1. Stato dei lavori Ottimizzazione dei wiggler di DA  NE Simona Bettoni

2. Outline  Method to reduce the integrated octupole in the wiggler of DA  NE  Analysis tools at disposal: → Multipolar analysis: I n (also vs x shift at the entrance) → Tracking: x (y) and x’ (y’) vs x (y) shift at the entrance (tools Tosca+Matlab)  Shifted poles & cut poles models  Axis optimization  Analysis of the results: → Multipolar analysis → Tracking → Comparison with the experimental data at disposal  In the future

3. Other methods to reduce the integrated octupole CURVED POLE MOVING MAGNETIC AXIS New method Reduction of the octupole around the beam trajectory in the region of the poles Compensation of the integrated octupole in each semiperiod Proposed by Pantaleo

4. Multipolar expansion of the field with respect to the beam trajectory 1.Determination of the beam trajectory starting from the measured data 2.Fit of B y between -3 cm and +3 cm by a 4º order polynomial in x centered in x T (z) = x T x T +3 cm x T -3 cm Beam trajectory (x T )

5. The integrated multipoles in periodic magnets Even multipoles → Odd multipoles → In a displaced system of reference: b A k → defined in the reference centered in O A (wiggler axis) b T k → defined in the reference centered in O T (beam trajectory) x’ y’ x y OAOA O T xTxT Left-right symmetry of the magnet Multipoles change sign from a pole to the next one Sum from a pole to the next one

6. Method to reduce the integrated octupole: displacement of the magnetic field WITHOUT POLE MODIFICATION In each semiperiod the particle trajectory is always on one side with respect the magnetic axis In each semiperiod the particle travels on both sides with respect to the magnetic axis Opportunely choosing the B axis is in principle possible to make zero the integrated octupole in each semiperiod WITH THE POLE MODIFICATION Octupole ↑

7. Optimization of the pole of the wiggler FC1-likeFC2-like Goals  Reduce as less as possible the magnetic field in the gap  Maintain the left-right symmetry FC 2FC 1

8. Analysis For each z fit of By vs x in the system of reference perpendicular to the beam trajectory

9. Cut poles model: analysis perpendicular to s I 3 calculated over the entire wiggler varies of more than a factor 2 if the analysis is performed perpendicular to s and not to z ! I FC = 693 A

10. Sector poles wiggler Cut the poles in z to have sector poles I 3 calculated over the entire wiggler perpendicular to z is 9.09 T/m 3 with respect to 4.13 T/m 3 of the analysis perpendicular to z I FC = 693 A

11. Shifted poles solution $ and field roll-off

12. Shifted poles model For the moment shifted the coils with the poles

13. Cut-shifted poles: the comparison of the field (at the same current = 550 A in FC) SHIFTED POLES CUT POLES

14. Cut-shifted poles: the comparison of the field (at the same current = 550 A in FC) With the shifted poles solution, the field roll-off is improved, therefore the shims can be eliminated maintaining more or less the same dependence of the solution on the x-shift at the entrance. Shim thick in cut poles solution = 1.15 mm x 2 = 2.3 mm/37 mm = 6 % gap By(z = 0, x = 0) SHIFTED POLES = By(z = 0, x = 0) CUT POLES +7.6%

15. Trajectory optimization Determined the best value of the current in HC to minimize the integral of By over z

16. Trajectory optimization  Exit angle = 8 x 10 -2 mrad  x-shift exit-entrance = 0.13 mm By integrated over z = 2 G.m

17. Tools analysis: multipoles with Tosca & Matlab TOSCA 1.Determination of the best beam trajectory (tracking Tosca) 2.For each z found By in the points on a line of ±3 cm around (x TR, 0, z TR,) and perpendicular to the trajectory 3.Fit of the By at each point of the line (Tosca) at steps of 1 mm (fit Matlab) MATLAB 1.Determination of the best beam trajectory (tracking Tosca/0 the integral of By over z) 2.For each found z points on a line of ±3 cm around (x TR, 0, z TR,) and perpendicular to the trajectory 3.Fit of the By at each point of the line at steps of 1 mm interpolated by Matlab

18. Tools analysis: tracking 1.Beam enters at several x 2.Tosca tracks the trajectory of each beam 3.Calculated the x exit-x TR NOM and x’exit in function of the x-shift at the entrance The curves are only to show the tool

19. Axis optimization For the moment used these codes to optimize the position of the axis

20. Multipoles Presence of spikes in my analysis

21. Multipoles Beam trajectory at fixed  z and parabolic interpolation in z

22. Spikes Spikes: solved problem

23. Axis optimization Minimized I 3 calculated in the entire wiggler 0.73 cm

24. Multipolar analysis: to summarize Multipolar analysis (entire wiggler) I 0 (T.m)I 1 (T)I 2 (T/m) I 3 (T/m 2 ) I 4 (T/m 3 ) Tosca (2 mm step)-1.17E-042.09-1.13 0.13 87.8 Tosca (1 cm step)4.6E-052.10-1.25 -0.98 211 Miro (2 mm step)1.87E-042.09-1.13 -1.01 95.0 Miro (1 cm step)1.08E-042.08-1.14 -1.32 101 To do the first optimization I used this technique

25. Analysis of the results: tracking (±3 cm) Beam enters from x = x TR NOM -3 cm to x = x TR NOM +3 cm at steps of 1 mm, where x TR NOM is the position of entrance of the nominal trajectory

26. Analysis of the results: tracking: the x exit (±3 cm) The fit is satisfactory already for the 3rth-4rth order Coefficient of the 3 rd order term = 13 m- 2

27. Analysis of the results: tracking: the x’ exit (±3 cm) The fit is satisfactory for the 3th-4th order Coefficient of the 3 rd order term = 10 rad/m 3

28. Analysis of the results: comparison with the experimental data I could compare the results only with the results of the experimental map at about 700 A Ho riscalato curva di Miro x_exit = x_exit MIRO -x_exit MIRO (x ENTR = 0)

29. Analysis of the results: comparison with the experimental data I could compare the results only with the results of the experimental map at about 700 A

30. Analysis of the results: tracking: the y exit

31. Analysis of the results: tracking: the y’ exit

32. Conclusions  Shifted poles - cut poles solution comparison:  The field roll-off is improved  no shim  increased B PEAK  Cheaper  At present:  improved the linearity zone of x and x’ with respect to the field map at dipsosal  In the future:  Shifted poles solution analysis:  Analysis of the field maps by Dragt, Mitchell and Venturini (the map considered the best one by us, one with the poles more centered and one with the poles more shifted)  Measurement of the field map of the wiggler at I = 550 A to have a real comparison with the results of the simulation (at LNF, at ENEA?)

33. Di scorta…

34. Situation in the present configuration (I = 693 A): x exit The fit is satisfactory for the 5rth-6rth order │Coefficient of the 3 rd order term │ >200 m -2

35. Situation in the present configuration (I = 693 A): x’ exit The fit is satisfactory for the 6rth order │Coefficient of the 3 rd order term │ ~600 rad/m 3

36. Trajectory optimization To determine the best value of I in HC for the several axis displacements

37. Fine!