Page 1
Jean Delayen
Center for Accelerator Science
Old Dominion University
and
Thomas Jefferson National Accelerator Facility
WAKEFIELDS, IMPEDANCES,
INSTABILITIES AND
HIGHER-ORDER MODES
USPAS@Rutgers
June 2015
Page 2
Outline
• Linear systems
• Longitudinal wakefield and impedance
• Transverse wakefield and impedance
• Regenerative beam breakup
• Single-pass cumulative beam breakup
• Multi-pass cumulative beam breakup
Page 3
Relativistic Particle In a Lossless Smooth Pipe
• All the fields are concentrated in a disk moving along with the particle – No wakefields
– No instability
• In order to get wakefields: – Non relativistic particles (outside this subject)
– Lossy walls (resistive wall instability)
– Non-uniform outer conductors (cavities, bellows,…)
Page 4
Wake Function Definition
A unit charge will generate electromagnetic fields
that will be experienced by a trailing (test) charge
Page 5
Wake Potential
Test charge
s
Charge q losing energy
s=0
The wake potential is the potential experienced by the test particle trailing the unit
charge
Page 6
Test charge
s
Charge q losing energy
s=0
The wake depends on:
Definition:
the wake potential W is the potential seen by a test particle following the unit charge
losing the E-H energy to the cavity.
W (position of charge q, position of the test charge, charge distribution q(z), shape of cavity, s)
Beam-Cavity Interaction
Page 7
The energy lost by the bunch to a mode n totally dissipates or/and radiates out of the
cavity before the next bunch enters the cavity (there is no build up effect).
When does it happen? tb
t
)nt
tnω(
en)0(Wn)t(W
tn << tb
Decay of the energy stored in mode n:
where: tn = ωn∙Qln is the decay time of mode n
Single passage:
Beam-Cavity Interaction
Page 8
The amount of energy lost by charge q to the cavity is:
ΔUq = k∙q2 for monopole modes (max. on axis)
ΔUq = k∙q2 for non monopole modes (off axis)
where k and k(r) are loss factors for the monopole and transverse modes respectively.
The induced E-H field (wake) is a superposition of cavity eigenmodes (monopoles and
others) having the En(r,φ,z) field along the trajectory.
4
)Q/R(k nn
n,||
p
For individual mode n and point-like charge:
Similar for other loss factors…….
Loss Factors
Note please the linac
convention of (R/Q)
definition.
Page 9
Two kind of phenomena can limit performance of a machine due to the beam induced HOM
power:
Beam Instabilities and/or dilution of emittance
Additional cryogenic power and/or overheating of HOM couplers output lines
Beam instabilities and/or dilution of emittance
Transverse modes (dipoles) causing emittance growth+ monopoles causing energy spread
This is mainly problem
in linacs: TESLA or ILC, CEBAF, European XFEL, linacs driving FELs.
Additional cryogenic power and/or overheating of HOM couplers output lines
Monopoles having high impedance on axis are excited by the beam and store energy which must
be coupled out of cavities, since it causes additional cryogenic load, and induces energy spread.
This is mainly problem
in high beam current machines: B-Factories, Synchrotrons, Electron cooling.
Beam-Cavity Interaction
Page 10
Linear Systems
• Impulsive response
• Transfer function
h t
t h t
i te i tZ e
( ) ( )
( ) ( )
and are related to each other through a Fourier Transform
i t
h t Z
Z e h t dtw
w
w+¥
-¥= ò
•
Page 11
Linear Systems (cont.)
• Causality
• For Resonant systems
( )
( ) ( )
0 0for
analytic and bounded for Im >0
(no pole in the upper half-plane)
h t t
Z w w
= <
( )
( )
is called a wake function
is called an impedance
h t
Z w
Page 12
Example: Single-Mode Cavity
2
2
11
4 2 2
r r rr r b c r
RRv v v i
Q Q Q Q Q
w w ww w w w a k+ + = = - = =
( ) ( )
2
sincos 0
4 1
( )0
2
0 0
t crc
r
tRe t t
Q Q
i t t v t Rt
Q
t
a www
d w
-ì é ùï - >ê úï -ê úë ûï
= = í=ï
ïï <î
( )0
1
i t
r
r
Ri i e Z
iQ
w www
w w
= =é ù
+ -ê úë û
Page 13
Longitudinal Wake Function, Impedance
Energy loss ΔW of a test particle with charge e, that follows
at a distance s, a point like bunch having total charge q = eN
( )
( ) ( )
( )
( ) ( ) ( )
, , ,
, ,
1, , ,
z=ct-s
-1Vis of dimension in MKS, cm in CGS
C
Since can be expressed as a function of time
is of dimension in
i s
i tc
W eq w s
cw s r dt E z r t
q
w
s ct w s r
Z r ds w s r e dt w t r ec
Z
www
+¥
-¥
+¥ +¥
-¥ -¥
D =
=-
=
= =
W
ò
ò òMKS, s/cm in CGS
Page 14
Longitudinal Wake Function, Impedance
• For narrow-band structures with cylindrical symmetry: sum over monopole modes
( )
( )
( )
21cos 0
2
10
4
0 0
1 1
4
2 2
n
t
Q
n n
n n
n
n n
nn nn n
n n
n n
Re t t
Q
Rw t t
Q
t
i RZ
i iQ
Q Q
w
w w
w
w ww w
w w w w
-ì æ ö>ï ç ÷è øï
ï æ öï= =í ç ÷è øïï <ïïî
é ùê úæ öê ú= +ç ÷è ø ê ú- + + +ê úë û
å
å
å
Page 15
Example: SLAC Structure
R. Ruth, SLAC-PUB-4948, April 1989
Page 16
EM Field of “Deflecting” Mode
Page 17
Transverse Wake Function, Impedance
• The transverse point wake function is defined as the integrated
transverse kick experienced by a test particle caused by the
transverse component of the radiated field of a point-like bunch
divided by the bunch offset (r0)
( ) ( )
( ) ( ) ( )
0
1, , ,
, ,
-2
2
V is of dimension in MKS, cm in CGS
C m
is of dimension /m in MKS, s/cm in CGS
z st
c
i s
i tc
vw t r dz E z r t
qr c
w
iZ r ds w s r e i dt w t e
c
Z
www
+¥
^ +-¥
^
^
+¥ +¥
^ ^ ^-¥ -¥
^
é ù=- + ´Hê ú
ë û
=- =-
W
ò
ò ò
Page 18
Transverse Wake Function, Impedance
• For narrow-band structures with cylindrical
symmetry: sum over dipole modes
( )( )
( )
22
2
1sin 0
2
0 0
1 1
4
2 2
n
n
t
Q
n n
n n
nn nn n
n n
n n
Re t t
w t c Q
t
i RZ
i ic Q
Q Q
w
w w
w ww w
w w w w
-
^
^
ì æ ö³ï ç ÷= è øí
ï<î
é ùê úæ öê ú= -ç ÷è ø ê ú- + + +ê úë û
å
å
Page 19
Example: SLAC Structure
R. Ruth, SLAC-PUB-4948, April 1989
Page 20
Resistive Wall Wake Functions and Impedances
1 131
2 220 2
1 11 1
2 22 20 0
3 3
( ) ( )1 (1 )10
4 2 2
( ) ( )1 (1 )1 10
2 2
:
for
for
Beam pipe radius
: Wall conductivit
ow t Z iZ Z
tz a z atc c
w t Z icZ Z ct
z a z at
a
ww
pp s s
w
p p wps s
s
^ ^
¶ ¶ -é ù é ùé ù= > =ê ú ê úê ú¶ ¶ë ûë û ë û
¶ ¶ -é ù é ùé ù é ù= > =ê ú ê úê ú ê ú¶ ¶ë û ë ûë û ë û
0
y
: Impedance of vacuum (377 )Z W
Page 21
Relationship Between Longitudinal
and Transverse Wake Functions
Panofsky – Wenzel Theorem:
( ) ( )
( ) ( )
0
0
, ,
, ,
cw t w t
t r
cZ Z
rw w
w
^ ^
^ ^
¶= Ñ
¶
= Ñ
r r
r r
Page 22
Scaling Laws
( ) ( )
( ) ( )
( )
2
22
1cos 0
2
1sin 0
2
is a geometrical property of a mode, independent of size or material;
it is a function only of shape
scale
n
n
n
t
Q
n n
n n
t
Q
n n
n n
Rw t e t t
Q
Rw t e t t
c Q
R
Q
w s
L
w
w
w w
w w
-
-
^
æ ö= >ç ÷è ø
æ ö= ³ç ÷è ø
æ öç ÷è ø
å
å
( )2 3s as scales as w t
Lw w^
Page 23
Regenerative Beam Breakup • Time instability
• A particle can be deflected inside a cavity to regions of higher field and
increased coupling.
• If the increase in transverse field due to one bunch is not compensated
sufficiently by the decay when the next bunch arrives, an instability
occurs.
• Threshold current 3
2
:
:
particle momentum
cavity length
th
pI
e Z L
p
L
p w
^
• 2Since thZ L I L-^ µ µ
Page 24
Single – Pass Cumulative Beam Breakup
• Instability in space, not in time.
• Initial offsets are amplified.
• Transient cumulative BBU can be much
larger than steady state.
• Unless exactly on resonance, steady state
behavior relatively insensitive to Q. Most
effective cure is to increase focusing.
Page 25
Equation of Transverse Motion
( ) ( ) ( ) ( )2
1 1 1 1
1, ,
Approximations:
- Cavitieshavenegligiblelength
- Cavitiesareelectromagneticallydecoupled
- Cavitiesandfocusingelementsaresolesourceof deflectingfields
- Disc
x d w F xz
bg k s z e z z z z s zbg s s -¥
é ù¶ ¶æ ö+ = -ç ÷ê úè ø¶ ¶ë û
ò
( )
/ , : linaclength, , : angular frequencyof deflecting field
couplingstrength todipo2
retedeflectingfieldsaresmoothedalongthelinac
- Beamislongitudinallyrigid
dss t
c
IZe
mc L
s z w wb
e sbg w
^
æ ö= = -ç ÷è ø
é ùé ù Gé ù= ê úê ú ê ú
ë ûë û ë û
ò
( ) ( ) ( ) ( )
( )
( )
2
2
0
0
lemode
/ : current formfactor, : wakefunction sin forsinglemode
0,0,
2: transverse shunt impedance
Q
i zL
zc
V
F I I w U e
E ze
x
d
z
w
b
z z z z z
e w
-
-
^
= =
¶
¶G =
ò
ò2
x xE
Page 26
Equation of Transverse Motion (Cont.) Example: Steady-state, coasting, delta function periodic beam
0
22
1 1 1 12
2
2 2
0
0,Steady State: 0, , 0
, ,
, function beam 1
2, ( ) ( )
, = M cos 0
For
ik
k k
k
iZt
k
xx x
z
x d w F x
F F e F
W Z w Z k w Z w t e dt
Z W Z
x x
2 2 sin single deflecting mode : 0
2cosh cos
2Q
Page 27
Resonance Function
Resonance function for Q=100.
2 p 4 p 6 p 8 p 10 pwt
-20
-15
-10
-5
5
10
15
20
Res
Page 28
Example: Elimination of BBU Instability
by Increasing Focusing
1000
0.2
10
Q
M
e
wt
=
=
=
= ¥
Page 29
Example: Transient BBU
Page 30
Distribution of Dipole Mode Frequency
• Assume the dipole mode frequencies are not identical along the linac but follow a probability density
• A distribution of dipole mode frequencies can be modeled by using this “modified” wakefunction
( )
( ) ( )
( ) ( ) ( )
0[ ]
1
2
ˆ 2
Define
Define a new "modified" wake function
iZt
g Z f Z
g t e g Z dZ
w t g t w t
w
p
p
+¥
-¥
= +
=
=
ò
( ) 0 around f w w
Page 31
Distribution of Dipole Mode Frequency
Example: Lorentzian Probability Density
( )( )
( )
( )0
0
2 2 22
0
1
2
0
0
1 1( )
2
ˆ ( ) ( ) sin
1 1 2
eff
t
tQ
f g ZZ
g t e
w t U t e t
Q Q
w
ww
w
w ww
p p ww w w
p
w
w
w
-D
æ öD- +ç ÷è ø
D D= =
+ D- + D
=
=
D= +
Page 32
Transverse Multipass BBU Instability
Page 33
Longitudinal Multipass Instability
Longitudinal HOMs Suppose HOM excited
– Get an energy error
– M56 converts to a phase error
– Phase modulation plus bunch beam spectrum
can generate sideband at HOM frequency
– Depending on the sideband phase, HOM may
be further excited
Page 34
Multipass BBU Instability
• Multi-pass BBU is an instability in time
• There is a threshold current above which the
instability occurs
• It is a form of regenerative instability (closed loop
system where a bunch experiences its own
wakefield)
• Very sensitive to Q. ost effective cure is to
decrease Q of dipole modes
Page 35
Instability Threshold
There is a well-defined threshold current that occurs when the power fed into the mode equals the mode power dissipation
An analytic expression that applies to all instabilities:
• For i,j = 1,2 or 3,4 and m HOM Transverse BBU
• For i,j = 5,6 and m || HOM Longitudinal BBU
• For i,j = 5,6 and m Fundamental mode Beam-Loading Instabilities
• l=1 for longitudinal HOMs and l=0 otherwise
(1)
/2
2
( / ) sin( / 2) m r m
im m
rth t Q
m rjm
p cI
e R Q Q k M t l eww p
-=
+
Page 36
HOM Coupling Measurement
Page 37
Beam Transfer Function
Page 38
TDBBU Simulation
• Short bunch simulation of Multibunch BBU assuming multiple cavity deflecting modes
• Multipass accelerators may be simulated
• Accelerators with several linac segments may be simulated
• Accelerators with accelerating passes and decelerating passes may be simulated
• Simulations include effects of differing path lengths from differing linac segments
• The current in successive bunches may be varied in a programmed manner
One iteration of code corresponds to one fundamental RF period
Page 39
10 mA, Below Threshold
Page 40
20 mA, Just Beyond Threshold
Page 41
30 mA, Above Threshold
Page 42
HOM Power Dissipation
High average current, short bunch length beams in srf cavities excite HOMs. Power in HOMs, primarily longitudinal:
PHOM = 2 k|| Q2 fbunch
For Iave= 100 mA, Q = 77 pC PHOM~ 160 W per cavity for k||=10.4 V/pC at z~ 0.6 mm
In the JLab IRFEL: Iave= 5 mA, PHOM~ 6 W
Fraction of HOM power dissipated on cavity walls depends on the bunch length
It can potentially limit Iave and Ipeak due to finite cryogenic capacity
Page 43
HOM Power Dissipation
The fraction of HOM power dissipated on cavity walls increases with HOM
frequency, due to Rs ~ ω2 degradation from BCS theory
Several models have been developed to address the high frequency
behavior of HOM losses
Models predict:
• Frequency distribution of HOM power
• Fraction of power dissipated on the cavity superconducting walls is
- a strong function of bunch length
- much less than the fundamental mode load
• High frequency fields propagate along the structure
Page 44
Frequency Distribution of HOM Losses
• ~20% of HOM losses (30 W) occur at frequencies 3.5 GHz
This power is typically extracted by input couplers and HOM
couplers and is absorbed in room temperature loads
• The remaining losses, at frequencies 3.5 GHz, will propagate along the
structure and be reflected at normal and superconducting surfaces
on-line absorbers are required
• Effect of losses in frequency range beyond the threshold for Cooper pair
breakup (750 GHz) in superconducting Nb has been investigated: the resulting
Q0 drop is negligible
Page 45
Coaxial Coupler Types
Coupling to the electric field Coupling to the magnetic field
Page 46
LEP II HOM Coupler
Page 47
LHC HOM Couplers
Page 48
HOM couplers limit RF-performance of sc cavities when they are placed on cells
no E-H fields at HOM couplers positions,
which are always placed at end beam tubes
The HOM trapping mechanism is similar to the FM field profile unflatness mechanism:
weak coupling HOM cell-to-cell, kcc,HOM
difference in HOM frequency of end-cell and inner-cell
f = 2385 MHz
That is why they
hardly resonate
together
f = 2415 MHz
Mode Trapping
Page 49
Trapping of Modes within Cavities HOM
To untrapp HOMs we can:
1) open both irises of inner cells and end-cells (bigger kcc,HOM) and keep shape of end cells
similar Example:
RHIC 5-cell cavity for the electron cooling:
fHOM = 1394 MHz fHOM = 1407 MHz
fHOM = 1403 MHz
Monopole mode kcc ,HOM = 6.7 %
The method causes (R/Q) reduction of fundamental mode, which in this application is less
relevant.
(Courtesy of R. Calaga and I.
Ben-Zvi)
Page 50
Trapping of Modes within Cavities HOM
2) tailor end-cells to equalize HOM frequencies of inner- and end-cells
The method works for very few modes but keeps the (R/Q) value high of the fundamental mode.
Example:
TESLA 9-cell cavity, which has two different end-cells (asymmetric cavity)
The lowest mode in the passband fHOM =
2382 MHz
The highest mode in the passband fHOM
= 2458 MHz
Page 51
3) one can also split a long structure in weakly coupled subsections to have space for HOM
couplers in mid of a structure.
Example: 2x7-cell instead of 14-cell structure (DESY)
2451 MHz, (R/Q) = 212 Ω
2453 MHz, (R/Q) = 230 Ω
E-H fields at HOM couplers positions, no
trapping
Trapping of Modes within Cavities HOM
High (R/Q) monopole mode trapped in the 14-cell structure
Page 52
HOM couplers and Beam Line Absorbers HOM
Waveguide HOM couplers
CE
BA
F/C
orn
ell 1
.5 G
Hz HOM ports
HOM loads Design (1982) works at present
in CEBAF both linacs with
Ibeam ~ 80µAx4 @ Eacc 7 MV/m
HOM power is very low. It can
be dissipated inside
cryomodule.
Design proposed by R. Rimmer (JLab)
750 MHz for 1A class ERLs
PAC2005
Design proposed by G. Wu (JLab)
1500 MHz for 100 mA class ERLs
LINAC2004
Page 53
HOM couplers and Beam Line Absorbers HOM
Waveguide HOM couplers, cont.
Design proposed by
T. Shintake and continued by K. Umemori (KEK)
1.3 GHz TESLA cavity, very good damping.
Proceedings ERL2005
Courtesy of KEK
Page 54
14. HOM couplers and Beam Line Absorbers HOM
Coaxial line HOM couplers
HERA 0.5 GHz Design (1985/86), 48 work still in HERA e-
ring cw operation Ibeam ~ 40 mA @ Eacc 2
MV/m
TM011 monopole modes with highest (R/Q)
damped in 4-cell cavity to Qext < 900 !!
Couplers are assembled in the LHe vessel
2 HOM couplers <PHOM> ~ few watts
output TESLA 1.3 GHz
TESLA HOM coupler is a simplified version of HERA HOM
couplers for pulse operation with DF of a few percent !!!!!
Couplers are assembled outside the LHe vessel !!
FM rejection filter
3 couplers PHOM ~ 100W
Page 55
HOM couplers and Beam Line Absorbers HOM
The TESLA –like HOM couplers are nowadays designed in frequency range: 0.8-3.9 GHz
Cs
Cf
Co
L1
L2
Ro
x1, z1 x2, z1
Page 56
HOM couplers and Beam Line Absorbers HOM
There is big progress in modeling (2D and 3D). Example: Modeling of HOM damping in TTF
9-cell structures by ACD-SLAC (Nov, 2004) . Very good agreement with the measured data !!!
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.60 1.64 1.68 1.72 1.76 1.80
Frequency (GHz)
Qe
xt
x-polarization y-polartization x-measured y-measured
Solid – Omega3P; Hollow – Measured (TDR)
E B
Page 57
HOM couplers and Beam Line Absorbers HOM
Increasing Duty Factor, one needs to improve cooling of HOM couplers.
SNS cavities: Linac DF =
6%
(Co
urt
es
y o
f O
ak
Rid
ge G
rou
p:
I. C
am
pis
i, S
an
g-H
o K
im)
No RF Tmax = 6.4 K With RF Tmax = 7.4
K
Page 58
HOM couplers and Beam Line Absorbers HOM
The main problem is heating of the output
line.
Heat coming via.
output cable from
outside.
Heating by residual
H field of the FM
Nb antenna loses superconductivity,
Nb, Cu antennae will warm when the RF
on
Three solutions to that problem are currently under investigation:
Page 59
HOM couplers and Beam Line Absorbers HOM
1. High heat conductivity feedthrough, ensuring thermal stabilization of Nb antenna
below the critical temperature (9.2 K) at 20 MV/m for the cw operation.
JLab R&D for the 12-GeV CEBAF
upgrade.
Al203 replaced with
sapphire
2. New HOM coupler design with hidden output antenna (JLab).
New HOM
coupler.
Old HOM
coupler.
Page 60
HOM couplers and Beam Line Absorbers HOM
3. New HOM coupler design without output capacitor (DESY).
The problem mentioned here looks very unimportant but following projects need a
solution to it:
12-GeV CEBAF upgrade, 4 GLS Daresbury, Elbe Rossendorf, BESSY Berlin, CW upgrade
of European XFEL, ERL Cornell…
Page 61
Beam Line Absorbers ; multi-cell cavities
HOM couplers and Beam Line Absorbers HOM
BNL e-cooling for RHIC (four 704 MHz cavities 54
MeV
ERL-Cornell, 310 TESLA 1.3 GHz cavities with modified end
cells
TESLA 6 HOM couplers/cavity + 2 Beam Line
Absorbers
