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

4

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

6

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

7

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!

8

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

9

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)

10

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

62

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

11

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

12

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

14

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

15

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)

16

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)

18

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

19

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

21

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

22

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

26

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

27

A Final Word

• Technology decision due 2004• Start of construction 2007+• First physics 2012++• There is still much to do!

WE NEED YOUR HELP

for

the Next Big Thing

hope you enjoyed the course

Nick, Andy, Andrei and PT