Linear Collider Damping Rings

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16 Νοε 2013 (πριν από 3 χρόνια και 4 μήνες)

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Linear Collider Damping Rings

Andy Wolski

Lawrence Berkeley National Laboratory


USPAS Santa Barbara, June 2003

2

300 m Main Damping Ring

3 Trains of 192 bunches

1.4 ns bunch spacing

231 m

Predamping Ring

2 Trains of 192 bunches

30 m Wiggler

30 m Wiggler

Injection and RF

Circumference
Correction and
Extraction

110 m

Injection
Line

110 m

Transfer
Line

90 m

Extraction
Line

Spin
Rotation

What do they look like?

NLC
Positron
Rings

3

Operating Cycle in NLC/JLC MDRs


Each bunch train is stored for three machine cycles


25 ms or 25,000 turns in NLC


Transverse damping time


4 ms


Horizontal emittance
×
1/50, vertical
×
1/7500

300 m Main Damping Ring

3 Trains of 192 bunches

1.4 ns bunch spacing

30 m Wiggler

30 m Wiggler

Injection and RF

Circumference
Correction and
Extraction

103 m

Injection
Line

160 m

Extraction
Line

Spin
Rotation

4

What do they look like?

TESLA Damping Rings

5

Performance Specifications

NLC MDR

TESLA e
+

Injected
γε

150 µm rad

10 000 µm rad

Extracted Horizontal
γε

3 µm rad

8 µm rad

Extracted Vertical
γε

0.02 µm rad

0.02 µm rad

Injected Energy Spread

1% full width

1% full width

Extracted Energy Spread

0.1% rms

0.13%

Extracted Bunch Length

4 mm

6 mm

Bunch Spacing

1.4 ns

20 ns

Bunches per Train

192

2820

Repetition Rate

120 Hz

5 Hz

6

Radiation Damping…


Longitudinal phase space


Particles perform synchrotron oscillations in RF focusing potential


Higher energy particles radiate energy more quickly in bends


At the equilibrium energy, the revolution period is an integer times the
RF period (the synchrotron principle…)


Transverse phase space


Particles perform betatron oscillations around the closed orbit


Radiation is emitted in a narrow cone centered on the
instantaneous

direction of motion


Energy is restored by the RF cavities
longitudinally


Combined effect of radiation and RF is a loss in transverse momentum


Damping time in all planes is given by:

0
0
0
2
T
U
E


J
7

…and Quantum Excitation


Radiation is emitted in discrete quanta


Number and energy distribution etc. of photons obey

statistical laws


Radiation process can be modeled as a series of “kicks” that
excite longitudinal and transverse oscillations

8

Synchrotron Oscillations




p
t


d
d

















E
U
U
T
T
E
eV
t
RF
s
RF
d
d
1
sin
d
d
0
0
0
0
0
d
d
2
d
d
2
2
2








s
E
t
t


Equilibrium orbit

Dispersive orbit

9

Longitudinal Damping



s
p
RF
RF
s
T
E
eV




cos
0
0
2






s
s
t
p
s
s
s
t
t
t
E
E


















sin
e
ˆ
cos
e
ˆ
2
4
0
0
0
2
2
I
I
T
U
E
E
E
E



J
J














s
k
I
s
I
d
2
1
d
1
1
2
4
2
2




Problem 1

Show that:




s
C
I
C
p
d
1
1
0
1
0



10

Quantum Excitation (Longitudinal)

δ

τ









0
1
1
1
1
1
1
sin
ˆ
sin
ˆ
cos
ˆ
cos
ˆ
E
u
s
p
s
p

























sin
ˆ
2
ˆ
ˆ
0
2
0
2
2
2
1
E
u
E
u



E
s
u
N
C
E
t



2
2
0
2
0
2
ˆ
2
d
1
d
ˆ
d



Including damping:

11

Equilibrium Longitudinal Emittance


We have found that:




From synchrotron radiation theory:






s
I
I
I
E
C
s
u
N
C
E
E
q
d
1
4
d
1
3
3
2
3
2
0
2
2
0



J
Problem 2

Find an expression for the equilibrium energy spread,

and show that:













t
equ
t





e
1
e
,
0
E
s
u
N
C
E
t



2
2
0
2
0
2
ˆ
2
d
1
d
ˆ
d



12


It is often more convenient to describe betatron oscillations
using action
-
angle variables:





The old variables are related to the new ones by:





The equations of motion take the simple form:

Betatron Oscillations: Action
-
Angle Variables



x
x
x
x
x
x
J















tan
2
2
2
2














cos
sin
2
cos
2





J
x
J
x
0
d
d
1
d
d


s
J
s


13

Damping of Vertical Oscillations


Radiation is emitted in a narrow cone (angle ~1/
γ) around
instantaneous direction of motion, so vertical co
-
ordinate and
momentum are not changed by photon emission


RF cavity changes longitudinal momentum, and hence the
vertical direction of motion:




Averaging over all betatron phase angles gives (per turn):




Hence the equation of motion is:


























p
p
y
p
p
p
p
p
p
p
y
y
y
y
y



1
1
1
1
J
E
U
J
0
0



J
T
E
U
t
J
0
0
0
d
d


Problem 3

Show this!

14

Damping of Horizontal Oscillations


When a photon is emitted at a point where there is some
dispersion, the co
-
ordinates with respect to the closed orbit
change:










0
1
0
1
E
u
x
x
E
u
x
x
J
T
E
U
J
T
E
U
I
I
t
J
x
0
0
0
0
0
0
2
4
1
d
d
J














Taking the energy loss to first order and averaging around the
ring, we find after some work:

15

Quantum Excitation of Betatron Motion


Let us now consider the second order effects. It is easy to
show that the change in the action depends to second order on
the photon energy as follows:





Averaging over the photon spectrum and around the ring, and
including the radiation damping gives:

2
2
2
0
2
2
1
η
β
η
αη
γη
E
u
J















H
H
x
x
x
x
q
x
x
x
J
I
I
C
J
s
u
N
C
E
t
J




2
2
2
d
2
1
d
d
2
5
2
2
0
2
0





J
H


s
I
d
3
5

H
16

Summary of Dynamics with Radiation





2
5
2
0
0
0
0
2
0
1
2
3
2
2
2
4
0
0
2
4
2
4
0
0
0
sin
cos
2
2
1
1
2
I
I
C
eV
U
T
E
eV
C
I
I
I
C
I
E
C
U
I
I
I
I
T
U
E
x
q
RF
s
s
p
RF
RF
s
p
s
p
E
q
E
y
x
E
E
y
y
x
x
J
J
J
J
J
J
J
J













































t
t
t
2
equ
2
inj
e
1
e
d
d





2
3
5
1
1
2
4
3
3
2
2
1
2
d
1
d
2
1
d
1
d
1
d









































2
H
H
s
I
x
B
B
k
s
k
I
s
I
s
I
s
I
y
17

The NLC TME Cell

Low dispersion and horizontal
beta function in the dipole

High field in dipole

Sextupoles at high dispersion
points, with separated betas

Vertical focusing in the dipole

Cell length
≈ 5 m

18

H Function in the NLC TME Cell

19

The TESLA TME Cell

Larger dispersion and horizontal
beta function in the dipole

Low field in dipole

Sextupoles at high dispersion points

No vertical focusing in the dipole

Cell length
≈ 15 m

20

NLC and TESLA TME Cells Compared


NLC


Compact cell to keep circumference as short as possible


High dipole field for greater energy loss, reducing wiggler length


Short dipole requires very low values for dispersion and beta function


Gradient in dipole field to improve transverse dynamics


TESLA


Circumference fixed by bunch train and kicker rise/fall time


Long dipole for larger momentum compaction, longer bunch


Optimum lattice functions at center of dipole:





Obtained by minimizing
I
5

for a ring without a wiggler


It is not usually possible to control the dispersion and beta function
independently

15
12
24
15
2
3
2
min
0
0
x
q
C
L
L
J










21

Two Simple Scaling Relationships

Problem 4

Show that for an isomagnetic ring with the lattice functions tuned

for minimum emittance:

3
1
0
cell
6
2
2
0
3
2
0
cell
6
2
2
0
cell
15
12
8
2
15
12
8



























q
e
q
e
C
L
c
em
C
B
C
L
c
em
C
B
N
22

Scaling Relationships Applied to the NLC

1 bunch train

6 bunch trains

γε
0

= 3 μm

L
cell

= 6 m

τ = N
train

1.6 ms

23

Scaling Relationships Applied to the NLC

2 bunch trains

6 bunch trains

γε
0

= 1 μm

L
cell

= 6 m

τ = N
train

1.6 ms

24

Damping Wiggler


A wiggler reduces the damping time by increasing the energy
loss per turn:





Wiggler must be located where nominal dispersion is zero,
otherwise there can be a large increase in the natural emittance


If horizontal beta function is reasonably small, wiggler can
significantly
reduce

the natural emittance (through reduced
damping time)


Drawbacks include possible detrimental effect on beam
dynamics

s
B
E
C
c
e
I
E
C
U
d
2
2
2
2
0
2
2
2
4
0
0







25

Types of Wiggler


A wiggler is simply a periodic array of magnets, such that the
field is approximately sinusoidal


Different technologies are possible:


Electromagnetic


Permanent magnet


Hybrid (permanent magnets driving flux through steel poles)


Choice of technology comes down to cost optimization for
given requirements on field strength and quality


Both TESLA and NLC damping rings have opted for hybrid
technology

26

Modeling the Dynamics in the Wiggler


Magnet design is produced using a standard modeling code


Field representation must be obtained in a form convenient for
fast symplectic tracking



















2
2
2
2
2
,
,
,
,
,
,
sin
sinh
cos
cos
cosh
cos
cos
sinh
sin
z
x
mn
y
z
mn
y
x
mn
y
z
mn
z
z
mn
y
x
mn
y
z
mn
y
x
mn
y
x
mn
x
k
n
k
m
k
z
nk
y
k
x
mk
k
nk
c
B
z
nk
y
k
x
mk
c
B
z
nk
y
k
x
mk
k
mk
c
B










27

Fitting the Wiggler Field

28

Tracking Through the Field


Using an appropriate field representation (that satisfies
Maxwell’s equations), one can construct a
symplectic
integrator
:




M

is an explicit function of the phase
-
space co
-
ordinates, and
satisfies the symplectic condition (so the dynamics obey
Hamilton’s Equations):



old
new
x
m
x




















0
1
1
0
T
S
x
m
M
S
M
S
M
j
i
ij
29

Dynamics in the NLC Wiggler

Horizontal Kicks and Phase Space

Vertical Kicks and Phase Space

30

Chromaticity


Chromaticity is the tune variation with energy


Quadrupole focusing strength gets smaller as particle energy
increases


It can easily be shown that:






Since beta functions peak at the focusing quadrupoles in the
appropriate plane, the natural chromaticity is always negative


Chromaticity is connected to beam instabilities


particles with large energy deviation cross resonance lines


some collective effects (e.g. head
-
tail instability) are sensitive to the
chromaticity












s
k
s
k
y
y
y
x
x
x
d
4
1
d
4
1
1
1










31

sextupole

k
1
=
x k
2

Correcting Chromaticity with Sextupoles

32

Dynamics with Sextupoles


Sextupoles can be used to correct chromatic aberrations…





…but introduce geometric aberrations and coupling:




It is important to keep the required strengths to a minimum by
designing the linear lattice functions for effective sextupole
location

x
Y
l
k
y
x
l
k
x










2
2
2
2
1






s
k
k
s
k
-
η
k
y
y
y
x
x
x
d
4
1
d
4
1
2
x
1
2
x
1









33

Dynamic Aperture


Geometric aberrations from sextupoles (and other sources)
distort the transverse phase space, and limit the amplitude
range of stable betatron oscillations

Horizontal phase space of NLC TME cell

Vertical phase space of NLC TME cell

34

Transverse and Longitudinal Aperture


Damping rings require a “large” dynamic aperture


Injected beam power ~ 50 kW average, and radiation load from any
significant injection losses will destroy the ring


Nonlinear distortion of the phase space may lead to transient emittance
growth from inability properly to match injected beam to the ring


For NLC Main Damping Rings, the target dynamic aperture is 15 times
the injected rms beam size


We also need a large momentum acceptance


Injected beam has a large energy spread


Particles may be lost from insufficient physical aperture in dispersive
regions, or through poor off
-
momentum dynamics


Particles within a bunch can scatter off each other, leading to a
significant change in energy deviation (Touschek Effect)


It is important to perform tracking studies with full dynamic
model and physical apertures

35

NLC Main Damping Ring Dynamic Aperture

Dynamic Aperture On
-
Momentum

δ= +0.005

δ=
-
0.005

15
×

Injected Beam Size

36

Longitudinal Acceptance


The longitudinal acceptance has three major limitations:


Poor off
-
momentum dynamics


Physical aperture in dispersive regions


RF bucket height


Off
-
momentum dynamics can be difficult to quantify


see previous slides


Physical aperture can be a significant limitation


1% momentum deviation in 1 m dispersion is a 1 cm orbit offset


RF bucket height comes from non
-
linearity of the longitudinal
focusing


Previous study of longitudinal dynamics assumed a linear slope of RF
voltage around the synchronous phase


Valid for small oscillations with synchronous phase close to zero
-
crossing

37

RF Bucket Height


The “proper” equations of longitudinal motion (without
damping) are:






These may be derived from the Hamiltonian:







s
RF
s
RF
p
T
E
eV
t
t








sin
sin
d
d
d
d
0
0




















RF
s
RF
s
RF
RF
p
T
E
eV
-
H
sin
cos
0
0
2
2
1















H
t
H
t
d
d
d
d
38

Longitudinal Phase Space


The Hamiltonian is a constant of the motion, which allows us
to draw a phase
-
space portrait

S

Stable fixed point

Unstable fixed point

Separatrix

S

V
RF























s
s
s
p
RF
RF
RF
T
E
eV







sin
2
cos
4
0
0
2

RF

39

Alignment Issues


The final luminosity of the collider is critically dependent on
the vertical emittance extracted from the damping rings


In a perfectly flat lattice, the lower limit on the vertical
emittance comes from the opening angle of the radiation


Gives about 10% of the specified values for NLC and TESLA


Magnet misalignments give the dominant contribution to the
vertical emittance


Quadrupole vertical misalignments


Vertical dispersion


Vertical beam offset in sextupoles


Quadrupole rotations and sextupole vertical misalignments


Couple horizontal dispersion into the vertical plane


Couple horizontal betatron oscillations into the vertical plane

40

Betatron Coupling


In a damping ring, the dominant sources of betatron coupling
are skew quadrupole fields


Normal quadrupoles have some “roll” about the beam axis


Sextupoles have some vertical offset with respect to the closed orbit


Particles with a horizontal offset get a vertical kick

Particle on

closed orbit

Particle with

horizontal

amplitude

Vertical kick

depends on

horizontal

amplitude

41

Effects of Betatron Coupling


In action
-
angle variables, the “averaged Hamiltonian” for a
coupled storage ring can be written:





The equations of motion are:



y
x
y
x
n
y
y
x
x
J
J
J
J
H
C










cos
~
2
0



































y
x
y
x
y
y
y
x
y
x
y
y
x
x
y
x
x
y
x
y
x
x
J
J
C
s
J
J
s
J
J
J
C
s
J
J
s
J


















cos
2
~
2
d
d
sin
~
d
d
cos
2
~
2
d
d
sin
~
d
d
0
0
42

Solutions to the Coupled Hamiltonian


The sum of the horizontal and vertical actions is conserved:





There are fixed points at:








With radiation, the actions will damp to the fixed points

0
d
d
0
0



s
J
J
J
J
y
x






























2
2
0
2
2
0
~
1
2
1
~
1
2
1


J
J
J
J
y
x
y
x





43

The Difference Coupling Resonance


The equilibrium emittance ratio is given by:





The measured tunes are given by:

2
2
2
4
~
~





x
y
J
J
2
2
2
1
2
1
0
~
d
d
2








s
C
y
x





44

What is the Coupling Strength?


We add up all the skew fields around the ring with an
appropriate phase factor:





k
s

is the skew quadrupole
k
-
value.


For a rotated quadrupole or vertically misaligned sextupole,
the equivalent skew fields are given by:




s
k
C
s
s
y
x
n
n
d
e
2
1
~
0
0
i













0
2
C
s
n
s
y
x
y
x
n














2
sin
1
k
k
s

y
k
k
s


2
45

Vertical Dispersion


In an electron storage ring, the vertical dispersion is typically
dominated by betatron coupling


Emittance ratios of 1% are typical


For very low values of the vertical emittance, vertical
dispersion starts to make a significant contribution


Vertical dispersion is generated by:


Vertical steering


vertically misaligned quadrupoles


Coupling of horizontal dispersion into the vertical plane


quadrupole rotations


vertical sextupole misalignments

46

Vertical Steering: Closed Orbit Distortion


A quadrupole misalignment can be represented by a kick that
leads to a “cusp” in the closed orbit





We can write a condition for the closed orbit in the presence of
the kick:




We can solve to find the distortion resulting from many kicks:



























x
x
x
x
x
x
x
x
x
x










sin
cos
sin
sin
sin
cos
M






















0
0
0
0
y
y
y
y
M
















s
s
s
s
s
s
s
y
y
y
y
y
y
y
d

cos
sin
2
1
1
1











47


The vertical dispersion obeys the same equation of motion as
the vertical orbit, but with a modified driving term:





We can immediately write down the vertical dispersion arising
from a set of steering errors:




Including the effect of dispersion coupling:

Vertical Steering: Vertical Dispersion










1
1
1
y
k
y



1
1
1






y
k
k
y
y
differentiate wrt
















s
s
s
y
k
s
s
s
y
y
y
y
y
y
y
d

cos
1
sin
2
1
1
1
1






































s
s
s
k
y
k
k
s
s
s
y
y
y
x
s
x
y
y
y
y
d

cos
1
sin
2
1
2
1
1
1
























48

Effects of Uncorrelated Alignment Errors


Closed orbit distortion from quadrupole misalignments:





Vertical dispersion from quadrupole rotation and sextupole
misalignment:





Vertical emittance generated by vertical dispersion:







2
1
2
2
2
sin
8
l
k
Y
y
y
y
q
y










2
1
2
2
2
sin
2
x
y
y
q
y
y
l
k











2
2
2
2
2
sin
8
x
y
y
s
y
y
l
k
Y





2
2





y
2
y
J
J
y
E
y

49

Examples of Alignment Sensitivities


Note:

Sensitivity values give the random misalignments that will
generate a specified vertical emittance. In practice, coupling
correction schemes mean that significantly larger
misalignments can be tolerated.

APS

SLS

KEK
-
ATF

ALS

NLC MDR

TESLA DR

Energy [GeV]

7

2.4

1.3

1.9

1.98

5

Circumference [m]

1000

288

140

200

300

17,000

γε
x

[µm]

34

23

2.8

24

3

8

γε
y

[nm]

140

70

28

20

19

14

Sextupole vertical [µm]

74

71

87

30

53

11

Quadrupole roll [µrad]

240

374

1475

200

511

38

Quadrupole jitter [nm]

280

230

320

230

264

76

50

Collective Effects


Issues of damping, acceptance, coupling are all
single particle

effects
-

they are independent of the beam current


Particles in a storage ring interact with each other (directly or
via some intermediary e.g. the vacuum chamber)


A wide variety of collective effects limit the achievable beam
quality, depending on the bunch charge or total current


The consequences of collective effects are


Phase space distortion and/or emittance growth


Particle loss


Damping rings have high bunch charges, moderate energies
and small emittance


Vulnerable to a wide range of collective effects


Too wide a subject to enter into here!