18th KEKB Review, March 3-5, 2014

Optics Issues
Y. Ohnishi
18th KEKB Review
KEK, March 3-5, 2014

KEKB Review 2013

R1: The committee recommends concentrating on developing very
detailed and robust procedures for correcting the closed orbit andlinear optics distortions caused by realistic field errors, especially in
difficult places like the local chromaticity correction section.
For the IR optics, an algorithm should be developed for the controlroom operational usage, aimed at improving the beam lifetime (andthe dynamic aperture) by numerous nonlinear corrections, inaddition to the blind downhill-simplex optimization.

R2: The Optics Group should continue to merge the IR and ringlattices with the intent to study optimized luminosity tuning with fullIR fields, full error tolerances, all correctors, and the needed small
vertical emittances.
2

Contents

IR design (Final focus system, corrector coils)
Optics correction and Dynamic aperture
Dynamic aperture with Beam-Beam effect (new issue)3IR design: Final focus system
IR design has been fixed. Especially, additional sextupole and skew
sextupole correctors to correct error fields induced by a coildisplacement in the radial direction.
skew sextupole

LER

octupole
QC
2LP

skew sextupole
octupole
QC
1LP

HER

LE
2
QC
octupole

E
R
2
C
Q

LE
1
QC
octupole

IP

E
R
1
C
Q

QC
1R
P
octupole+
skew sextupole

sextupole

QC
2R
P

skew sextupole
Ideal lattice is OK ....

How to adjust these nonlinear correctors ?

4

Optics Correction
andDynamic Aperture5

Machine error for optics correction

Assumption of machine error
Δxrms (µm)Δyrms (µm)Δθrms (µrad)(ΔΚ/Κ)rms

Dipole00100

3.5x10-4

Quadrupole

100

100

100

7x10-4

Sextupole

100

100

0

1.3x10-3

QCS

100

100

0

0

BPM*

75

75

1000

-

*BPM jitter error of 2 µm (rms) is included for an averaged mode.
Misalignment is based on the measurement at KEKB. (realistic error)
We evaluate the beam quality after corrections of these machine error on
the computer simulation.
6

Vertical emittance after optics correction

ideal
lattice#Entries#Entries

Optics correction is based on the measurement of orbit response. (quainttechnique)
Machine error is corrected by using dipole, quadrupole, and skew quadrupole
correctors.
Closed orbit distortion, X-Y coupling, dispersions, and beta functions arecorrected.
#samples: 100 (different seed numbers)
ideal
lattice

The vertical emittance can be corrected down to a few pm,
which satisfy the luminosity performance.
H. Sugimoto7

Dynamic aperture with machine error

Dynamic aperture after correcting the machine error
HER

transverse aperturetransverse aperture

LER

momentum aperturemomentum aperture

blue plots: averaged value of 100 samples
bars:
standard deviation
dashed line: ideal latticeH. Sugimoto8

Correction of nonlinear optics

Even though COD, X-Y coupling, dispersions, and beta functions
are corrected, dynamic aperture can not be recovered by these
corrections perfectly.
In order to recover the dynamic aperture, sextupole, skew sextupole,and octupole optimizations are necessary.
So, how to adjust these nonlinear correctors ?
We have 135 one-pass BPMs for each ring. Information from onepass BPMs with rf-frequency shift gives us chromatic phaseadvance and chromatic X-Y coupling. Sextupole and skew
sextupoles can be optimized by using these measurement in principle.

9

Derivation of phase advanceBeam position is measured by one-pass BPM.
x1 (n) =2J 1 cos ⇥(n)⌅(n) = 2⇥ n + ⇤0x2 (n) =2J 2 cos{⇥(n) + ⇥21 }n: turn number

Averaged values of x1, x2 and x1x2:< x1 (n)2 > = J 1< x2 (n)2 > = J 2< x1 (n)x2 (n) > = J

1 2 cos ⇥21

Phase advance between two BPMs:
21 = cos1< x1 (n)x2 (n) >< x1 (n)2 >< x2 (n)2 >two-hold ambiguity:⇥21⇥21 or 2

10

Measurement of phase advance
Correlation
betweenneighbor BPMs

no error(ideal)Correlation between x(i)and x(i+1) at a BPM
gives a betatron tune

< x1 (n)2 > = J 1< x2 (n)2 > = J 2< x1 (n)x2 (n) > = J

1 2 cos ⇥21

positionresolution
100 µm11

Measurement of chromatic phase-advance
Example: one-pass BPM at QC1LP and QC2LP
Measurement of chromatic phase-advance with rf-frequency shift(-0.1<δ<+0.1 %)

Δp/p0

x

=

Δp/p0

0.407

y

= 0.12712

Chromatic phase-advanceIn case that QC1LP has sextupole error filed of B3/B2 = 0.1 %Deviation of chromatic phase-advance between neighbor one-pass
BPMs
LER
x
designxydesignyIP

13Derivation of X-Y coupling
physicalcoordinate

x
px
y
py

=

µ
0
r1
r3

0
µ
r2
r4

µ2 + (r1 r4< x2 > = µ2 < u2 > +r42 < v 2 >< xpx > = µ2 < upu >

u
pu
v
pv

normalcoordinate

r2 r3 ) = 1µ2 < u2 >r1 r2 < p2v > +(r1 r4 + r2 r3 ) < vpv >r3 r4 < v 2 >2r1 r3 < vpv > +r12 < p2v >We can solve theequations< xy >< xpx >
< xpy >
< px py >< x2 >< xpx > < ypy >0< p2y >

< xpx > + < ypy >< p2x >
< p2y >
0

where

r2
r1
0
µ

2r2 r4 < vpv > +r22 < p2v >< p2x > = µ2 < p2u > +r32 < v 2 >

=

r4
r3
µ
0

=

1
µ

r1
r2
r3
r4

µ2 < upu >µ2 < p2u >Need to estimatepx and py
using two BPMs0
< y2 >< x2 >< xpx > + < ypy >< y2 >
0< xpx > < ypy >< p2x >
14

Measurement of chromatic X-Y coupling
Example: one-pass BPM at QC1LP
Measurement of X-Y coupling with rf-frequency shift (-0.1<δ<+0.1 %)
r1

r3

Δp/p0

=

=

72.7
r2

= 28.6r4= 56.0

142.5

Δp/p0
15

Chromatic X-Y coupling
In case that QC1LP has skew error filed of A3/B2 = 0.1 %Deviation of chromatic X-Y coupling at one-pass BPMs
LER
r1
r2
r3
r4

IP

rotatable sextupoles(skew sextupole)16

Correction of chromatic effects (1/2)
If we can detect any deviation of chromatic phase-advance from theideal lattice, we can correct them by using the sextupole corrector
between QC1R and QC2R and normal sxtupoles(54 families).
If we can detect any deviation of chromatic X-Y coupling from theideal lattice, we can correct them by using the skew sextupole
correctors at QC1/QC2 and 12 families of rotatable sextupoles(LER
only) or 10 families of skew sextupoles in HER.
We need to check sensitivity of the measurement and whether
dynamic aperture can be recovered or not.
These measurements and corrections are used for making an initialcondition.
Adiabatic tuning is still necessary during beam operation(day-by-day
tuning). Downhill-simplex method and/or new algorithm arenecessary.
17Correction of chromatic effects (2/2)
The orbit does not pass along a center at a sextupole magnet ingeneral.
If we change field strength of a sextupole, quadrupole and skew
quadrupole fields are induced as a side effect.
In order to preserve the optics before the change, quadrupole
correctors neighbor of the sextupole and a skew quadrupole-likecorrector at the sextupole should be adjusted properly.
In the case of a rotatable sextupole, the situation becomes more
complicated. Additional variables are a rotation angle and K-value tocontrol a normal and skew element.
Under developing a realistic procedure for the control-room
operational usage.
R1: The committee recommends concentrating on developing very detailed and robust
procedures for correcting the closed orbit and linear optics distortions caused by realistic
field errors, .... still the relevant statement now
18

Dynamic Aperture
with Beam-Beam Effect19

Beam-beam effect for large horizontal orbit
The horizontal orbit(deviation from beam axis) is translated into thelongitudinal displacement in the nano-beam scheme.

beam axisxz=
test particle

x

x

2 x

y(

z) =

Δx

z

high vertical beta →y(yz) = 48 mm >>bbyy ( z)yzΔzx = 30 x0y +z2

z = 3.6 mm

y = 0.27 mm~factor of 180

Particles with a large horizontal orbit are kicked by beam-beam at high verticalbeta region if there is a vertical orbit. Consequently, the vertical betatronoscillation increases due to the vertical beam-beam kick. The transverse
aperture decreased, which implies small dynamic aperture.

Beta function at colliding pointy(Enhancement
factorβy /βy*

z)
⇤
y

20σx*
10σx*

=1+

✓

x
2 x y⇤◆2

σx* = 10 μm

40σx*
30σx*

nominalβy*21

Difficulty in the Nano-Beam scheme
w/o beam-beam
LER

with beam-beam (W-S)
LER
aperturelost

2Jy= 0.27 %2Jx

Transverse aperture is reduced significantly.
HER

2Jy= 0.25 %2Jx

HERaperturelost

22Tracking simulation for single particle
Initial orbit is 10 sigmas in the horizontal direction and 0
for the vertical direction

Δx (mm)blue: no beam-beam
red: with beam-beam
Horizontal betatron oscillation is stablefor both case.

#turnΔy (mm)The vertical oscillation exists for the case w/o beam-beam,
since there is a X-Y coupling.

Vertical betatron oscillation is stablefor beam-beam effect.
The amplitude is slightly large.

#turn

23

Tracking simulation: beam-beam effectInitial orbit is 15 sigmas in the horizontal direction and 0
for the vertical direction

Δx (mm)blue: no beam-beam
red: with beam-beam
Horizontal betatron oscillation is stablefor both case.

#turnΔy (mm)The vertical oscillation exists for the case w/o beam-beam,
since there is a X-Y coupling.

Vertical betatron oscillation is unstablefor beam-beam effect.

#turn

24

Tune survey: No beam-beam effectLER(νx, νy)=(44.53, 46.57)2νx+2νs=int.

HER(νx, νy)=(45.53, 43.57)

Single-beam operation (no beam-beam effect)
Lighter color indicates larger dynamic aperture (only for on-momentum).
Nominal working point is .53 for the horizontal and .57 for the vertical direction.

25

LER tune survey: Beam-beam effect
LER: w/o beam-beam

LER: with beam-beamνx+2νy=int.

Zoom

There is a good region nearhalf integer resonance forthe vertical tune.
Chromaticity correction
becomes very difficult nearhalf integer.
26

Better working point for Touschek lifetime ?
Luminosity (W-S model)

D. Zhou

w/o lattice nonlinear

Touschek lifetime (LER)

H. Sugimoto

Candidates are (44.53,46.54) and (44.65,46.535)
L~6x1035
L~8x1035

27

Optimization of DA with beam-beam effect(νx, νy)=(44.65, 46.535)(νx, νy)=(44.53, 46.54)
L~8x1035

L~6x1035

LER

Δx/σx

Δx/σx

LER

H. Sugimoto

Δε/σε

Δε/σε

Optimization is done by sextupoles, skew sextupoles, and octupoles.
Touschek lifetime is improved up to 230 sec. Still short lifetime.

28

Crab-waist scheme
to mitigate DA reduction with
Beam-Beam effect ?
Ideal crab-waist has a potential to mitigate this effect. (only solution)
But a real crab-waist consists of sextupoles has a serious issue.

29

Dynamic aperture with beam-beam and ideal CW
Stability of an initial amplitude in the horizontal and vertical plane.
Initial momentum deviation is zero.
(synchrotron motion is included.)

Ideal LER lattice

injection
aperture

with Beam-Beamwith ideal CW

with Beam-Beamaperture
limit

aperture
limit
Ideal crab-waist is a map of fBB

fCW (+ )fBB fCW (

fCW ( ) : px

px +

2

p2y , y)yaperturelimit
1=
tan 2 xxpy

30

Crab-waist optics in LER
sler_1689_cw2a.sad

K2=13.68 [1/m2]
x = 12.522
y = 13.25

K2=13.68 [1/m2]
x = 12.522
y = 13.25

x = 8.5 m

x = 8.5 m

y = 200 m

y = 200 m

OHO

NIKKO

Crab-waist
sextupole
31

Dynamic aperture in realistic crab-waist

Crab-waist sextupole reduces dynamic aperture
under the influence of beam-beam effect.
x0

LERwith Beam-Beamx

2Jy= 0.27 %2Jx

K2 = 0K2/K2,nominal = 50 %0

K2/K2,nominal = 75 %K2/K2,nominal = 100 %32

Dynamic aperture in realistic crab-waist

Crab-waist sextupole reduces dynamic aperture
without beam-beam effect significantly.
x0

LER
No Beam-Beamx

2Jy= 0.27 %2Jx

K2 = 0K2/K2,nominal = 50 %HSX =9871x 3 + 6x py2

The x3 term does not affect DA.

0

K2/K2,nominal = 75 %K2/K2,nominal = 100 %33

Simple IR and crab waist (1/2)
Simple IR: No solenoid, B1=0, A1=0, A2=0, no offset and rotation of
QC1/2, Bn and An (n>3) are non-zero
A. Morita

Dynamic aperture depends on the location of CW sextupole, realistic
location does not work. There are QC1 and QC2 between IP and
CW sextupole.
Ratio of βy to βx at CW sextupole should be larger than 10.
34

Simple IR and crab waist (2/2)
Simple IR: No solenoid, B1=0, A1=0, A2=0, no offset and rotation of
QC1/2, no multipoles, QCs have B2 only.
A. Morita

CW sextupole
in NIKKO and OHO

QC1 and QC2 between
CW I-cell

QC1 only between
CW I-cell

No QCs between
CW I-cell

Dynamic aperture still depends on the location of CW sextupole,even though QCs are the simple quadrupoles.
Nonlinear maxwellian fringe of QCs and kinematic term at drift
space between CW I-cell reduce the dynamic aperture significantly.
35

Summary
We checked the performance of optics corrections with a realistic
machine error. correction of COD , X-Y coupling, dispersions, beta
functions.
The vertical emittance can be recovered to be almost that of the ideal
lattice.
However, dynamic aperture can not be recovered by the traditional
optics corrections.
New technique is necessary to fix the dynamic aperture issue.
Dynamic aperture with beam-beam is still the serious issue.
Crab-waist is a possible cure, but nonlinear Maxwellian fringe andkinematic term between CW sextupole and IP reduce the dynamic
aperture. No solution so far.
px1
2 2
2 2
2
2
x2 = x1 +

ps1

l

ps =

pt /c

m c

px

py
36

The longest journey begins with a single step.
千里の道も一歩から。

37

Commissioning and Lattice
Preparation

38

Lattice preparation
phase

sub-phaselattice, commissioningIb (mA)

Phase 1.1

wiggler off, device check,
optics tuning, vacuum scrub.

< 30-100Phase 1.2

wiggler on, circumference,
optics tuning< 30

high emittance for vacuum
scrubbing (LER)

500-1000

Phase 1.4

optics tuning (low emittance)< 30

Phase 2.1

vertical beta* = 80 mm,
optics and injection tuning

< 30

:

:

Phase 2.x

vertical beta* = 2.2 mm,
optics and luminosity tuning

1000/800

Phase 3.1

vertical beta* = 2.2 mm,
optics and luminosity tuning

1000/800::

ultimate beta*,
optics and luminosity tuning

3600/2600

Phase 1
Phase 1.3

Phase 2Phase 3::Phase 3.x

IR status

No QCSNo Belle II

QCS
Belle II
w/o VXD

QCS
Belle II
with VXD(Physics Run)

39

Emittance change in arc
Emittance of arc cell can be changed by field strength of quadrupols.
Range is from 2 to 12 nm in LER and from 5 to 17 nm in HER,
respectively.

nominal αpnominal αp40LER: Emittance change in wiggler section
Emittance can be adjusted by optics in Nikko section(No cavity).
Only adjust field strength of the quadrupole magnets
Emittance becomes 20* nm in case of 1 m dispersion at midpoint thestraight section.
LER: Nikko* without intra-beam scattering

41

Phase 2.1 LER (1/2)
Phase 3.xx = 32 mm

Phase 2.1

y = 0.27 mm

x = 400 mm
y = 80 mm

LER

LER

beam size

σx = 1100 μm at QC2
rb = 35 mm at QC2
rb/σx = 31 (linear calc.)beam size

σx = 350 μm at QC2
rb = 35 mm at QC2
rb/σx = 100 (linear calc.)
KEKB

x = 590 mm
y = 5.9 mm
42Phase 2.1 LER (2/2)

y = 80 mm

Δy (mm)LER

QC1P: misalignment
ΔyQC1P = 100 µm (opposite)Δy (mm)x = 400 mm+-5 mmArc: < 5 mm

The vertical orbit distortion is less than 10 mmwithout any corrections. We can find the orbit.

43

Machine parameters with QCS
Pilot run

UltimatePhase 2.x

Phase 3.x

Parameters

symbolEnergyE#Bunches

nb

Emittance

εx

2.2

5.2

Coupling

εy/εx2Hor. beta at IP

βx*Ver. beta at IP

unit

LER

HER

LER

HER

4

7.007

4

7.007

GeV

3.2

4.6

nm

2

0.27

0.28

%

128

100

32

25

mm

βy*

2.16

2.4

0.27

0.30

mm

Bunch current

Ib

1.0

0.8

3.6

2.6

A

Beam-beamξy0.02400.0257

0.088

0.081

Hor. beam size

σx*

16.8

22.8

10

11

µm

Ver. beam size

σy*

308

500

48

62

nm

Luminosity

L

2500

2500

1x10348x1035cm-2s-1
Y. Funakoshi
44Misalignment of quadrupole coils in QC1 and QC2
Sextupole field

Skew sextupole field

y

y

IP
x
QC1L

QC1R

+
+
- sign of + error field +Touschek Lifetime (sec)Δr=35 μm induces 0.1 % of B3/B2.
B3 ~7 Gaussxx

without any corrections

b3/b2 (%)
45

Sextupole error field: normal and skew
Touschek lifetime for 16 combinations of field error in 4 QCs
Dynamic aperture is optimized by using normal sextupoles(108) and skew
sextupoles(24) in the arc.
LER
LER

sextupole error

skew sextupole error

Sextupole error can be recovered by using normal sextupoles(in the arc),
however, skew sextupoles error can not be corrected for enough level.
Position dependence of the skew corrector coilis stronger than that of normal.

Designideal lattice

Scan effective corrector position

QC2R

QC1R IPw/ error

QC1L QC2L

Skew sextupole corrector must be installedin the vicinity of the error source.
46

IR optics in LER
X-LCC corrects QC2 chromaticity and Y-LCC corrects QC1 chromaticity locally.
beta functions

horizontal dispersion
vertical dispersion{

coupling parameters

X-LCC

Y-LCC

2
QC2 QC1strongsextupoles47

IR optics in HER
X-LCC corrects QC2 chromaticity and Y-LCC corrects QC1 chromaticity locally.
beta functions

horizontal dispersion
vertical dispersion{

coupling parametersstrongsextupoles

X-LCC

Y-LCC

2
QC2 QC1

48