emission by a Filament Electron Beam6
5
4
3
2
1
0
x101
5
302520151050
keVEnergy
Flux in a Pinhole
n=1
n=3
n=2 1EE nN∆
≈
1.0
0.8
0.6
0.4
0.2
0.0
T
-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15
mAngle
Flux in a Pinhole
0
∆θ ≈1
γ nN
n : Harmonic number
N : Number of Periods
γ =1
1− v2
c 2
=E
mc2
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• Until now a Filament mono-energetic electron beam has been assumed.
• What happens if the beam presents a finite emittance (size and divergence) and finite energy spread ?
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Filament e- Beam
e- Beam withFinite Divergence
1Nnγ
∼
'zσ∼
'xσ∼
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400
300
200
100
x109
2520151050
keV
Filament Mono energetic Beam
ESRF Emittance and Energy Spread(4, nm, 0.02 nm, 0.1% )
Electron BeamEnergy = 6 GeVCurrent = 200 mA UndulatorPeriod = 35 mmK = 2.2Length = 3.2 m Collection 10 x 10 µm
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Broadening of the Harmonics by the Electron Emittance
200x10 9
150
100
50
0
Phot
/s/0
.1%
bw
20keV151050
Photon Energy
Electron BeamEnergy = 6 GeVCurrent = 200 mAVert. Emitt= 0.04 nmEnergy Spread = 0.1 % UndulatorPeriod = 35 mmK = 2.2Length = 3.2 m Collection 10 x 10 µm
Horiz Emitt.= 2 nm
8 nm 4 nm
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Broadening of the Harmonics by Electron Energy Spread
160x10 9
140
120
100
80
60
40
20
0
Phot
/s/0
.1%
bw
20keV151050
Photon Energy
Electron BeamEnergy = 6 GeVCurrent = 200 mAHor,z. Emitt. = 4 nmVert. Emitt= 0.04 nm UndulatorPeriod = 35 mmK = 2.2Length = 3.2 m Collection 10 x 10 µm
Energy Spread = 0 %0.1 %
0.2 %
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• To Make Optimum use of the Undulators, The magnet Lattices of synchrotron light sources are optimized to produce the smallest emittance and smallest energy spread of the electron beam possible.
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3.0x10 15
2.5
2.0
1.5
1.0
0.5
0.0
Phot
/s/0
.1%
bw
8000eV600040002000
Photon Energy
Electron BeamEnergy = 6 GeVCurrent = 200 mAHoriz. Emitt. = 4 nmVert. Emitt= 0.04 nmEnergy Spread = 0.1 % UndulatorPeriod = 35 mmK = 2.2Length = 3.2 m
8 x 8 mm
5 x 5 mm
3 x 3 mm
1 x 1 mm
Collecting Undulator Radiation in a variable Aperture
F1 F3
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Maximum Spectral Flux On-axis on odd harmonics
1.0
0.8
0.6
0.4
0.2
0.0
Qn(
K)
543210
K
n =1n =3 n =5
n =7
Fn [Ph /sec/ 0.1%] = 1.431 1014 N I[A ] Qn (K )
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Brilliance ( or Brightness)
Bn =Fn
(2π)2ΣxΣ'x Σ zΣ' z
Σx = σx2 + λnL
(2π)2
Σ'x= σ 'x2 + λn
2L
Electron beam Single electron emission
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Brilliance vs Photon Energy
1018
1019
1020
1021
1022
Phot
/s/0
.1%
bw/m
m2 /m
r2
3 4 5 6 7 8 910keV
2 3 4
Photon Energy
UndulatorPeriod = 35 mmK = 2.2Length = 5 m
UndulatorPeriod = 26 mmK = 1.2Length = 5 m
ESRF Energy = 6 GeVCurrent = 200 mAEmittances = 4 & 0.02 nmLow Beta Straight
Bn =Fn
(2π)2ΣxΣ'x Σ zΣ' z
Σx = σx2 + λnL
(2π)2
Σ'x= σ 'x2 + λn
2L
with
n = 1
n = 3n = 5
n = 1
n = 3
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200x1012
150
100
50
0
Phot
ons/s
ec/0
.1%
35302520151050
600x1012
500
400
300
200
100
0
Phot
ons/
sec/
0.1%
35302520151050
1 .0x1015
0.8
0.6
0.4
0.2
Phot
ons/
sec/
0.1%
3 5302520151050
Angle Integrated Flux
K=0.5 K=1
K=2 For Large K, the angle integrated spectrum from an Undulator tends toward that of a bending magnet x 2N=> Such Devices are called Wigglers
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Technology
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Technology of Undulators and Wigglers
• The fundamental issue in the magnetic design of a planar undulator or wiggler is to produce a periodic field with a high peak field B and the shortest period λ0 within a given aperture (gap).
• Three type of technologies can be used :– Permanent magnets ( NdFeB , Sm2Co17 )– Room temperature electromagnets – Superconducting electromagnets
Gap
Period
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Current Equivalent of a Magnetized Material
M
0
[ ]Air coil with Surface Current Density[A/m] rB Tµ
≅
⇔
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Periodic Array of Magnets
⇔
0
0
2
0
2 [ ]Surface Current Density[A/m]
4 [ ]or Current Density[A/m ]
r
r
B T
B T
µ
µ λ
≅
≅
0λ
2r 0
2r 0
Examples:B =1 T , λ =20 mm Equiv. Current Density=160A/mm !!
B =1 T , λ =400 mm Equiv. Current Density=8 A/mm
⇒
⇒
~ 95 % of Insertion Devices are made of Permanent Magnets !!
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Permanent Magnet Undulator
Magnet (NdFeB, Sm2Co17,...)
Pole(Steel)
Pure Permanent MagnetHybrid
0.1
2
3
4
5
6789
1
2
Mag
netic
Fie
ld [T
]
1.00.80.60.40.2Gap / λ0
Magnet Volume = 2 N λ03
Lx Magnet = 2 λ0 First Harmonic Peak Field
Hybrid (Vanadium Permendur)
Pure Permanent Magnet
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Support StructureStepper Motors90˚ Gear Box
Springs
Ball Bearing Leadscrew
Upper Girder
Lower Girder
Height & TiltAdjustment
Welded Framework
Guiding Rails
Mechanical Stop
Dovetail to FixMagnetic Assemblies
Must Handle :Magnetic Force : 1-20 TonsGap Resolution : < 1 µmParallellism < 20 µm
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• For a permanent magnet undulator , shrinking all dimensions maintains the field unchanged. The peak field Bp ~ Br F(gap/period)
• Benefits of using small gaps Insertion Devices :– Decrease the volume of material (cost driving) ~ gap3
– The lower the gap, the higher the energy of the harmonics of the undulator emission => the lower the electron energy required to reach the same photon energy
• The most advanced undulators have magnet blocks in the vacuum with an operating magnetic gap of 4-6 mm
Undulators are Fundamentally Small Gap Devices
202 (1 )2 2
Kλλγ
= + 0 0 2eBwith K
mcλ
=π
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In Vacuum Permanent Magnet Undulators
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Application : Build a pure permanent magnetundulator with NdFeB Magnets (Br = 1.2 T)
8.210.315.2
Fundamental [keV]@ 6 GeV
8.27.36.0
ElectronEnergy [GeV]
Fund = 15.2 keV
282215
Period[mm]
0.38150.49100.725
B [T]
Gap [mm]
Undulator with K=1
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Electro-Magnet Undulator
Current Densities < 5-20 A/mm2
Lower fieldthan permanent magnetFor small period / gap
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Superconducting Wigglers
- High field : up to 10 T => Shift the spectrum to higher energies- Complicated engineering & High costs
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Magnetic Field Errors in Permanent Magnet Insertion Devices :
• Field errors originate from :– Non uniform magnetization of the magnet blocks (poles).– Dimensional and Positional errors of the poles and magnet blocks. – Interaction with environmental magnetic field (iron frame, earth field,…)
• Important to use highly uniform magnetized blocks– perform a systematic characterization of the magnetization – Perform a pairing of the blocks to cancel errors– Still insufficient …
• Two main type of field errors remain– Multipole Field Errors (Normal and skew dipole, quadrupole, sextupole,…). – Phase errors which reduce the emission on the high harmonic numbers – Further corrections :
• Active steerers• Shimming
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Shimming
• Mechanical Shimming : – Moving permanent magnet or iron pole vertically or horizontally– Best when free space and mechanical fixation make it possible.
• Magnetic Shimming : – Add thin iron piece at the surface of the blocks– Reduce minimum gap and reduce the peak field
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Magnetic shims
Phase Shim
Phase Shim
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Field Integral and Multipole Shimming
Horizontal DeflectionQuadrupoleSextupole …
Vertical DeflectionSkew QuadrupoleSkew sextupole …
Gap/2 [mm]
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Tp Tp+1 Tp+2
Harmonic # 4
Harmonic # 1
The Phase shimming consists of a set of local magnetic field corrections, which make Tp always identical.
2
2
0 (1 )2
pcT
Kγ
λ=
+
Tp varies from one pole to the next due to period and peak field fluctuations
Tp : time distance between successive peaks
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Phase Shimming and the single electron spectrum
500
400
300
200
100
0
x101
5
6050403020100-10keV
BeforeShimming
AfterShimming
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• I hope that this short introduction has incited your curiosity in the broad and exciting field that is Insertion Devices.
• Bibliography – J.D. Jackson, Classical Electrodynamics, Chapter 14, John Wiley– K.J. Kim, Characteristics of Synchrotron Radiation, AIP Conference Proceedings 184, vol.
1 p567 (American Institute of Physics, New York, 1989).– R.P. Walker , CAS - CERN Accelerator School: Synchrotron Radiation and Free Electron
Lasers, Grenoble, France, 22 - 27 Apr 1996 , CERN 98-04 p 129.– H. Onuki, P. Elleaume “Undulators, Wigglers and their Applications”, Taylor and Francis,
2003, ISBN 0 415 28040 0.– A. Hofman, “The Physics of Synchrotron Radiation”, Cambridge University Press, 2003.
ISBN 0 521 30826 7– J. A. Clarke , “The Science and Technology of Undulators and Wigglers, Oxford Science
Publications, 2004. ISBN 0 19 850855 7– S. Russenschuck, “Electromagnetic Design and Mathematical Optimization Methods
in Magnet Technology”, ebook available from “http://russ.home.cern.ch/russ/”