Chengkun Huang UCLA

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

2. Collaborations 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

3. + + + + + + + + + + + + + + + + + + + + + + - + + + + + + + + + + + + + + + + + + + + + + - + + + + + + - - Linear focusing field Uniform accelerating field Plasma/Laser Wakefield Acceleration Focusing force Fr - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + - - - - - - - - - - - + - - + + + + + - + + + + + - + + + + + + + - - - - + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - - - - - Fz - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Accelerating force

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

5. Particle-In-Cell simulation Computational cycle (at each step in time) New particle position and momentum Particle pusher Deposition weight to grid • Maxwell’s equations for field solver • Lorentz force updates particle’s position and momentum Lorentz Force t Field solver The PIC method makes the fewest physics approximations And it is the most computation intensive:

6. Challenge in PIC modeling Typical 3D high fidelity PWFA/LWFA simulation requirement *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.

7. 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

8. Quasi-static Approximation • Equations of motion: s = z is the slow “time” variable  = ct - z is the fast “time” variable • Quasi-static or frozen field approximation Maxwell equations in Lorentz gauge Reduced Maxwell equations

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

10. 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.

11. Implementation Ponderomotive guiding center approximation: Big 3D time step Plasma evolution: Maxwell’s equations Lorentz Gauge Quasi-Static Approximation

12. Benchmark with full PIC code 100+ CPU savings with “no” loss in accuracy

13. The Energy Doubling Experiment Simulations suggest “ionization-induced head erosion” limited further energy gain. Nature, Vol. 445, No. 7129, p741 Etching rate :

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

15. Laser wakefield simulation

16. Laser wakefield simulation J. Vieira et. al., IEEE Transactions on Plasma Science, vol.36, no.4, pp.1722-1727, Aug. 2008. 12TW 25TW

17. ne=0 ne≈1014 cm-3 2mm •Ideal Plasma Lens in Blow-Out Regime e- 2mm •Plasma Lens with Aberrations, Halo e+ FOCUSING OF e-/e+ • OTR images ≈1m from plasma exit (x≠y) • Single bunch experiments • Qualitative differences

18. x-size x-size y-size y-size Experiment/Simulations: Beam Size x0=y0=25µm, Nx=39010-6, Ny=8010-6m-rad, N=1.91010 e+, L=1.4 m Downstream OTR Simulations Experiment • Excellent experimental/simulation results agreement! • The beam is ≈round with ne≠0 P. Muggli et al., PRL 101, 055001 (2008).

19. y-core y-core x-halo x-halo y-halo y-halo x-core x-core Experiment/Simulations: Halo formation x0≈y0≈25 µm, Nx≈39010-6, Ny≈8010-6m-rad, N=1.91010 e+, L≈1.4 m Experiment Simulations • Very nice qualitative agreement • Simulations to calculate emittance P. Muggli et al., PRL 101, 055001 (2008).

20. 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. 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.

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

22. 3 orders of magnitude Simulation shows much less hosing Hosing in the blow-out regime Hosing for an intense beam Tail Head Ion Channel Parameters: Ipeak = 7.7 kA

23. Hosing in the blow-out regime • Previous hosing theory : • Based on fluid analysis and equilibrium geometry • 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 • Electrons 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.

24. A new hosing theory Perturbation theory on the relativistic equation of motion is developed The results include the mentioned two effects: where Solution: 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.

25. (1) (2) (3) (4) Verification Adiabatic, relativistic Adiabatic, non-relativistic Non-adiabatic, non-relativistic Non-adiabatic, relativistic C. Huang et. al., Phys. Rev. Lett. 99, 255001 (2007)

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

27. Ez  rb Beam profile design for PWFA-LC 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/dξ=-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)

28. Box size 1000x1000x272 Grids 1024x1024x256 Plasma particle 4 / cell Beam particle 8.4 E6 x 3 Time step 60 kp-1 Total step 520 PWFA-LC simulation setup Simulation of the first and the last stages of a 19 stages 0.5TeV PWFA Physical Parameters Numerical Parameters

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

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

31. LWFA design with externally injected beam W. Lu et. al., Phys. Rev. ST Accel. Beams 10, 061301 (2007) Theory predicts EP, QP1/2 P=15TW P=30TW P=60TW

32. LWFA design with externally injected beam

33. 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.

34. plasma response 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

35. beam 1 2 3 4 Initial plasma slab solve plasma response solve plasma response solve plasma response solve plasma response solve plasma response update beam update beam update beam update beam update beam With pipelining: Each section is updated when its input is ready, the plasma slab flows in the pipeline. Pipelining: scaling QuickPIC to 10,000+ processors Initial plasma slab beam Without pipelining: Beam is not advanced until entire plasma response is determined

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

37. Performance in pipeline mode Feng et al, submitted to JCP • 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.

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