Emittance Preservation in Electron Accelerators

Nick Walker (DESY)

CAS – Triest – 10/10/2005

What’s so special about ‘emittance preservation in electron machines’ ?

– Theoretically, very little• it’s all basically classical mechanics

– Fundamental difference is Synchrotron Radiation• Careful choice of lattice for high-energy arcs can mitigate

excessive horizontal emittance growth• Storage rings: Theoretical Minimum Emittance (TME) lattices

– Electrons are quickly relativistic (v/c 1)• I will not discuss non-relativistic beams

– Another practical difference: e± emittances tend to be significantly smaller (than protons)

• High-brightness RF guns for linac based light sources• SR damping (storage ring light sources, HEP colliders)

generate very small vertical emittances (flat beams)

Critical Emittance

• HEP colliders– Luminosity

• Light sources (storage rings)– Brightness

• SASE XFEL– e.g. e-beam/photon beam overlap condition

* *

1

x y x y

Lb b ee

µ

( )( )( )( )2 2 2 2' '

1

/ /x x r x x r y y r y y r

Beb s e b s eb s e b s

µ+ + + +

' 4r r

ls s

p=radiation emittance:

4l

ep

<

Typical Emittance Numbers

ILC

Typical Emittance Numbers

ILC

10-14

10-11

ILC @250GeV11

14

2 10 m

6 10 m

x

y

e

e

-

-

= ´

= ´

5

8

10 m

3 10 m

x

y

ge

ge

-

-

æ ö= ÷ç ÷ç ÷ç ÷ç = ´ ÷çè ø

EURO XFEL@20 GeV

112.5 10 mre-= ´

Back to Basics:Emittance Definition

i ipdq cnte= =òÑLiouville’s theorem:

Density in phase space is conserved (under conservative forces)

4 2 0 2 4

6

4

2

0

2

4

6

8

p

q

4 2 0 2 4

6

4

2

0

2

4

6

8

Back to Basics:Emittance Definition

Statistical Definition

2nd-order moments:

2

2

22 2

'

' '

' '

x xx

xx x

x x xx

s

e s

æ ö÷ç ÷ç= ÷ç ÷ç ÷÷çè ø

=

= -

RMS emittance is not conserved!

x

'x

4 2 0 2 4

6

4

2

0

2

4

6

8

Back to Basics:Emittance Definition

Statistical Definition

Connection to TWISS parameters

2(1 )/

eb eas

ea e a b

s e

æ ö- ÷ç ÷ç= ÷ç ÷- +ç ÷è ø

=

x

'x

Some Sources of (RMS) Emittance Degradation

• Synchrotron Radiation• Collective effects

– Space charge– Wakefields (impedance)

• Residual gas scattering• Accelerator errors:

– Beam mismatch • field errors

– Spurious dispersion, x-y coupling • magnet alignment errors

Which of these mechanisms result in ‘true’ emittance growth?

High-Energy Linac

• Simple regular FODO lattice– No dipoles

• ‘drifts’ between quadrupoles filled with accelerating structures

• In the following discussions:– assume relativistic electrons

• No space charge• No longitudinal motion within the bunch

(‘synchrotron’ motion)

Linearised Equation of Motion in a LINAC

Remember Hill’s equation: ''( ) ( ) ( ) 0y s K s y s+ =

Must now include effects of acceleration:

Include lattice chromaticity (first-order in pp):

adiabatic damping

'( )'(''( ) ( ) ( )

(0)

)sy s

sy s K s y s

gg

+ + =

[ ]'( )''( ) '( ) ( ) ( )1

)) 0

((

sy s y s K s y s

ssd

gg

+ -+ =

And now we add the errors…

(RMS) Emittance Growth Driving Terms

[ ] ( )'( )''( ) '( ) 1 ( ) ( )( ) ( ) 0

( ) q

sy s y s s K ys sy s

sg

dg

+ + -- =

“Dispersive” effect from quadrupole offsets

quadrupole offsets

'( )''( ) '( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )q q

sy s y s K s y s

s

K s y s s K s y s s K s y s

gg

d d

+ +

=- + +

put error source on RHS (driving terms)

trajectory kicksfrom offset quads

dispersive kicksfrom offset quads

dispersive kicks from coherent -oscillation

Coherent Oscillation

5 10 15 20 25 30 35

1

0.5

0.5

1

'( )''( ) '( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )q q

sy s y s K s y s

s

K s y s s K s y s s K s y s

gg

d d

+ +

=- + +

Just chromaticity repackaged

Scenario 1: Quad offsets, but BPMs aligned

BPM

Assuming:

- a BPM adjacent to each quad

- a ‘steerer’ at each quad

simply apply one to one steering to orbit

steererquad mover

dipole corrector

Scenario 2:Quads aligned, BPMs offset

BPM

1-2-1 corrected orbit

one-to-one correction BAD!

Resulting orbit not Dispersion Free emittance growth

Need to find a steering algorithm which effectively puts BPMs on (some) reference line

real world scenario: some mix of scenarios 1 and 2

Dispersive Emittance Growth

2 222

20

( ) ( )1 1

11

fRMSBPM

i

y s saa

gdee e b a g

b g-é ùæ öD ê ÷ úçµ -÷çê ú÷ç ÷ úû

µ- è øêë

After trajectory correction (one-to-one steering)

scaling of lattice along linac

Reduction of dispersive emittance growth favours weaker lattice(i.e. larger functions)

Wakefields and Beam Dynamics

• bunches traversing cavities generate many RF modes.

• higher-order (higher-frequency) modes (HOMs) can act back on the beam and adversely affect it.

• Separate into two time (frequency) domains:– long-range, bunch-to-bunch– short-range, single bunch effects (head-tail

effects)

Long Range Wakefieldstb

( , ) ( , ) ( , )V t I t Z t

Bunch ‘current’ generates wake that decelerates trailing bunches.

Bunch current generates transverse deflecting modes when bunches are not on cavity axis

Fields build up resonantly: latter bunches are kicked transversely

multi- and single-bunch beam break-up (MBBU, SBBU)

wakefield is the time-domain description of impedance

Transverse HOMs/ 22

( ) sin( )n nt Qnn

n n

k cW t e t

wake is sum over modes:

kn is the loss parameter (units V/pC/m2) for the nth mode

Transverse kick of jth bunch after traversing one cavity:

1

/ 2

1

2sinn n

ji t Qi i i

j i bi i n

y q k cy e i t

E

where yi, qi, and Ei, are the offset wrt the cavity axis, the charge and the energy of the ith bunch respectively.

10 100 1000 10000 100000.0.001

0.01

0.1

1

10

100

10 100 1000 10000 100000.0.001

0.01

0.1

1

10

100

Detuning

abs.

wak

e (V

/pC

/m)

abs.

wak

e (V

/pC

/m)

time (ns)

no detuningHOMs cane be randomly detuned by a small amount.

Over several cavities, wake ‘decoheres’.

with detuningEffect of random 0.1%detuning(averaged over 36 cavities).

next bunch

Still require HOM dampers

Effect of Emittance

vertical beam offset along bunch train

Multibunch emittance growth for cavities with 500m RMS misalignment

Single Bunch Effects• Analogous to low-range wakes• wake over a single bunch• causality (relativistic bunch): head of bunch

affects the tail• Again must consider

– longitudinal: effects energy spread along bunch– transverse: the emittance killer!

• For short-range wakes, tend to consider wake potentials (Greens functions) rather than ‘modes

Transverse Wakefields

dipole mode:offset bunch – head generates trailing E-field which kicks tail of beam

y

sIncrease in projected emittanceCentroid shift

Transverse Single-Bunch Wakes

When bunch is offset wrt cavity axis, transverse (dipole) wake is excited.

1000 500 0 500 10000

2000

4000

6000

8000

V/pC/m2

z/mm

headtail

‘kick’ along bunch:

( ) ( ' ) ( ) ( ; )( )b

z z

qy z W z z z y s z dz

E z

Note: y(s; z) describes a free betatronoscillation along linac (FODO) lattice(as a function of s)

2 particle modelEffect of coherent betatron oscillation

- head resonantly drives the tail2

1 1 0y k y

head eom (Hill’s equation):

tail eom:head

tail

1 0( ) ( ) sin ( )y s a s s

solution:

resonantly driven oscillator

22 2 1

' 22 z

beam

qW

y k y yE

BNS DampingIf both macroparticles have an initial offset y0 then particle 1 undergoes a sinusoidal oscillation, y1=y0cos(kβs). What happens to particle 2?

2 0

'cos sin

2z

beam

W qy y k s s k s

k E

Qualitatively: an additional oscillation out-of-phase with the betatron term which grows monotonically with s.

How do we beat it? Higher beam energy, stronger focusing, lower charge, shorter bunches, or a damping technique recommended by Balakin, Novokhatski, and Smirnov (BNS Damping) curtesy: P. Tenenbaum (SLAC)

BNS DampingImagine that the two macroparticles have different betatron frequencies, represented by different focusing constants kβ1 and kβ2

The second particle now acts like an undamped oscillator driven off its resonant frequency by the wakefield of the first. The difference in trajectory between the two macroparticles is given by:

2 1 0 2 12 22 1

' 11 cos cosz

beam

W qy y y k s k s

E k k

curtesy: P. Tenenbaum (SLAC)

BNS DampingThe wakefield can be locally cancelled (ie, cancelled at all points down the linac) if:

2 22 1

' 11z

beam

W q

E k k

This condition is often known as “autophasing.”

It can be achieved by introducing an energy difference between the head and tail of the bunch. When the requirements of discrete focusing (ie, FODO lattices) are included, the autophasing RMS energy spread is given by:

2

2

'1

16 sincellE z

beam beam

LW q

E E

curtesy: P. Tenenbaum (SLAC)

Wakefields (alignment tolerances)

bunch

0 km 5 km 10 km

head

head

headtailtail

tail

accelerator axis

cavities

y

tail performsoscillation

2 22 ( ) ( )

11f

lattice cavi

Qs

Wsy

a

ageb

e e ab g

g^

é ùæ öD ÷çê úµ - D÷ç ÷ê úç ÷ úµ

è øêë ûfor wakefield control, prefer stronger focusing (small)

Back to our EoM (Summary)y

s

z

[ ]'

0

( ; ) ( ) ( ; )

( ) ( )'( )

''( ; ) '( ; ) ( ) ( ; ) ( ; ) ( ) ( )( )

1 ( ; ) ( ; ' ) ( ') ( ; ') '( )

q

q

z z

s z K s y s z

K s y ss

y s z y s z K s y s z s z K s y ssQ

s z W s z z z y s z dzs m

d

gdg

d lg

¥

^=

+

+ + = -

+ - -ò

orbit/dispersive errors

transverse wake potential(V C-1 m-2)

long. distribution

Preservation of RMS emittance

Quadrupole Alignment Cavity Alignment

Wakefields

Dispersion

controlof

Transverse

Longitudinal

E/E

beam loading

Single-bunch

Some Number for ILC

RMS random misalignments to produce 5% vertical emittance growth

BPM offsets 11 mRF cavity offsets 300 mRF cavity tilts 240 rad

Impossible to achieve with conventional mechanical alignment and survey techniques

Typical ‘installation’ tolerance: 300 m RMS 2300

5% 3800%!!11

æ ö÷ç´ »÷ç ÷è ø

Use of Beam Based Alignment mandatory

YjYi

Ki

yj

gij

1

j

j ij i i ji

y g K Y Y

Basics Linear Optics Revisited

0

34 ( , )j

iij

j y

yg

y

R i j

linear system: just superimpose oscillations caused by quad kicks.

thin-lens quad approximation: y’=KY

1

j

j ij i i ji

y g K Y Y

=- ×y Q Y

= × +Q G diag(K) I

Original Equation

Defining Response Matrix Q:

Hence beam offset becomes

Introduce matrix notation

21

31 32

41 42 43

0 0 0 0

0 0 0

0 0

0

g

g g

g g g

G

G is lower diagonal:

Dispersive Emittance GrowthConsider effects of finite energy spread in beam RMS

( ) ( )1

K

Q G diag Ichromatic response matrix:

latticechromaticity

dispersivekicks0

34 34 346

( ) (0)

( ) (0)R R T

GG G

dispersive orbit: ( ) ( ) (0) Δy Q Q Y

What do we measure?BPM readings contain additional errors:

boffset static offsets of monitors wrt quad centres

bnoise one-shot measurement noise (resolution RES)

0BPM offset noise 0 0

0

y

y

y Q Y b b R y y

fixed fromshot to shot

random(can be averaged) launch condition

In principle: all BBA algorithms deal with boffset

BBA usingDispersion Free Steering (DFS)

– Find a set of steerer settings which minimise the dispersive orbit

– in practise, find solution that minimises difference orbit when ‘energy’ is changed

– Energy change: • true energy change (adjust linac phase)• scale quadrupole strengths

DFS( ) (0)

E E

E E

E

E

Δy Q Q Y

M Y

1 Y M Δy

Problem:

Solution (trivial):

Note: taking difference orbit y removes boffset

Unfortunately, not that easy because of noise sources:

noise 0 Δy M Y b R y

DFS example

300m randomquadrupole errors

20% E/E

No BPM noise

No beam jitter

m

m

DFS exampleSimple solve

1 Y M Δy

original quad errors

fitter quad errors

In the absence of errors, works exactly

Resulting orbit is flat

Dispersion Free

(perfect BBA)

Now add 1m random BPM noise to measured difference orbit

DFS exampleSimple solve

1 Y M Δy

original quad errors

fitter quad errorsFit is ill-conditioned!

DFS example

m

m

Solution is still Dispersion Free

but several mm off axis!

DFS: Problems• Fit is ill-conditioned

– with BPM noise DF orbits have very large unrealistic amplitudes.

– Need to constrain the absolute orbitT T

2 2 2res res offset2

Δy Δy y y

minimise

• Sensitive to initial launch conditions (steering, beam jitter)– need to be fitted out or averaged away

0R y

DFS exampleMinimise

original quad errors

fitter quad errors

T T

2 2 2res res offset2

Δy Δy y y

absolute orbit now

constrained

remember

res = 1m

offset = 300m

DFS examplem

m

Solutions much better behaved!

Orbit not quite Dispersion Free, but very close

DFS practicalities• Need to align linac in sections (bins), generally

overlapping.• Changing energy by 20%

– quad scaling: only measures dispersive kicks from quads. Other sources ignored (not measured)

– Changing energy upstream of section using RF better, but beware of RF steering (see initial launch)

– dealing with energy mismatched beam may cause problems in practise (apertures)

• Initial launch conditions still a problem– coherent -oscillation looks like dispersion to algorithm.– can be random jitter, or RF steering when energy is changed.– need good resolution BPMs to fit out the initial conditions.

• Sensitive to model errors (M)

Orbit Bumps

• Localised closed orbit bumps can be used to correct– Dispersion– Wakefields

• “Global” correction (eg. end of linac) can only correct non-filamented part– i.e. the remaining linear correlation

• Need ‘emittance diagnostic’– Beam profile monitors– Other signal (e.g. luminosity in the ILC)

I’ll Stop Here

The following slides were not shown due to lack of time

Emittance Growth: Chromaticity

Chromatic kick from a thin-lens quadrupole:

' ; /x K x p pd dD = º D

2 2

2 2 2 2 2

' '

' '

x x

xx xx

x x K xd

®

®

® +

2nd-order moments:

Chromatic kick from a thin-lens quadrupole:

' ; /x K x p pd dD = º D

RMS emittance:

2 2 2

22 2 2 2,0

2 2 2 4 4,0 ,0

' 'x

x

x RMS x x

x x xx

K x

K

e

e d

e d b e

= -

= +

= +

2 2 4 2,0

,0

12

xRMS x x

x

Ke

d b eeD

»

Emittance Growth: Chromaticity

Synchrotron Radiation

Synchrotron Radiation

Synchrotron Radiation

Synchrotron Radiation( )/ /E E E ug w dD = - ºh

Synchrotron Radiation( )/ /E E E ug w dD = - ºh

Synchrotron Radiation

16( )R s

16

2 2 216

( ) ( )

( )

i i

i i

x L R s u

x R s ug

d

d

D =

D =

s L=

( )/ /E E E ug w dD = - ºh

52 11

3

[ ]4.13 10 [ ]

[ ]E GeV

u smm

dr

-» ´ D

Synchrotron Radiation

22 5 26

30

( )'

( )

L

s

R sx C E ds

sg r=D = ò

22 5 16

30

( )( )

L

s

R sx C E ds

sg r=D = ò

5 16 2630

( ) ( )'

( )

L

s

R s R sx x C E ds

sg r=D D = ò

We have ignored the mean energy loss(assumed to be small, or we have taken some suitable average)

11 2 54.13 10 [ ]C m GeVg- -» ´

1222 2

5 16 26 16 263 3 30 0 0

( ) ( ) ( ) ( )( ) ( ) ( )

L L LR s R s R s R sC E ds ds ds

s s sg ger r r

é ùæ öê ÷úçD = - ÷çê ú÷ç ÷è øê úë ûò ò ò

phase space due to quantum excitation

Synchrotron RadiationWhat is the additional emittance when our initial beam has a finite emittance? 0e

geD

0 ge e e= +D

Quantum emission is uncorrelated, so we can add 2nd-order moments

When quantum induced phase space and original beam phase space are ‘geometrically similar’, just add emittances.

Synchrotron RadiationWhat is the additional emittance when our initial beam has a finite emittance? 0e

geD

0 ge e e= +D

( )

0

0

g

g

b a b as e e

a g a g

b ae e

a g

æ ö æ ö- -÷ ÷ç ç÷ ÷= +Dç ç÷ ÷ç ç- -÷ ÷ç çè ø è ø

æ ö- ÷ç ÷= +D ç ÷ç- ÷çè ø

ellipse shape

Synchrotron RadiationWhen quantum induced phase space and initial beam phase space are dissimilar, an additional (cross) term must be included.

( )

0

2 2 20 0

12 2

2

g g

gg g

g g g g g

b a b as e e

a g a g

e s e e e e gb aa g b

æ ö æ ö- -÷ ÷ç ç÷ ÷= +Dç ç÷ ÷ç ç- -÷ ÷ç çè ø è ø

é ù= = +D + D - +ê ú

ê úë û

Initial beam ellipse Q.E. beam ellipse

Beam Mismatch (Filamentation)Matched beam (normalised to unit circle)

Mismatched beam (-mismatch) rotates with nominal phase advance along beamline

-beat along machine (but emittance remains constant)

xu

bº

'x xv

a bb

+º

Beam Mismatch (Filamentation)

Mismatched beam (-mismatch) rotates with nominal phase advance along beamline

-beat along machine (but emittance remains constant)

Finite energy spread in beam + lattice chromaticity causes mismatch to “filament”

Emittance growth

xu

bº

'x xv

a bb

+º

Beam Mismatch (Filamentation)

[ ]

0 00

20

T T

0

0 0 0 0

1 01

(1 )

cos( ) sin( )( )

sin( ) cos( )

1( ) ( )

12

2

fil

fil

R

dp

a bb

b a

s e aa

b

j jj

j j

s j s j jp

s g b gb aa e

æ ö÷ç ÷ç= ÷ç ÷÷çè ø

æ ö- ÷ç ÷ç ÷ç= ÷+ç ÷ç ÷-ç ÷÷çè ø

æ ö÷ç ÷ç= ÷ç ÷-ç ÷è ø

= × × × ×

= + -

ò

M

R M M R

Normalisation matrix (matched beam)

Mismatched beam

Phase space rotation

0 0 02 ; 0; 1.2filb b a a e e= = = =

Fully filamented beam

Longitudinal Wake( )W z ctConsider the TESLA wake potential

3

V( ) 38.1 1.165exp 0.165

pC m 3.65 10 [m]

sW z

, ( ) ( ) ( )bunch

z z

W z W z z z dz

wake over bunch given by convolution:

4 2 0 2 4

20

15

10

5

0

V/p

C/m

z/z

300μmz

headtailaverage energy loss:

, ( ) ( )b bunchE q W z z dz

((z) = long. charge dist.)

For TESLA LC: 46 kV/mE

RMS Energy Spread

4 2 0 2 4

10

8

6

4

26

0 2 4 6 8 10 12

0.3

0.4

0.5

0.6

0.7

0.8

0.9

z/z

rms E

/E (

ppm

)E

/E (

ppm

) RF

wake+RF

(deg)

accelerating field along bunch:

, 0( ) ( ) cos(2 / )b bunch RFE z q W z E z

Minimum energy spread along bunch achieved when bunch rides ahead of crest on RF.

Negative slope of RF compensates wakefield.

For TESLA LC, minimum at about ~ +6º

RMS Energy Spread