Evaluation of X‐Y coupling /Requirements on correction
Y. Ohnishi
The 14th KEKB Accelerator Review Committee
10/Feb/2009

Y. Ohnishi / KEK, KEKB Review 20091

Outline• Introduction– Estimation of vertical beam size from luminosity– Simulation with machine error (see Apendix)• Chromaticity ‐ Turn‐by‐turn BPM analysis ‐• Twiss‐chromaticity at IP• R‐chromaticity at IP• Method of a X‐Y coupling measurement• Summary of X‐Y coupling• Miscellaneous
Y. Ohnishi / KEK, KEKB Review 20092

Specific luminosity
Estimation of vertical beam sizeINTRODUCTION

Y. Ohnishi / KEK, KEKB Review 20093

Lspec/nb (x1030 cm-2s-1 mA-2)

Specific luminosity€beam-beamsimulation

0.9/0.007 m
crab on

βx*=0.8 m k=1%
w/ crab
βx*=1.5 m k=1%

1.5/0.0059 m
crab off

1.5/0.0059 m
crab on0.8/0.0059 m
crab on0.59/0.0065 m
crab offLsp =βx*=1.5 m k=1.3%
βx*=1.5 m
k=0.5/0.3%
w/o crab

Lnb
RL=Ib,LER Ib, HER 4 πe 2 f 0 σ *xσ *y

I bunch(LER) x I bunch(HER) mA2
Y. Ohnishi / KEK, KEKB Review 20094

Estimation of vertical beam sizewith dynamic effectVertical Beam Size (µm)νx,LER/HER = 0.508/0.510
νy,LER/HER = 0.590/0.590

σx*LER=σx*HER σy*LER=σy*HER
0.59/0.0065 m
crab off

0.8/0.0059 m
crab on0.9/0.007 m
crab on

1.5/0.0059 m
crab off

k=0.5 %

1.5/0.0059 m
crab onI bunch(LER) x I bunch(HER) mA2
Y. Ohnishi / KEK, KEKB Review 20095

Comparison of vertical beam size
• Estimated value from luminosity– 1.1 µm at 0.6 mA2 ‐> 1.6 µm at 1.5 mA2– increased by ~50 %– k=εy/εx= 0.5 % at the low bunch current products.

• SRM measurement– 1.8/1.8 µm at 0.6 mA2 ‐> 2.8/2.4 µm at 1.5 mA2– increased by ~50 %• Blowup rate is similar each other except for absolutevalues.
• The coupling parameter, k is small at the small bunch
current ?.
Y. Ohnishi / KEK, KEKB Review 20096

Chromaticity
Twiss‐chromaticity
R‐chromaticity
Method of X‐Y coupling measurement
Summary of X‐Y coupling

TURN‐BY‐TURN BPM ANALYSIS AT KEKB
EXPERIMENTS

Y. Ohnishi / KEK, KEKB Review 20097

Chromaticity
Measured by single-pass BPMs using a horizontal kicker
Date: 2008/Dec/14

Horizontal Chromaticity : ξx = 2.56

αp = 3.31x10-4

νx

LER

model (SAD)Δp/p0
Δf = -400 ~ +400 Hz, 100 Hz step
Y. Ohnishi / KEK, KEKB Review 2009

ΔfΔp= −α pf0p08

Twiss‐chromaticity : Measurements
Date: 2008/Dec/14

LER

model (SAD)

βx,IP (m)αx,IPLER

model (SAD)

Δp/p0

Δp/p0

α x,IP = −0.016

β x,IP = 1.29 m∂α x,IP= 15.2∂δ∂β x,IP= 1.4∂δ
Y. Ohnishi / KEK, KEKB Review 20099

R‐chromaticity : Measurements
Date: 2008/Dec/14

LER
model (SAD)r4,IPr3,IP (m-1)

LER

model (SAD)

Δp/p0

Δp/p0

r3,IP = 0.473 m −1

r4,IP = −0.112∂r3= −100 m −1∂δ

∂r4= −243∂δ+10 units

Y. Ohnishi / KEK, KEKB Review 2009

-2 units10

R‐chromaticity : Measurements (cont’d)LER

r2,IP (m)r1,IPLER
model (SAD)

model (SAD)

Δp/p0

Δp/p0r1,IP = −0.047

r2,IP = −0.066 m∂r1= 8.2∂δ-150 units∂r2= −21 m∂δ-15 units

Accuracy of BPM is not enough for r1 and r2. (V-mode is necessary.)
Y. Ohnishi / KEK, KEKB Review 200911

X‐Y coupling (1)Transformation from a decoupled coordinate to a physical coordinate: x  µ   x' =  0
 y   −r1   y'  −r30µ−r2−r4−r2  X  r1  X '
0  Y  µ  Y ' µ 2 + ( r1r4 − r2 r3 ) = 1€

When Y=0 and Y’=0 (H‐mode):€

X = 2J x β x cosψ xx = µXx'= µX 'y = −r1 X − r2 X 'y'= −r3 X − r4 X'

r4−r3µ0X'= −2J x(α x cosψ x + sinψ x )βxThis induces a vertical betatron oscillation.
€Y. Ohnishi / KEK, KEKB Review 200912

X‐Y coupling (2)[1]y = −r1 X − r2 X'
2J x 2J x= −r1 2J x β x + r2
α x  cos ψ x + r2sin ψ xβxβx= A cos ψ x + B sin ψ x=

€A2 + B2 sin(ψ x + φ )A = −r1x ampµ+ r2x' ampµPhase is important.

sin θ = y amp sin φx' amp
B = r2cos θ = y amp cos φµyx'
r1 = −µ amp sin φ + r2 amp sin θx ampx amp€r2 = µy amp cos φx' amp cos θWhen αx=0 at IP, sinθ=0 and cosθ=1.

€ X'= − 2Jx (1+ α x2 ) sin(ψ x + θ )βxsin θ =€αx1+ α x2cos θ =11+ α x2xamp, x’amp, yamp, φ, θ can be obtained
from single-pass BPMs.
The ratio of amplitudes andphase provides
an information of X-Y coupling.

Y. Ohnishi / KEK, KEKB Review 2009€

Asin
φ=


A2 + B2
B
cos φ =
A2 + B213

X‐Y coupling (3)[2]y'= −r3 X − r4 X'

2J x 2J x=  −r3 2J x β x + r4α x  cosψ x + r4sinψ xβxβx= C cosψ x + Dsinψ x= C 2 + D 2 sin(ψ x + ϕ ) Phase is important.

€C = −r3x ampµ+ r4x' ampµsin θ = y' amp sin ϕ€
r4 = µy' amp
x'sin ϕ + r4 amp sin θx ampx ampy' amp cos ϕx' amp cos θWhen αx=0 at IP, sinθ=0 and cosθ=1.

€ X'= − 2Jx (1+ α x2 ) sin(ψ x + θ )βxx' amp
D = r4cos θ = y' amp cosϕµr3 = −µsin θ =€αx1+ α x2cos θ =11+ α x2

xamp, x’amp, y’amp, ϕ, θ can be obtained
from single-pass BPMs.
The ratio of amplitudes andphase provides
an information of X-Y coupling.

Y. Ohnishi / KEK, KEKB Review 2009€

Csinϕ=C 2 + D2Dcos ϕ =
C 2 + D2

14

Reconstruction of phase space
Phase space (x,x’,y,y’) at IP is reconstructed by using neighborsingle-pass BPMs (location: QCS-L and QCS-R)

ML

 m L,11
 x
 x'  =  m L,31
 m R ,11
 y

 
 y' IP  m R ,31m L,12m L,32m R,12m R,32MRm L,13m L,33m R ,13m R ,33−1ML, MR is a transfer matrix
from IP to the single-pass
BPM. The model is used.

m L,14   x L   m L,34   y L m R,14   x R   m R,34   y R QCS

Y. Ohnishi / KEK, KEKB Review 200915

Experiments

Date: 2008/Dec/14

LERLERα x = 0.01
β x = 1.3 m€

LER

Beam oscillation is induced by a horizontal kicker.
Phase space plots at IP reconstructed by twoBPMs(QCS-L and QCS-R)
α x = tan θ
βx =x amp 1x 'amp cosθ

Ratio of amplitudes and phase

Horizontal tune is near half integer.
Y.€Ohnishi / KEK, KEKB Review 200916

x (mm)H-modeTurn‐by‐turn data at IPLERLER

Date: 2008/Dec/14

x = µX

FFT

induced by horizontal kickerx’ (mrad)€ νxx'= µX '€y’ (mrad)y (mm)

r1 and r2 is small or lessresolutiony'= −r3 X − r4 X'y' β ∝ 1 β y sin(2πν y n + ψ y 0 )turns
Y. Ohnishi / KEK, KEKB Review 2009€

νy€

17

Twiss‐chromaticity : Measurements

model (SAD)

βx,IP (m)αx,IPDate: 2008/Dec/12

HER

HERΔp/p0HERmodel (SAD)Δp/p0

α x,IP = 0.59

β x,IP = 1.90 m∂α x,IP= −23∂δ∂β x,IP= 54 m
∂δLarge alpha, something wrong !.

€(BPMs at QC1LE and QC1RE are used.)€

Y. Ohnishi / KEK, KEKB Review 200918

R‐chromaticity : Measurements
Date: 2008/Dec/12

HER

r4,IPr3,IP (m-1)

HER

model (SAD)

model (SAD)

Δp/p0

Δp/p0

r3,IP = 0.0012 m −1

r4,IP = −0.035∂r3= −74 m −1∂δ

∂r4= −51∂δ
Y. Ohnishi / KEK, KEKB Review 200919

R‐chromaticity : Measurements
Date: 2008/Dec/12

HER
model (SAD)

r2,IP (m)r1,IPHER

model (SAD)

Δp/p0

Δp/p0

r1,IP = −0.324

r2,IP = −0.408 m∂r1= 48∂δ∂r2= 63 m
∂δAccuracy of BPM is not enough for r1 and r2. (V-mode is necessary.)
Y. Ohnishi / KEK, KEKB Review 200920

Summary of X‐Y coupling• x(n), x’(n), y(n), y’(n) are fitted by a linear combination of cosine andsine with an exponential damping. Resol. at each BPM is ~100 µm.
– Model transfer matrix is used in this analysis.

1000 turns are used or analysis.

• Twiss parameters, αx and βx can be obtained from x(n) and x’(n):X'(n) = −sin θ =2J x1+ α x2 ) sin(2πν x n + ψ x 0 + θ ) exp(−Γn )(βxαx1+ α x2cos θ =11+ α x2

α x = tan θx
1β x = amp
x'amp cos θ– If the location is IP, αx is zero in the model.
€• Uncertainty about µ. If r1 and r2 is small, µ is 1 approximately.
2µ + ( r1r4 − r2 r3 ) = 1€€

X amp = x amp

y' ampr3 = −sin ϕx amp

X' amp = x' amp

y' amp(α x = 0)r4 =x' amp

cos ϕ

Y. Ohnishi / KEK, KEKB Review 2009

The ratio of amplitudes andphase provides
an information of X-Y coupling.
21

Summary of X‐Y coupling (cont’d)• The global X‐Y coupling measurement is sensitive to r1 and r2, butless sensitive to r3 and r4.
x = µXx'= µX 'y = −r1 X − r2 X 'y'= −r3 X − r4 X'€

The correction makes y small as much as possible
which is induced by a horizontal kick.

– The X‐Y coupling, r3 and r4 at IP are large in LER(14/Dec/2008), which are
estimated by single‐pass BPMs. Accuracy is not enough for r1 and r2. If r1 andr2 are well corrected, σy/σx should be small. However σ’y/σ’x might be large.

• R‐chromaticity can be measured by changing rf frequency which issimilar to chromaticity measurements.
• IP tilt knob can correct r3 and r4.
– Skew sextupoles are needed to correct the R‐chromaticity.
Y. Ohnishi / KEK, KEKB Review 200922References• D. Sagan and D. Rubin, PHYSICAL REVIEW SPECIAL TOPICS ‐ACCELERATORS AND BEAMS, VOLUME 2, 074001 (1999) .
• Y. Ohnishi, Y. Funakoshi, K. Mori, E. Perevedentsev, M. Tanaka, M.
Tejima, M. Tobiyama, Measurement of xy coupling using turn‐by‐turn BPM at KEKB, EPAC2000.
• D. Rubin, “AC” dispersion measurement, ILCDR08.

Y. Ohnishi / KEK, KEKB Review 200923

LER damping rate
Head‐Tail damping
Smearing effect due to nonlinearity

MISCELLANEOUS

Y. Ohnishi / KEK, KEKB Review 200924

LER Damping RateMeasured by single-pass BPMs

Radiation damping rate (1/turn) : Γ = (2.48±0.33)x10‐4 (meas)/ 2.47x10‐4 (model)
LER

Γ1Chromaticity
ξx = 2.56

Ib (mA)
Y. Ohnishi / KEK, KEKB Review 2009

*DCCT25

€€

Head‐Tail Damping (LER)
π β Ib Z ⊥ξΓHT =(E b /e)α p(1/turn)

Y(2S)

R
Cβ ==
ν β 2πν βΓ=Γrad+ΓHT

LER transverse impedance:

Z ⊥ = 17 ± 2 kΩm
Y. Ohnishi / KEK, KEKB Review 200926

Smearing effect due to nonlinearlity
Measured by single-pass BPMs

LER
Γ2 (1/turn2)x(n)= exp{−Γ1n − Γ2 n 2 }x(0)(2πσ νx )Γ =222(1/turn2)

less dependence on bunch current
Ib (mA)€Γ2 (1/turn2)

LER

Jx/εx = 11Jx (m)σνx = 2.25x10‐5 << Γ1=2.47x10‐4
10 times smaller than radiation damping
Y. Ohnishi / KEK, KEKB Review 200927

synchro-beta
resonance2νx+2νs=integer

LERLERΓ2 (1/turn2)

Γ1 (1/turn)Nonlinearity at off‐momentum

νxΔp/p0x(n)= exp{−Γ1n − Γ2 n 2 }x(0)Γ1 (1/turn)

LER(2πσ νx )
Γ =222(1/turn2)€Δp/p0

Y. Ohnishi / KEK, KEKB Review 200928

APPENDIX

Y. Ohnishi / KEK, KEKB Review 200929

KEKB HER optics: Beta05_14_2008_22:07:34i
εx = 24 nm
βx* = 0.9 m
βy* = 5.9 mm
νx = 44.5138
νy = 41.590

SIMULATION WITH MACHINE ERROR

Y. Ohnishi / KEK, KEKB Review 200930

Machine Errors
σΔx,rms (µm)σΔy,rms (µm)σΔθ,rms (mrad)

σΔK/K,rms

Quad100

meas.*2meas.*2

1x10‐4Skew Quad100meas.*2

meas.*2

1x10‐4

Sextupole

100

extrapolation+1000.1

1x10‐4BPM*1

100extrapolation+100‐‐Steering

100

extrapolation+1000.1‐*1) BPM jitter error : σΔx, Δy,rms = 2 µm*2) Measured by MG group31

Machine Error and Optics Correction : Simulations
Orbits, X-Y couplings, dispersions, beta functions are corrected.
HER-13 mm !

IP

32

Sample of X‐Y coupling correction
Bump height
Skew Q

HER: simulation

Δy1,Δy2Δy3,Δy4Δy5,Δy6Δy7,Δy8Δy9,Δy10Δy11,Δy12
r1 = .0019268
r2 = -.001642
r3 = -.136334
r4 = .663629
Δyrms= 15.8 micron

33

Emittance ratio : Simulations
100 samples: Each sample has a different random seed.
HER

Beam size is less than 1 µm at IP

HER

Emittance ratio is less than 0.5 %.

After the optics correction, the optics seems to be better condition.

Y. Ohnishi / KEK, KEKB Review 200934

X‐Y coupling at IP : Simulations
100 samples: Each sample has a different random seed.
HER

HER

averagestandard deviation

r1*

0.000860.00105

3 units*r2* (m)0.001250.00122

3 unitsr3* (1/m)‐0.01217

0.15403

6 units

r4*0.21266

0.54331

23 units*IP tilt knob

Largeresiduals

The r1 and r2 are well corrected, however r3 and r4 are scattered.
Y. Ohnishi / KEK, KEKB Review 200935

εx = 18 nm νx = 0.508

Horizontal
beam size (µm)

Horizontal
beta (m)Emittance (nm)Dynamic Effectweak dependence of tune shift

Beam-Beam Tune Shift
Y. Ohnishi / KEK, KEKB Review 200936

Dynamic Beta

Vertical beta function (mm)βy* = 5.9 mm νy = 0.59

ξyY. Ohnishi / KEK, KEKB Review 200937

Beam size measured by SRM
green: vertical beam size / red: lifetime

Vertical beam size ~2.8/2.4 µm for LER/HER(Crab ON, 1.5 mA2).
c.f. Estimated value is 1.6 µm
Y. Ohnishi / KEK, KEKB Review 200938