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18.11.2014
1
Particle-in-Cell Codes for plasma-based
particle acceleration
A. Pukhov
Institute for Theoretical Physics I
University of Dusseldorf, Germany
• Relativistic plasmas, acceleration and the simulation tools
• Explicit PIC codes
• Lorentz boost
• Quasi-static approximation
• Hybrid methods
Outline
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2
Atomic physics
Plasma physics
Relativistic plasmas, eA>mc2
Ultra-relativistic optics, eA>Mpc2
Field ionization: B
2
a
eE =
Physics of vacuum, 22~ mceE Cλ⋅
Laser technology and new physics
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1010
1015
1020
1025
1030
1960 1970 1980 1990 2000 2010
La
ser
Inte
nsi
ty, W
/cm
2 ELI, ExaWatt
CPA technology
iZEST, ZetaWatt
Important relativistic laser-plasma parameters
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Dimensionless laser amplitude2mc
eAa =
relativistic when a ≈ 1 ↔ Iλ2=1.37×1018 W µm2/cm2
S-number caN
NS = Gordienko & Pukhov
Phys. Plasmas 12, 043109 (2005)
Critical plasma density 2
2
0c
4 e
mN
π
ω= ~ 1021 cm-3
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Virtual Laser Plasma Lab
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Plasma or neutral gas
A. P u kho v, J. P la sm a Ph ysi cs , vol . 61, p a r t 3, p . 425 (1999).
Gas of an arbitrary element can be used.
The code VLPL is written in C++, object oriented,
parallelized using MPI for Massively Parallel performance
109 particles and 108 cells can be treated
Advanced physics & numerics
• Inelastic processes
• Radiation damping
• QED effects
• hybrid hydro model
• quasi-static approximationtc
ctc
∂
∂−=×∇
+∂
∂=×∇
BE
jE
B
1
41 π
2
2
)(1
mcp
c
dt
d
+=
×+=
γ
γBpE
p
Fields Particles
A. Pukhov, J. Plasma Phys. 61, 425 (1999)
The standard PIC time step cycle
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Depositj and ρ
Advancefields
Particles push
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Equations to solvein full electromagnetic codes
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Ampere’s law jBE
ctc
π41−×∇=
∂
∂
Faraday’s law EB
×−∇=∂
∂
tc
1
Poisson’s eq. πρ4=⋅∇ E
0=⋅∇ BNo magnetic
dipoles
Dynamic
equations
Static equations
Charge continuityand locality of the equations
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Continuity Eq.
jBE
ctc
π41−×∇=
∂
∂Ampere’s law
0=⋅∇+∂
∂j
t
ρ
Poisson’s eq. πρ4=⋅∇ E
0=⋅∇ BNo magnetic
dipoles Global initial
conditions
Local equations
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Locality of the equations
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Only particles within the radius
of cτ communicate to each other
at every time step τ.
This allows for efficient
parallelization using
domain decomposition
Distribution function and kinetic equation
18.11.2014 pukhov@tp1.uni-duesseldorf.de 10
Sttfmc
qtf
mt
tf=⋅∇
×++⋅∇+
∂
∂),(),(
),(pr,B
pEpr,
ppr,pr
γγγ
N-particles distribution function: ),...,,( 11 NNN tF p,rp,r
Nearly ideal plasma:
single particle distribution function:),( pr,tf
How to solve this equation?
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Eulerian approach, FDTD“Vlasov codes”
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Very inefficient:
a lot of empty phase space
has to be processed.
However, temperature effects
may be described more carefully.
Low noise.
May be subject
to numerical diffusion.
“Finite elements” approach, integrating along characteristics
18.11.2014 pukhov@tp1.uni-duesseldorf.de 12
stmc
qdt
dFB
pE
p+
×+=
γ
γmdt
d pr=
Characteristics of the Vlasov Eq.
coincide with the equations of motion
Thus, we just push
the numerical macroparticles
in self-consistent
electromagnetic fields
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Sampling phase space:numerical macroparticles
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Computationally efficient:
only filled parts of phase space
have to be processed.
However, temperature effects
are poorly described,
or require very many particles.
Noisy as N1/2.
Maxwell solver: Yee latticeThe conservative scheme
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Yee lattice and the continuity equation
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The Yee solver conserves div E and rot E.
In Coulomb gauge, e.g.:
;1
||tc ∂
∂−−∇=+= ⊥
AEEE ϕ ;|| ϕ−∇=E
πρϕ 42 −=∇However, we don’t solve the Poisson Eq.
Rather, we solve jBE
ctc
π41−×∇=
∂
∂
We must define the currents correctly
to satisfy the continuity Eq0=⋅∇+
∂
∂j
t
ρ
Numerical macro-particles
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We define the charge density at the centers of the grid cells
( )∑ −= ++++++n
nkjinkji SW rr 2/1,2/1,2/12/1,2/1.2/1
ρρρ
The macro-particles may have arbitrary shapes S.
The most common shapes are bricks
( ) ( ) ( ) ( )zSySxSS zyx
ρρρρ =r
( )2
||,||
21 ii
i
iii r
rrS
∆<
∆−=ρ
x∆
y∆
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Pushing macro-particles in a cell
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We assume the second order numerical scheme.
Thus, the particle trajectory within the time step τ is a straight line.
yδ
( )∫∫
∫∫∆+
+++
Ω
Ω
−Ω=
Ω=
rr
r
rrr
VJ
dSWd
dtSWd
kji 2/1,2/1,2/1
0
ρρ
τρρτ
xδ
Pushing macro-particles in a cell
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For the simple brick shape, the equations are:
( )( )
( )xyyx
z
kji
xzzx
y
kji
yzzy
x
kji
baazWJ
baayWJ
baaxWJ
+=
+=
+=
++
++
++
ρ
ρ
ρ
δ
δ
δ
,2/1,2/1
2/1,,2/1
2/1,2/1,
zyx
xxb
ra
,,,
12
1
|5.0|21
=
=
∆
∆+−=
βα
δδ βααβ
α
ααα
2/12/11 4 +++ −×∇+= nnnnc JBEE πττ
Leap-frog field pusher:
Villacenor, Buneman CPC 69, 306 (1992)
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Pushing macro-particles across cells
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In its motion within one single time step, the particle can cross
several cells. The algorithm must handle all these cases.
yδ
xδ
Interpolating fields to the particle positions
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One can interpolate the fields to the centers for the cells first
and then interpolate them to the particle center
in the same way as one deposits charges
yE yExE
xEThis scheme “conserves”
momentum but not energy
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Interpolating fields to the particle positions
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Alternatively, one can interpolate the electric fields
in the same way as one deposits currents
yE yExE
xE
This scheme “conserves”
energy, but not momentum
When the particle crosses
the cell boundaries,
all conservations fail
Energy (in-)conservation in PIC
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The total energy: ( ) ( )∫∑ ++−=Vn
n dVBEcm 222
8
11
πγE
Debye length resolved: h=0.5rD Cold plasma: h=10-3rD
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Numerical dispersion in the Yee Maxwell solver
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Numerical waves do not propagate with the speed of light.
Even in vacuum.
The standard Yee dispersion relation is:
2sin
1
2sin
1
2sin
1
4
ˆ
2sin
1 2
2
2
2
2
2
2
2
2
zz
z
yy
y
xx
x
p hk
h
hk
h
hk
h+++=
ωωτ
τ
The stability condition is:222
2
2
111
4
ˆ1
zyx
p
hhh+++>
ω
τ
Numerical dispersion in the Yee Maxwell solver
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cτ = 0.01h
ω
k
Numerical frequency is
lower than the real one
this leads to lower phase
velocity
cτ = 0.99h
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Numerical dispersion in the Yee Maxwell solver
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cτ = 0.01h
ω
k
Waves with
the shortest wavelengths
are very slow!
Small time steps
lead to a bad dispersion
cτ = 0.99h
Implementation of a PIC codeSingle processor optimization
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The central problem
of a PIC code:
particles must
interact with
the array of cells
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Implementation of a PIC codeSingle processor optimization
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The linked list of
particles with the
bases in the cells
help to optimize
the cache and
memory usage
Implementation of a PIC codeParallelization: domain decomposition
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Implementation of a PIC codeParallelization: domain decomposition
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Algorithm locality:
only guard cells must
communicate between
processors.
No global
communications
Possibility for load balancing
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The adaptive partitioning
according to the number
of particles per processor
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PIC codes for plasma-based accelerationBubble regime
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A.Pukhov & J.Meyer-ter-Vehn, Appl. Phys. B, 74, p.355 (2002)
The multi-scale problem
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The plasma acceleration has several very disparate scales
1. Small scale: laser wavelength λ or plasma wavelength λp
2. Medium scale: driver length
3. Large scale: acceleration length LA=104…107 λ, λp
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Energy gain and simulation time
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Energy gain and simulation time
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Energy gain and simulation time
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Energy gain and simulation time
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Energy gain and simulation time
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Numerical Cerenkov Instability
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Numerical Cerenkov Instability
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Numerical Cerenkov Instability
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Numerical Cerenkov Instability persists even for dispersion-free solvers
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Numerical Cerenkov Instability persists even for dispersion-free solvers
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Numerical Cerenkov Instability persists even for dispersion-free solvers
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Numerical Cerenkov Instability persists even for dispersion-free solvers
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Numerical Cerenkov Instability is mitigated by filtering
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Electron acceleration in a channel:towards high quality acceleration
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A channel helps to
moderate the accelerating
field and adjust the laser
depletion length
A region of constant
accelerating field appears
where monoenergetic
acceleration is possible
A Pukhov et al. 2014 preprint arXiv:1408.0155, accepted in PRL
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Electron acceleration in a channel:towards high quality acceleration
18.11.2014 pukhov@tp1.uni-duesseldorf.de 47
3D PIC simulation
in a Lorentz-boosted frame
Energy gain: 24 GeV
Laser pulse: 140 J, 16 fs
Plasma: 3x1017 1/cm3.
Acceleration distance: 100 cm
A Pukhov et al. 2014 preprint arXiv:1408.0155, accepted in PRL
Electron acceleration in a channel:Field Reversed Bubble
18.11.2014 pukhov@tp1.uni-duesseldorf.de 48
The transverse electric field reverses
Uniform plasma Deep channel
A Pukhov et al. 2014 preprint arXiv:1408.0155, accepted in PRL
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Electron acceleration in a channel:towards high quality acceleration
18.11.2014 pukhov@tp1.uni-duesseldorf.de 49
Analytical methods to bridge the scales gap
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First-principles PIC codes are universal but computationally
expensive. Efficient analytical methods exist to handle the
multi-scale problems.
1. Envelope approximation for the laser removes the laser
wavelength λ scale
2. Quasi-static approximation explicitly separates the fast
coordinate ζ=z-ct and the slow evolution time τ=t
Any approximation means some physics is neglected though…
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Envelope approximation for the laser
18.11.2014 pukhov@tp1.uni-duesseldorf.de 51
When the laser pulse amplitude changes slowly on the laser
wavelength scale, the envelope approximation can be used
LL
n
mc
qik
tcAA
γ
π
ς 2
22
0
42=
∇+
∂
∂+
∂
∂
This approximation excludes, e.g. sharp fronts of laser pulses.
Particle motion in ponderomotive approximation
18.11.2014 pukhov@tp1.uni-duesseldorf.de 52
The laser pulse acts on particles then via its ponderomotive
force only
This approximation works only when the particle excursion
length is smaller than the laser pulse focal spot.
22
||2
LAm
q
mcq
dt
d∇−
×+=
γγB
pE
p
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Ponderomotive guiding center PIC
18.11.2014 pukhov@tp1.uni-duesseldorf.de 53
D.F.Gordon et al. IEEE TPS 28, 1224 (2000)
Fields are separated in low frequency and high frequency
components. The LF is treated by standard PIC, the HF is
treated in envelope approximation
Depositj and ρ
AdvanceLF fields
Lorentzpush
Calculate<nq2/γm>
Advance HF fields
Ponderomotivepush
+
Quasi-static approximation
18.11.2014 pukhov@tp1.uni-duesseldorf.de 54
We separate the fast coordinate ζ=z-ct
and the slow evolution time τ=t.
It is assumed that the driver does not change during the time it
passes its own length
ζ∂
∂<<
∂
∂
t
Ez
z
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Quasi-static approximationIntegral of motion
18.11.2014 pukhov@tp1.uni-duesseldorf.de 55
Particles have an additional integral of motion in the QS
approximation:
Where the wake potential is
and we use the axial gauge
so that
1=+−= ψγ qpH z
zA−= ϕψ
2/ψϕ =−= zA
γ
ψς qv
dt
dz
−−=−=
11
Quasi-static approximationField equations
18.11.2014 pukhov@tp1.uni-duesseldorf.de 56
We change variables to the fast coordinate ζ=z-ct
and the slow evolution time τ=t in Maxweel Eqs.:
and neglect
the time derivatives
in (1) and (2), so that
Thus, the quasi-static approximation cannot treat radiation
anymore!
)4(
)3(0
)2(
)1(
ρ=⋅∇
=⋅∇
−∂=×∇
∂+=×∇
E
B
BE
EjB
t
t
τ∂
ξ∂−=∂ t
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Quasi-static approximationField equations
18.11.2014 pukhov@tp1.uni-duesseldorf.de 57
It is possible to write the quasi-static equations on the fields
only (K.V.Lotov 2007):
⊥⊥⊥⊥ ∇−∇=∇ jE ||
2 ρ
jB ×−∇=∇⊥2
⊥⊥⊥ ⋅∇=∇ j||
2E
Alternatively, one can use potentials.
In all cases, the quasi-static equations are not local anymore in
the transverse planes. This is the result of removing the wave
behavior. Instead of the wave equation, we obtain a set of
elliptic equations on the fields.
Pushing particles
18.11.2014 pukhov@tp1.uni-duesseldorf.de 58
In quasi-static codes, one seeds a single layer of particles at the
leading edge of the simulation box and pushes them through
the driver
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Quasi-static approximationEquations of motion
18.11.2014 pukhov@tp1.uni-duesseldorf.de 59
We push the particles along the ζ coordinate
( )ς
ς
ς d
dv
dt
d
d
d
dt
dz
ppp1−==
so that
( )ς
ς
ς d
dv
dt
d
d
d
dt
dz
rrrv 1−===
∇−
×+
−= 2
2
||1
1L
z
Aq
mcq
vd
d
γγςB
pE
p
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Compare quasi-static qs-VLPL3D vs explicit VLPL3D
18.11.2014 pukhov@tp1.uni-duesseldorf.de 61
Ne E||
ExampleCompare quasi-static VLPL3D vs explicit
18.11.2014 pukhov@tp1.uni-duesseldorf.de 62
E B
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Efficiency of the PIC codes
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Explicit PIC effort (number of operations): zyxop NNNNN τ=
pzyx hhhc λλλτ <<<<<< ,,,/ 00
Lorentz-boost: relativistic gain up to2γ
Quasi-static codes
gain in performance as
pzypxp hhhc λλλλτ <<<<<< ,,,/ 0
2
( )30/ λλp
Hybrid PIC/hydro codes
18.11.2014 pukhov@tp1.uni-duesseldorf.de 64
The kinetic PIC description is rather expensive
Alternative: Computational Fluid Dynamics (CFD)
• Euler’s equations, Navier-Stokes equations or variants are
solved on a grid
• Very established field of research - large number of solvers
available
• Faster and more efficient
• Less physics: cannot e.g. model wave breaking
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Hybrid PIC/hydro codes
18.11.2014 pukhov@tp1.uni-duesseldorf.de 65
Hybrid PIC/hydro codes
18.11.2014 pukhov@tp1.uni-duesseldorf.de 66
Tuckmantel et al., IEEE TPS 38 (2012)
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Hybrid PIC/hydro codes
18.11.2014 pukhov@tp1.uni-duesseldorf.de 67
Tuckmantel et al., IEEE TPS 38 (2012)
Hybrid PIC/hydro codes
18.11.2014 pukhov@tp1.uni-duesseldorf.de 68
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Hybrid PIC/hydro codes
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www.cern.ch/awake
Hybrid PIC/hydro codes
18.11.2014 pukhov@tp1.uni-duesseldorf.de 70T Tückmantel, A Pukhov Journal of Computational Physics 269, 168-180 (2014)
18.11.2014
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Hybrid PIC/hydro codes
18.11.2014 pukhov@tp1.uni-duesseldorf.de 71T Tückmantel, A Pukhov Journal of Computational Physics 269, 168-180 (2014)
Hybrid PIC/hydro codes
18.11.2014 pukhov@tp1.uni-duesseldorf.de 72
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Hybrid PIC/hydro codes
18.11.2014 pukhov@tp1.uni-duesseldorf.de 73T Tückmantel, A Pukhov Journal of Computational Physics 269, 168-180 (2014)
Hybrid PIC/hydro codes
18.11.2014 pukhov@tp1.uni-duesseldorf.de 74T Tückmantel, A Pukhov Journal of Computational Physics 269, 168-180 (2014)
18.11.2014
38
Hybrid PIC/hydro codes
18.11.2014 pukhov@tp1.uni-duesseldorf.de 75T Tückmantel, A Pukhov Journal of Computational Physics 269, 168-180 (2014)
Hybrid PIC/hydro codes
18.11.2014 pukhov@tp1.uni-duesseldorf.de 76
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Summary
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PIC codes are the established tools for plasma simulation
Lorentz boost can bridge the gap of scales in plasma acceleration
still keeping the first principles. Unfortunately, it is subject to
numerical instabilities.
Quasi-static codes use analytic approaches to separate the scales
Hybrid hydro-PIC codes help to improve the numerical
dispersion and stability
Recommended