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Lecture 4 Wakefields
Dr G Burt
Lancaster University
Engineering
Beam-Wave Coupling
• Like acceleration,
beam-wave coupling
can only occur when
the RF is resonant with
the beam.
10.0
9.0
frequency (GHz)
• Wakefields occur
where ever there is
coupling between the
beam and RF at any
frequency or mode.
8.0
TM110
TE111
dipole 3
dipole 4
dipole 5
dipole 6
light line
7.0
6.0
5.0
4.0
3.0
0
• This means either the
wave must have a
phase velocity equal to
the speed of the beam
or the beam must only
experience the RF a
fraction of the time.
50
100
150
Phase Advance (degrees)
In multi-cell cavities or in waveguide a dispersion
diagram can be used to find resonances.
Resonances occur where ever the light line (Phase
Advance=kzL =wL/vbeam) crosses the modes
dispersion line (ie phase velocity=beam velocity).
Generation of RF Current
A
A bunch of electrons
approaches a resonant
cavity and forces the
electrons to flow away
from the bunch.
The negative potential
difference causes the
electrons
to
slow
down and the energy
is absorbed into the
cavity
B
C
The lower energy electrons
then pass through the cavity
and force the electrons
within the metal to flow back
to the opposite side
Pancake effect
• Due to lorentz contraction the electric fields of the bunch
are almost entirely perpendicular to the bunch.
• This means wakefields cannot affect charges in front of
itself only behind.
Bunch Spectrum
• A charged bunch can induce wakefields over a wide spectrum given
by, fmax=1/T. A Gaussian bunch length has a Gaussian spectrum.
 w2 z 2 
exp  
2 
2
c


• On the short timescale (within the bunch) all the frequencies induced
can act on following electrons within the bunch.
• On a longer timescale (between bunches) the high frequencies
decay and only trapped low frequency (high Q) modes participate in
the interaction.
Coupling Impedance
Narrowband Impedance
Broadband Impedance
Impedance
Cut-off
frequency
The fourier transform of a wakefield is the coupling impedance. It has three

regions (shown above).
Z||  w 
 W (t ) exp  iwt  dt
||

The broadband impedance region doesn’t look like it has much of an effect but
it covers a huge frequency spectrum so it’s integral can dwarf the narrowband
region.
Fundamental Theorem of Beam
Loading
• A charge, q, traverses a cavity and induces a voltage, Vc, in a mode
of the cavity.
• What proportion of Vc is seen by the charge?
• If the voltage in the mode was initially zero and has voltage Vq when
the charge leaves the cavity, the average voltage is Vq/2. This is the
voltage seen by the charge.
• This is known as the fundamental theorem of beam loading.
• If a cavity has an initial voltage, Vc, in a mode and a bunch passes
and induced an additional voltage Vq, the bunch will loose energy
1


U q  q  Vc cos    Vq cos   
2


Mode Excitation
If we have a mode in a cavity with initial voltage Vc and a bunch traverses the
cavity inducing an additional, superimposed, voltage Vq, The modal energy is
Ui 
Vc
2w
2
R
Q
i
Vc e  Vq e
Uf 
R
2w
Q
And by conservation of energy
i 2
VcVq cos      Vq
U c  U f  U i 

R
R
w
2w
Q
Q
U q  U c  0
As Vq is proportional to q we can group the terms in powers of q and q2. The
q2 term gives
Vq 2
1
q Vq cos   
R
2
2w
Q
As the power of q term must balance  must be 2p times an integer, hence
R iwt
Vq  qw e  qWz
Q
2
Short range wakes
Short range wakes are the
wakefields acting inside
the bunch, where the
charge in the head induce
wakes that act on the tail
of the bunch.
For short length bunches,
this is dominated by the
narrowband impedance.
The wake is not a step for a real bunch (shown above) as the charge
density varies as a Gaussian.
Single Bunch Wake
A mode excited by a single bunch will
decay exponentially with time due to
ohmic heating and external coupling.
Wake (V)
The single bunch will excite
several modes each with
different beam coupling and
damping rates.
Wz 

all mod es
w
 wt 
R
cos wt  exp  

Q
 2Q 
12
1.
10
1.
10 11
1.
10 10
1.
10 9
1.
10 8
0.01
0.1
1
10
Bunch Separation km
100
Mode Indices
Dipole modes
Dipole mode have a transverse
magnetic and/or transverse electric
fields on axis. They have zero
longitudinal field on axis. The
longitudinal electric field increases
approximately linearly with radius
near the axis.
Electric
Magnetic
Wakefields are only induced by the
longitudinal electric field so dipole wakes are
only induced by off-axis bunches.
Once induced the dipole wakes can apply a
kick via the transverse fields so on-axis
bunches can still experience the effect of the
wakes from preceding bunches.
Panofsky-Wenzel Theorem
If we rearrange Farday’s Law (   E   dB )and integrating along z we
dt
can show
 E  z, t 

c  dzB  z ,  c   c  dz  dt 
   Ez  z , t  

z


0
0
t0
L
z
L
c
z
Inserting this into the Lorentz (transverse( force equation gives us
 dE  z , t 

z
z
dz
E
z
,

cB
z
,

c
dz
dt


E
z
,
t

 z 
0    c   c   0 t  dz

0
L
L
z
c
for a closed cavity where the 1st term on the RHS is zero at the limits of the
integration due to the boundary conditions this can be shown to give
L
ic mV||
V    dz  Ez  z , c  ~ 
w0
w rm
ic
z
This means the transverse voltage is given by the rate of change of the
longitudinal voltage
 m
V  r 
2
c 2 V||  r 
2
R

 2m 3
Q
2wU
2r w U
Multibunch Wakefields
• For multibunch wakes, each bunch induces the
same frequencies at different amplitudes and
phases.
• These interfere to increase or decrease the
fields in the cavity.
• As the fields are damped the wakes will tend to
a steady state solution.
Long Range Wakefields
• The long range wakes are found by summing over all modes and all
bunches
 wt  
R1
Wz   2w    cos wt  exp  

Q
2
2
Q
all mod es
allbunches



 wt 
R( m)
W    2c
sin wt  exp  

Q
2
Q
all mod es allbunches


• This is known as the sum wake.
• The kick from a dipole mode can then be found using the equation
below
eq
x '  mrleadW
E
Long Range Transverse Wakes
Horizontal kick for 4 offset.
Vertical kick for 4 offset.
0.60
2.00
9-cell
9-cell
0.50
1.50
Bunch angle nrad
Bunch angle nrad
0.40
1.00
0.50
0.30
0.20
0.10
0.00
0
50
100
150
200
250
300
0.00
0
50
100
150
200
250
-0.10
-0.50
Bunch number
• The long range wakefield is a
sum of damped periodic
oscillations and hence converges
to a finite value.
Bunch number
• The time taken to converge to
this value is dependant on the
modal Q factors.
300
Resonances
• As you are summing the contribution to the wake
from all previous bunches, resonances can
appear. For monopole modes we sum
 cos(nw ) exp( n
w
2Q
)
• Hence resonances appear when w 
n
2p
n
• It is more complex for dipole modes as the sum
is
w
 sin( nw ) exp( n 2Q )
n
• This leads to two resonances at +/-some Δfreq
from the monopole resonant condition.
Effect of frequency errors
Cavity
Deflected beam
The effect of frequency errors can be
estimated by introducing small
variations in the bunch spacing.
X’
Undeflected beam
X’
Wakefield Kick
2 10
Tolerance
As can be seen if
the beam hits a
resonance with a
HOM the
wakefields increase
significantly.
1.5 10
-9
1 10
-9
5 10
-0.1
-0.05
-9
-10
0.05
% variation in bunch separation
0.1
Damping
• As the wakes from each bunch add together it is
necessary to damp the wakes so that wakes from only a
few bunches add together.
• The smaller the bunch spacing the stronger the damping
is required (NC linacs can require Q factors below 50).
• This is normally achieved by adding external HOM
couplers to the cavity.
• These are normally quite complex as they must work
over a wide frequency range while not coupling to the
operating mode.
• However the do not need to handle as much power as
an input coupler.
Coaxial HOM couplers
HOM couplers can be represented by equivalent circuits. If the coupler couples
to the electric field the current source is the electric field (induced by the beam in
the cavity) integrated across the inner conductor surface area.
I
Cs
R
If the coaxial coupler is bent at the tip to produce a loop it can coupler to the
magnetic fields of the cavity. Here the voltage source is the induced emf from the
time varying magnetic field and the inductor is the loops inductance.
V
L
R
Loop HOM couplers
Inductive stubs to probe couplers can be added for impedance matching to the
load at a single frequency or capacitive gaps can be added to loop couplers.
L
L
I
Cs
R
I
Cs
R
Cf
Also capacitive gaps can be added to the stub or loop inductance to make
resonant filters.
1
wc 
LCs
The drawback of stubs and capacitive gaps is that you get increase fields in the
coupler (hence field emission and heating) and the complex fields can give rise
to an electron discharge know as multipactor (see lecture 6).
As a result these methods are not employed on high current machines.
F-probe couplers
Capacative
gaps
F-probe couplers are a type of co-axial
coupler, commonly used to damp HOM’s in
superconducting cavities.
Their complex shapes are designed to give
the coupler additional capacitances and
inductances.
Output
antenna
The LRC circuit can be used to
reduce coupling to the operating
mode (which we do not wish to
damp) or to increase coupling at
dangerous HOM’s.
Log[S21]
Inductive
stubs
These additional capacatances and
inductances form resonances which can
increase or decrease the coupling at specific
frequencies.
frequency
Waveguide Couplers
Waveguide HOM couplers allow higher
power flow than co-axial couplers and
tend to be used in high current systems.
They also have a natural cut-off
frequency.
They also tend to be larger than co-axial
couplers so are not used for lower
current systems.
waveguide 2
To avoid taking the waveguides through the
cryomodule, ferrite dampers are often placed in
the waveguides to absorb all incident power.
waveguide 1
w2/2
w1/2
Choke Damping
load
choke
cavity
For high gradient accelerators, choke mode
damping has been proposed. This design uses a
ferite damper inside the cavity which is shielded
from the operating mode using a ‘choke’. A Choke
is a type of resonant filter that excludes certain
frequencies from passing.
The advantage of this is simpler (axiallysymmetric) manufacturing
Beampipe HOM Dampers
For really strong HOM damping we can place ferrite
dampers directly in the beampipes. This needs a
complicated engineering design to deal with the heating
effects.