Orbits, Optics and Beam Dynamics in PEP-II

Yunhai Cai

Beam Physics DepartmentSLAC

March 6, 2007ILC damping ring meeting at Frascati, Italy

Normalized Coordinates

)2sin()2cos()2sin()2sin()2sin()2cos(

)2cos()2sin()2sin()2cos(

01

,10

1

1

ARAM

R

AA

To normal coordinate

Back to physical coordinate

R-Matrix in Terms of Lattice Functions

Mab aa

b

Rab

Aa-1

Ab

1 aabbab ARAMIn the “normalized ring”, the R-matrix is simplya rotation with the phase advance.

Closed Orbit in the “Normalized Ring”

1)cot()cot(1

21)(

)(,,,

)(,

1

111

1111

1

RI

RIxxxRAxxA

xAAxMAAA

MIxxxM

Trivial to generalized to higher dimension since (I-R) is block diagonal.

Closed Orbits

)]cos()[sin(1)cos(

)sin(2

)sin()cos(

)sin(2

1)cot(

2

abb

abb

abba

ab

aba

a

Closed orbit at position (b) in the“normalized ring”

Closed orbit at position (a) of the kick in the “normalized ring”

Closed orbit at position (b) in the physical ring

Perturbation of a Closed Orbit Due to a Kick

11)( aabbb ARIRAx

11)( aaa ARIAx

Closed orbit at location b is given:

Here the kick is at location a. In particular,

One can check directly,

aaa xxM

Definition of Coupling Parameters

,0

0

2

1

gIwwgI

uu

gIwwgI

M

Given one-turn matrix M, we can decouple it with a symplectic transformation:

where u1 and u2 can be parameterized as if no coupling case and w is asymplectic matrix:

.

,sincossin

sinsincos

,sincossin

sinsincos

2221

1211

22222

222222

11111

111111

wwww

w

u

u

There are ten independent parameters. Bar notes symplectic conjugate. g2=1-det(w).

Coupled Lattices

JJAAwwwwg

ggwww

gwww

wwwgg

wwwg

A

T

121122211

22

2

1

22

1

121122

21

12

1

111112

2

11

2

221211

11

1

2

12

2

2222121

,)(1

0

0

Presentation of A is far from unique!!! There are eight independentparameters.

To the “Normalized Ring”

22

2

2

222212

2

221211

22

12

2

11

2

111112

1

1211221

1

1

1

12

1

22

1

1

0

0

ggwwww

gww

wwwwgg

wwg

A

Horizontal Kick:

)]sin()cos()[(sin2

)]sin()cos()[(sin2

)}cos()])(([

)sin()]()({[sin2

)cos(sin2

221222222212

22

2

111211111112

11

1

222222122222121212

222222121222221212

222

111

11

abaabaaaa

a

bb

abbabbbbb

b

aa

b

abbbbbaaaaba

abbbbbaaaaab

ba

abba

bab

wwwg

wwwg

y

wwwwww

wwwwww

ggx

Comparison to Simulation in the LER of PEP-II

Difference Between the Numerical and Analytical Solutions

Coherently Excited Betatron Motion and Turn-by-Turn data

• Beam excited at eigen frequency in x or y

• Equilibrium reached due to radiation damping or decoherence

• Take turn-by-turn reading at all beam position monitors up to 1024 turns

• The phase advances between the beam position monitors can be accurately measured

J. Borer, C. Bovet, A. Burns, and G. Morpurgo, Proc. The 3rd EPAC, p1082 (1992)

In addition, Four Eigen Orbits Extracted Using FFT

• These orthogonal orbits are the Fourier transforms of the turn-by-turnreadings of beam position monitors at the driving frequency. Since the peak in the spectrum can be located accurately, they can be measured precisely as well.

horizontal vertical

real

imaginary mode 1

real

imaginary

mode 2

R-Matrix Elements Derived from Four Orthogonal Orbits

where a and b are indices for the locations of the beam position monitors, Q12 and Q34 are global invariance of the

orbits. For general orbits, the relationship is much more complicated.

34344312122134

34344312122132

34344312122114

34344312122112

/)(/)(

/)(/)(

/)(/)(

/)(/)(

QyyyyQyyyyR

QyxyxQyxyxR

QxyxyQxyxyR

QxxxxQxxxxR

babababaab

babababaab

babababaab

babababaab

BPM Gains and Couplings

ab

ab

ab

ab

ay

by

ayx

by

ay

byx

ayx

byx

axy

by

ax

by

axy

byx

ax

byx

ay

bxy

ayx

bxy

ay

bx

ayx

bx

axy

bxy

ax

bxy

axy

bx

ax

bx

ab

ab

ab

ab

gggggggg

gggggggg

34

32

14

12

34

32

14

12

RRRR

RRRR

where gx, gy are gains and xy, yx are cross-coupling between x and y.

Measured xyg

yxg

yxy

xyx

yx

Beam

Beating correction for the High Energy Ring

measured

prediction

implemented

Coupling Correction in the HER

Before

After

Dispersion Corrections in the HER

Before

After

Source that Generates the Vertical Emittance in the HER

Luminosity for Tilted Gaussian Beams

))(/())((

,))((2/

,)(sin1/

2222222212

222200

2120

bbaababae

bbaafNNL

eLL

Hour-glass effects:

2*

00

,2/

),(2/

aba

bKaeLLF

zy

bh

Beam-Beam Scan at Low Beam Currents

Comparison to the MeasurementMeasured Calculated

(mrad) -10.0 -17.73 -17.73

a (microns) 154 175 139

b (microns) 6.43 4.89 5.62

Lsp (1030cm-2s-

1mA-2)5.40 5.33 5.74

Dynamic beta and emittance and hour-glass effect are not included.

Dynamic beta and emittance and hour-glass effect are included.

Chromatic Optics for the HER• Measured chromatic optics and dynamic aperture in

HER– Excellent agreement between measurements and LEGO

model in the chromatic optics– Improvement of understanding of nonlinear dynamics

including sextupoles

PEP-2 LER Dynamic Aperture Simulation

> 10 aperture atp/p = 0 p/p = 5 = .00355

• Single beam dynamic aperture versus tune and p/p. • Realistic MIA machine model, * = 36 / 0.8 cm.• Tune space near half-integer is limited by resonances, especially 2x – ns , and chromatic tune spread.• Best aperture at tunes .522 < x < .530, y > .574.• Better compensation of the 2nd order chromatic y tune shift is needed.

best aperture 2x-2s

x+y-3s

p/p

y

x2x-2s

2x-s

x+y-4s

Dynamic Aperture Near Half Integer

• There is a dynamic aperture near half integer only after a correction to the paraxial approximation is added into LEGO.

seed 2544 seed: 834

Conclusion• Turn-by-tune data from beam position monitors are very

useful for constructing precision model and improving machine optics.

• Directly minimizing the sources (bending magnets) that generates the vertical (second-mode) emittance could be a very effective method to achieve the smallest emittance in storage rings.

• We find an analytical formula for the change of closed orbit in coupled lattice by a kick. It could be used to understand the coupling in the machine or to speed up the ORM fitting.

• We have used optics models not only to improve the linear optics but also to study the nonlinear beam dynamics in the machine. The study has shown the model has some predictive power as well.

Acknowledgements

• Thanks to my colleagues and collaborators who have contributed to this talk:– John Irwin, Yiton Yan, Yuri Nosochkov– J. Yocky, P. Raimondi