90°-Cavities With Improved
Inner-Cell HOM Properties
Shannon Hughes
Advisor: Valery Shemelin
Introduction
• Ideal cavities have geometry
for working π-mode
frequency
• Real cavities have many
minor defects…
– Frequency can be different
than intended
• Non-propagating frequency
→ trapped higher-order
modes
– Trapped HOMs can’t get to
damping couplers, so their
energy can’t be removed –
has negative effect on beam
quality
Goal: Stop trapped modes from occurring.
Introduction
How do we avoid trapped modes?
• All frequencies within each dipole-mode
bandwidth propagate.
– Broader bandwidths → fewer non-propagating
modes, so less likelihood of trapping
• Bandwidths can be broadened by modifying
elliptic arc parameters (i.e. geometry)
– Need to find geometry that yields widest
bandwidths
Programs Used
• SLANS/SLANS2
–
–
–
–
–
creates meshes
calculates frequencies
plots electric fields
SLANS → monopole mode
SLANS2 → dipole modes
• TunedCell
– wrapper program for SLANS/SLANS2
– calculates figures of merit (e, h, etc)
– writes half-cell geometries for each
set of elliptic arc parameters
• MathCAD
– fits curves to data using splines
– generates random numbers (for
Monte Carlo technique)
– plots data, and a lot more
Geometry
• A cell is made up of two
elliptic arcs (AB and ab)
connected by a line l, as
shown in the half-cell figure
• Many figures of merit
determined by elliptic arc
parameters (A, B, a and b)
• α = cell wall slope angle
• Three types of cells – nonreentrant, 90°, and
reentrant
• Non-reentrant: α > 90
• Reentrant: α < 90
Geometry
Half-Cell Mesh
Single-Cell Mesh
Geometry
Six-Cell Mesh
Why 90°-Cavities?
•
•
•
•
•
Frequency vs Phase Shift for
Fundamental Mode
Red → α ≤ 90°
Blue → α > 90°
ERL: - - - - - - - TESLA:
Greater difference between
0- and π-mode → larger
bandwidth (B0 = f π - f0)
Geometries with α ≤ 90° dominate the lower part of both
graphs, tending to have the broadest bandwidths for a given e.
e = Epk/2Eacc
Why 90°-Cavities?
Cell-to-Cell Coupling vs Cell Wall Slope
Angle for Fundamental Mode
• Multiple cells per cavity
– cells must work well together
• Higher k → better coupling
Geometries with α ≤ 90° tend to have the highest k
values for a given e.
Why 90°-Cavities?
• Best acceleration gradient
comes from
h vs α for e = 1
– minimizing peak magnetic
field (Hpk)
– maximizing accelerating field
(Eacc)
• So minimizing h = Hpk/42Eacc
yields best acceleration
gradient
• 95% of overall decrease in h
occurs from α = 105° to α =
90°
Geometries with α ≤ 90° tend to have the lowest h
values for a given e.
Why 90°-Cavities?
• Geometries with α ≤ 90° tend to have the best
h, k and B0 values for a given e.
• Reentrant cavities (α < 90°) have some
practical problems
– Difficult to remove water/chemicals during
cleaning
– Difficult to fabricate properly
• 90°-cavities do not share these problems
90°-cavities can be used for small-angle
benefits without reentrant drawbacks.
Why 90°-Cavities?
• Other groups interested in
90°-cavities
– Examples: LL, Ichiro, LSF, NLSF
• Our minimized h vs e values
just as good or better than
these others
Ichiro
51: the goal gradient (MV/m) for the
9-cell low-loss “Ichiro” cavity
h vs e
Higher-Order Modes
Frequency vs Phase Shift for 7
Dipole Modes
• Graph shows frequencies of
first seven dipole modes in
initial 90°-cavity
– Focus on these because we
limit maximum frequency to 4
GHz
• Some bands very broad, some
very narrow
• Is it possible to broaden these
bands?
– How much can these bands be
broadened?
• e and h are limited to 5%
increase
• α must remain at 90°
→a=L-A
Broadening One Mode
• For 3rd dipole mode, 90°-cavity
bandwidth is narrow
– Especially compared with
TESLA and ERL!
• How much can this particular
bandwidth be broadened?
• Several broadening methods
using geometry
– Changing A incrementally
– Changing A, B, and b in the
direction of the gradient of
increasing B3
– Changing only B and b in the
direction of the gradient of
increasing B3
Frequency vs Phase Shift for 3rd
Dipole Mode
Changing A Incrementally
B3 vs ΔA
• Of all the elliptic arc
parameters, changing A
has the biggest impact on
B3
• A changed incrementally
– B and b held at initial
values
– a held at a = L – A
– Stopped when h increased
by 5%
• B3 increased from 12.025
MHz to 68.181 MHz
Changing A, B and b
• Derivatives of B3 with
respect to A, B and b
were calculated
– Used to create a 3-D
gradient vector with length
k in direction of increasing
B3
• k increased until h
increased 5%
• B3 increased from 12.025
MHz to 75.747 MHz
B3 vs k
Changing B and b
B3 vs k
• Idea: changing A affects h too
much
– Changes stopping too soon
because of h
– B and b have less effect on h →
change just these two
• B and b derivatives used to
create 2-D gradient vector
with length k in direction of
increasing B3
• k increased until e increased
5%
– h increased less but e increased
more!
• B3 increased from 12.025 MHz
to 49.237 MHz
Broadening One Mode
• All three methods successfully broadened the
3rd Dipole mode
• Changing A, B and b as a 3-D gradient → most
successful method
– B3 grew 6 times wider!
It is possible to significantly increase the bandwidth of
one dipole mode of a 90°-cavity with limits on e and h
by modifying only the elliptic arc parameters.
Broadening All Modes
Next step: increase net bandwidth of all
seven modes
• Need to maximize goal function:
• Monte Carlo Method
1.
2.
3.
4.
5.
Derivatives taken for each Bn with
respect to each elliptic arc parameter
(EAP)
Equations created predicting change in
Bn for change in EAPs (assuming linear
dependence of Bn on EAP)
10,000 random numbers generated
from a set range for each EAP →
10,000 values for each Bn prediction
EAPs maximizing predicted G without
exceeding e or h limit recorded
Prediction tested
Monte Carlo Casino
Broadening All Modes
• Predictions become much
less accurate after range
amplitude exceeds 1.0
– So different by range
amplitude of 5.0 that
calculations were stopped
– Maybe derivatives
continue to change with
range → must be
recalculated for every
increase of 1.0?
• G was increased by
20.881 MHz when the
range amplitude was 5.0
ΔG vs Range of Random Numbers
Broadening All Modes
ΔG vs Range of Random Numbers
• In this case, derivatives
were recalculated for
each step of 1.0 in range
• Predicted and actual
values are closer but
differences more erratic
• G was increased by
20.100 MHz when the
range amplitude was 5.0
– Slightly less than when
derivatives were kept the
same!
Broadening All Modes
Frequency vs Phase Shift for 7
Dipole Modes
• Both Monte Carlo approaches
successfully increased the net
bandwidth of all seven modes
– Leaving derivatives the same →
better results than recalculating at
each step
• Small increase compared with
initial G, but final value still better
than ERL or TESLA
G
90°, initial
90°, final
ERL
TESLA
1102.673
1123.554 1120.941 1111.875
It is possible to increase the net
bandwidth of a 90°-cavity with
limits on e and h by using a Monte
Carlo technique to modify elliptic
arc parameters.
Final: dashed
Initial: solid line
Brillouin light lines : dotted
Sixth Dipole Mode
• A special case: B6 = f π – fπ/4,
not |f π - f0| as with all
other modes
– When general calculation is
applied to this mode → B6 is
half what it should be
– If half- or single-cell
geometries are used for
calculation, correct
bandwidth is overlooked
• Multicell cavity must be used!
• More accurate bandwidth
formula necessary for
future broadening of bands
– B = fmax – fmin ?
Frequency vs Phase Shift for 6th
Dipole Mode
Conclusion
• Several successful ways to reduce trapped
modes by broadening bandwidth were
determined
– A single mode was broadened significantly using a
3-D gradient vector to modify elliptic arc
parameters
– Net bandwidth was broadened using a Monte
Carlo random number technique
Acknowledgements
I would like to thank my advisor Valery Shemelin
for his help and guidance throughout this
project. Thanks also to everyone who made the
CLASSE REU program possible. This work was
supported by the NSF REU grant PHY-0849885.