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

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?

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.

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).

Parallel/Gaussian X-Y Profiles Parallel beam Gaussian beam

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Gaussian 2Dline Results Using Gaussian beam distribution gives larger variations between forward and reverse beams (2Dline model, 0 mm): Emittance: +0.1% x,rms (0.00833 to 0.00834  mm mrad), +0.3% y,rms (0.00833 to 0.00836  mm mrad). Size: +1 nm xrms (1.62326 to 1.62327 mm), +1 nm yrms (1.62346 to 1.62347 mm). Divergence: +280 nrad x’rms (1.72808 to 1.72836 mrad), +760 nrad x’rms (1.72881 to 1.72957 mrad).

Gaussian tree2D Results Similar results for SCtree2D model (0 mm): Emittance: +0.1% x,rms (0.00833 to 0.00834  mm mrad), +0.3% y,rms (0.00833 to 0.00836  mm mrad). Size: +2 nm xrms (1.62326 to 1.62328 mm), -2 nm yrms (1.62346 to 1.62344 mm). Divergence: +270 nrad x’rms (1.72808 to 1.72835 mrad), +760 nrad x’rms (1.72881 to 1.72957 mrad).

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.