90°-Cavities With Improved Inner-Cell HOM Properties

Shannon HughesAdvisor: 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 – non-

reentrant, 90°, and reentrant • Non-reentrant: α > 90• Reentrant: α < 90

Geometry

Half-Cell Mesh Single-Cell Mesh

GeometrySix-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?h vs α for e = 1• Best acceleration gradient

comes from– 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 problems90°-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

h vs e

Ichiro 51: the goal gradient (MV/m) for the 9-cell low-loss “Ichiro” cavity

Higher-Order ModesFrequency 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

Frequency vs Phase Shift for 3rd Dipole 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

Changing A Incrementally

• 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

B3 vs ΔA

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

• 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

B3 vs k

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 ModesNext step: increase net bandwidth of all seven modes• Need to maximize goal function:

• Monte Carlo Method1. Derivatives taken for each Bn with respect

to each elliptic arc parameter (EAP)2. Equations created predicting change in Bn

for change in EAPs (assuming linear dependence of Bn on EAP)

3. 10,000 random numbers generated from a set range for each EAP → 10,000 values for each Bn prediction

4. EAPs maximizing predicted G without exceeding e or h limit recorded

5. Prediction testedMonte 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

• 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!

ΔG vs Range of Random Numbers

Broadening All 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

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.

90°, initial 90°, final ERL TESLA

G 1102.673 1123.554 1120.941 1111.875

Frequency vs Phase Shift for 7 Dipole Modes

Final: dashedInitial: solid lineBrillouin 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.