Application of Frequency Map Analysis to Storage Rings in China

Jiao Yi

Institute of High Energy Physics Chinese Academy of Sciences, Beijing, China

Outline

Shanghai Synchrotron Radiation Facility (SSRF)

Super-periodic structural resonance (SSR)

Beijing Electron Positron Collider Upgrade Project (BEPCII)

Synchro-betatron resonance

Beijing Advanced Photon Source (BAPS)

Multi-objective genetic algorithms (MOGA) method

),(),(),(: 0022

yxyx QQorvvyxRR Frequency map:

SSRF, China

SOLEIL, France

Frequency map analysis (FMA)

Courtesy of L. Nadolski, SOLEIL

(1) (2) 2 (1) (2) 2( ) ( )x x y yD v v v v

1/42

SSRF: A third generation light source

Beam energy GeV 3.5

Circumference m 432

Super-period No. 4

Natural horizontal emittance x0

nm rad 3.90(7.98*)

Number of cells 20

Straight sections: lengthnumber

m 12.046.516

Betatron tunes Qx/Qy 22.22/11.32

Natural chromaticities x/y

-54.3/-18.3

Main parameters of SSRF storage ring

Twiss parameters in one super-period

Dominative resonance:

3Qx2Qy = 44

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Motion near the resonance 3Qx2Qy=44

Tune vs. horizontal amplitudeFor particle with different coordinates 0 0 0 0( , 0, 0, 0)x x y y Tracking with AT.

It is a Super-periodic Structural Resonance (SSR).

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Linear super-periodic structural resonances-I

Linear magnetic field errors lead to first or second order super-periodic structural resonances.

0.96 0.98 1 1.02 1.04

0.96

0.98

1

1.02

1.04

0.99 0.995 1 1.005 1.01u v 0.99 1.01

2.5

5

7.5

10

12.5

15

17.5

20

x&ym

0.99 0.995 1 1.005 1.01u v 0.99 1.01

-0.2

0

0.2

0.4

0.6

0.8

Q x22&Q y11

Necktie diagram for SSRF Black star: nominal working point (22.22, 11.32)

Qx=22 Qx=24

Qy=12

Qy=10

Qx=23v=Kd / kd0

u=Kf / kf0

Blue: horizontal

Red ：Vertical

If Q is closed to stop band, the off momentum particle dynamics optimization will be difficult.

K source: off-momentum particles (1+) p0

0 0 00 0 0

1(1 ) (1 )

(1 )y y yB B Be e

K KB x p x p x

Stopbands: Q = M N /2, M is the super-period No.

Qy=11

4/42

Linear super-periodic structural resonances-II

The first and second order super-periodic structural resonance (SSR) is

/ 2Q M l with l Ν

M is super-period number of the lattice. The resonance is first order when l is even and second order when l is odd.

How about nonlinear magnetic fields (sextupoles, most strong nonlinear components in light source)?

For modern acclerators with complex lattice structure, necktie diagram still works well. Some integer or half integer resonance exhibits as “stopband” in Kf - Kd space, implying stronger effect than others. “…The formation of the structural resonance stopband comes from the harmonic number of the super-periodic resonance in the lattice configuration. …”

Courtesy of S.X. Fang and Q. Qin, HEP & NP 30(9) 880

How about higher order resonances with the same harmonics as the linear SSR?

For SSRF, the resonance 3Qx2Qy = 44 has the same harmonic with the 2Qx = 44 second order SSR stop band

5/42

Linear SSR by sextupoles

(22.04, 11.24) (23.04, 11.24)

Ideal SSRF lattice, with sextupoles as the only one nonlinear source.

Qx =22 is second order SSR, Qx =23 is not linear SSR.

6/42

Sextupoles & SSR

11.36

B

A

C

off-momentum DA vs. QxOn-momentum DA vs. Qx

A, B (~0.1): 2Qx = 44

C(~0.04) : 3Qx 2Qy = 44

Coutesy of S.Q. Tian, SSRF

7/42

Higher order SSR vs. Working point

Five work points, (22.12, 11.17), (22.16, 11.23), (22.22, 11.32), (22.26, 11.38) and (22.32, 11.47) along the resonance 3Qx2Qy = 44.

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5(5)

5 5 5 ,, , , ,

| ,abcd ea b c d e

f g t C abcd e

W.P. Resonance location Trapped particles NO. (%)

Average diffusion rate <D>On x axis (y

= 0 mm)On the line x = 20 mm

(22.12, 11.17)

(7.3, 0.0) (20, 12.3) 1.51 -3.48

(22.16, 11.23)A

(9.5, 0.0) (20, 13.7) 14.43 -4.70

(22.16, 11.23)B

(10.2, 0.0) (20, 12.3) 1.79 -3.54

(22.22, 11.32)

(9.1, 0.0) (20, 13) 1.64 -3.76

(22.26, 11.38)

(10.5, 0.0) (20, 10.2) 1.04 -3.53

2 2

2 2

1 1( , ) ( / / )

2 21 1

( , ) ( / / )2 2

x x xx x xy y

y y yy y xy x

Q x y C x C y

Q x y C y C x

Identify resonance with FMA1, Average diffusion rate (resonance strength) 2, Resonance location in (x, y) space 3, Resonance-trapped particle NO.

Lie Method calculation (LieMath) for contrast

1, Resonance amplitude term

2, Resonance location2.5 5 7.5 10 12.5 15 17.5 20

xmm2.5

5

7.5

10

12.5

15

17.5

ymm

The position of the resonance

Pruple-(22.12,11.17)Pruple-(22.12,11.17) black-(22.16,11.23)Ablack-(22.16,11.23)AYellow-(22.16,11.23)BYellow-(22.16,11.23)B Red -(22.22,11.32)Red -(22.22,11.32) Green- (22.26,11.38)Green- (22.26,11.38) Blue- (22.32,11.44)Blue- (22.32,11.44)

Higher order SSR vs. Working point (cont.)

9/42

Resonance location comparison

Resonance strength comparison

Higher order SSR vs. Working point (cont.)

W.P. further away from the SSR stopband 2Qx = 44,

Resonance affects the particle with larger initial amplitudes.

W.P. further away from the SSR stopband 2Qx = 44,

Resonance strength becomes weaker ((22.16, 11.23)A is an exception, with low tune diffusion while much more particles are trapped into the resonance).

10/42

/ 2QR M l with l Ν

x ymQ nQ k M satisfies

x y QmQ nQ s R with s N

Namely, the harmonic number of the resonance is the multiple of both M and RQ, the resonance will be HOSSR.

SSR is structural resonance with specific harmonic number, even with high order, it can have relatively large effects on beam dynamics.

Y. Jiao, S.X. Fang, “High Order Super-periodic Structural resonance”, EPAC08

Higher order SSR & tune optimization

If the lattice is M periodic, i.e., composed of M identical sectors (this is the case for a perfect machine), the first and second order SSR is

The resonance is first order when l is even and second order when l is odd. If the working point near the first or second order SSR stopband and the nearby structural resonance

SSR diagram near the SSRF W.P.

11/42

Super-period No. = 4

SSR diagram nearby the W. P.

SSR stopband nearby W.P.：

Qy = 8, 2Qx = 36

HOSSR nearby W.P.:a. 7Qx =168; b. 5Qx + 2Qy = 3×36; c. 3Qx + 4Qy=11×8; d. 3Qx + 3Qy=10×8; e. Qx + 6Qy=4×8; f. 4Qx 3Qy=6×8

Application of higher order SSR, SOLEIL

Frequency maps in the following slides, mostly come from paper: L. Nadolski and J. Laskar, Review of single particle dynamics for third generation light sources through frequency map analysis, Phys. Rev. ST AB 6, 114801 (2003)

a

b

c

d

e

f

12/42

Modify the sextupoles strength to fold the frequency map thus avoid 5Qx + 2Qy = 3×36。

Application of higher order SSR, SOLEIL (cont.)

The 7th-order coupling resoance, 5vx + 2vy 4×27=0, reached for the horizontal amplitude x =24mm.Courtesy of L. Nadolski, SOLEIL

13/42

Move W. P. right to avoid 7Qx = 16×8。

Application of higher order SSR, SOLEIL (cont.)

7 16 8xQ

New W.P. (18.30, 8.38)

Courtesy of L. Nadolski, SOLEIL…the new working pint is (vx, vy) = (18.30, 8.38), in order that the horizontal tune never cross the 7th-order resonance 7vx 4×32=0…

14/42

ESRF Super-period No.=16 ， W.P. = (36.44 ， 14.39), far away from first or second order SSR stopband.

On-momentum particle can cross the integer resonance Qx = 36 without loss.

Application of higher order SSR, ESRF

Courtesy of L. Nadolski, SOLEIL

15/42

Circumference = 196m ， super-period No. = 12 ， W.P. = (14.25 ， 8.18), far away from SSRs.

On-momentum particle can cross the integer resonance Qy = 8 without loss.

Application of higher order SSR, ALS

Courtesy of L. Nadolski, SOLEIL

16/42

Circumference = 72m ， super-period NO.=4 ， W.P.= （ 4.72 ， 1.70 ） , close to SSR stopband2 4yQ

The dominative resonance Qx + 2Qy = 4×2 is a third order HOSSR.

Application of higher order SSR, Super-ACO

Courtesy of L. Nadolski, SOLEIL…Globally the beam dynamics is mainly dominated by this coupoled resonance vx

+ 2vy -2×4=0. In its vicinity, a particle repidly escapes to unbonded motions…

17/42

BEPCII: a high luminosity double-ring collider

e+ e

18/42

BEPCII high luminosity mode lattice

Qx near 0.5 provides the highest luminosity

Parameter Unit Collision mode

Beam energy GeV 1.89

Circumference m 237.53

RF voltage MV 1.5

Qx/Qy/Qs 6.51/5.58/ 0.034

Natural chromatity 10.7/21.0

Horizontal natural emittance

nm rad 141

x y (IP) m 1 / 0.015

Coutesy of Dr. Y. Zhang

Twiss functions and main parameters along the ring

19/42

p / p=0 p / p= 0.6%

p / p= 0.6%On- and off-momentum DAs

FMA on the high luminosity mode

RF and radiation are turned on while tracking.

20/42

FMA on the high luminosity mode (cont.)

The off-momentum FMs are not folded as the on-momentum case and the tune footprints cover the range Qx (6.514, 6.516) with a high diffusion rate or even particle loss.

The FMs with p /p =0, ±0.6% together

On-momentum FM

Off-momentum FM

Synchro-betatron resonance: 2Qx Qs = 13

1, Reduce the anharmonic terms (dQx/dx2,

dQx/dy2, dQy/dy2) from (73, 77, 208) to (70, 68, 198) .

2, Reduce the effect of the synchro-betatron resonance

The growth time of the resonance 2Qx - Qs = 13 is:

HfeDkfn

ixnxnsn

xn

ˆˆ10

20

H along the whole ring is calculated and minimized by fine-tuning all the sextupole strengths. For example, we reduce the H from 19.7 to 3.1 for = 0.3%.

21/42

0

5

10

15

20

25

0 5 10 15 20 25

x / s x

y/s

y

Reduce anharmonicity

Minimize the growth time of resonance

before optimization

The increase of size of DA with errors tracking with SAD

Optimizing the high luminosity mode

Solid line square: collision DA requirement Dashed line square: injection DA requirement

22/42

Resonance confirmed in commissioning

Luminosity scan in tune space, 8th May, 2009.

Electron ring Positron ring

Beam lost when Qx is nearby 6.515 Luminosity falls down in tune range Qx (6.515, 6.520)

Qx

Qy Qy

Qx

23/42

Beijing Advanced Photon Source (BAPS)

BAPS

A third generation light source, Energy: 5Gev, Circumference: ~1200m, Low emittance: ~1nm

PEP-X

PEP-X

24/42

Beijing Advanced Photon Source (BAPS)Parameters Unit Value

Energy E GeV 5

Circumference C m 1218.4

Revolution time μs 4.06

Number of cells 50

Standard structure DBA

Super-period number 5

Straight sections: length*number m 514.6456.4

Beta functions in the middle of straight sections x/y/Dx

16.5/3.8/01.88/1.63/0

Emittance x/y (with wiggler)

nm.rad 1.12/0.01(0.5/0.005)

Natural energy spread 0.094%

Momentum compaction 0.00007

RF voltage MV 6

RF frequency MHz 499.982

Harmonic number 2032

Bunch length ps/mm 8.7/2.6

Damping time x/y/z ms 15.2/15.2/7.6

Beam current mA 200~300

W.P.: (64.28, 29.20)

Phase advance per cell: (1.307, 0.588)

Natural chromaticity per cell: (-5.54, -1.31)

Twiss functions through one super-period of Storage ring

25/42

0 5 10 15 20 25-10

0

10

20

30

40

50

Tw

iss

fun

ctio

ns

[m]

S [m]

x [m]

y [m]

50*x [m]

BAPS sextupoles arrangement

Two groups of chromaticity correction sextupoles (SD/SF) Seven groups of harmonic correction sextupoles (SYF/SZF/SWD/SZD/SXD/SXF/SYD)

SXFSXDSD

SFSDSYF

SZF SZDSWD SYD

26/42

-0.03 -0.02 -0.01 0 0.01 0.02 0.0363.5

63.6

63.7

63.8

63.9

64

64.1

64.2

64.3

64.4

64.5

Qx

p/p

Qx & Qy vs p/p

-0.03 -0.02 -0.01 0 0.01 0.02 0.0328.5

28.6

28.7

28.8

28.9

29

29.1

29.2

29.3

29.4

29.5

Qy

BAPS—Dynamics with no harmonic sextupoles

Large Q/p/p0 ~0.5

Large Q/Jx,y

Small DA for all p/p0

W.P. 64.28, 29.20

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

3

3.5

4

x[mm]

y[m

m]

Dynamic Aperture

dp/p=0%dp/p=+1%dp/p=+2%dp/p=+3%dp/p=-1%dp/p=-2%dp/p=-3%

27/42

-0.03 -0.02 -0.01 0 0.01 0.02 0.0364

64.05

64.1

64.15

64.2

64.25

64.3

64.35

64.4

64.45

64.5

Qx

p/p

Qx & Qy vs p/p

-0.03 -0.02 -0.01 0 0.01 0.02 0.0329

29.05

29.1

29.15

29.2

29.25

29.3

29.35

29.4

29.45

29.5

Qy

BAPS—OPA optimization

OPA code based on Hamilton resonance theroy

W.P. 64.35, 29.20

-10 -8 -6 -4 -2 0 2 4 6 8 100

1

2

3

4

5

6

7

8

x[mm]

y[m

m]

Dynamic Aperture

dp/p=0%dp/p=+1%dp/p=+2%dp/p=+3%dp/p=-1%dp/p=-2%dp/p=-3%

DA still small

Qx nonlinear increase with Jx

Qx with p/p still large

-6 -4 -2 0 2 4 60.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

x (mm)

nux

& n

uy

nuxnuy

28/42

-15 -10 -5 0 5 10 150

2

4

6

8

10

12

x[mm]

y[m

m]

Dynamic Aperture

dp/p=0%dp/p=+1%dp/p=+2%dp/p=+3%dp/p=-1%dp/p=-2%dp/p=-3%

-0.03 -0.02 -0.01 0 0.01 0.02 0.0364.1

64.15

64.2

64.25

64.3

64.35

64.4

Qx

p/p

Qx & Qy vs p/p

-0.03 -0.02 -0.01 0 0.01 0.02 0.0329.1

29.15

29.2

29.25

29.3

29.35

29.4

Qy

BAPS—MOGA optimization

Multi-objective genetic algorithms

Seven Variables: harmonic sextupole strength

Aim value: larger DA size and smaller tune variation over p/p~(-0.03, 0.03)

Initial random seeds (104)

Calculate aim value

Choose the best ones Selection, crossing, mutation

New random seeds

The final best result(s)

New generation

W.P. 64.28, 29.20

D. Robin et al, PRSTAB, 11, 024002 (2008)

29/42

BAPS—three approaches comparison

No harmonic sextupoles With OPA code With MOGA

Q vs P

(-

0.03~0.03)

Qx max 0.56 0.25 0.23

Qy max 0.10 0.03 0.012

DA for

p/p =

X max 1.8 mm 5 mm 10.5 mm

Y max 3.2 mm 4 mm 7.5 mm

DA for

p/p = 3%

X max 2 mm 3.8 mm 6.5 mm

Y max 2.4 mm 3 mm 6.2 mm

DA for

p/p = 3%

X max 2.4 mm 7 mm 10 mm

Y max 3.1 mm 6 mm 9 mm

Can we find a lattice with small emittance and relatively small natural chromaticity at the same time?

How can we make sure the lattice we find is the best one?

30/42

BAPS—linear optics scanning with MOGA

Limitions:

Structure: 48 standard DBA cells

Straight section length: not small than 6.4m

Circumference: not larger than 1180m

3.2m>x/y at the center of the straight section>1.5m

Twelve Variables:

Drift lengths, quadrupole gradients

Attention:

Phase advance per cell

Natural chromaticity per cell

Circumference

Emittance

Twiss function at the middle of the straight section and the entrance of the dipole

DM(3.2m)+D0

QM1 QM4QM2 QM3 QM1QM2QM3

B

QM4QM5QM5

B

D0 +DM(3.2m)

D1 D2 D3 D4 D5 D6 D5 D4 D3 D2 D1

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Stable solution-generation I

32/42

Stable solution-generation II

33/42

Stable solution-generation III

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Stable solution-generation IV

Most stable solutions exist in range of Qx~ (0, 1.5) per cell. With the Qx per cell changes from 0 to 1.5, the available low emittance decreases, and the available small natural chromaticity increases, as we expected.

35/42

Stable solution-detail information

Emittance 1.0nm, Qx, min ~0.9, x ~ (0.7, 2.0) per cell

36/42

Stable solutions fulfilling achromatic conditions

Achromatic conditions:

Dx=Dx’=0 at the entrance of dipole (numerically smaller than 0.01)

Solutions fulfilling achomatic condition exist only in Qx ~(0.9,1.5)

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3.5

4

4.5

5

5.5

6

6.5

7

7.5

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

x (m)

x

x & x at the entrance of dipole

Dx, |Dx’|<0.01

Stable solutions fulfilling achromatic conditions II

3.5

4

4.5

5

5.5

6

6.5

7

7.5

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

4

x (m)

x

x &

x at the entrance of dipole

Dx, |Dx’|<0.01

4.5

5

5.5

6

6.5

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

x & x at the entrance of dipole

x (m)

x

Dx,|Dx’|<

Dx, |Dx’|<10

For =1.2, MOGA vs. theoretical calculation

22 2 2 2

0 0 0 0 0200

4 5 2 32 0 0 0

2 2 20

1( (1 ) 4 4 )

4

1 (1 )( )

4 20 3

L

q

q

sC s s ds

L

L L LC

L

When achromatic conditions are satisfied for DBA cells,

L: dipole length; : bending radius; 0, 0: beta functions at the entrance of the dipole; : energy factor; Cq=3.8319×10-13m

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Stable solutions fulfilling achromatic conditions III

If <2nm, Qx>1.2, x>2.65 per cell

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Dx=Dx’=0 at the entrance of dipole (numerically smaller than 10-4)

Stable solutions fulfilling achromatic conditions IV

Emittance vs. x per cell

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2 nm is a turning point. Above 2 nm, emittance changes a lot, the corresponding available small chromaticity changes a little. Below 2 nm, with emttiance decreases a little, the corresponding available small chromaticity increases very quickly.

Dx=Dx’=0 at the entrance of dipole smaller than 10-4。

1100 1110 1120 1130 1140 1150 1160 1170 11801

1.1

1.2

1.3

1.4

1.5

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Circumference (m)

N,x

(nm

)

2.53.03.54.04.55.05.5

Stable solutions fulfilling achromatic conditions V

Emittance vs. circumference for specific x

With 48 achromatic DBA cells, it is hard to reach 1nm with natural chromaticity smaller than 5.5.

To reach ~1.3nm,

the available small x is about 4.5,

the available small circumference is about 1120m.

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Stable solutions fulfilling achromatic conditions VI

Lattice based on MOGA linear optics scanning

Super-period No. is 4.

The circumference is 1186.7m.

The tune per cell is (1.326, 0.628).

The natural chromaticity per cell is (-4.5, -1.63);

The emittance is 1.33nm.

42/42

Half matching + half standard cell

Summary

Frequency map analysis is very helpful in nonlinear optimization.

Super-periodic structural resonances have non-negligible effects on beam dynamics for the third generation light sources.

Synchro-betatron resonance is an important source of dynamic aperture limitation for the BEPCII colliding mode lattice.

FMA together with MOGA show large potential in design and optimization of light source nowadays.

Additional slides

Tune is closed to stopband

Beta function variation rate will be large

Particles with different momentum will “experience” different optics, the

natural chromaticity will be different too

Chromaticity correction Effective range will be small, for a little p/p,

tune may change a lot, cause asymmetry of dynamic aperture (DA)for different p/p.

If too closed to the stopband, the particle amplitude will increase quickly, finally lost

The nonlinear correction effective range may be small, thus possibly bad results

for nonzero momentum DA。

First and second order SSR mechanism

Experimental evidence for SSRs ?

Two best experimental machines:SSRF, does not start FMA experiments till now

SOLEIL, design changed. W.P. from (18.28, 8.38) to (18.2, 10.3)

The W.P. just locates on the SSR resonance

SSR diagram nearby the SOLEIL W.P.

Experimental evidence for SSRs ?

6 8 10x yQ Q

W. P.：（ 18.2 ， 10.3） (18.2 ， 10.305)

The SSR resonance’s effect is obvious, but the resonance is not very strong.

Experimental evidence for SSRs ?

However, the W.P. is moved again to (18.202, 10.317) due to optimization of the injection efficiency.

Courtesy of Nadji, SOLEIL

HOSSR Qx + 6Qy = 80

The tune footprints for on-momentum case of bare lattice cover small area in tune space, do not cross the resonance.

The experiment results do not accord well with the model.

DO NOT HAVE STRONG

EXPERIMENTAL EVIDENCE TILL

NOW!