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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