chengkun huang ucla quasi-static modeling of beam/laser plasma interactions for particle...
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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 p
nE 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 B
c
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 limitTotal 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 limitTotal time of simulation per GeV
stage (node-hour)
Full EM PIC ~0.05 t< 0.05 0-1 ~ 500000
Ponderomotive
Guiding center PIC
~0.05c/p t< 0.05p-1 ~1500
Quasi-static PIC
~0.05c/p t < 0.05r ~ 10
1
3
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
4
c
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 swhere f is any
f 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 c
d qFor plasma electrons
d 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
10 .
.
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 10
b 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
Laser wakefield simulationLaser wakefield simulation
J. Vieira et. al., IEEE Transactions on Plasma Science, vol.36, no.4, pp.1722-1727, Aug. 2008.
Laser wakefield simulationLaser wakefield simulation
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
Tra
nsve
rse
Siz
e (
µm
)
ne (1014 cm-3)
0
500
1000
1500
2000
0 1 2 3 4 5 6
P390by80BeamSizes6
Tra
nsve
rse
Siz
e (
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
Ch
arg
e F
ractio
n
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
Ch
arg
e F
ractio
n
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 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 ),( 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
|xb|
(mm
) 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 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 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
d
d(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(») =nbR2
b
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 Parameters
Drive 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
OperationIF, ID, EX, MEM,
WBPlasma/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
solve plasma response
update beam
solve plasma response
update beam
solve plasma response
update beam
solve plasma response
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 1Stage 1Stage 1
Stage 3Stage 3Stage 3Stage 3
Stage 4Stage 4Stage 4Stage 4
Stage 2Stage 2Stage 2Stage 2
Step 2 Step 4Step 3
Plasma Plasma updateupdate
Plasma Plasma updateupdate
BeamBeamupdateupdateBeamBeam
updateupdate
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
co
mp
uta
tio
n s
peed
up non-pipeline
Pipeline(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