1. Click to edit Master subtitle style 1 Reconstruction of HOM spectrum for XFEL 3 rd Harmonic module by Generalised Scattering Matrix technique. Nirav Joshi, Roger Jones, Accelerator Physics Division, The University of Manchester Liangliang Shi, Nicoleta Baboi DESY, Hamburg, Germany Dr. Nirav Joshi, The University of Manchester @ EuCARD2 Group Meeting, 18/02/2016

2. Click to edit Master subtitle style 2 Cascading 8-Cavities of ACC39 Beam Pipe End Cup With HOM Coupler 8 X Two-Mid Cups End Cup withHOM And Fundamental Couplers Cavity-1 with 9 cells Cavity-2 Cavity-1 Cavity-1 and Cavity-2 connected through beam pipe FLASH and Eu-XFEL use third harmonic accelerating modules with eight 9-cell cavities connected in chain. EM simulation of a cavity, and the whole module, is a very large computational problem, which takes lot of time and requires larger computational facility. A cascading technique is developed, in which the large structure is divided in smaller building blocks. S-parameters of each block is simulated separately, and cascaded to reconstruct the large structure. Advantages of Cascading technique: To study the effect of structural change in any part of the large structure, only that building block needs to be re-simulated, not the whole large structure. Smaller structure can be simulated with higher accuracy and in shorter time duration, improving turn around time. Performance of the whole structure can be realised using normal workstation computer, without requiring larger special purpose high performance computational facility.

3. S-parameters of two compatible modules can be cascaded to calculate the resultant S-parameters of the whole module. A cascading code is developed in Python, which reads S-parameters of unit-modules exported from HFSS or CST, and cascaded them to recreate larger structures If more than one modes are calculated at ports, then each S value in the equation will be a matrix of all modes. Possible problem of matrix blow up can arise in the inversion terms. Cascading: principle 3 Module-R Port-1Port-2 A1RA1R A2RA2R B1RB1R B2RB2R Module-L Port-1Port-2 A1LA1L A2LA2L B1LB1L B2LB2L

4. Cascading: Wide Narrow Wide structure 4 Port-1Port-2 Port-1Port-2 Port-1 Port-2 Simpler WNW structure cascaded, and compared to full structure simulation, using HFSS. All ports simulated for 25 modes. Mode 1: TE 11, was coupled to Mode 7: TE 21, in reflection and transmission S 11 S 21

5. 5 Cascade Cavity-1 from unit modules 8 X Two-Mid Cups Port-1 C1H1 Port-2 C1H2 Port-3 C1F Port-4 Port-5 Cavity-1 Port-1 Port-3 Port-4 Port-2 Port-1 Port-3 Port-2 Port-1Port-2 StructureHFSS Version-15 (Trial) Mesh (tets)Time (Hours) Start coupler7301941 Mid cups3132028 End coupler6095833 Full Cavity-1234,175204 (9days) Cascading unit cells on a laptop computer took less than 2 minutes, for 2000 frequency steps. Cascading is 50% faster even for even a single cavity structure S 11 S 21

6. 6 FLASH Cavity-1 Up-stream coupler: Min/Max( S-magnitude(over whole F range) ) HIGHLY COUPLED MODES: Reflection 0.8 (S(1:1,1:1)) [] = -0.18220 MEDIUM COUPLED MODES: 0.2 Transmission>0.4 (S(2:1,1:1)) [] = 0.52545 (S(2:2,1:1)) [] = 0.47118 (S(2:7,1:1)) [] = 0.49276 (S(2:8,1:1)) [] = 0.43153 (S(3:2,1:1)) [] = 0.45775 LOW COUPLED MODES: 0.6 Transmission>0.01 (S(2:3,1:1)) [] = 0.03989 (S(2:15,1:1)) [] = 0.03836 (S(2:16,1:1)) [] = 0.03623 Effect of number of equator modes 2:1 TE11-X 2:2 TE11-Y 2:7 TM11-X 2:8 TM11-Y 2:15 TE21-X 2:16 TE21-Y2:17 TM02-X2:18 TM02-Y

7. 7 Effect of adding 2 nd cavity Beam Pipe P1 P4 P7 P5 P3 P6 P8 C1C2 P2 P1 P2P3 P4P5 P3P2 P1 P4 P5 P1P2 Cascaded C1Cascaded C2 S 22 S 21 S 61

8. 8 Cascading complete 8-Cavity chain on XFEL Cascading all 8 cavities from 6 unit modules, over the frequency band (2000 frequency steps), on a laptop computer took only 3m:30s (2:00 load module files + 1:30 cascade). P1 C1 P3F P4 P2 C1 P7F P5 P6 P8 C3 P10FP9 C4 P13F P11 P12 P14 C5 P16F P15 C6 P19F P17 P18 P20 C7 P22F P21 C8 P25F P23 P24 P26 S 22 S 21

9. 9 Q calculation: Cavity-1 (single cavity), DB1 Reflection parameter for the cavity can be calculated from circuit analogy as follows. Where, At resonance, Where as the S21 can be fitted using Lorentzian function

10. 10 Q calculation: Cavity-1 (single cavity), DB1 Reflection parameter for the cavity can be calculated from circuit analogy as follows. In case of S11, the formula to calculate Q, using 3dB bandwidth method modifies to

11. 11 Q calculation: Cavity-1 (single cavity), DB1 Using resonant circuit analogy, following equations can be derived to fit the reflection and transmission S-parameter curves. The code automatically detect peaks, and individual peak is fitted separately, and are summed to create total S 21, which does not agree with the simulated plot. The modes are degenerated due to asymmetric orientation of HOM couplers. All the peaks has to be fitted, simultaneously for accurate Q-calculation.

12. 12 Q calculation: Cavity-1 (single cavity), DB1 Peak frequencies from S11 and S22 are used as an initial guess for fitting. Peak-3 has been degenerated in to 4 peaks to re-create non-lorentzian shape. Minimising the multiple peaks together, reduces the error considerably. Minimization is sensitive to initial guess value, and will be optimized further for best result.

13. 13 Q calculation: Cavity-1 (single cavity), DB2 Peak frequencies from S11 and S22 are used as an initial guess for fitting. Each peak has been degenerated in to 2 peaks to re-create non-lorentzian shape. Minimising the multiple peaks together, reduces the error considerably. Minimization is sensitive to initial guess value, and will be optimized further for best result.

14. 14 Q calculation: Cavity-1 (single cavity), DB2 The higher frequency modes in DB2, as non lorentzian response curve. Those peaks can be fitted accurately by degenerating in to many different peaks, but that has no physical justification. Eigen mode simulations of cavity-1 are to be solved to compare and verify number of modes and Q values.

15. Benefits and Applications of Cascading Mode spectrum of a complete 8-cavity chain can be recreated from simulations of unit-models on a normal workstation, with higher accuracy, in considerably shorter duration. Fast turn around increases productivity considerably. 6 X Two-Mid Cups Port-1 Port-3 Port-4 Port-2 Port-1 Port-3 Port-2 Port-1Port-2 To study fabrication error, only need to simulate one unit-model of Mid-cells with errors, and cascade it at different positions to recreate error in any single mid- cell, or multiple mid-cells along the cavity chain. Same simulated erroneous cell be cascaded at different position and effect on different modes can be studied in no time. StructureHFSS Version-12HFSS Version-15 + HPC (Trial) Supports Parallel processing Performance gain from Version-15 Mesh & Frequency steps Simulation TimeMesh & Frequency steps Simulation Time Full Cavity-1132690 Tet-elements (Degraded mesh) 3MHz step size (coarse) (334 frequency points) 223 Hours (10 days, 60 days for 2000 ) 234175 Tet-elements 0.5MHz step size (2000 steps) 204 Hours (9days) 6 times faster Up stream Coupler with End Cup 33420Tet-elements (Degraded mesh) 3MHz step size (coarse) (334 frequency points) 52 hours (6 days for 2000 steps???) 73019Tet-elements 0.5MHz step size (2000 steps) 33 Hours~8 times faster Down stream coupler with End-cup 27932Tet-elements (Degraded mesh) 3MHz step size (coarse) (334 frequency points) 48 hours (6 days for 2000 steps???) 60958 Tet-elements 0.5MHz step size (2000 steps) 28 Hours~8 times faster

16. Summary and Future work Summary A code has been developed to cascade the S-Matrices of unit cell, to reconstruct larger structure response. Different discretization schemes has been studied, with effect of meshing on accuracy of cascading. Cascaded S-matrix results for a single cavity structure agreed very well with simulation of a whole structure. The application of code has been extended to reconstruct response of whole 8-cavity module mode spectrum. The studied showed considerable coupling between cavities, which has to be studied in details. Stainless steel bellow has also been divided and simulated for coarser frequency steps size. To include it in cascading the finer frequency steps near peaks in different bands has to be further simulated. Further Simulation Studies Calculate Quality factor of HOMs, in different cavities. Simulate fabrication errors in multi-cavity structure and study the effects on Eigen modes. Experimental Measurements: Measure S-parameters between different ports of cavity chain and recreate those measurements using cascading simulations.