1. 90°-Cavities With Improved Inner-Cell HOM Properties Shannon Hughes Advisor: Valery Shemelin

2. 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.

3. 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

4. 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

5. 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

6. Geometry Half-Cell MeshSingle-Cell Mesh

7. Geometry Six-Cell Mesh

8. 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 (B 0 = f π - f 0 ) Geometries with α ≤ 90° dominate the lower part of both graphs, tending to have the broadest bandwidths for a given e. e = Epk/2Eacc

9. 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.

10. Why 90°-Cavities? h vs α for e = 1 Best acceleration gradient comes from – minimizing peak magnetic field (H pk ) – maximizing accelerating field (E acc ) So minimizing h = H pk /42E acc 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.

11. Why 90°-Cavities? Geometries with α ≤ 90° tend to have the best h, k and B 0 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.

12. 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

13. 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

14. Broadening One Mode Frequency vs Phase Shift for 3 rd Dipole Mode For 3 rd 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 B 3 – Changing only B and b in the direction of the gradient of increasing B 3

15. Changing A Incrementally Of all the elliptic arc parameters, changing A has the biggest impact on B 3 A changed incrementally – B and b held at initial values – a held at a = L – A – Stopped when h increased by 5% B 3 increased from 12.025 MHz to 68.181 MHz B 3 vs ΔA

16. Changing A, B and b Derivatives of B 3 with respect to A, B and b were calculated – Used to create a 3-D gradient vector with length k in direction of increasing B 3 k increased until h increased 5% B 3 increased from 12.025 MHz to 75.747 MHz B 3 vs k

17. 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 B 3 k increased until e increased 5% – h increased less but e increased more! B 3 increased from 12.025 MHz to 49.237 MHz B 3 vs k

18. Broadening One Mode All three methods successfully broadened the 3 rd Dipole mode Changing A, B and b as a 3-D gradient → most successful method – B 3 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.

19. Broadening All Modes Next step: increase net bandwidth of all seven modes Need to maximize goal function: Monte Carlo Method 1.Derivatives taken for each B n with respect to each elliptic arc parameter (EAP) 2.Equations created predicting change in B n for change in EAPs (assuming linear dependence of B n on EAP) 3.10,000 random numbers generated from a set range for each EAP → 10,000 values for each B n prediction 4.EAPs maximizing predicted G without exceeding e or h limit recorded 5.Prediction tested Monte Carlo Casino

20. 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

21. 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

22. 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°, initial90°, finalERLTESLA G1102.6731123.5541120.9411111.875 Frequency vs Phase Shift for 7 Dipole Modes Final: dashed Initial: solid line Brillouin light lines : dotted

23. Sixth Dipole Mode A special case: B 6 = f π – f π/4, not |f π - f 0 | as with all other modes – When general calculation is applied to this mode → B 6 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 = f max – f min ? Frequency vs Phase Shift for 6 th Dipole Mode

24. 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

25. 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.