Post on 08-Jan-2016
description
Double RF system at IUCF
Shaoheng Wang
06/15/04
Contents
1. Introduction of Double RF System
2. Phase modulation Single cavity case Double cavity case
3. Voltage modulation Single cavity case Double cavity case
Introduction: Double RF system
: Harmonic number
: RF peak voltages
: RF phase of syn. particle
: Orbit angle
: Synchrotron tune at zero
amplitude for primary cavity
s
ss
VV
hh
21
21
21
,
,
,
)(
)(
222
11
ss
ss
h
h
h2
V2
1s22 s
Primary cavity
Secondary harmonic
cavityh1
V1
s
s
Synchronous particle
Other particles in the bunch
Particle bunch
Introduction: Why Double RF system
• By reducing the voltage gradient at the bunch position, it will also increase the bunch length. Hence, lower the space charge effect.
• There is an increase in the spread of synchrotron frequencies within the bunch. This spread can help in damping coherent instabilities such as the longitudinal coupled bunch instabilities through an effect known as Landau damping which come from the non-linearity of the voltage along the bunch .
)sin()sin()( 0 ns nnkVV The voltage seen by the beam with a double RF system is
0)(cos0
20
V
V s
The equation of synchrotron motion
HY
s
s
2
0
22
0
ˆ
cos
),(2
00
2 )(1
),( dVV
Y s
Make the integration
To maximize the bunch length, the first derivative of V should vanish at the center of the bunch.
To avoid having a second region of phase stability close by, the second derivative of V must also vanish.
sn
sn
nkn
nnk
sinsin
coscos2
The peak value along a given trajectory
Introduction: Working conditions
Computed distribution in synchrotron tune
t
t
dHT
2
1
2)(
In the phase space, along the H contour,
Period:
Single RF System
Zero Gradient
Qs
density
Reduced voltage slope
Shifting
Nonlinearity Spread
Introduction: Synchrotron tune spread
Introduction: IUCF cooler ring
Injection
Cavity 1
Cavity 2
Introduction: Experiments at IUCF
86.0 factor slip phase
679 frequency n synchrotro
6 ,3 number harmonic
03168.1 frequency revolution
protons MeV 45
21
0
Hzf
hh
MHzf
syn
The bunched beam intensity was found to increase by about a factor of 4 in comparison with that achieved in operating only the primary rf cavity at same rf voltage.
Introduction: Equations of motion
]}sin))([sin()sin{(sin 211
22
1
21 sssss
s
h
h
V
VP
P
0211
1
2 E
eVh
p
phP
s
s
: Normalized momentum deviation
: Synchrotron tune at zero amplitude for primary cavity
Contribution from primary cavity
Contribution from secondary cavity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Introduction: Hamiltonian
]}sin)())(cos([cos
sin)()cos{(cos2
1
21122
111
2
sssss
ssss
s
hhh
r
U
UPH
1
2
1
2
h
hh
V
Vr
: Ratio of the amplitude of RF voltages
: Ratio of harmonic numbers
0 0.2 0.4 0.6 0.8
1 1.2 1.4 1.6
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
phi
shape of the potential
Primary cavity onlyDouble RF system
Flattened potentialPotential shape example
)2cos1(2
cos1 r
U
r = 0 r = 0.5
Flattened potentialPotential shape example
)2cos1(2
cos1 r
U
-0.5
0
0.5
1
1.5
2
-3 -2 -1 0 1 2 3
phi
r > 0.5
At the equilibrium state, the particle distribution follows the shape of the potential
)2cos1(2
cos12
as simplified is hamiltonia the,0 ,2When
2
21
rPH
h
ss
ss
Synchrotron tune: Hamiltonian
)2sin(sin
rH
P
PP
H
s
s
02sinsin2 rs
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-4 -3 -2 -1 0 1 2 3 4
P
phi
r<0.5
r > 0.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-4 -3 -2 -1 0 1 2 3 4
P
phi
r < 0.5
P
P
phi
Synchrotron tune: action-angle variables
ˆ
ˆ
1 :Action PdJ
1
:n tunesynchrotro
E
JQs
ˆˆ2
ˆ2
''
:becomes variableangle The
')'(),( :function generatingWith
P
dQd
E
P
J
E
J
F
dPJF
s
s
Synchrotron tune: synchrotron tune
ˆ and , 5.0hen w,
11
2
ˆ and , 5.0or
,5.0when ,
12
21221
222
120
40
20
b
lu
u
b
s
s
rkKtt
tr
r
r
kKt
trtr
Q
integral elliptical theof moduli are ,)21(2)21(
)21(1
kind.first theof integral elliptical complete theis )(),2
1arccos(2
ˆ,2
sin2
sinarcsin2,2
tan,2
ˆtan where,
22
240
20
200
1
22,,0
u
lu
b
uub
llu
lu
t
ttk
trtr
trtk
kKr
tt
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-4 -3 -2 -1 0 1 2 3 4
P
phi
r<0.5
r > 0.5
r < 0.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-4 -3 -2 -1 0 1 2 3 4
P
phi
P
P
phi
Synchrotron tune: graph
(deg) ̂
ss
Q/
Synchrotron tune variation w.r.t. phase amplitude for a double RF system.
Synchrotron tune compare with experiments
J.Y. Liu et al, Phys. Rev. E 50, 3349 (1994)
The synchrotron tune spread is maximized at r=0.5 for a given bunch area, which is provided to Landau damping.
JdJdQQ ss /
The effective tune spread is given by:
Synchrotron tuneCompare with one RF cavity system
Synchrotron tune measured as a function of phase amplitude at IUCF.
M. Ellison et al, PRL 70, 591, 1993
Synchrotron tune: Synchrotron phase space measurement
• Synchrotron phase is measured with phase detectors.– By comparing the bunch arrival time with the RF cavity wave.
• Momentum deviation is measured according to the dispersion relation:
• FFT on to get synchrotron tune
p
pDx xco
n
tinB
lB eeNlTteNtI )(
00),(
Synchrotron tune: Phase shifter
To have a certain phase amplitude, a phase shifter is used.
1. The RF signal is split into two channels, one of them is 90 degrees shifted.
2. Each of these two channels is multiplied with a signal proportional to sine or cosine of intended phase shift.
3. They are combined again. )ˆcos(
)ˆsin()2
cos(
)ˆcos()cos(
)2
cos(
)cos(
thA
thA
thA
thA
thA
Phase modulation Phase Modulation Signal
With this phase modulation, the phase variation will be given by :
)sin( ma
amplitude modulation theis
tune,modulation theis where0
a
mm
d
ad
d
d m ))sin(( 0
Consider a sinusoidal RF phase modulation:
cavity. thearrives particle when phase RF dunperturbe is where 0
Single cavity: Equations of motion
mms
s
p
PaP
UHH
cos2
sin22
22
0
sin
cos
s
mms
P
aP
Hence, the equations of motion are given by :
The corresponding Hamiltonian is :
Perturbation potential
Single cavity expressed with action-angle variables
can be transformed into Action-angle coordinates0H ),( J
In this transformation, old coordinates are expressed as function of ,J
,P
Further more, can be expanded in Fourier harmonics series of
),( JP
Perturbation potential can be expressed as:
)]3cos()3[cos(128
2
)cos()cos(2
2/3
mmm
mmmp
Ja
JaU
Note: since is an odd function of , only odd harmonics exist.
P
When is close to , stationary phase condition exists for a parametric resonance term. All non-resonance terms can be neglected.
sn 12 m
Single cavity: dipole mode
)cos(216
2/12
mss
seff JaJ
JH
When n=0 case, or dipole mode, is considered, the approximate effective Hamiltonian is:
The effective Hamiltonian is dependent. We can go to resonance rotating frame to find the independent Hamiltonian.
Single cavity: resonance rotating frame
IF m )(2
JIm ,
)cos(216
)(~ 2
IaI
IH ssms
In the phase space, the structure of resonant islands can be characterized by fixed points, which satisfy conditions:
0
0
I
With the generating function:
We can realize the transformation:
And get the new Hamiltonian in resonance rotating frame:
Single cavity: Poincare surface of section
)cos(2 mI
)si
n(2
m
I : Outer SFP
: Inner SFP
: UFP
: Seperatrix phase axis crossings
2,1g
g
g
g
c
b
a
])4(16
31[
:when
3/2asc
cm
Single cavity: bifurcation
)cos(2 mI
)si
n(2
m
I
cWhen goes to from below, the fixed points move as arrows showmWhen = , and coincide. This is the bifurcation point, beyond which, only exists
m c bg cgag
cs
ms
3/14/ agb
3/12 4/ ag
3/11 4/ ag
3/14/ aga
3/14/ agc
Single cavity: experiments
)cos(2 mI
)si
n(2
m
I
Double cavities: Equations of motion
mms
s
p
Par
P
UHH
cos)2cos1(2
cos12
2
0
)2sin(sin
cos
rP
aP
s
mms
Hence, the equations of motion are given by :
The corresponding Hamiltonian is :
Perturbation potential
Double cavities: Perturbation analysis
in
nn
inn PeJgeJgP
2
1)( with ,)(
can expressed with action-angle variables of the unperturbed . can be expanded in Fourier series on ,
motion. particle perturb
tocoherently scontribute termresonance the, when Obviously,
n.Hamiltonia dunperturbe theofenergy theis )( where
)]cos()[cos()()(
becomesn Hamiltonia the),,( of In terms
nonzero. isn odd with only , offunction oddan is Since
0n
n
JE
nnJgaJEH
J
gP
m
nmnmnm
n
pU
0H P
Double cavities: Parametric resonances
0)(')(
)12,,2,1,0(,
:solved becan points fixed here, From
cos)(n
)(
:isn Hamiltonia new average timeThe
and ,
:become variablesangle-action new The
)(),( :function generating
usingby frame rotating resonance ation to transformMake
m
2
Igan
IQ
nlln
nIgaIIEH
nnJI
nnIIW
nmm
s
nm
nm
nm
Double cavities: numerical simulation (1)
)2sin(sin2
)2cos(22
111
1
nnsnn
mmnnn
r
na
Simulations are based on the difference equations:
with:
4108
5.2
s
oa
Double cavities: numerical simulation (2)
5.0 and 5.0th section wi of surfaces Poincare rs
m
Double cavities: bifurcation
5.0 and 3 toclose th section wi of surfaces Poincare rs
m
30.2s
m
32.2s
m
35.2s
m
Double cavities: wave structure
Figure (a) shows two beamlets obtained about 15 ms after the phase modulation was turned on, and Fig (b) shows the final beam profile captured after 25 ms, showing a wave structure. The beam profiles were extended from a half length of about 10 ns to 50 ns without beam loss.
Voltage modulation: Single cavity
111
1
sinsin12
2
nnmsnn
nsnn
0sinsin12 ms With the dot corresponds to the time derivative wrt θ.
The equation of motion can be derived from the Hamiltonian:
cos1sin
cos12
1
1
20
10
ms
ss
H
H
HHH
Unperturbed Hamiltonian:
Perturbation:
Action of the Unperturbed Hamiltonian:
dJ2
1
2
0 256
1
16
11 JJJJEH s
Synchrotron tune:
256
3
81
2
2JJ
EKJ
EJQ s
s
ss
Complete elliptic integral of the first kind
Single cavity: Action-angle variable
ˆ
',', dJJGGenerating function:
n
nmnmns nnJgH sinsin2
11 0
choose
deJg inn cos1
2
1is zero except for n=even with *
nn gg
RF voltage modulation contributes only even-order harmonics to the perturbation H1
J
JG
,
Single cavity: rotating frameJ
nnnF m ~
22
Generating function:
,~,
~~~cos~~~~
JHnJgJn
JEH nsm
~cos~~~~
nJgJn
JEH nsm
Including both ±n terms, the resonance Hamiltonian:
The time-averaged Hamiltonian:
~2cos4
~~
16
~
2
~ 2 JJJH ssm
s
2cos482
2sin2
ssms
s
J
JJ
When n=2:
For simplicity, the tilde notation is dropped:
22 if0
22 if2
218
ssm
ssm
s
m
SFPJ
22
22 if0
22 if2
218
ssm
ss
ssm
s
m
UFPJ
Fixed points: and 0with
2
3 and
2with
Single cavity: Fixed point
Single cavity: experiments
• The bunch was kicked longitudinally, all particles then were captured and dampened into one attractor, see fig.
• At the same time, rf voltage modulation was applied.
• A total of 16000 points at intervals of 50 revolutions, i.e. 800000 orbital revolutions, was recorded.
• Poincare surfaces in resonance processing frame, see fig, the particle damping paths and the island structure were clearly observed
Single cavity: experiments
Single cavity: beam profile
The profile of the beam in a single pass.
Double cavity: Hamiltonian
2sinsinsin1 rms
s
0
whereV
Vm
msm
mss
E
rH
cos2
1cos1
2cos12
cos1sin12
1
2
2
l
lnlin
nn
inn JgJgeJGeJG
22
2
1)( with ,)(
Gn is not zero only for even harmonics.
0
coscos2
1cos
nnmnmnsm nnJGJEJEH
Inn
IW nm
,2
nnIJ nm and ,
,,cos
2
1IHnGI
nEΗ ns
m
Time dependent part
Double cavity: Fixed point
2cos
2cos
2
ˆsin
2sin
21
4 0
02
2
2
n
K
un
K
nKun
q
qKQG
n
n
s
sn
KKeqk
,kk
kFuK
'2
2
0
and , 2
ˆsin1
2
1 modulus with kindfirst
theoffunction elliptical incomplete theand complete the
ly respective are 21
arccos and where
0'2
1
,,3,2,1,0 ,
:aren Hamiltonia averaged time theof points fixed The
FPnsm
FPs
FP
IGn
IQ
lln
Double cavity: numerical simulation
nnnmsnn
nsnn
rn
111
1
2sinsin2sin12
2
:equations differencefor performed weressimulation numerical
Double cavity: Bifurcation point
sm 4.1sm 3.1 sm 55.1
Bifurcation point
Conclusion
• The benefits of double RF system– Longer bunch, less space charge effect– Landau damping from synchrotron tune
spread
• Resonance structure when system is under phase and voltage modulation
reference
A. Hoffman,
SY Lee,
M. Ellison
JY Liu,
D. Li,
H. Huang
…