Chengkun HuangUCLA

Quasi-static modeling of beam/laser plasma interactions for particle accelerationZhejiang University07/14/2009

V. K. Decyk, M. Zhou, W. Lu, M. Tzoufras, W. B. Mori, K. A. Marsh, C. E. Clayton, C. Joshi

B. Feng, A. Ghalam, P. Muggli, T. Katsouleas (Duke)

I. Blumenfeld, M. J. Hogan, R. Ischebeck, R. Iverson, N. Kirby, D. Waltz, F. J. Decker, R. H. Siemann

J. H. Cooley (LANL), T. M. Antonsen

J. Vieira, L. O. Silva

Collaborations

Accelerating forcePlasma/Laser Wakefield AccelerationFocusing force++++++++++++++++++++++++++++++----------------------------------Fr----------------------------------------------------------------------------------------------------------++++++------------Fz

Plasma Accelerator ProgressAccelerator Moores Law

Maxwells equations for field solverLorentz force updates particles position and momentum New particle position and momentum Lorentz ForceParticle pusherweight to gridtComputational cycle (at each step in time)Particle-In-Cell simulation The PIC method makes the fewest physics approximations And it is the most computation intensive:

Field solverDeposition

*These are rough estimates and represent potential speed up. In some cases we have not reached the full potential. In some cases the timing can be reduced by lowering the number of particles per cell etc. Challenge in PIC modelingTypical 3D high fidelity PWFA/LWFA simulation requirement

PWFAFeatureGrid size limitTime step limitTotal time of simulation per GeV stage (node-hour)*Full EM PIC ~0.05c/pt< 0.05p-11500Quasi-static PIC~0.05c/pt

Quasi-static ModelThere are two intrinsic time scales, one fast time scale associated with the plasma motion and one slow time scale associated with the betatron motion of an ultra-relativistic electron beam. Quasi-static approximation eliminates the need to follow fast plasma motion for the whole simulation.Ponderomotive Guiding Center approximation: High frequency laser oscillation can be averaged out, laser pulse will be repre-sented by its envelope.

Caveats: cannot model trapped particles and significant frequency shift in laser

Quasi-static or frozen field approximationMaxwell equations in Lorentz gaugeReduced Maxwell equations Equations of motion: s = z is the slow time variable = ct - z is the fast time variableQuasi-static Approximation

Equations for the fields Gauge equation Conserved quantity of plasma electron motion Conserved quantityHuang, C. et al. J. Comp. Phys. 217, 658679 (2006).

ImplementationThe driver evolution can be calculated in a 3D moving box, while the plasma response can be solved for slice by slice with the being a time-like variable.

Ponderomotive guiding center approximation: Big 3D time stepPlasma evolution: Maxwells equations Lorentz Gauge Quasi-Static ApproximationImplementation

Benchmark with full PIC code100+ CPU savings with no loss in accuracy

The Energy Doubling Experiment Simulations suggest ionization-induced head erosion limited further energy gain.Nature, Vol. 445, No. 7129, p741Etching rate :

Laser wakefield simulation

Laser wakefield simulation

J. Vieira et. al., IEEE Transactions on Plasma Science, vol.36, no.4, pp.1722-1727, Aug. 2008.Laser wakefield simulation12TW25TW

FOCUSING OF e-/e+ OTR images 1m from plasma exit (xy) Single bunch experiments Qualitative differences

Experiment/Simulations: Beam Sizex0=y0=25m, Nx=39010-6, Ny=8010-6 m-rad, N=1.91010 e+, L=1.4 m Downstream OTR Excellent experimental/simulation results agreement!SimulationsExperiment The beam is round with ne0P. Muggli et al., PRL 101, 055001 (2008).

x0y025 m, Nx39010-6, Ny8010-6 m-rad, N=1.91010 e+, L1.4 m Very nice qualitative agreement Simulations to calculate emittanceExperimentSimulationsP. Muggli et al., PRL 101, 055001 (2008).Experiment/Simulations: Halo formation

Electron hosing instability is the most severe instability in the nonlinear ultra-relativistic beam-plasma interaction.Electron hosing instability could limit the energy gain in PWFA, degrade the beam quality and lead to beam breakup. Hosing InstabilityElectron hosing instability is caused by the coupling between the beam and the electron sheath at the blow-out channel boundary. It is triggered by head-tail offset along the beam and causes the beam centroid to oscillate with a temporal-spatial growth.

Linear fluid theoryThe coupled equations for beam centroid(xb) and channel centroid(xc) (Whittum et. al. 1991):

whereSolution:

Hosing in the blow-out regimeParameters:Hosing for an intense beamHeadTailIon ChannelIpeak = 7.7 kA

Previous hosing theory :Based on fluid analysis and equilibrium geometryOnly good for the adiabatic non-relativistic regime, overestimate hosing growth for three other cases: adiabatic relativistic, non-adiabatic non-relativistic, non-adiabatic relativistic. Two effects on the hosing growth need to be includedElectrons move along blow-out trajectory, the distance between the electron and the beam is different from the charge equilibrium radius.Electrons gain relativistic mass which changes the resonant frequency, they may also gain substantial P// so magnetic field becomes important.Hosing in the blow-out regime

Perturbation theory on the relativistic equation of motion is developed cr, c represent the contribution of the two effects to the coupled harmonic oscillator equations. In adiabatic non-relativistic limit, cr=c= 1. Generally crc< 1 for the blow-out regime, therefore hosing is reduced.The results include the mentioned two effects:A new hosing theorywhereSolution:

VerificationC. Huang et. al., Phys. Rev. Lett. 99, 255001 (2007)

a 19 Stages PWFA-LC with 25GeV energy gain per stagePWFA-based linear collider concept

To achieve the smallest energy spread of the beam, we want the beam-loaded wake to be flat within the beam.

Formulas for designing flat wakefield in blow-out regime (Lu et al., PRL 2006; Tzoufras et al, PRL 2008 ):We know when rb=rb,max, Ez=0, dEz/d=-1/2. Integrating Ez from this point in +/- using the desirable Ez profile yield the beam profile. For example, Rb=5, Ez,acc=-1, =6.25-( - 0)

Beam profile design for PWFA-LC

Simulation of the first and the last stages of a 19 stages 0.5TeV PWFAPhysical ParametersNumerical ParametersPWFA-LC simulation setup

Drive beamTrailing beamBeam Charge (1E10e-)0.82 + 3.651.62Beam Length (micron)13.4 + 44.722.35Emittance (mm mrad)10 / 62.962.9Plasma density (1E16 cm-3)5.66Plasma Length (m)0.7Transformer ratio1.2Loaded wake (GeV/m)45 GeV/m

475 GeV stage25 GeV stageenvelope oscillationEngery depletion;Adiabatic matchingHosings = 0 ms = 0.23 ms = 0.47 ms = 0.7 ms = 0 ms = 0.23 ms = 0.47 ms = 0.7 mMatched propagationSimulations of 25/475 GeV stages

s = 0 ms = 0.47 ms = 0.7 mEnergy spread = 0.7% (FWHM) Energy spread = 0.2% (FWHM) longitudinal phasespace25 GeV stage475 GeV stageSimulation of 25/475 GeV stages

LWFA design with externally injected beamTheory predicts EP, QP1/2 P=15TW P=30TW P=60TWW. Lu et. al., Phys.Rev. ST Accel. Beams 10, 061301 (2007)

LWFA design with externally injected beam

Summary

By taking advantage of the two different time scales in PWFA/LWFA problems, QuickPIC allows 100-1000 times time-saving for simulations of state-of-art experiments. QuickPIC enables detail understanding of nonlinear dynamics in PWFA/LWFA experiments through one-to-one simulations and scientific discovery in plasma-based acceleration by exploring parameter space which are not easily accessible through conventional PIC code.

Exploiting more parallelism: Pipelining Pipelining technique exploits parallelism in a sequential operation stream and can be adopted in various levels.Modern CPU designs include instruction level pipeline to improve performance by increasing the throughput.In scientific computation, software level pipeline is less common due to hidden parallelism in the algorithm.We have implemented a software level pipeline in QuickPIC.Moving Window

Instruction pipelineSoftware pipelineOperandInstruction streamPlasma sliceOperationIF, ID, EX, MEM, WBPlasma/beam updateStages5 ~ 311 ~(# of slices)

beamInitial plasma slabWithout pipelining: Beam is not advanced until entire plasma response is determinedPipelining: scaling QuickPIC to 10,000+ processors

More detailsStep 1Step 2Step 4Step 3Plasma updateBeamupdatePlasma slice Guard cell Particles leaving partitionComputation in each block is also parallelized TimeStage

Performance in pipeline mode Fixed problem size, strong scaling study, increase number of processors by increasing pipeline stagesIn each stage, the number of processors is chosen according to the transverse size of the problem. Benchmark shows that pipeline operation can be scaled to at least 1,000+ processors with substantial throughput improvement.Feng et al, submitted to JCP

Basic Quasi-Static mode

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computation speed up non-pipeline

Pipeline(4procs/group)

Modeling Externally Injected Beam in Laser Wakefield AccelerationDue to photon deceleration, verified with 2D OSIRISWe need plasma channel to guideToo low for ultrarelativistic blowout theory to work

pwfa, 16x16x16 c/wp, 4 ppc, 2 micro sec/ particle/step***