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