R. Bartolini, John Adams Institute, 19 November 20101/30 Electron beam dynamics in storage rings Synchrotron radiation and its effect on electron dynamics

R. Bartolini, John Adams Institute, 19 November 2010 1/30

Electron beam dynamics in storage rings

Synchrotron radiation and its effect on electron dynamics

Lecture 9: Synchrotron radiation (11 Nov)

Lecture 10: Radiation damping (12 Nov)

Lecture 11: Radiation excitation (18 Nov)

Lecture 12: Undulators and Wigglers (19 Nov)

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Contents

Radiation emitted by undulators and wigglers

Types of undulators and wigglers

Effects on electron beam dynamics

Conclusions

R. Bartolini, John Adams Institute, 19 November 2010

3/30

Undulators and wigglersPeriodic array of magnetic poles providing a sinusoidal magnetic field on axis:

),0),sin(,0( 0 zkBB u

ztK

ctxtK

tr uu

zuu ˆ)2cos(

16ˆsin

2)(

2

2

mc

eBK u

20

Undulator parameter

Constructive interference of radiation emitted at different poles

Solution of equation of motions:

nd uu cos

21

2

11

2

2

Kz

22

2

2 21

2

K

nu

n


4/30

From the lecture on synchrotron radiation

Continuous spectrum characterized by c = critical energy

c(keV) = 0.665 B(T)E2(GeV)

eg: for B = 1.4T E = 3GeV c = 8.4 keV

(bending magnet fields are usually lower ~ 1 – 1.5T)

Quasi-monochromatic spectrum with peaks at lower energy than a wiggler

undulator - coherent interference K < 1Max. angle of trajectory < 1/

wiggler - incoherent superposition K > 1Max. angle of trajectory > 1/

bending magnet - a “sweeping searchlight”

2

2 21

2 2u u

n

K

n n

2

2

[ ]( ) 9.496

[ ] 12

n

u

nE GeVeV

Km


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Radiation integral for a linear undulator (I)The angular and frequency distribution of the energy emitted by a wiggler is computed again with the radiation integral:

2

)/ˆ(2

0

223

)ˆ(ˆ44

dtennc

e

dd

Wd crnti

Using the periodicity of the trajectory we can split the radiation integral into a

sum over Nu terms

2)1(2

22/

2/

)/ˆ(2

0

223

...1)ˆ(ˆ44

u

u

u

Niiic

c

crnti eeedtennc

e

dd

Wd

)(

2

res

where

22

2

2 21

2)(

Ku

res)(

2)(

resres

c


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Radiation integral for a linear undulator (II)

),,()(4 0

2223

KFNL

c

Ne

dd

Wdn

res

22/

2/

)/ˆ(0

0

)ˆ(ˆ),,(

c

c

crntin dtennKF

= - n res()

))(/(sin

))(/(sin

)( 22

2

res

res

res N

NNL

The sum on generates a series of sharp peaks in the frequency spectrum

harmonics of the fundamental wavelength

The radiation integral in an undulator or a wiggler can be written as

The integral over one undulator period generates a modulation term Fn which

depends on the angles of observations and K


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Radiation integral for a linear undulator (II)

e.g. on axis ( = 0, = 0): )0,0,()0(4 0

2223

KFNLc

Ne

dd

Wdn

res

2

2

1

2

12

22

)()()2/1(

)0,0,(

ZJZJ

K

KnKF nnn )2/1(4 2

2

K

nKZ

Only odd harmonic are radiated on-axis;

as K increases the higher harmonic becomes stronger

Off-axis radiation contains many

harmonics


8/30

Angular patterns of the radiation emitted on harmonics

Angular spectral flux as a function of frequency for a linear undulator; linear polarisation solid, vertical polarisation dashed (K = 2)

2

12

2

21

Ku

Fundamental wavelength emitted

by the undulator

21

2

2nd harmonic, not emitted on-axis !

31

2

3rd harmonic, emitted on-axis !


9/30

Synchrotron radiation emission from a bending magnet

Dependence of the frequency distribution of the energy radiated via synchrotron emission on the electron beam energy

No dependence on the energy at longer

wavelengths

3

2

3

ccc

Critical frequency

3

2

3

cc

Critical angle

3/11

cc

Critical energy


10/30

Undulators and wigglers (large K)

For large K the wiggler spectrum becomes similar to the bending magnet spectrum, 2Nu times larger.

Fixed B0, to reach the bending magnet critical wavelength we need:

Radiated intensity emitted vs K

K 1 2 10 20

n 1 5 383 3015


11/30

Undulator tuning curve (with K)

Brightness of a 5 m undulator 42 mm period with maximum K = 2.42 (ESRF) Varying K one varies the wavelength emitted at various harmonics

(not all wavelengths of this graph are emitted at a single time)

high K

low K

mc

eBK u

20

Undulator parameter

K decreases by opening the gapof the undulator

(reducing B)


12/30

Spectral brightness of undulators of wiggler

Comparison of undulators for a 1.5 GeV ring for three harmonics (solid dashed and dotted) compared with a wiggler and a bending magnet (ALS)


13/30

Diamond undulators and wigglerSpectral brightness for undulators and wigglers in state-of-the-art

3rd generation light sources


14/30

Summary of radiation characteristics of undulators or wiggler

Undulators have weaker field or shorter periods (K< 1)

Produce narrow band radiation and harmonics

Intensity is proportional to Nu2

Wigglers have higher magnetic field (K >1)

Produce a broadband radiation

Intensity is proportional to Nu


15/30

Type of undulators and wigglers

Electromagnetic undulators: the field is generated by current carrying coils; they may have iron poles;

Permanent magnet undulators: the field is generated by permanent magnets Samarium Cobalt (SmCo; 1T) and Neodymium Iron Boron (NdFeB; 1.4T); they may have iron poles (hybrid undulators);

APPLE-II: permanent magnets arrays which can slide allowing the polarisation of the magnetic field to be changed from linear to circular

In-vacuum: permanent magnets arrays which are located in-vacuum and whose gap can be closed to very small values (< 5 mm gap!)

Superconducting wigglers: the field is generated by superconducting coils and can reach very high peak fields (several T, 3.5 T at Diamond)


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Electromagnetic undulators (I)

Period 64 mm

14 periods

Min gap 19 mm

Photon energy < 40 eV (1 keV with EM undulators)

HU64 at SOLEIL:

variable polarisation electromagnetic undulator


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Electromagnetic undulators (II)

Depending on the way the coil power supplies are powered it can generate linear H, linear V or circular polarisations


18/30

Permanent magnet undulators

Halback configuration

hybrid configuration

with steel poles


19/30

In-vacuum undulatorsU27 at Diamond

27 mm, 73 periods 7 mm gap, B = 0.79 T; K = 2


20/30

Apple-II type undulators (I)

HU64 at Diamond; 33 period of 64 mm; B = 0.96 T; gap 15 mm; Kmax = 5.3


21/30

Apple-II type undulators (II)

Four independent arrays of permanent magnets

Diagonally opposite arrays move longitudinal, all arrays move vertically

Sliding the arrays of magnetic pole it is possible to control the polarisation of the radiation emitted


22/30

Superconducting Wigglers

Superconducting wigglers are used when a high

magnetic field is required

3 - 10 T

They need a cryogenic system to keep the coil

superconductive

Nb3Sn and NbTi wires

SCMPW60 at Diamond

3.5 T coils cooled at 4 K

24 period of 64 mm

gap 10 mm

Undulator K = 21


23/30

000

)cos( xctkk

Kx zu

zu

Equations of motion in an undulator

• Assume electrons travelling on the s axis

• In case of a linear wiggler with By 0

)ctksin(cK

v 0zux

0v y

The solution reads ( = constant)

)2cos(

421

2

11v

2

22

2zk

KKc uz

0yy

00zu0zu

2

2

0z z)ctk2sin(k8

Kctz

0,zkcos,0BB u0• Paraxial approximation (small angular deflection)

Max amplitude of oscillations K

)cos(0 zkvc

eB

dt

dpuz

x

0dt

dp y

)cos(0 zkvc

eB

dt

dpux

z


24/30

Closed Orbit errors induced by an undulator

The integral of the magnetic field seen in the nominal trajectory path must be zero, otherwise the undulator induces an overall angular kick or an overall offset to orbit

End poles and trim coils are used to ensure that

0dsBL

0

y 0'dsBdsL

0

s

0

y First field integral

(angle)

second field integral

(offset)


25/30

Roll-off of transverse magnetic field

0,cos,00 zkBB w0xB

)cos()cosh( zkykBB wwwy

)sin()sinh( zkykBB wwwz

-40 -20 0 20 400.0

0.2

0.4

0.6

0.8

1.0

X (mm)

BY

(T)

Field Profile of U23 In-Vac

A more realistic analytical expression for the magnetic field of an undulator with a finite pole transverse width is given by:

The magnetic fields in real structures exhibit an even more complicated transverse dependence.

e.g. numerically computed field roll-off for an in-vacuum undulator (U23) at Diamond

Linear focussing and non-linear term appear in the equations of

motion and have to be integrated numerically.


26/30

Quadrupole effect of an undulator (I)

Expanding the generic trajectory (x, y) as

Analytical calculation of the motion can be still performed by keeping the lowest order in y in the expansion of the magnetic field around the nominal oscillatory trajectory

0xB

)cos(2

)(1)cos()cosh(

2

zkyk

BzkykBB ww

wwwwy

)sin()sin()sinh( zkykBzkykBB wwwwwwz

An undulator behaves as a focussing quadrupole in the vertical plane and as a drift in the horizontal plane, to the lowest order in the deviation form the reference trajectory.

refr xxx refr yyy

0'' rx ru

r yK

y2

2''

and averaging over one undulator period we end up with


27/30

Quadrupole effect of an undulator (II)

ywy

y

LKQ

2

Unlike a true vertically focussing quadrupole, an ideal undulator does not have a corresponding defocussing effect in the horizontal plane in first order; in the horizontal plane it may have a weak defocussing due to the finite width of the magnetic poles;

The quadrupole associated to the undulator generated a tune shift and a beta-beating

yyy

y

y

Q

LK

2sin2

The quadrupole strength is proportional

to B2 and 1/E2

22

2

12

mc

eBKK

uy

e.g. Diamond Superconducting wiggler

Qy = 0.012

/ ~ 10% SCMPW60


28/30

Modification of Emittance and energy spread with undulators

2

3

0

221

1

234

1

u

ID

u

ID

xx

L

HHL

Term dependent on dispersion in IDs and bending

Additional damping

Depending on the dispersion at the ID one or the other term will dominate

Wigglers and undulators will affect the equilibrium emittance and the energy spread of the electron beam

2/1

2

3

0

221

1

234

1

u

ID

u

ID

L

L

While the emittance can either increase or

decrease the energy spread generally

increases with ID field


29/30

Effect of undulators and wigglers on beam dynamics and cures

Principal effects of undulators and wigglers on beam dynamics

Closed orbit distortion

Betatron Tune shift

Optics variation ( - beating)

Dynamic aperture reduction

Variation of damping times; Emittances; Energy spread

Remedies improving field qualities

Correction of the field integral + Trim coil for closed orbit distortion

Wide transverse gap (reduced roll-off) for linear optics

“Magic fingers” to decrease the multipole component of the wiggler

Remedies using beam optics methods

Feed forward tables for trim coil orbit corrections

Local correction of optics functions (alpha matching schemes, LOCO)

Non-linear beam dynamics optimisation with wiggler


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Summary and additional bibliography

Undulators and Wigglers enhance synchrotron radiation

Undulators produce a narrow band series of harmonics

Wigglers produce a broadband radiation

Radiation can have linear or elliptical polarisation

Undulators and wiggler perturb the beam dynamics in the storage ring

Field quality must be excellent

Effective correction schemes for orbit and linear optics are available

R.P. Walker: CAS 98-04, pg. 129

A. Ropert: CAS 98-04, pg. 91

P. Elleaume in Undulators, Wigglers and their applications, pg. 69-107


Documents

R. Bartolini, John Adams Institute, 19 November 20101/30 Electron beam dynamics in storage rings Synchrotron radiation and its effect on electron dynamics