Optics Correction
Hiroshi Sugimoto (KEK)for SuperKEKB Optics GroupContents
1. Correction of XY Coupling and Vertical Dispersion
2. Correction of Betatron Function
3. Full Optics Correction

Introduction••

Correction of beam optics errors will be a critical factor in SuperKEKB.
Innovative optics measurement & correction schemes are probably necessary at
high-current beam operation.
i.e. Horizontal & vertical (XY) coupling measurement with single-pass BPMs.
(Ref. Ohnishi’s talk @ the 16th KEKB review)•On the other hand, KEKB based optics measurement using closed orbit
distortion (COD) will be employed at early stage of the commissioning andlow-current and non-collision operation.
• It is informative to investigate how effective is KEKB based scheme.

Today’s Report
Optics Correction with KEKB based scheme
1. XY-Coupling and Vertical Dispersion
2. Betatron Function
3. Full Optics Correction
02/20/2012

The 17th KEKB Accelerator Review Committee

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XY-Coupling and Vertical Dispersion

Measurement & Correction Scheme• XY-coupling measurement & correction.
1. XY-coupling is estimated by observing the verticalleakage orbits associated with horizontal kick.
2. Suppress the vertical leakage orbit by skew correctors.

• Vertical dispersion measurement
Use vertical orbit response with RF frequency change.
RF frequency

Orbit change%y"y # $ f 0& Phase slip factor
2%fFrequency change

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The 17th KEKB Accelerator Review Committee

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Correctors••

Skew winding of sextupole magnet will be available.
Symmetric / asymmetric excitation of skew-corrector pare can be used asorthogonal correctors for coupling and vertical dispersion.
Symmetric Excitation(XY-coupling corrector)

Asymmetric Excitation(Vertical-dispersion corrector)

Couplingparameters

Verticaldispersion
Skew-fieldSDSD"
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The 17th KEKB Accelerator Review Committee

5Simulation•Assumptions- No orbit drift in time.
- Zero-current (ignore intra-beam scattering in emittance evaluation).
- No error source in the IR region.

Error Source
Vertical misalignment[µm]

Rotation error[µrad]

Normal quadrupoles

0100

Normal sextupoles

1000Sextupoles(local chromaticity)

500All errors are Gaussian distributed.

Skew correctors are adjusted by measured optics and response matrix of themodel lattice with singular value decomposition.
02/20/2012

The 17th KEKB Accelerator Review Committee

6Simulation Example

No BPM Jitter!

Vertical leakage orbit

Before Correction"y1[mm]

"y 2!

"y 3!

"#y[m]!
02/20/2012IP"y = 69 pm

The 17th KEKB Accelerator Review Committee

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Simulation Example (No BPM Error)
After Correction"y1 [mm]!

"y 2 [mm]!

"y 3 [mm]!

"#y [m]!

"y = 69 pm
02/20/2012" y = 1.5 pm

The 17th KEKB Accelerator Review Committee

8Vertical Emittance and BPM Jitter Error• 100 simulations with different seed of random numbers.
Emittance before correction"!
*DesignTarget('+,-./0(+,-./0

Jitter Error 0 µm
Jitter Error 2 µm
Jitter Error 5 µm)

)
'&%&%
$

$

#

#""!

!

Emittance after 1 correction

!*!

"!!

#!!

$!!

%!!

&!!

12/34567893//5.427:;9<'!!

(!!

*!

"

#$%&'()!*

12/34567893//5.427:;9<• Corrections of XY-coupling and vertical dispersion
effectively reduce vertical emittance.
02/20/2012

The 17th KEKB Accelerator Review Committee

9Vertical Emittance and BPM Jitter Error• 100 simulations with different seed of random numbers.
Emittance before correction"!
*DesignTarget('+,-./0(+,-./0

Jitter Error 0 µm
Jitter Error 2 µm
Jitter Error 5 µm)

)
'&%&%
$

$

#

#""!

!

After 5 corrections

!*!

"!!

#!!

$!!

%!!

&!!

12/34567893//5.427:;9<'!!

(!!

*!

"#$%&'()!*

12/34567893//5.427:;9<• Iterative correction is effective especially when BPM
jitter error is large
02/20/2012

The 17th KEKB Accelerator Review Committee

10Vertical Emittance and BPM Rotation Error• 100 simulations with different seed of random numbers.
Vertical Emittance after 5 corrections#!

DesignTarget*)+,-./0(0.1 mrad
0.3 mrad0.6 mrad

'"
&%
$#!

!

"

#!

#"

$!

12/34567893//5.427:;9<• BPM rotation error should be less than 0.5 mrad forthe target vertical emittance.
02/20/2012

The 17th KEKB Accelerator Review Committee

11Dynamic Aperture (DA)• 100 simulations with different seed of random numbers.
2J x

Before Correction(" x + " y )%#!

Design%$%$"#"#"$"$

#

#

$!"#

!"$!#

$#

After Correction

%#

"$"#

$!"#

!"$!#

2J x /" z$2J x /" z#"$"#• DA is dramatically improved by the optics correction.
!
02/20/2012!
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Betatoron Function

Beta Measurement

Ref. A. Morita et al, PRSTAB 10, 072801 (2007)•Analyze CODs induced by steering kicks.

•COD at the i th BPM induced by j th steering kick in two forms.
"x ij ="k jcos #$ X % S sin #$ Y ) X + ( cos #$ Y + S sin #$ X )Y(2sin #$xSjij

Sjx

Mi

Sjxij

Sjx

i

Mx

() ! Form 1"k=cos #$ X % S sin #$ Y ) X + ( cos #$ Y % S sin #$ X )Y
(2 sin #$!
G ( X ,Y ) ! Form 2
X" #F X iM ,YiMj!

xMiijx i

MSjSjSjx i

Mijx

MiSjxM Si

M,Scos $ iM , S
i

YiM , S " # iM , S sin $ iM , S!

•Find {X iM , YiM} and {X Sj!, Y jS}!

(Sij " sign # iM $ # Sj)

which reproduce measured beam positions !{ "x ij meas}•!

!

The solution is obtained by iteratively minimize two quadratic functions.
!

() %S1 X iM ,YiM "i, j

02/20/2012(#x ij meas $ F X iM ,YiM)

2(!

) %

S2 X Sj ,Y jS "(#x ij meas $ G X Sj ,Y jS)2i, j

The 17th KEKB Accelerator Review Committee

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Beta Measurement @ SuperKEKB HER No BPM Jitter
Exact
Measured"# x / # x!

!

!

"# y / # y" x # " xdesign" y # " ydesignIP!

• This scheme inevitably has two ambiguous.
1. Scaling of beta function{" iM , " Sj } ! { # " iM , #$1 " Sj }02/20/2012

2. Phase origin{" iM , " Sj } ! {" iM + # , " Sj + #}The 17th KEKB Accelerator Review Committee

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Beta Measurement @ SuperKEKB HER No BPM Jitter
Exact
Measured" ( #$ x / $ x )!

!

" #$ y / $ y()!

" # x $ # xdesign()()" # y $ # ydesign

IP!

• To avoid the ambiguous
we use difference between neighborhood BPMs in the optics correction.
02/20/2012

The 17th KEKB Accelerator Review Committee

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Simulation of Beta Correction•

Assumptions- No orbit drift in time.
- Zero-current (ignore intra-beam scatteringin emittance calculation).
- No error source in the IR region.
- Exact values of perturbed tunes are known.

•Error Source (Gaussian distributed)"K1= 5 # 10 $4
K1

All normal quadrupoles•

Beta Measurement- Use 6 horizontal and 6 vertical orbit kicks.
- Orbit amplitude at Arc section is about < 1mm.

•Corrector
all quadrupoles (fudge factor of the electric power supplies)!

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Beta Correction @ SuperKEKB HER

No BPM Jitter

BeforeAfter"# x / # x!

"# y / # y!

" x # " xdesign" y # " ydesign

!

After

Before!

"# x$ 22%#x"# y$ 33%#y"# x$ 0.9%#x%2%4"# x = $1.3 %10$4 "# y = 8.2 $10"# x = $6.5 %10$3 "# y = 4.1$10!

02/20/2012

The 17th KEKB Accelerator Review Committee!

"# y$ 0.5%
#y!

18!

Global Beta Beat After Correction (SuperKEKB HER)
• 100 simulations with different seed of random numbers.
BPM Jitter 2 micron•

Typically, global beta beat for 2 micro bpm jitter is within ~5% forhorizontal and within ~10% for vertical betatron motion.

02/20/2012

The 17th KEKB Accelerator Review Committee

19Full Optics Correction

Measurement Method and Corrector Knob
• All measurements are based on COD or its response.
Measurement MethodCorrector KnobCOD

BPM reading

Horizontal/Vertical Steerings

XY Coupling

COD response associatedwith horizontal steering kickSkew winding of sextupoles

Vertical
Dispersion

COD response associatedwith RF frequency change

Skew winding of sextupoles

Betatron
Function&Phase

COD response associatedwith steering kickquadrupole (fudge factor)

Horizontal
Dispersion

COD response associatedwith RF frequency change

Horizontal mover offocusing sextupoles

02/20/2012

The 17th KEKB Accelerator Review Committee

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Simulation Condition
Error Source
Horizontal
misalignment[µm]

Verticalmisalignment[µm]

Rotation
error[µrad]

ΔK/K

Quadrupoles

100

100

100

2.5*10-4

Sextupoles

100

10000•

Assumptions- No orbit drift in time.
- Zero-current (ignore intra-beam scattering in emittance calculation).
- No error source in the IR region.
- Exact values of perturbed tunes are known.

••

Beam optics is unstable if we introduce the error sources at once.
Thus, we gradually increase the error amplitude with the optics corrections.

02/20/2012

The 17th KEKB Accelerator Review Committee

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Emittance• 100 simulations with different seed of random numbers.
85*.+4/0123+../-4516937'Target%Jitter Error 5 µm
Jitter Error 2 µm
Jitter Error 0 µm$!
#&!"#!"#$!"!

!"!$!"$!"$$!"%

Design()*+,)-./0123+../-4516-37• Iterative correction with adjusting correction parameters isnecessary to improve vertical emittance.
• Reached vertical emittance is sensitive to the correction
parameters especially when BPM jitter is large.
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The 17th KEKB Accelerator Review Committee

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Beta Beat After Optics Correction••

2 micro BPM Jitter

100 simulations for different random seeds.

Vertical beta beat is typically within ~10%, and ~5% for horizontal direction.

02/20/2012

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Dynamic Aperture

2 micro BPM Jitter

20 samples

Design2J x(" x + " y )!

2J z /"z• On-momentum aperture is improved to the design level.
• Optimization of higher!order multipole magnets is necessary toimprove off-momentum aperture (in progress)
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The 17th KEKB Accelerator Review Committee

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Summary and …
• COD based scheme is applied to SuperKEKB HER lattice.
-Iterative corrections is necessary to reduce vertical emittance.
BPM rotation should be less than 0.5 mrad for the target vertical emittance.
In typical case, global beta beat is within ~5% for horizontal and within~10% for vertical betatron motion.
After full optics corrections, we can reach the target vertical emittance.
Achieved vertical emittance is sensitive to the correction parameters
especially when BPM error is large.
On-momentum DA is improved to the design level.
As for off-momentum DA, it is expected to be improved by tuning higher
order multipole fields, but not confirmed yet.

• To be studied-More detailed study on optics measurement using single-pass BPM.
Error source in IR region.
Orbit drift

02/20/2012

and so on…

Thank you for listening!

The 17th KEKB Accelerator Review Committee

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