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Chengkun HuangUCLA

Quasi-static modeling of beam/laser plasma interactions for particle

acceleration

Zhejiang 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

CollaborationsCollaborations

Accelerating force

Plasma/Laser Wakefield AccelerationPlasma/Laser Wakefield Acceleration

Uniform accelerating field Linear focusing field

,max 30.96 /z pnE mc e V cm

cm Focusing force

++++++++++++++ ++++++++++++++++

----- --- ----------------

--- -------Fr

--

-------- ------- -------------------- - -

-

---- - -- ---

------ -

- -- ---- - - - --

---- - -- - - - -

-- --

- -- - - - --

---- - ----

------

+ + + + + + + + + + ++ + + + + + + + + + + + + + ++ + + + + + + + + + + + + + ++ + + + + + + + + + + + + + +

-

- --

--- --

-- ---- -- -- --- FFzz

RAL

LBL Osaka

UCLA

E164X

Current Energy Frontier

ANL

E167

LBL

Plasma Accelerator ProgressPlasma Accelerator Progress“Accelerator Moore’s Law”

ILCSLAC

• 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

Computational cycle (at each step in time)

Particle-In-Cell simulationParticle-In-Cell simulation

The PIC method makes the fewest physics approximations And it is the most computation intensive:

dpi

dtq E

vi Bc

Field solver

x i,pi

Deposition

n,m , jn,m

x i,pi

n,m , jn,m ,En,m ,Bn,m

En,m,Bn,m

x , t x (Courant Condition)

*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 modelingChallenge in PIC modelingTypical 3D high fidelity PWFA/LWFA simulation requirement

PWFA

Feature Grid size limit Time step limit Total time of simulation per GeV stage (node-hour)*

Full EM PIC ~0.05c/p t< 0.05p-1 1500

Quasi-static PIC ~0.05c/p

t<0.05-1

=4 (20)

LWFA

Feature Grid size limit Time step limit Total time of simulation per GeV stage (node-hour)

Full EM PIC ~0.05 t< 0.05 0-1 ~ 500000

PonderomotiveGuiding center

PIC

~0.05c/p t< 0.05p-1 ~1500

Quasi-static PIC

~0.05c/p t < 0.05r ~ 10

13

0.05 2 p 1

Quasi-static ModelQuasi-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

22

2 2

22

2 2

1 4( )

1( ) 4

c t c

c t

A J 2

2

4

4c

A Jquasi-static: 0s

Maxwell equations in Lorentz gauge Reduced Maxwell equations

1 ( , , , ) 1 1 ( , , , ) 1 , ( , , , ) ( , , , )

2 2 ( ), ~ , ~ .

b p

i i i b pp

f x y s f x y s where f is anyf x y s s L f x y s L

field quantity A or or E or B and L Lk k

Equations of motion:

//

:

: [ ( ) ]

b b b

e e e

e

d qFor beam electrons

ds c cd qFor plasma electronsd c V c

P VE B

P VE B

s = z is the slow

“time” variable

= ct - z is the fast

“time” variable

Quasi-static ApproximationQuasi-static Approximation

//

//

//

1 ,

ˆ ˆ ˆ ( )

Ec t

A z z z

B B

A AE

B A A

Equations for the fields

1 0 .

.c t

This is used to advance

A A

Gauge equation

2// /// 1 / , .e e ep mc q mc where A

Conserved quantity of plasma electron motion

Conserved quantityConserved quantity

Huang, C. et al. J. Comp. Phys. 217, 658–679 (2006).

ImplementationImplementation

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

ImplementationImplementation

ct z

Benchmark with full PIC codeBenchmark with full PIC code

100+ CPU savings with “no” loss in accuracy

The Energy Doubling ExperimentThe Energy Doubling Experiment

Simulations suggest “ionization-induced head

erosion” limited further energy gain. Nature, Vol.

445, No. 7129, p741

10

17 3

42 , 1.7 10 , 10

15 , 2.7 10b r

z p

GeV N m

m n cm

Etching rate :

Laser wakefield simulationLaser wakefield simulation

QuickPIC simulation for LWFA in the blow-out regime

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


12TW

25TW

FOCUSING OF e-/e+FOCUSING OF e-/e+

0 50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

0 50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

0 50 100 150 200 250 300 350

50

100

150

200

250

30050 100 150 200 250 300 350

50

100

150

200

250

300

e-

e+

ne=0 ne≈1014 cm-3

2mm

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

0

500

1000

1500

2000

0 0.5 1 1.5 2

PE390by80resulysOTR6

Tran

sver

se S

ize

(µm

)

ne (1014 cm-3)

0

500

1000

1500

2000

0 1 2 3 4 5 6

P390by80BeamSizes6

Tran

sver

se S

ize

(µm

)

ne (1014 cm-3)

Experiment/Simulations: Beam SizeExperiment/Simulations: Beam Size

x0=y0=25µm, 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 ne≠0

y-sizex-size

y-sizex-size

P. Muggli et al., PRL 101, 055001 (2008).

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

P390by80peakHalo6LowE

Cha

rge

Frac

tion

ne (1014 cm-3)

x0≈y0≈25 µm, Nx≈39010-6, Ny≈8010-6 m-rad, N=1.91010 e+, L≈1.4 m

• Very nice qualitative agreement• Simulations to calculate emittance

Experiment Simulations

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

P390by80PeakHaloFraction6

Cha

rge

Frac

tion

ne (1014 cm-3)

x-halo y-halox-core y-core

x-halo y-halox-core y-core

P. Muggli et al., PRL 101, 055001 (2008).

Experiment/Simulations: Halo formationExperiment/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 InstabilityHosing 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.

Beam centroid ),( sxb

),( sxc

Channel centroid

Equilibrium ion channel

Linear fluid theoryLinear fluid theory

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

s2xb k

2 xb k2 xc,

2 xc 02xc 0

2xb .

where

kp p c , k kp 2, 0 kp / 2

Solution:

Hosing in the blow-out regimeHosing in the blow-out regime

3160 100.2 cmn

9.25/ 0 nnb

mL 5.2011.0tilt

Parameters:

Hosing for an intense beam

Head Tail

Ion Channel

3 orders of magnitude

0.0001

0.001

0.01

0.1

1

10

0 0.5 1 1.5 2 2.5|x

b| (m

m)

s (m)

Linear fluidmodel prediction

Actual hosing growth

Simulation shows much less hosing

Ipeak = 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 included1. 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 regimeHosing in the blow-out regime

2 xc 0

2xc 02xb

Perturbation theory on the relativistic equation of motion is developed

2(1 )2

1(1 )2(dr d)2 (1 )2

dd

(1 )dr d (Er vzB ) Fr

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 theoryA new hosing theory

2 20( ) ( )c b cx c x x

c(») = cr(»)cÃ(») = nbR2b

r20

11+Ã0where

Solution:

VerificationVerification

(1) (2)

(3) (4)

Adiabatic, non-relativistic

Adiabatic, relativistic

Non-adiabatic, non-relativistic

Non-adiabatic, relativistic

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

a 19 Stages PWFA-LC with 25GeV energy gain per stage

PWFA-based linear collider conceptPWFA-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/dξ=-1/2. Integrating Ez from this point in +/- using the desirable Ez profile yield the

beam profile.

Ezrb

For example, Rb=5, Ez,acc=-1, =6.25-( - 0)

Beam profile design for PWFA-LCBeam profile design for PWFA-LC

Simulation of the first and the last stages of a 19 stages 0.5TeV PWFA

Physical ParametersPhysical Parameters Numerical ParametersNumerical ParametersDrive beam Trailing beam

Beam Charge (1E10e-)

0.82 + 3.65

1.62

Beam Length (micron)

13.4 + 44.7

22.35

Emittance (mm mrad)

10 / 62.9 62.9

Plasma density (1E16 cm-3)

5.66

Plasma Length (m)

0.7

Transformer ratio

1.2

Loaded wake (GeV/m)

45 GeV/m

Beam particle

8.4 E6 x 3

Time step 60 kp-1

Total step 520

Box size 1000x1000x272

Grids 1024x1024x256

Plasma particle

4 / cell

PWFA-LC simulation setupPWFA-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 m s = 0.7 m

s = 0 m s = 0.23 m s = 0.47 m s = 0.7 m

Matched propagation

Simulations of 25/475 GeV stages Simulations of 25/475 GeV stages

s = 0 m s = 0.47 m s = 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 Simulation of 25/475 GeV stages

LWFA design with externally injected beamLWFA design with externally injected beam

Theory predicts P, QP1/2

P=15TW P=30TW P=60TW

W. Lu et. al., Phys.Rev. ST Accel. Beams 10,

061301 (2007)

LWFA design with externally injected beamLWFA design with externally injected beam

SummarySummary

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

plasma response

Instruction pipeline Software pipeline

Operand Instruction stream Plasma slice

Operation IF, ID, EX, MEM, WB

Plasma/beam update

Stages 5 ~ 31 1 ~(# of slices)

beam

solve plasma response

update beam

Initial plasma slab

Without pipelining: Beam is not advanced until entire plasma response is determined


update beam


update beam


update beam


update beam

beam

1 2 3 4

With pipelining: Each section is updated when its input is ready, the plasma slab flows in the pipeline.

Initial plasma slab

Pipelining: scaling QuickPIC Pipelining: scaling QuickPIC to 10,000+ processorsto 10,000+ processors

More detailsMore details

Step 1

Stage 1Stage 1

Stage 3Stage 3

Stage 4Stage 4

Stage 2Stage 2

Step 2 Step 4Step 3

Plasma Plasma updateupdate

BeamBeamupdateupdate

Plasma slice Guard cell Particles leaving partition

Computation in each block is also parallelized

Time

Stage

Basic Quasi-Static mode

0

2

4

6

8

10

12

14

0 50 100 150number of processor used

com

puta

tion

spe

ed u

p non-pipelinePipeline(4procs/group)

Performance in pipeline modePerformance 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.

256 256 1024 cells 64 64 1024 cells

Feng et al, submitted to JCP

Modeling Externally Injected Beam in Modeling Externally Injected Beam in Laser Wakefield AccelerationLaser Wakefield Acceleration

Due to photon deceleration, verified with 2D OSIRIS

We need plasma channel to guide

Too low for ultrarelativistic blowout theory to work

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