1. Time-Reversed Particle Simulations In GPT (or “There And Back Again”)
Simon Jolly
Imperial College
FETS Meeting, 12/10/05

2. Time-reversed Simulations
GPT only has capacity to run time forwards in simulations.
To make comparisons with “downstream” emittance measurements, need to find a way of running time backwards.
Create “reverse” simulation by making divergence negative ie. all angles are inverted:
Is this a realistic assumption to make?
Does it produce realistic results?

3. Backwards Simulations
“Time-reversed” (backwards) technique tested in the following way:
Create beam and track forwards 300mm;
Invert transverse velocity (angle) of each particle and reverse longitudinal profile: equivalent to a reflection in X-Y plane;
Re-insert “reversed” beam into GPT and track forward another 300mm.
“Reverse” beam a second time and compare to original model at t=0.

4. Simulation Parameters
2 different beam models used:
“Parallel” beam - circular, uniform beam; xrms = yrms = 5mm, x’ = y’ = 0, z = 0, 35keV, 60mA, 100% SC, E = 0, 10,000 particles.
Gaussian beam - xrms = yrms = 1.6mm, x’rms = y’rms = 1.7mrad, x,rms = y,rms = 8.3x10-3  mm mrad, z = 0, 35keV, 60mA, 100% SC, E = 0, 10,000 particles.
2 different space charge models used: 2Dline and tree2D (“reverse” simulation tests SC model accuracy).

5. Parallel/Gaussian X-Y Profiles
Parallel beam
Gaussian beam

6. Parallel Beam Trajectories (1)
Forward trajectories: Z-X, tree2D model

7. Parallel Beam Trajectories (2)
Reverse trajectories: Z-X, tree2D model

8. Parallel Beam: tree2D X-Y (1)
Difference between transverse positions at 0mm of forward and reverse beams: X-Y, tree2D model

9. Parallel Beam: tree2D X-Y (2)
Difference between transverse positions at 0mm of forward and reverse beams: X-Y, tree2D model

10. Parallel Beam: tree2D X’-Y’
Difference between transverse angles at 0mm of forward and reverse beams:
X’-Y’, tree2D model

11. Parallel Beam Trajectories (3)
Forward trajectories: Z-X, 2Dline model

12. Parallel Beam: 2Dline X-Y
Difference between transverse positions at 0mm of forward and reverse beams: X-Y, 2Dline model

13. Parallel Beam: 2Dline X’-Y’
Difference between transverse angles at 0mm of forward and reverse beams:
X’-Y’, 2Dline model

14. Parallel Beam SC Models (1)
Difference between forward trajectories (Z-X) for tree2D and 2Dline space charge models

15. Parallel Beam SC Models (2)
Difference between transverse positions at 0mm (X-Y) for tree2D and 2Dline space charge models

16. Gaussian Beam: Forward (1)
Forward trajectories: Z-X, tree2D model

17. Gaussian Beam: Forward (2)
Forward trajectories: Z-X, 2Dline model

18. Gaussian Beam: Reverse
Reverse trajectories: Z-X, 2Dline model

19. Gaussian Beam: tree2D X-Y
Difference between transverse positions at 0mm of forward and reverse beams: X-Y, tree2D model

20. Gaussian Beam: 2Dline X-Y
Difference between transverse positions at 0mm of forward and reverse beams: X-Y, 2Dline model

21. Gaussian Beam: tree2D X’-Y’
Difference between transverse angles at 0mm of forward and reverse beams:
X’-Y’, tree2D model

22. Gaussian Beam: 2Dline X’-Y’
Difference between transverse angles at 0mm of forward and reverse beams:
X’-Y’, 2Dline model

23. Gaussian Beam: Z-X (1)
Longitudinal particle position at 0mm for reverse beam: Z-X, 2Dline model

24. Gaussian Beam: Z-X (2)
Longitudinal particle position at 0mm for reverse beam (enhanced): Z-X, 2Dline model

25. Gaussian Trajectory Diff (1)
Difference between forward trajectories (Z-X) for tree2D and 2Dline space charge models

26. Gaussian Trajectory Diff (2)
Difference between forward trajectories (Z-X) for tree2D and 2Dline space charge models (enhanced)

27. Gaussian Angle Diff (1)
Difference between forward angles (Z-X’) for tree2D and 2Dline space charge models

28. Gaussian Angle Diff (2)
Difference between forward angles (Z-X’) for tree2D and 2Dline space charge models (enhanced)

29. Gaussian Beam: 600mm (1)
Trajectories for reverse Gaussian beam tracked for 600mm: Z-X, 2Dline model

30. Gaussian Beam: 600mm (2)
Angle trajectories for reverse Gaussian beam tracked for 600mm:
Z-X’, 2Dline model

31. Gaussian 2Dline Results
Using Gaussian beam distribution gives larger variations between forward and reverse beams (2Dline model, 0 mm):
Emittance: +0.1% x,rms ( to  mm mrad), +0.3% y,rms ( to  mm mrad).
Size: +1 nm xrms ( to mm), +1 nm yrms ( to mm).
Divergence: +280 nrad x’rms ( to mrad), +760 nrad x’rms ( to mrad).

32. Gaussian tree2D Results
Similar results for SCtree2D model (0 mm):
Emittance: +0.1% x,rms ( to  mm mrad), +0.3% y,rms ( to  mm mrad).
Size: +2 nm xrms ( to mm), -2 nm yrms ( to mm).
Divergence: +270 nrad x’rms ( to mrad), +760 nrad x’rms ( to mrad).

33. Conclusions
Space charge models are accurate enough to run “reverse” simulations in GPT.
Space charge models get worse with increasing angle:
From Pulsar: “We have no solid mathematical proof, but it seems to us that as long as the typical angle with respect to the z-axis times the 'thickness (in z)' of the bunch is less than the radius, all is fine.”
Inaccuracies clear from simulation results, but not large enough to affect RMS beam parameters.