90-Cavities With Improved Inner-Cell HOM Properties

90-Cavities With Improved Inner-Cell HOM PropertiesShannon HughesAdvisor: Valery ShemelinIntroductionIdeal cavities have geometry for working -mode frequencyReal cavities have many minor defectsFrequency can be different than intendedNon-propagating frequency trapped higher-order modesTrapped HOMs cant get to damping couplers, so their energy cant be removed has negative effect on beam quality

Goal: Stop trapped modes from occurring.

IntroductionHow do we avoid trapped modes?All frequencies within each dipole-mode bandwidth propagate.Broader bandwidths fewer non-propagating modes, so less likelihood of trappingBandwidths can be broadened by modifying elliptic arc parameters (i.e. geometry)Need to find geometry that yields widest bandwidthsPrograms UsedSLANS/SLANS2creates meshescalculates frequenciesplots electric fieldsSLANS monopole modeSLANS2 dipole modesTunedCellwrapper program for SLANS/SLANS2calculates figures of merit (e, h, etc)writes half-cell geometries for each set of elliptic arc parametersMathCAD fits curves to data using splinesgenerates random numbers (for Monte Carlo technique)plots data, and a lot more

GeometryA cell is made up of two elliptic arcs (AB and ab) connected by a line l, as shown in the half-cell figureMany figures of merit determined by elliptic arc parameters (A, B, a and b) = cell wall slope angleThree types of cells non-reentrant, 90, and reentrant Non-reentrant: > 90Reentrant: < 90

GeometryHalf-Cell MeshSingle-Cell Mesh

Geometry

Six-Cell MeshWhy 90-Cavities?Frequency vs Phase Shift for Fundamental ModeRed 90Blue > 90ERL: - - - - - - - -TESLA: Greater difference between 0- and -mode larger bandwidth (B0 = f - f0)

Geometries with 90 dominate the lower part of both graphs, tending to have the broadest bandwidths for a given e.

e = Epk/2EaccWhy 90-Cavities?Cell-to-Cell Coupling vs Cell Wall Slope Angle for Fundamental ModeMultiple cells per cavitycells must work well togetherHigher k better coupling

Geometries with 90 tend to have the highest k values for a given e.

Why 90-Cavities?h vs for e = 1Best acceleration gradient comes from minimizing peak magnetic field (Hpk) maximizing accelerating field (Eacc)So minimizing h = Hpk/42Eacc yields best acceleration gradient95% of overall decrease in h occurs from = 105 to = 90

Geometries with 90 tend to have the lowest h values for a given e.

Why 90-Cavities? Geometries with 90 tend to have the best h, k and B0 values for a given e.Reentrant cavities ( < 90) have some practical problemsDifficult to remove water/chemicals during cleaningDifficult to fabricate properly90-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-cavitiesExamples: LL, Ichiro, LSF, NLSFOur minimized h vs e values just as good or better than these others

h vs e

Ichiro51: the goal gradient (MV/m) for the 9-cell low-loss Ichiro cavityHigher-Order ModesFrequency vs Phase Shift for 7 Dipole ModesGraph shows frequencies of first seven dipole modes in initial 90-cavityFocus on these because we limit maximum frequency to 4 GHzSome bands very broad, some very narrowIs it possible to broaden these bands?How much can these bands be broadened?e and h are limited to 5% increase must remain at 90 a = L - A

Broadening One ModeFrequency vs Phase Shift for 3rd Dipole ModeFor 3rd dipole mode, 90-cavity bandwidth is narrowEspecially compared with TESLA and ERL!How much can this particular bandwidth be broadened?Several broadening methods using geometryChanging A incrementallyChanging 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 IncrementallyOf all the elliptic arc parameters, changing A has the biggest impact on B3A changed incrementallyB and b held at initial valuesa held at a = L AStopped when h increased by 5%B3 increased from 12.025 MHz to 68.181 MHz

B3 vs A

Changing A, B and bDerivatives of B3 with respect to A, B and b were calculatedUsed 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 bIdea: changing A affects h too much Changes stopping too soon because of hB and b have less effect on h change just these twoB 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 ModeAll three methods successfully broadened the 3rd Dipole modeChanging A, B and b as a 3-D gradient most successful methodB3 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 modesNeed to maximize goal function:

Monte Carlo MethodDerivatives 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 predictionEAPs maximizing predicted G without exceeding e or h limit recordedPrediction tested

Monte Carlo CasinoBroadening All ModesPredictions become much less accurate after range amplitude exceeds 1.0So different by range amplitude of 5.0 that calculations were stoppedMaybe 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

G vs Range of Random Numbers

Broadening All ModesIn this case, derivatives were recalculated for each step of 1.0 in rangePredicted and actual values are closer but differences more erraticG was increased by 20.100 MHz when the range amplitude was 5.0Slightly less than when derivatives were kept the same!

G vs Range of Random Numbers

Broadening All ModesBoth Monte Carlo approaches successfully increased the net bandwidth of all seven modesLeaving derivatives the same better results than recalculating at each stepSmall 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, initial90, finalERLTESLAG1102.6731123.5541120.9411111.875Frequency vs Phase Shift for 7 Dipole Modes

Final: dashedInitial: solid lineBrillouin light lines : dotted

Sixth Dipole ModeA special case: B6 = f f/4, not |f - f0| as with all other modesWhen general calculation is applied to this mode B6 is half what it should beIf half- or single-cell geometries are used for calculation, correct bandwidth is overlookedMulticell cavity must be used!More accurate bandwidth formula necessary for future broadening of bandsB = fmax fmin ?

Frequency vs Phase Shift for 6th Dipole ModeConclusionSeveral successful ways to reduce trapped modes by broadening bandwidth were determinedA single mode was broadened significantly using a 3-D gradient vector to modify elliptic arc parametersNet bandwidth was broadened using a Monte Carlo random number techniqueAcknowledgementsI 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.