Introduction to Insertion Devices - CERN...P. Elleaume, CAS, Trieste October 2 - 14, 2005 2/46 What...

P. Elleaume, CAS, Trieste October 2 - 14, 2005 1/46

Introduction to Insertion DevicesPascal ELLEAUME

European Synchrotron Radiation FacilityGrenoble, France


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,…)


Table of Content

• Beam Dynamics

• Radiation

• Technology


Beam Dynamics


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


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


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

⎛ ⎞= ⎜ ⎟

⎝ ⎠


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.


Radiation from IDs


Electron bunchSynchrotron light

Electrons

Bending Magnet

1γ


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=


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


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]


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

= =< > − +


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


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


• 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 ?


Filament e- Beam

e- Beam withFinite Divergence

1Nnγ

∼

'zσ∼

'xσ∼


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


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


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 %


• 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.


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


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 )


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


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


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


Technology


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


Current Equivalent of a Magnetized Material

M

0

[ ]Air coil with Surface Current Density[A/m] rB Tµ

≅

⇔


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 !!


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


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


• 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


In Vacuum Permanent Magnet Undulators


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


Electro-Magnet Undulator

Current Densities < 5-20 A/mm2

Lower fieldthan permanent magnetFor small period / gap


Superconducting Wigglers

- High field : up to 10 T => Shift the spectrum to higher energies- Complicated engineering & High costs


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


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


Magnetic shims

Phase Shim

Phase Shim


Field Integral and Multipole Shimming

Horizontal DeflectionQuadrupoleSextupole …

Vertical DeflectionSkew QuadrupoleSkew sextupole …

Gap/2 [mm]


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


Phase Shimming and the single electron spectrum

500

400

300

200

100

0

x101

5

6050403020100-10keV

BeforeShimming

AfterShimming


• 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/”

Introduction to Insertion Devices - CERN...P. Elleaume, CAS, Trieste October 2 - 14, 2005 2/46 What...

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Transcript of Introduction to Insertion Devices - CERN...P. Elleaume, CAS, Trieste October 2 - 14, 2005 2/46 What...