Introduction to Insertion Devices
Pascal ELLEAUME European Synchrotron Radiation Facility Grenoble, France
P. Elleaume, CAS, Trieste October 2 - 14, 2005 1/46 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,…)
P. Elleaume, CAS, Trieste October 2 - 14, 2005 2/46 Table of Content
• Beam Dynamics
•Radiation
• Technology
P. Elleaume, CAS, Trieste October 2 - 14, 2005 3/46 Beam Dynamics
P. Elleaume, CAS, Trieste October 2 - 14, 2005 4/46 Electron Trajectory in an Insertion Device
Consider Ortogonal Frame Oxzs Lorentz Force Electron velocity: v= (, vxzs v , v ) dv γ mevB=× Electron position :(,,) R= x z s dt dvx Magnetic field :(,,) B= Bx Bzs B =>mevBvB = −()s zzs − 1 γ dt Define :γ = v2 1− c2
Assume: vxz , v <<≈ v s c
vs() e s x =− Bsds(')' cmcγ ∫−∞ z e ss' x() s=− Bz ('') s ds ''' ds mcγ ∫∫−∞ −∞
and similar expression for vsz ( ) and zs ( ) P. Elleaume, CAS, Trieste October 2 - 14, 2005 5/46 Electron Trajectory in a Planar Sinusoidal Undulator s Consider B=π( 0, B0 sin(2 ), 0 ) λ0 v Ks x =πcos(2 ) c γλ0 v z = 0 c λ v s 1 22 s =−1(1cos(2))2 +K π c 2γ γλλ 0 Ks x −π0 sin(2 ) 2 π 0 eB λ with KBTmm == 00 0.0934 [ ] λ [ ] 2πmc 00 K is a fundamental parameter called : Deflection Parameter
Example : ESRF, Energy= 6GeV , Undulator λ0 = 35 mm, B0 = 0.7 T K λ K => K = 2.3, = 200 µrad, 0 = 1.1 µm !! γ 2π γ P. Elleaume, CAS, Trieste October 2 - 14, 2005 6/46 Bx = 0 zs BB= cosh(2ππ )cos(2 ) dv z 0 λ λ ππ00 γ mevB=× zs dt BBs =− 0 sinh(2 )sin(2 ) λ00λ Undulator Field Satisfying Lorentz Force Equation Maxwell Equation
2nd Order in γ −1 dx2 = 0 ds2 2 2 dz1 ⎛⎞eB λ z 2 2 γπ00 λ 1 ⎛⎞eB 2 =− ⎜⎟sinh(4 ) −()Kzz 0 ds24 mc Kz = ⎜⎟ ⎝⎠ π 0 2 ⎝⎠γ mc
2 11⎛⎞eB A vertical Field Undulator is Vertically Focusing ! ==K ds 0 L ∫ z ⎜⎟ Fmcz ID 2 ⎝⎠γ
P. Elleaume, CAS, Trieste October 2 - 14, 2005 7/46 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.
P. Elleaume, CAS, Trieste October 2 - 14, 2005 8/46 Radiation from IDs
P. Elleaume, CAS, Trieste October 2 - 14, 2005 9/46 Bending Magnet
Synchrotron light Electron bunch
1 γ
Electrons
P. Elleaume, CAS, Trieste October 2 - 14, 2005 10/46 Critical Energy of Bending Magnet Radiation
Electric Field in the Time Domain Angular Flux in Frequency Domain
3 600x10 140x10 12 500 120 ρ 400 tB≈≈γ 2 100 AB γ 3c 300 80
200 60 Electric Field 40 100 Photons/s/.1%/mrad2 20 0 0 20 40 60 80 100 120 140x10 -12 0 0 20 40 60 80 100 120 Distance E c Photon Energy [keV] Computed for 6 GeV, I = 200 mA, B =1 tesla
3 33hcγ he 2 2 Ec ==γ B E []0.665keV= E[] GeV B[] T 44πρ π m c
P. Elleaume, CAS, Trieste October 2 - 14, 2005 11/46 Undulator
Source Points
1.5x10 6
A C 1.0 B D
0.5
0.0
e- -0.5 Electric Field BA-1.0 D -1.5 C 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4x10 -9 Distance [m] λ0 Electron trajectory Electric Field in Time Domain
P. Elleaume, CAS, Trieste October 2 - 14, 2005 12/46 Electric Field and Spectrum vs K
1.5x10 6
2 8x10 15 1.0
0.5 6
0.0 K=1 4 -0.5 Electric Field -1.0 2
-1.5 -9 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4x10 Photons/sec/0.1%/mrad 0 Distance [m] 0 20 40 60 80 100 Photon Energy [keV]
7x10 15 2x10 6
2 6
1 5
4 0 K=2 3
Electric Field -1 2
1 -2 Photons/sec/0.1%/mrad
-9 0 0.0 0.5 1.0 1.5 2.0 2.5x10 0 20 40 60 80 100 Distance [m] Photon Energy [keV]
Computed for 6 GeV, I=200 mA, 35 mm period P. Elleaume, CAS, Trieste October 2 - 14, 2005 13/46 Fundamental Wavelength of the Radiation Field
λ0 cosθ 2x10 6
1 A θ C 0
e- Electric Field -1
-2
0.0 0.5 1.0 1.5 2.0 2.5x10 -9 Distance [m] λ0 λλ θ λ λ λ t ==00 AC <>v 1 K 2 s c(1−+ (1 )) λ 222γ θ γ 2 λ K 2 =−≅−−cosct (1 ) 0 ≅0 (1++γ 22θ ) 00AC 2 1 K 2 222 (1−+ (1 )) λ 222γ P. Elleaume, CAS, Trieste October 2 - 14, 2005 14/46 Wavelength of the Harmonics
λ γ K 2 λ =++0 (1γθ22 ) n 22n 2
In an equivalent manner , the energy En of the harmonics are given by
9.5 nE 2 [ GeV ] E[] keV = n K 2 λ [](1mm ++γθ22 ) 0 2
th λnn,E : Wavelength , Energy of the n harmonic n=1, 2, 3, ..: Harmonic number
0 λ: Undulator period E= mc2 : Electron Energy γ K : Deflection Parameter = 0.0934 B00 [ T ] [ mm ] : Angle between observer direction and e − beam P. Elleaume, CAS, Trieste October 2θ - 14, 2005 15/46
λ Undulator Emission by a Filament Electron Beam
6 Flux in a Pinhole 5 ∆ E 1 n=1 n=2 ≈ 4 EnN
15 3 x10
2 n=3
1
0 0 5 10 15 20 25 Energy30 keV
n : Harmonic number 1 1.0 Flux in a Pinhole ∆θ ≈ γ nN N : Number of Periods 0.8
0.6
T 1 E γ = = 2 2 0.4 v mc 1− 2 0.2 c
0.0 -0.15 -0.10 -0.05 0.000 0.05 0.10Angle 0.15 P. Elleaume, CAS, Trieste October 2 - 14,m 2005 16/46 • 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 ?
P. Elleaume, CAS, Trieste October 2 - 14, 2005 17/46 1 ∼ Filament γ Nn e- Beam
' e- Beam with ∼ σ z Finite Divergence ' ∼ σ x
P. Elleaume, CAS, Trieste October 2 - 14, 2005 18/46 Electron Beam Energy = 6 GeV Filament Current = 200 mA
400 Mono energetic Beam Undulator Period = 35 mm K = 2.2 Length = 3.2 m
300 Collection 10 x 10 µm
9 ESRF Emittance x10 and Energy Spread 200 (4, nm, 0.02 nm, 0.1% )
100
0 5 10 15 20 25 keV
P. Elleaume, CAS, Trieste October 2 - 14, 2005 19/46 Broadening of the Harmonics by the Electron Emittance
200x10 9 Electron Beam Energy = 6 GeV Horiz Emitt.= 2 nm Current = 200 mA 4 nm Vert. Emitt= 0.04 nm 150 Energy Spread = 0.1 % 8 nm Undulator Period = 35 mm K = 2.2 100 Length = 3.2 m
Collection
Phot/s/0.1%bw 10 x 10 µm
50
0 0 5 10 15 20keV Photon Energy P. Elleaume, CAS, Trieste October 2 - 14, 2005 20/46 Broadening of the Harmonics by Electron Energy Spread
Energy Spread = 0 % 9 Electron Beam 160x10 0.1 % Energy = 6 GeV Current = 200 mA 140 0.2 % Hor,z. Emitt. = 4 nm Vert. Emitt= 0.04 nm
120 Undulator Period = 35 mm 100 K = 2.2 Length = 3.2 m
80 Collection 10 x 10 µm Phot/s/0.1%bw 60
40
20
0 0 5 10 15 20keV Photon Energy
P. Elleaume, CAS, Trieste October 2 - 14, 2005 21/46 • 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.
P. Elleaume, CAS, Trieste October 2 - 14, 2005 22/46 Collecting Undulator Radiation in a variable Aperture
3.0x10 15 Electron Beam Energy = 6 GeV Current = 200 mA F1 Horiz. Emitt. = 4 nm F 2.5 Vert. Emitt= 0.04 nm 3 Energy Spread = 0.1 %
Undulator 2.0 Period = 35 mm 8 x 8 mm K = 2.2 Length = 3.2 m
1.5 5 x 5 mm Phot/s/0.1%bw
1.0 3 x 3 mm
0.5 1 x 1 mm
0.0 2000 4000 6000 8000eV Photon Energy
P. Elleaume, CAS, Trieste October 2 - 14, 2005 23/46 Maximum Spectral Flux On-axis on odd harmonics
14 Fn [Ph /sec/ 0.1%] = 1.431 10 N I[A] Qn (K )
1.0
0.8 n =1 0.6 n =3 n =5 (K) n
Q 0.4
0.2 n =7
0.0 0 1 2 3 4 5 K P. Elleaume, CAS, Trieste October 2 - 14, 2005 24/46 Brilliance ( or Brightness)
Fn Bn = 2 (2π) ΣxΣ'x ΣzΣ'z
λ L Σ = σ 2 + n x x (2π)2 λ Σ' = σ '2 + n x x 2L
Electron beam Single electron emission
P. Elleaume, CAS, Trieste October 2 - 14, 2005 25/46 Brilliance vs Photon Energy
1022 F ESRF B = n Energy = 6 GeV n 2 Current = 200 mA (2π) ΣxΣ'x Σ zΣ'z 1021 Emittances = 4 & 0.02 nm 2 n = 1 Low Beta Straight /mr 2 with n = 3 1020 n = 1 2 λnL Undulator Σx = σx + 2 Period = 35 mm (2π) K = 2.2 Phot/s/0.1%bw/mm 19 10 Length = 5 m n = 3 2 λn Undulator n = 5 Σ'x = σ 'x + Period = 26 mm K = 1.2 2L Length = 5 m 1018 3 4 5 6 7 8 9 2 3 4 10keV Photon Energy
P. Elleaume, CAS, Trieste October 2 - 14, 2005 26/46 Angle Integrated Flux
12 200x10 12 600x10
500 150 K=0.5 K=1 400
100 300
Photons/sec/0.1% 200 50 Photons/sec/0.1% 100
0 0 5 10 15 20 25 30 35 0 0 5 10 15 20 25 30 35
1.0x1015 For Large K, the angle integrated spectrum 0.8 K=2
0.6 from an Undulator tends toward that of a
0.4
Photons/sec/0.1% bending magnet x 2N 0.2 => Such Devices are called Wigglers
0 5 10 15 20 25 30 35
P. Elleaume, CAS, Trieste October 2 - 14, 2005 27/46 Technology
P. Elleaume, CAS, Trieste October 2 - 14, 2005 28/46 Technology of Undulators and Wigglers
Gap
Period • 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
P. Elleaume, CAS, Trieste October 2 - 14, 2005 29/46 Current Equivalent of a Magnetized Material
M ⇔
B []T Air coil with Surface Current Density[A/m] ≅ r µ 0
P. Elleaume, CAS, Trieste October 2 - 14, 2005 30/46 Periodic Array of Magnets
λ0 ⇔
2[]BT Surface Current Density[A/m] ≅ r µ 0
2 4[]BTr or Current Density[A/m ] ≅ µλ Examples: 0 0 2 B=1T,r0λ =20mm⇒ Equiv. Current Density=160A/mm !! 2 B=1T,r0λ =400mm⇒ Equiv. Current Density=8 A/mm
~ 95 % of Insertion Devices are made of Permanent Magnets !!
P. Elleaume, CAS, Trieste October 2 - 14, 2005 31/46 Permanent Magnet Undulator
Hybrid Pure Permanent Magnet
Pole (Steel)
2 3 Hybrid Magnet Volume = 2 N λ0 Lx Magnet = 2 λ Magnet (NdFeB, Sm Co ,...) (Vanadium Permendur) 0 2 17 First Harmonic 1 9 Peak Field 8 7 6 5 4 Pure Permanent Magnet
3 Magnetic Field [T]
2
0.1 0.2 0.4 0.6 0.8 1.0
Gap / λ0 P. Elleaume, CAS, Trieste October 2 - 14, 2005 32/46 Support Structure Stepper Motors 90û Gear Box
Must Handle : Magnetic Force : 1-20 Tons Springs Guiding Rails Gap Resolution : < 1 µm Parallellism < 20 µm
Upper Girder
Ball Bearing Mechanical Stop Leadscrew
Lower Girder Dovetail to Fix Magnetic Assemblies
Welded Framework
Height & Tilt Adjustment P. Elleaume, CAS, Trieste October 2 - 14, 2005 33/46 Undulators are Fundamentally Small Gap Devices
• 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 2 λ λ K eB00λ =+0 (1 ) with K = 2 2πmc 22γ
• The most advanced undulators have magnet blocks in the vacuum with an operating magnetic gap of 4-6 mm
P. Elleaume, CAS, Trieste October 2 - 14, 2005 34/46 In Vacuum Permanent Magnet Undulators
P. Elleaume, CAS, Trieste October 2 - 14, 2005 35/46 P. Elleaume, CAS, Trieste October 2 - 14, 2005 36/46 Application : Build a pure permanent magnet
undulator with NdFeB Magnets (Br = 1.2 T)
Undulator with K=1
Gap B Period Fundamental [keV] Electron [mm] [T] [mm] @ 6 GeV Energy [GeV] Fund = 15.2 keV 5 0.72 15 15.2 6.0 10 0.49 22 10.3 7.3 15 0.38 28 8.2 8.2
P. Elleaume, CAS, Trieste October 2 - 14, 2005 37/46 Electro-Magnet Undulator
Current Densities < 5-20 A/mm2
Lower field than permanent magnet For small period / gap
P. Elleaume, CAS, Trieste October 2 - 14, 2005 38/46 Superconducting Wigglers
- High field : up to 10 T => Shift the spectrum to higher energies - Complicated engineering & High costs
P. Elleaume, CAS, Trieste October 2 - 14, 2005 39/46 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
P. Elleaume, CAS, Trieste October 2 - 14, 2005 40/46 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
P. Elleaume, CAS, Trieste October 2 - 14, 2005 41/46 Magnetic shims
Phase Shim
Phase Shim
P. Elleaume, CAS, Trieste October 2 - 14, 2005 42/46 Field Integral and Multipole Shimming
Horizontal Deflection Vertical Deflection Quadrupole Skew Quadrupole Sextupole … Skew sextupole …
Gap/2 [mm]
P. Elleaume, CAS, Trieste October 2 - 14, 2005 43/46 cγ 2 T : time distance between successive peaks T = p p K 2 λ(1+ ) 0 2
Tp varies from one pole to the next due to period and peak field fluctuations
Harmonic # 1 T Tp p+1 Tp+2
Harmonic # 4
The Phase shimming consists of a set of local magnetic field corrections,
which make Tp always identical.
P. Elleaume, CAS, Trieste October 2 - 14, 2005 44/46 Phase Shimming and the single electron spectrum
500 After Shimming
400 Before Shimming 300
15 x10 200
100
0
-10 0 10 20 30 40 50 60 keV
P. Elleaume, CAS, Trieste October 2 - 14, 2005 45/46 • 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/”
P. Elleaume, CAS, Trieste October 2 - 14, 2005 46/46