Chengkun Huang UCLA Quasi-static modeling of beam/laser plasma interactions for particle acceleration Zhejiang University 07/14/2009

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 force Plasma/Laser Wakefield Acceleration Uniform accelerating field Linear focusing field Focusing force FrFr FzFzFzFz

RAL LBL Osaka UCLA E164X Current Energy Frontier ANL E167 LBL Plasma Accelerator Progress “Accelerator Moore’s Law” ILC SLAC

Maxwell ’ s equations for field solver Lorentz force updates particle ’ s position and momentum New particle position and momentum Lorentz Force Particle pusher weight to grid tt Computational 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 solver Deposition

* 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 modeling Typical 3D high fidelity PWFA/LWFA simulation requirement PWFA FeatureGrid size limitTime step limit Total time of simulation per GeV stage (node-hour)* Full EM PIC ~0.05c/  p  t< 0.05  p Quasi-static PIC ~0.05c/  p  t<0.05   -1 = 4 (20) LWFA FeatureGrid size limitTime step limit Total time of simulation per GeV stage (node-hour) Full EM PIC ~0.05  t< 0.05  0 -1 ~ Ponderomotive Guiding center PIC ~0.05c/  p  t< 0.05  p -1 ~1500 Quasi-static PIC ~0.05c/  p  t < 0.05  r ~ 10

Quasi-static Model There 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 approximation Maxwell equations in Lorentz gauge Reduced Maxwell equations  Equations of motion: s = z is the slow “time” variable  = ct - z is the fast “time” variable Quasi-static Approximation

 Equations for the fields  Gauge equation  Conserved quantity of plasma electron motion Conserved quantity Huang, C. et al. J. Comp. Phys. 217, 658–679 (2006).

Implementation The 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 step Plasma evolution: Maxwell’s equations Lorentz Gauge Quasi-Static Approximation Implementation

Benchmark with full PIC code 100+ 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, p741 Etching rate :

Laser wakefield simulation QuickPIC simulation for LWFA in the blow-out regime

Laser wakefield simulation

J. Vieira et. al., IEEE Transactions on Plasma Science, vol.36, no.4, pp , Aug Laser wakefield simulation 12TW 25TW

FOCUSING OF e-/e+ e-e- e+e+ n e =0n e ≈10 14 cm -3 2mm Ideal Plasma Lens in Blow-Out Regime Plasma Lens with Aberrations, Halo  OTR images ≈1m from plasma exit (  x ≠  y )  Single bunch experiments  Qualitative differences

Experiment/Simulations: Beam Size  x0 =  y0 =25µm,  Nx =390  10 -6,  Ny =80  m-rad, N=1.9  e +, L=1.4 m Downstream OTR  Excellent experimental/simulation results agreement! Simulations Experiment  The beam is ≈round with n e ≠0 y-size x-size y-size x-size P. Muggli et al., PRL 101, (2008).

 x0 ≈  y0 ≈25 µm,  Nx ≈390  10 -6,  Ny ≈80  m-rad, N=1.9  e +, L≈1.4 m  Very nice qualitative agreement  Simulations to calculate emittance ExperimentSimulations x-haloy-halo x-core y-core x-haloy-halo x-core y-core P. Muggli et al., PRL 101, (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 Instability Electron 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.

Beam centroid Channel centroid Equilibrium ion channel Linear fluid theory The coupled equations for beam centroid(x b ) and channel centroid(x c ) (Whittum et. al. 1991): where Solution:

Hosing in the blow-out regime Parameters: Hosing for an intense beam Head Tail Ion Channel 3 orders of magnitude Simulation shows much less hosing I peak = 7.7 kA

Previous hosing theory : 1.Based on fluid analysis and equilibrium geometry 2.Only 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 included 1.Electrons move along blow-out trajectory, the distance between the electron and the beam is different from the charge equilibrium radius. 2.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 c r, c  represent the contribution of the two effects to the coupled harmonic oscillator equations. In adiabatic non-relativistic limit, c r =c  = 1. Generally c r c  < 1 for the blow-out regime, therefore hosing is reduced. The results include the mentioned two effects: A new hosing theory where Solution:

Verification (1)(2) (3)(4) Adiabatic, non-relativistic Adiabatic, relativistic Non-adiabatic, non-relativistic Non-adiabatic, relativistic C. Huang et. al., Phys. Rev. Lett. 99, (2007)

a 19 Stages PWFA-LC with 25GeV energy gain per stage PWFA-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 r b =r b,max, E z =0, dE z /dξ=- 1/2. Integrating E z from this point in +/-  using the desirable E z profile yield the beam profile. Ez rbrb For example, R b =5, E z,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 PWFA Physical Parameters Numerical Parameters Drive beamTrailing beam Beam Charge (1E10e -) Beam Length (micron) Emittance (mm mrad) 10 / Plasma density (1E16 cm -3 ) 5.66 Plasma Length (m) 0.7 Transformer ratio 1.2 Loaded wake (GeV/m) 45 GeV/m Beam particle8.4 E6 x 3 Time step60 k p -1 Total step520 Box size1000x1000x272 Grids1024x1024x256 Plasma particle 4 / cell PWFA-LC simulation setup

475 GeV stage 25 GeV stage envelope oscillation Engery depletion; Adiabatic matching Hosing s = 0 m s = 0.23 m s = 0.47 ms = 0.7 m s = 0 m s = 0.23 m s = 0.47 ms = 0.7 m Matched propagation Simulations of 25/475 GeV stages

s = 0 ms = 0.47 ms = 0.7 m Energy spread = 0.7% (FWHM) Energy spread = 0.2% (FWHM) longitudinal phasespace 25 GeV stage 475 GeV stage Simulation of 25/475 GeV stages

LWFA design with externally injected beam Theory predicts  P, Q  P 1/2 P=15TW P=30TW P=60TW W. Lu et. al., Phys. Rev. ST Accel. Beams 10, (2007)

LWFA design with externally injected beam

Summary  By taking advantage of the two different time scales in PWFA/LWFA problems, QuickPIC allows 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 plasma response Instruction pipelineSoftware pipeline OperandInstruction streamPlasma slice Operation IF, ID, EX, MEM, WB Plasma/beam update Stages5 ~ 311 ~(# of slices)

beam solve plasma response update beam Initial plasma slab Without pipelining: Beam is not advanced until entire plasma response is determined solve plasma response update beam solve plasma response update beam solve plasma response update beam solve plasma response update beam beam With pipelining: Each section is updated when its input is ready, the plasma slab flows in the pipeline. Initial plasma slab Pipelining: scaling QuickPIC to 10,000+ processors

More details Step 1 Stage 1 Stage 3 Stage 4 Stage 2 Step 2Step 4Step 3 PlasmaupdatePlasmaupdateBeamupdateBeamupdate Plasma slice Guard cell Particles leaving partition Computation in each block is also parallelized Time Stage

Performance in pipeline mode Fixed problem size, strong scaling study, increase number of processors by increasing pipeline stages In 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

Modeling Externally Injected Beam in Laser Wakefield Acceleration Due to photon deceleration, verified with 2D OSIRIS We need plasma channel to guide Too low for ultrarelativistic blowout theory to work