1
An Introduction to thePhysics and Technologyof e+e- Linear Colliders
Lecture 9: a) Beam Based Alignment
Nick Walker (DESY)
DESY Summer Student Lecture31st July 2002USPAS, Santa Barbara, 16th-27th June, 2003
Emittance tuning in the LET
• LET = Low Emittance Transport– Bunch compressor (DR→Main Linac)– Main Linac– Beam Delivery System (BDS), inc. FFS
• DR produces tiny vertical emittances (γεy ~ 20nm)
• LET must preserve this emittance!– strong wakefields (structure misalignment)– dispersion effects (quadrupole misalignment)
• Tolerances too tight to be achieved by surveyor during installation
⇒ Need beam-based alignmentmma!
2
YjYi
Ki
yj
gij
1
j
j ij i i ji
y g K Y Y=
= − −
∑
Basics (linear optics)
0
34 ( , )j
iij
j y
ygy
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=
= − −
∑
YQy ⋅−=
Idiag(K)GQ +⋅=
Original Equation
Defining Response Matrix Q:
Hence beam offset becomes
Introduce matrix notation
21
31 32
41 42 43
0 0 0 00 0 0
0 00
gg gg g g
=
GG is lower diagonal:
3
Dispersive Emittance Growth
Consider effects of finite energy spread in beam δRMS
( ) ( )1
δ δδ
= ⋅ + + KQ G diag Ichromatic response matrix:
latticechromaticity
dispersivekicks0
34 34 346
( ) (0)
( ) (0)R R Tδ
δδ
δ δ=
∂= +
∂= +
GG G
dispersive orbit: [ ]( ) ( ) (0)yδ δ
δ≈ = − − ⋅∆yη 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 σRES)
0BPM offset noise 0 0
0
yy
= − ⋅ + + + ⋅ = ′
y Q Y b b R y y
fixed fromshot to shot
random(can be averaged
to zero)launch condition
In principle: all BBA algorithms deal with boffset
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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
5
BBA
• Dispersion 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
• Ballistic Alignment– Turn off accelerator components in a given section, and
use ‘ballistic beam’ to define reference line– measured BPM orbit immediately gives boffset wrt to
this line
DFS
( ) (0)E EE E
EE
∆ ∆ = − − ⋅ ∆ ≡ ⋅
∆y Q Q Y
M Y
1−= ⋅Y M ∆y
Problem:
Solution (trivial):
Note: taking difference orbit ∆y removes boffset
Unfortunately, not that easy because of noise sources:
noise 0= ⋅ + + ⋅∆y M Y b R y
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DFS example
300µm randomquadrupole errors
20% ∆E/E
No BPM noise
No beam jitter
µm
µm
DFS example
Simple solve
1−= ⋅Y M ∆y
original quad errors
fitter quad errors
In the absence of errors, works exactly
Resulting orbit is flat
⇒Dispersion Free
(perfect BBA)
Now add 1µm random BPM noise to measured difference orbit
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DFS example
Simple solve
1−= ⋅Y M ∆y
original quad errors
fitter quad errorsFit is ill-conditioned!
DFS example
µm
µm
Solution is still Dispersion Free
but several mm off axis!
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DFS: Problems
• Fit is ill-conditioned– with BPM noise DF orbits have very large unrealistic
amplitudes.– Need to constrain the absolute orbit
T T
2 2 2res res offset2σ σ σ
⋅ ⋅+
+∆y ∆y y y
minimise
• Sensitive to initial launch conditions (steering, beam jitter)– need to be fitted out or averaged away
0⋅R y
DFS example
Minimise
original quad errors
fitter quad errors
T T
2 2 2res res offset2σ σ σ
⋅ ⋅+
+∆y ∆y y y
absolute orbit now
constrained
remember
σres = 1µm
σoffset = 300µm
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DFS example
µm
µm
Solutions much better behaved!
! Wakefields !
Orbit not quiteDispersion 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 β-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)
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Ballistic Alignment
• Turn of all components in section to be aligned [magnets, and RF]
• use ‘ballistic beam’ to define straight reference line (BPM offsets)
• Linearly adjust BPM readings to arbitrarily zero last BPM
• restore components, steer beam to adjusted ballistic line
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BPM, 0 0 offset, noise,i i i iy y s y b b′= + + +
Ballistic Alignment
∆bi ∆qi
Lb
quads effectively aligned to ballistic reference
angle = αi
ref. line
with BPM noise
62
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Ballistic Alignment: Problems
• Controlling the downstream beam during the ballistic measurement– large beta-beat– large coherent oscillation
• Need to maintain energy match – scale downstream lattice while RF in ballistic
section is off• use feedback to keep downstream orbit
under control
An Introduction to thePhysics and Technologyof e+e- Linear Colliders
Lecture 9: b) Lessons learnt from SLC
Nick Walker (DESY)
DESY Summer Student Lecture31st July 2002USPAS, Santa Barbara, 16th-27th June, 2003
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Lessons from the SLC
IP Beam Size vs Time
0
1
2
3
4
5
6
7
8
9
10
1985 1990 1991 1992 1993 1994 1996 1998
Year
Bea
m S
ize
(mic
rons
)
0
1
2
3
4
5
6
7
8
9
10
σx ∗
σy
(mic
rons
2 )
SLC Design(σx ∗ σy)
σX
σY
σX ∗ σy
New Territory in Accelerator Design and Operation
• Sophisticated on-line modeling of non-linear physics.
• Correction techniques expanded from first-order (trajectory) to include second-order (emittance), and from hands-on by operators to fully automated control.
• Slow and fast feedback theory and practice.
D. Burke, SLAC
The SLC1980 1998
taken from SLC – The End Game by R. Assmann et al, proc. EPAC 2000
note: SLC was a single bunch machine (nb = 1)
13
SLC: lessons learnt
• Control of wakefields in linac– orbit correction, closed (tuning) bumps– the need for continuous emittance measurement
(automatic wire scanner profile monitors)
• Orbit and energy feedback systems– many MANY feedback systems implemented over the life time of
the machine– operator ‘tweaking’ replaced by feedback loop
• Final focus optics and tuning– efficient algorithms for tuning (focusing) the beam size at the IP– removal (tuning) of optical aberrations using orthogonal knobs.– improvements in optics design
• many many more!
The SLC was an 10 year accelerator R&D project that also did some physics ☺
The Alternatives
nm
×10-8m
MW
MW
×1033 cm-2s-1
GHz
1.2345σy*
1443γεy
175195233140PAC
4.96.95.811.3Pbeam
21201434L
30.011.45.71.3f
CLICJLC-X/NLCJLC-CTESLA
2003 Ecm=500 GeV
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Examples of LINAC technology
9 cell superconducting Niobium cavity for TESLA (1.3GHz)
11.4GHz structure for NLCTA
(note older 1.8m structure)
Competing Technologies: swings and roundabouts
SLC(3GHz)
RF frequencyCLIC(30GHz)
TESLA
(1.3GHz SC)
NLC(11.4GHz)
higher gradient = short linac ☺higher rs = better efficiency ☺High rep. rate = GM suppression ☺smaller structuredimensions = high wakefields Generation of high pulse peakRF power
long pulse low peak power ☺large structure dimensions = low WF ☺very long pulse train = feedback within train ☺SC gives high efficiency ☺Gradient limited <40 MV/m = longer linac low rep. rate bad for GM suppression (εy dilution) very large unconventional DR
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An Introduction to thePhysics and Technologyof e+e- Linear Colliders
Lecture 9: c) Summary
Nick Walker (DESY)
DESY Summer Student Lecture31st July 2002USPAS, Santa Barbara, 16th-27th June, 2003
The Luminosity Issue
,
BSRF RFD
cm n y
PL H
Eδηε
∝ zyβ σ≈
• high RF-beam conversion efficiency ηRF
• high RF power PRF
• small normalised vertical emittance εn,y
• strong focusing at IP (small βy and hence small σz)• could also allow higher beamstrahlung δBS if willing to
live with the consequences• Valid for low beamstrahlung regime (ϒ<1)
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High Beamstrahlung Regime
BSbeam
,y nL P
δε
∝low beamstrahlung regime ϒ<<1:
32
BSbeam
,z y nL P
σδ
ε∝high beamstrahlung regime ϒ>>1:
*y zβ σ≈with
Pinch Enhancement
,
BSRF RFD
cm n y
PL H
Eδηε
∝
( )
2 2( )
D D y
b e z b e zzy
beam y x y y x
H H D
N r N rD
fσ σσ
γσ σ σ γσ σ
=
= ≈ ≈+
Trying to push hard on Dy to achieve larger HD leads to single-bunch kink instability detrimental to luminosity
17
The Linear Accelerator (LINAC)
• Gradient given by shunt impedance:– PRF RF power /unit length– rl shunt impedance /unit length
• The cavity Q defines the fill time:– vg = group velocity, – ls = structure length
• For TW, τ is the structureattenuation constant:
• RF power lost along structure (TW):
( ) ( )z RF lE z P z r=
2 / SW2 Q/ / TWfill
s g
Qt
l vω
τ ω
= =
2, ,RF out RF inP P e τ−=
2RF z
b zl
dP Ei E
dz r= − −
power lost to structure beam loading
ηRF
would like RS to be as high as possible
sR ω∝
The Linear Accelerator (LINAC)
For constant gradient structures:
unloaded
av. loaded
( )20 1u lV r P L e τ−= − unloaded structure voltage
2
beam 2
1 212 1l u l
eV V r Lie
τ
τ
τ −
−
= − − −
loaded structure voltage(steady state)
( )3
220
opt 2
11 (1 2 )l
ePir L e
τ
ττ
−
−
−=
− +optimum current (100% loading)
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The Linear Accelerator (LINAC)
Single bunch beam loading: the Longitudinal wakefield
700 kV/mz bunchE∆ ≈NLC X-band structure:
The Linear Accelerator (LINAC)
Single bunch beam loading Compensation using RF phase
wakefield
RF
Total
φ = 15.5º
RMS ∆E/E
<Ez>
φmin = 15.5º
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Transverse Wakes: The Emittance Killer!
∆tb
( , ) ( , ) ( , )V t I t Z tω ω ω=
Bunch current also 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 breakup (MBBU, SBBU)
Damped & Detuned Structures
NLC RDDS1bunch spacing
Slight random detuning between cells causes HOMs to decohere.
Will recohere later: needs to be damped (HOM dampers)
HOM2Qtω
∆ ≈∆
20
Single bunch wakefieldsEffect of coherent betatron oscillation
- head resonantly drives the tail
head
tail
22
22
0 head
tail
hy
tt wf h
d y k ydsd y k y w yds
+ =
+ = −
2cell
BNS 2
116 sin ( )
z LW qE β
σδπν
⊥′≈
Cancel using BNS damping:
Wakefields (alignment tolerances)
bunch
0 km 5 km 10 km
head
head
headtailtail
tail
accelerator axis
cavities
∆y
tail performsoscillation
RMS
3
1 Z
z
EY NWf EN
δ β
β
⊥−
∝
∝
higher frequency = stronger wakefields
-higher gradients
-stronger focusing (smaller β)
-smaller bunch charge
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Damping Rings
2 /( ) DTf eq i eq e τε ε ε ε −= + −
final emittance equilibriumemittance
initial emittance(~0.01m for e+)
damping time
wiggler
wiggler
ρ
Lwig
4
arcEE Cγ ρ
∆ =
6 2 2wig wig1.27 10 (T) (GeV) (m)E B E L−∆ ≈ ×
train
rep
28DnE
P fγ
τ = ≤
train b bC n n t c= ∆
Bunch Compression
• bunch length from ring ~ few mm• required at IP 100-300 µm
RF
z
∆E/E
z
∆E/E
z
∆E/E
z
∆E/E
z
∆E/E
long.phasespace
dispersive section
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The linear bunch compressor
,0 2
,0 ,0
1RF RF z c uc RF c
RF z RF z
k V E EV FE k k
σ δ δδσ σ
≈ ⇔ = = −
2 22 1c u
c c u cu
F Fδ δ
δ δδ
+= ⇔ = −conservation of longitudinal
emittance
RF cavity
initial (uncorrelated) momentum spread: δuinitial bunch length σz,0compression ration Fc=σz,0/σzbeam energy ERF induced (correlated) momentum spread: δcRF voltage VRFRF wavelength λRF = 2π / kRFlongitudinal dispersion: R56
see lecture 6
The linear bunch compressor
2,0 ,0
56 2 2 2 2
1c z zRF RF
u u
z k VRF E Fδ σ σδ
δ δ δ
= − = − =
chicane (dispersive section)
Two stage compression used to reduce final energy spread
COMP #1 LINAC COMP #2F1 F2∆EE0
( ) 01 2
0
0
0
f i
T i
EF FE E
EFE E
δ δ
δ
= ⋅ ⋅ + ∆
= + ∆
56
0 wiggler0 arc
R<
>
23
Final FocusingIP
FD
Dx
sextupoles
dipole
0 0 00 1/ 0 00 0 00 0 0 1/
mm
mm
=
RL*
( )
FD *
FD *
SX
2 2SX
( )
( )
yyL
xxL
y S x y
x S x y
δ
δ ηδ
ηδ
ηδ
′∆ = −
+′∆ = −
′∆ = +
′ ∆ = − + −
*
1SLη
=chromatic correction
Synchrotron Radiation effects
IP
FD
ηxdipole
L*
LB
5B
33 *5ee
2
Lη'Lγλr19
E∆E
≈
2*2y 2
y*2y
∆σ ∆EWσ E
≈
FD chromaticity + dipole SR sets limits on minimum bend length
24
Final Focusing: Fundamental limits
Already mentioned that
At high-energies, additional limits set by so-called Oide Effect:synchrotron radiation in the final focusing quadrupoles leads to a beamsize growth at the IP
zyβ σ≥
( )1 57 71.83 e e nr Fσ ε≈minimum beam size:
occurs when ( )2 37 72.39 e e nr Fβ γε≈
independent of E!
F is a function of the focusing optics: typically F ~ 7(minimum value ~0.1)
0 500 1000 1500 2000
- 1
- 0.5
0
0.5
1
0 500 1000 1500 2000
- 1
- 0.5
0
0.5
1 100nm RMS random offsets
sing1e quad 100nm offset
LINAC quadrupole stability*
,1 1
** sin( )
Q QN N
Q i i i Q i ii i
ii i i
y k Y g k Y g
g γ β β φγ
= =
= ∆ = ∆
= ∆
∑ ∑
* 2*2 2 2
,*1
sin ( )QN
i Q i i iji
Yy k
βγ β φ
γ =
∆= ∆∑
for uncorrelated offsets
2 22 2
*2,
0.32
j Q QY
y y n
y N k β γσ
σ ε ∆≈ ≤
take NQ = 400, εy ~ 6×10−14 m, β ~ 100 m, k1 ~ 0.03 m−1 ⇒ ~25 nm
Dividing by and taking average values:
*2 * *, /y y nσ β ε γ=
25
Beam-Beam orbit feedback
use strong beam-beam kick to keep beams colliding
see lecture 8
IP
BPM
θbb
FDBK kicker
∆y
e−
e+
Generally, orbit control (feedback) will be used extensively in LC
Beam based feedback: bandwidth
0.0001 0.001 0.01 0.1 1
0.050.1
0.5
1
510
f / frep
g = 1.0g = 0.5g = 0.1g = 0.01
f/frep
Good rule of thumb: attenuate noise with f<frep/20
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Ground motion spectra
[ ]
( , )
1( , ) ( , ) 1 cos( )
P k
L P k kL dk
ω
ρ ω ωπ
+∞
−∞
= −∫
2D power spectrum
measurable relativepower spectrum
Both frequency spectrum and spatialcorrelation important for LC performance
Long Term Stability
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1 1 10 100 1000 10000 100000 1000000
time /s
rela
tive
lum
inos
ity
1 hour 1 day1 minute 10 days
No Feedback
beam-beam feedback
beam-beam feedback +
upstream orbit control
understanding of ground motion and vibration spectrum important
example of slow diffusive ground motion (ATL law)
see lecture 8