accelerator physics topic vii coupled bunch effects
DESCRIPTION
Accelerator Physics Topic VII Coupled Bunch Effects. Joseph Bisognano Engineering Physics & Synchrotron Radiation Center University of Wisconsin-Madison. Coupled Bunch Instabilities. We have discussed instabilities internal to a single bunch of charged particles - PowerPoint PPT PresentationTRANSCRIPT
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
1
UW Spring 2008
Accelerator Physics
Accelerator PhysicsTopic VII
Coupled Bunch Effects
Joseph Bisognano
Engineering Physics &
Synchrotron Radiation Center
University of Wisconsin-Madison
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
2
UW Spring 2008
Accelerator Physics
Coupled Bunch Instabilities• We have discussed instabilities internal to a single bunch of charged
particles• Typically in a storage ring or linear accelerator there are trains (finite or
cw) of bunches separated by nanoseconds to maybe milliseconds• Say we have a resonant structure at 300 MHz, with an angular frequency
of 2(300) 2 GHz• If it has a Q of 20,000 (typical of Cu), its fields ring for 20,000/2 GHz=10
microsecond; if the Q were 2 109 more typical of superconducting RF, the ringing would last a full second
• So a sequence of bunches can talk to each other through resonant structures
• Whereas low Q impedances have a large bandwidth and can “see” the peak current, these high Q structures have a narrow bandwith and only see the average current.
• In other words, broadband impedances generate peak current limitations in accelerators, narrowband impedances generate average current limitations
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
3
UW Spring 2008
Accelerator Physics
Bunch Spectrum
ondistributi of smoothness
on dependent rolloff withoff, rolls I
lengthRn for butII n,
smallfor bunch, length finite a For
tnIeI
eQfnTtQtI
Then function.- a be bunch the Let ring. storage
a in gcirculatin bunch singlea Consider
n
n
nn
tin
n n
tin
/2,2
)(cos2
)()(
0
01
00
00
0
0
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
4
UW Spring 2008
Accelerator Physics
Robinson Instability
•
spectrumcurrent the in
nsoscillatio on synchrotrinclude wenow But
seenveyou' loading beam just is This
Q
withQ
tRIV
resonance, lfundamenta the excites current this If
II
frequency, RF than ly shorterconsiderab be must
bunch sincel,fundamentacavity RF at particle In
r
r
r
sc
n
tan;
21
)cos(
2
0
00
0
Following A. Hoffman, CERN77-13
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
5
UW Spring 2008
Accelerator Physics
Robinson/cont.
))sin(()cos(2
)]cos)(sin()[cos(2
)sin();cos(
)cos(2
'
00000
0000
00
00
tItI
tttII
nsoscillatio amplitude smallFor
tt
with tII
systemRF 1h an have weassume sLet
nsoscillatio on synchrotrundergoing will wholea as
bunch the offset, an given is centroid bunch If
s
sb
sss
b
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
6
UW Spring 2008
Accelerator Physics
Robinson/cont.
2
0
2
000
02
00
02
00
0
)2(1
)sin((cos
)2(1
)sin((cos[
2
)2(1
))sin((
)2(12
))sin((
)cos(
r
ss
r
ssT
r
ss
r
ss
0
Q
tR
Q
tRIV
yields term phase in The
yinstabilit generate can see, shall weas voltage,
resistive phase, in the shifts;frequency causes
only impedance the of part reactive The
Q
tRIV
Q
tRIV
voltages other generate however, terms, sidebandThe
loading beam just is term t2I the Now
R+ R-
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
7
UW Spring 2008
Accelerator Physics
Robinson/cont.
])[(2
sin
]sin)[(2
)()()()(ˆ
0
0
00
00
s
T
ss
sT
tin
RRI
V
and t But
tRRI
V
is problems our for term varying slowThe
afterall sampledaliased, gets Voltage
etVnTttVtV
seesbunch the ,revolution
each sampleV(t) voltage localized a Given
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
8
UW Spring 2008
Accelerator Physics
Robinson/cont.
0cos
)(2
)(2
cos(2
2
0
0
20
00
0
s
s
s
s
s
VRRI
Eh
dtd
and
RRI
Ve
dtd
satisfies Energy
Damping or antidamping
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
9
UW Spring 2008
Accelerator Physics
Robinson Conclusions
0cos
0cos
cos)(
4
/2cos
/1/1;cos
0
0
20
0
22
0
s
s
s
s
ss
ts
t
0, ,transition boveA
0, ,transition Below
VRRI
eEVh
te
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
10
UW Spring 2008
Accelerator Physics
Robinson Stability Condition
r0 r 0
Above transition
Below transition
- + - +
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
11
UW Spring 2008
Accelerator Physics
Coupled Bunch Instabilities
yiyix
N
m
tih
n
n
N
mtin
b
N
m n
thN
mtin
b
N
m n
tin
b
N
m nmb
exJe using
eJeItI
eItI
Let;eIntItI
raccelerato circular a in bunches N Consider
msm
msm
m
)(
])([)(
)(
0)2)(()(
sin
1
)()2
(
1
))sin(2
(
01
))((
10
00
00
0
phase definition change
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
12
UW Spring 2008
Accelerator Physics
Coupled Bunch/cont.
spacingequal for integerN since
yinstabilit Robinson multibunch a to scorrespond This
same the ifvely constructi add signals
bunch all and m2mn/N2 N,n When-
spectrumGeneral
before as n at I current
a have we1N h,n 1,μ For
rf
m
s0b
)(
.
0
0
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
13
UW Spring 2008
Accelerator Physics
General Phase Relationship
phase in m(- 4n
m(- 3n
m(- 2n
m(- 1n
bunches 4 for E.g.,
)mod(2 mN
n
or ,coherently sumterms M If
n line for Nmn
phase overall 1; at Look
4
4
4
4
m
m
mm
)
)
)
)
2
,2
/2
8
6
4
2
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
14
UW Spring 2008
Accelerator Physics
Normal Modes N=4
,4,......,8,-4,0n
1,......,-7,-3,n
2,......,-6,-1,n
3,......,-5,-1,n
-4,......,8,4,0,n
,......,7,3,-1n
,......,6,2,-2n
,......,5,1,-3n
4
3
2
1
1
4
3
2
1
1
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
15
UW Spring 2008
Accelerator Physics
Spectrum/cont.
-4 -3 -2 -1 0 1 2 3 4
4 4 3 1 2 2 1 3 4 4 3 1 2 2 1 3 4 4
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
16
UW Spring 2008
Accelerator Physics
Growth Rates
)(
2
0
0
)(
/4
);)(()
m
s
bs
s-p
m
ZeE
NiI
1-N0,...,m mpNZm(pNZ
to alproportion term aby replaced RR have we
Robinson, for did weas analysis similara Following
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
17
UW Spring 2008
Accelerator Physics
Fixes
mode particular a to coupling lessen
to character spectralchange :ionconfigurat Bunch
worksabove softgrowth/shi lower
:damping mode order Higher
work)factories-B (makes feedback Active
spread".synch.freqrate/shift growth
:dampingLandau
damping rate growth
:damping radiation on Synchrotr
"
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
18
UW Spring 2008
Accelerator Physics
Mode Coupling
yinstabilit yielding meet, can
modes quadrupole and dipole the of sfrequencie The
potential the
flatteningby bunch the lengthen to used iscavity
harmonic higher a whenlimiting become can This
modes 1 include to need weI.e.,
developing coupling-mode ofy possibilit the
is there ies,instabilit bunch single withAs
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
19
UW Spring 2008
Accelerator Physics
Mode Coupling at SRC
0 2 4 6frequency (kHz)
sign
al (
dB)
-110
-100
-90
-80
-70
-60
-50
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
2
3
4
5
freq
uenc
y (k
Hz)
analytic model experiment simulation
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
20
UW Spring 2008
Accelerator Physics
Transverse Phenomena
yinstabilit tail-head cause can which
ty chromatici from mode in shiftis feature New
coherence for conspire
phases make weif spectrummode a have weAgain
iittni
hhnJaItI
nt
th
taItI
tI current Dipole
jmjmjms
jmjmN
m n
particlesn
jdipolebD
jm
jmjms
jmN
m n
particlesn
jdipolebD
D
bunch
bunch
,
)])()(exp[
)))(()(
)2)((
))sin((cos)(
)(
,,,0
,2
,
1
)(
1
,
,,
,2
01
)(
1
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
21
UW Spring 2008
Accelerator Physics
Transverse Coupling
offset rolloff1/ sonecessary ionhybridizat
withterm, leading r in Linear
term leading r in Constant
ekrJ as suchfunctions
Bessel of sumsare solutionsequation MaxwellSince
kVki
qcp
Eic
qF
c)(v Theorem Wenzel-Panofsky Recall
2
i
n
nacctzt
1
0
,)(
;
22
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
22
UW Spring 2008
Accelerator Physics
Deflecting Modes
:pattern field withcan dipole
but deflect, tdon' modes 0 So
1
Particle on axis doesn’t see Ez , doesn’t deposit energy
Particle off axis can excite mode through Ez
But deflection is constant through derivative of Ez
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
23
UW Spring 2008
Accelerator Physics
Resonant Wakefield
s
s
psb
s
s
twiss2
Q
rexciteexcite
test
ZeE
IZic
spNZZ
xRE
IZiexx
simplyhave
webunch, point"" a of motion rigid For
klZ Zring Storage
erqQlcZ
qcp
r
/4
2
2/
sin2
2
0
2/
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
24
UW Spring 2008
Accelerator Physics
Beam Breakup in Linear Accelerators• In a linac there the higher order cavity modes produce the same
basic resonant self-interaction, both longitudinal and transverse• For relativistic linacs, the longitudinal motion is more “frozen” than
in a storage ring, which has bending. So transverse effects are often the limiting factor in linacs
• For transverse effects, the primary difference in the dynamics is number of times the same bunch sees a given cavity HOMs– Straight linac: once, amplification– Recirculated linac: several times, instability with finite threshold– Storage ring: infinite times, zero threshold unless some form of
damping present• In linacs, these effects are call Beam Breakup
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
25
UW Spring 2008
Accelerator Physics
Regenerative Beam Breakup• Basic mechanism: a train of bunches excites a transverse deflecting mode of a
single cavity
• Feedback loop
– Say, HOM has small excitation
– Even a bunch perfectly aligned on axis will receive a transverse kick
– If energy is low and structure long, a significant deflection will occur while the bunch is in the cavity
– The offset bunch is now in a region of longitudinal electric field and can deposit energy into mode
– Go to next bunch
– We have a feedback loop that can go unstable unless the cavity losses (more with lower Q) exceed the gain of the loop
– An honest instability
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
26
UW Spring 2008
Accelerator Physics
Regenerative Beam Breakup
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
27
UW Spring 2008
Accelerator Physics
Threshold Condition
lRecE
I Il/cQ with
yields whichequal, whenthreshold out,energy is Pl/c
inenergy gives Qcm
elPRPR
clPR
lmcet
V
trPRcl
E PRlm
ettr
PRlm
emF
dtdv
PRle
F
Recp
Power
klZR rE withHOM
a withucturecavity str singlea Consider
th
z
z
/6
,623
)(;2
)(
;
)(
/;
3
3
2
2
2
2
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
28
UW Spring 2008
Accelerator Physics
Cumulative BBU Amplification
cavity first in offset be beam let and cavities, of series
a through passing beam bunched a Consider
1 2 3 4 5
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
29
UW Spring 2008
Accelerator Physics
Cumulative BBU/cont.• Cavity 1: Bunch will coherently excite cavity, later
bunches will receive transverse kick• Cavity 2: Bunch will enter cavity 2 with an extra offset;
cavity 2 experiences an enhanced excitation• Cavity N: DITTO• Overall, initial offset causes growing excitation of
subsequent cavities which can increase offset downstream: Amplification
• Since there is no closure of loop, there is no instability as such
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
30
UW Spring 2008
Accelerator Physics
Cumulative Beam Breakup
• Typically bunching frequency and transverse HOM frequency are not harmonically related
• So, there can be a large transient, but the equilibrium excitation can be rather small. For a pulsed linac, however, the transient can cause beam loss, limiting currents to ~100 mA
• For CW operation with equally spaced bunches, the excitation settles down to a DC value that can be steered away
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
31
UW Spring 2008
Accelerator Physics
Multipass Beam Breakup
• A “new” feature of SRF linacs is the possibility of recirculation, and even energy recovery
• SRC allows CW operation and the beam can pass through the linac several times
• The “cumulative” beam breakup amplifier now has its feedback loop closed and at high enough gain there can be instability
• Limited the first generation of SRF linaces to 10 microamps average currents when HOM Q’s were in the 10,000,000 range
• In some ways it’s a combination of cumulative and regenerative BBU
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
32
UW Spring 2008
Accelerator Physics
Multipass BBU Mechanism
• Displaced bunch excites a HOM
• Following bunches deflected
• Recirculation optics transforms kick into a displacement
• Displaced bunch further excites HOM in same cavity
• Again threshold occurs when excitation rate exceeds damping rate
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
33
UW Spring 2008
Accelerator Physics
Beam Breakup Mechanism
Initial noise excitation of cavity mode kicks particle bunch
On subsequent pass,bunch enters off axis and coherently excitescavity mode
cavity
Beam on pass n+1
beam on pass n
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
36
UW Spring 2008
Accelerator Physics
Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
37
UW Spring 2008
Accelerator Physics
Multipass BBU Theory/cont.
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
38
UW Spring 2008
Accelerator Physics
Multipass BBU Theory/cont.
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
39
UW Spring 2008
Accelerator Physics
Multipass BBU Theory/cont.
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
40
UW Spring 2008
Accelerator Physics
Multipass BBU Theory/cont.
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
41
UW Spring 2008
Accelerator Physics
Multipass BBU Theory/cont.
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
42
UW Spring 2008
Accelerator Physics
Multipass BBU Theory/cont.
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
43
UW Spring 2008
Accelerator Physics
Multipass BBU Theory/cont.
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
44
UW Spring 2008
Accelerator Physics
Multipass BBU Theory/cont.
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
45
UW Spring 2008
Accelerator Physics
Simulation: transient and steady state below threshold (cumulative-like)
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
46
UW Spring 2008
Accelerator Physics
Simulation: instability
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
47
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
48
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
49
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
50
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
51
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
52
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
53
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
54
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
55
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
56
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
57
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
58
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
59
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
60
UW Spring 2008
Accelerator Physics
Longitudinal Multipass BBU Theory
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
61
UW Spring 2008
Accelerator Physics
General Scaling with Q• For a single cavity, the threshold scales like 1/Q• For several cavities at the same resonance frequency,
the threshold scales like 1/Q times weighted sum over the transport optics
• But HOMs have a distribution in frequency from construction errors which– Decreases the peak value of the weighted sum– Changes the 1/Q dependence to something more
typically like 1/Q until frequency spread is so large that the cavity modes don’t overlap significantly
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
62
UW Spring 2008
Accelerator Physics
Typical Frequency Distribution Scaling
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
63
UW Spring 2008
Accelerator Physics
Typical Q Scaling
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
64
UW Spring 2008
Accelerator Physics
Energy Recovery Linacs
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
65
UW Spring 2008
Accelerator Physics
Current Limits• Typically, a storage ring light source will store a
few hundred milliamps• Progress in electron source development make
CW guns at the 100 milliamp level reasonable to talk about with emittance performance comparable and even better than storage rings
• Progress in HOM damping has made current limits at the 100 milliamp level obtainable
• So, an energy recovery linac should be able to produce storage ring levels of current with better emittance
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
66
UW Spring 2008
Accelerator Physics
Is It Worth Recirculating an ERL DriverMore Than Once
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
67
UW Spring 2008
Accelerator Physics
Energy Recovery Linac (ERL)
superconducting
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
68
UW Spring 2008
Accelerator Physics
Issues• Good
– Save money: SRF, RF, cryo– Beams of different energy right there
• Bad– Costs of more magnets– Beam breakup– Coherent synchrotron radiation (CSR)
• emittance growth• energy spread growth
– CSR instability– Space charge – Weaker focusing: head tail could be worse– RF constraints: off phase choices– Layout of compressors
• Just different– Site constraints: shorter and wider
• Multipass harder, but are limits limiting and are the savings significant
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
69
UW Spring 2008
Accelerator Physics
One up/one down Current Limits with Arc Optics Variation
Threshold vs. Phase Advance
672.12
528.13
386.59354.86
425.63
750.21
481.76
401.23389.03
459.8
672.12
0
100
200
300
400
500
600
700
800
900
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
Phase Advance [2*pi]
Th
res
ho
ld C
urr
en
t (m
A)
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
70
UW Spring 2008
Accelerator Physics
Two up/two down Current Limits with Arc Optics Variation
300
200
100
mA
J. J. Bisognano
Topic Seven: Coupled Bunch Effects
71
UW Spring 2008
Accelerator Physics
Homework Problems Topic VII• Topic VII-1 Consider a storage ring with 4 equally spaced bunches.
– a) Derive an eigenvalue problem for longitudinal coupled bunch motion for 4 point-like bunches undergoing small oscillations. Assume the instability is excited by a single higher order mode at some frequency. (Proceed as follows: calculate the voltage produced by an individually oscillating bunch. Second, calculate the perturbatin induced by this voltage on any of the four bunches.)
– b) Show that the eigenmodes reduce to those discussed in lecture
– c)Discuss what happens if one of the bunches is missing; i.e., there is a gap.
• Topic VII-2 Consider a charge Q passing rQ off axis through a cavity and a charge e passing time behind
– For a transverse deflecting mode, write an expression for the energy loss of the charge Q and the transverse deflection of charge e if it were on axis, and also if it were at rQ
– In physical terms, why would isochronous transport in a two-total-pass recirculated linac prevent longitudinal beam breakup? What would be the analogous transport constraint to prevent transverse beam breakup? What are the limitations of such solutions?