A. Pukhov Institute for Theoretical Physics I University of Dusseldorf, Germany Particle-in-Cell Codes for plasma-based particle acceleration Outline Relativistic plasmas, acceleration and the simulation tools Explicit PIC codes Lorentz boost Quasi-static approximation Hybrid methods 1
Laser technology and new physics 10 30 Physics of vacuum, ee λ ~ mc C izest, ZetaWatt Laser Intensity, W/cm 10 5 10 0 10 15 10 10 Ultra-relativistic optics, ea>m p c Relativistic plasmas, ea>mc Plasma physics Atomic physics CPA technology e Field ionization: E = a 1960 1970 1980 1990 000 010 B ELI, ExaWatt 11/18/014 pukhov@tp1.uni-duesseldorf.de 3 Important relativistic laser-plasma parameters ea a = mc Dimensionless laser amplitude relativistic when a 1 Iλ =1.37 10 18 W µm /cm ω m 4πe 0 Critical plasma density N c = ~ 10 1 cm -3 S-number S = N an c Gordienko & Pukhov Phys. Plasmas 1, 043109 (005) 18.11.014 pukhov@tp1.uni-duesseldorf.de 4
A. P u kho v, J. P la sm a Ph ysi cs, vol. 61, p a rt 3, p. 45 (1999). Virtual Laser Plasma Lab A. Pukhov, J. Plasma Phys. 61, 45 (1999) Plasma or neutral gas Gas of an arbitrary element can be used. The code VLPL is written in C++, object oriented, parallelized using MPI for Massively Parallel performance 10 9 particles and 10 8 cells can be treated Advanced physics & numerics Fields 1 E 4π B = + j c t c 1 B E = c t Particles dp = qe + dt γ = 1+ p q cγ p B ( mc) Inelastic processes Radiation damping QED effects hybrid hydro model quasi-static approximation 11/18/014 pukhov@tp1.uni-duesseldorf.de 5 The standard PIC time step cycle Deposit j and ρ Advance fields Particles push 18.11.014 pukhov@tp1.uni-duesseldorf.de 6 3
Equations to solve in full electromagnetic codes Ampere s law Faraday s law Poisson s eq. No magnetic dipoles 1 E 4π = B j c t c 1 B = E c t E = 4πρ B = 0 } Dynamic equations } Static equations 18.11.014 pukhov@tp1.uni-duesseldorf.de 7 Charge continuity and locality of the equations Continuity Eq. Ampere s law Poisson s eq. No magnetic dipoles ρ + j = 0 t 1 E 4π = B j c t c E = 4πρ B = 0 } } Local Global equations initial conditions 18.11.014 pukhov@tp1.uni-duesseldorf.de 8 4
Locality of the equations Only particles within the radius of cτ communicate to each other at every time step τ. This allows for efficient parallelization using domain decomposition 18.11.014 pukhov@tp1.uni-duesseldorf.de 9 Distribution function and kinetic equation N-particles distribution function: F t, r,p,..., r, p ) N ( 1 1 N N Nearly ideal plasma: single particle distribution function: f ( t, r, p) f ( t, r,p) p + t mγ r q p f ( t, r,p) + E + B γ mc γ How to solve this equation? p f ( t, r,p) = St 18.11.014 pukhov@tp1.uni-duesseldorf.de 10 5
Eulerian approach, FDTD Vlasov codes 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. 18.11.014 pukhov@tp1.uni-duesseldorf.de 11 Finite elements approach, integrating along characteristics Characteristics of the Vlasov Eq. coincide with the equations of motion Thus, we just push the numerical macroparticles in self-consistent electromagnetic fields dp p = q E B F dt + mc γ + dr p = dt mγ st 18.11.014 pukhov@tp1.uni-duesseldorf.de 1 6
Sampling phase space: numerical macroparticles 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 N 1/. 18.11.014 pukhov@tp1.uni-duesseldorf.de 13 Maxwell solver: Yee lattice The conservative scheme 18.11.014 pukhov@tp1.uni-duesseldorf.de 14 7
Yee lattice and the continuity equation The Yee solver conserves div E and rot E. In Coulomb gauge, e.g.: 1 A E = E + E = ϕ ; E = ϕ; c t However, we don t solve the Poisson Eq. Rather, we solve 1 E 4π = B j c t c We must define the currents correctly to satisfy the continuity Eq ϕ = 4πρ ρ + j = 0 t 18.11.014 pukhov@tp1.uni-duesseldorf.de 15 Numerical macro-particles We define the charge density at the centers of the grid cells ρ ( r ) ρ ρ i+ 1/. j+ 1/, k + 1/ = Wn S ri + 1/, j+ 1/, k + 1/ n The macro-particles may have arbitrary shapes S. The most common shapes are bricks y x S S ρ ρ i ρ ρ ρ ( r) = S ( x) S ( y) S ( z) ( r ) i = x n y ri, z ri < 1 i i 18.11.014 pukhov@tp1.uni-duesseldorf.de 16 8
Pushing macro-particles in a cell We assume the second order numerical scheme. Thus, the particle trajectory within the time step τ is a straight line. δy δx Jτ = = Ω Ω dω dω τ 0 VW r+ r r W ρ ρ S S ρ ρ dt ( r r) i+ 1/, j+ 1/, k + 1/ dr 18.11.014 pukhov@tp1.uni-duesseldorf.de 17 Villacenor, Buneman CPC 69, 306 (199) Pushing macro-particles in a cell J J J For the simple brick shape, the equations are: r ρ aα = 1 = δxw a a + b x i, j+ 1/, k + 1/ y i+ 1/, j, k + 1/ z i+ 1/, j+ 1/, k = δyw = δzw ρ ρ ( ) y ( a a + b ) x ( a a + b ) Leap-frog field pusher: E n+ 1 x = E z z y n yz xz xy + cτ B n+ 1/ b αβ = α, β = 4πτJ 1 1 δx δx α α { x, y, z} n+ 1/ + 0.5 β α α 18.11.014 pukhov@tp1.uni-duesseldorf.de 18 9
Pushing macro-particles across cells In its motion within one single time step, the particle can cross several cells. The algorithm must handle all these cases. δy δx 18.11.014 pukhov@tp1.uni-duesseldorf.de 19 Interpolating fields to the particle positions 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 Ey E x E x Ey This scheme conserves momentum but not energy 18.11.014 pukhov@tp1.uni-duesseldorf.de 0 10
Interpolating fields to the particle positions Alternatively, one can interpolate the electric fields in the same way as one deposits currents Ey E x E x Ey This scheme conserves energy, but not momentum When the particle crosses the cell boundaries, all conservations fail 18.11.014 pukhov@tp1.uni-duesseldorf.de 1 Energy (in-)conservation in PIC The total energy: 1 E = mnc ( γ 1) + ( E + B ) n 8π V dv Debye length resolved: h=0.5r D Cold plasma: h=10-3 r D 18.11.014 pukhov@tp1.uni-duesseldorf.de 11
Numerical dispersion in the Yee Maxwell solver Numerical waves do not propagate with the speed of light. Even in vacuum. The standard Yee dispersion relation is: sin τ ωτ ˆ p = ω 4 1 + sin h kxh 1 + sin h 1 + sin h 1 x y y x 1 ˆ p 1 1 1 The stability condition is: > ω + + + τ 4 h h h y k h x y z z kzh z 18.11.014 pukhov@tp1.uni-duesseldorf.de 3 Numerical dispersion in the Yee Maxwell solver ω cτ = 0.99h cτ = 0.01h Numerical frequency is lower than the real one this leads to lower phase velocity k 18.11.014 pukhov@tp1.uni-duesseldorf.de 4 1
Numerical dispersion in the Yee Maxwell solver ω cτ = 0.01h cτ = 0.99h Waves with the shortest wavelengths are very slow! Small time steps lead to a bad dispersion 18.11.014 pukhov@tp1.uni-duesseldorf.de 5 k Implementation of a PIC code Single processor optimization The central problem of a PIC code: particles must interact with the array of cells 18.11.014 pukhov@tp1.uni-duesseldorf.de 6 13
Implementation of a PIC code Single processor optimization The linked list of particles with the bases in the cells help to optimize the cache and memory usage 18.11.014 pukhov@tp1.uni-duesseldorf.de 7 Implementation of a PIC code Parallelization: domain decomposition 18.11.014 pukhov@tp1.uni-duesseldorf.de 8 14
Implementation of a PIC code Parallelization: domain decomposition Algorithm locality: only guard cells must communicate between processors. No global communications 18.11.014 pukhov@tp1.uni-duesseldorf.de 9 Possibility for load balancing The adaptive partitioning according to the number of particles per processor 18.11.014 pukhov@tp1.uni-duesseldorf.de 30 15
PIC codes for plasma-based acceleration Bubble regime A.Pukhov & J.Meyer-ter-Vehn, Appl. Phys. B, 74, p.355 (00) 18.11.014 pukhov@tp1.uni-duesseldorf.de 31 The multi-scale problem The plasma acceleration has several very disparate scales 1. Small scale: laser wavelength λ or plasma wavelength λ p. Medium scale: driver length 3. Large scale: acceleration length L A =10 4 10 7 λ, λ p 18.11.014 pukhov@tp1.uni-duesseldorf.de 3 16
Energy gain and simulation time 18.11.014 pukhov@tp1.uni-duesseldorf.de 33 Energy gain and simulation time 18.11.014 pukhov@tp1.uni-duesseldorf.de 34 17
Energy gain and simulation time 18.11.014 pukhov@tp1.uni-duesseldorf.de 35 Energy gain and simulation time 18.11.014 pukhov@tp1.uni-duesseldorf.de 36 18
Energy gain and simulation time 18.11.014 pukhov@tp1.uni-duesseldorf.de 37 Numerical Cerenkov Instability 18.11.014 pukhov@tp1.uni-duesseldorf.de 38 19
Numerical Cerenkov Instability 18.11.014 pukhov@tp1.uni-duesseldorf.de 39 Numerical Cerenkov Instability 18.11.014 pukhov@tp1.uni-duesseldorf.de 40 0
Numerical Cerenkov Instability persists even for dispersion-free solvers 18.11.014 pukhov@tp1.uni-duesseldorf.de 41 Numerical Cerenkov Instability persists even for dispersion-free solvers 18.11.014 pukhov@tp1.uni-duesseldorf.de 4 1
Numerical Cerenkov Instability persists even for dispersion-free solvers 18.11.014 pukhov@tp1.uni-duesseldorf.de 43 Numerical Cerenkov Instability persists even for dispersion-free solvers 18.11.014 pukhov@tp1.uni-duesseldorf.de 44
Numerical Cerenkov Instability is mitigated by filtering 18.11.014 pukhov@tp1.uni-duesseldorf.de 45 A Pukhov et al. 014 preprint arxiv:1408.0155, accepted in PRL Electron acceleration in a channel: towards high quality acceleration 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 18.11.014 pukhov@tp1.uni-duesseldorf.de 46 3
A Pukhov et al. 014 preprint arxiv:1408.0155, accepted in PRL Electron acceleration in a channel: towards high quality acceleration 3D PIC simulation in a Lorentz-boosted frame Energy gain: 4 GeV Laser pulse: 140 J, 16 fs Plasma: 3x10 17 1/cm 3. Acceleration distance: 100 cm 18.11.014 pukhov@tp1.uni-duesseldorf.de 47 A Pukhov et al. 014 preprint arxiv:1408.0155, accepted in PRL Electron acceleration in a channel: Field Reversed Bubble The transverse electric field reverses Uniform plasma Deep channel 18.11.014 pukhov@tp1.uni-duesseldorf.de 48 4
Electron acceleration in a channel: towards high quality acceleration 18.11.014 pukhov@tp1.uni-duesseldorf.de 49 Analytical methods to bridge the scales gap 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. 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 18.11.014 pukhov@tp1.uni-duesseldorf.de 50 5
Envelope approximation for the laser When the laser pulse amplitude changes slowly on the laser wavelength scale, the envelope approximation can be used ik c t 0 + + ς A L 4πq = mc n γ A L This approximation excludes, e.g. sharp fronts of laser pulses. 18.11.014 pukhov@tp1.uni-duesseldorf.de 51 Particle motion in ponderomotive approximation The laser pulse acts on particles then via its ponderomotive force only dp p q = q dt E + B mc γ γm A L This approximation works only when the particle excursion length is smaller than the laser pulse focal spot. 18.11.014 pukhov@tp1.uni-duesseldorf.de 5 6
D.F.Gordon et al. IEEE TPS 8, 14 (000) Ponderomotive guiding center PIC Fields are separated in low frequency and high frequency components. The LF is treated by standard PIC, the HF is treated in envelope approximation Deposit j and ρ Advance LF fields Lorentz push + Calculate <nq /γm> Advance HF fields Ponderomotive push 18.11.014 pukhov@tp1.uni-duesseldorf.de 53 Quasi-static approximation E z z We separate the fast coordinate ζ=z-ct and the slow evolution time τ=t. t << ζ It is assumed that the driver does not change during the time it passes its own length 18.11.014 pukhov@tp1.uni-duesseldorf.de 54 7
Quasi-static approximation Integral of motion Particles have an additional integral of motion in the QS approximation: H = γ p + qψ =1 z Where the wake potential is and we use the axial gauge so that ψ = ϕ A z ϕ A =ψ / = z dς 1 qψ = vz 1 = dt γ 18.11.014 pukhov@tp1.uni-duesseldorf.de 55 Quasi-static approximation Field equations 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 (), so that t = ξ τ B = j + E E = B B = 0 E = ρ () (3) (4) Thus, the quasi-static approximation cannot treat radiation anymore! t t (1) 18.11.014 pukhov@tp1.uni-duesseldorf.de 56 8
Quasi-static approximation Field equations It is possible to write the quasi-static equations on the fields only (K.V.Lotov 007): E = ρ j E = j B = j 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. 18.11.014 pukhov@tp1.uni-duesseldorf.de 57 Pushing particles 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 18.11.014 pukhov@tp1.uni-duesseldorf.de 58 9
30 Quasi-static approximation Equations of motion 18.11.014 pukhov@tp1.uni-duesseldorf.de 59 We push the particles along the ζ coordinate ( ) ς ς ς d d v dt d d d dt d z p p p 1 = = so that ( ) ς ς ς d d v dt d d d dt d z r r r v 1 = = = + = 1 1 L z A q mc q v d d γ γ ς B p E p 11/18/014 pukhov@tp1.uni-duesseldorf.de 60
Compare quasi-static qs-vlpl3d vs explicit VLPL3D Ne E 18.11.014 pukhov@tp1.uni-duesseldorf.de 61 Example Compare quasi-static VLPL3D vs explicit E B 18.11.014 pukhov@tp1.uni-duesseldorf.de 6 31
Efficiency of the PIC codes Explicit PIC effort (number of operations): N op 0 / c, hx << λ0 = N τ << λ, h, h << λ y τ N z x N y N z p Lorentz-boost: relativistic gain up to γ Quasi-static codes gain in performance as τ << λ / cλ, 0 h << λ, h, h << λ p ( λ / λ ) 3 p 0 x p y z p 18.11.014 pukhov@tp1.uni-duesseldorf.de 63 Hybrid PIC/hydro codes 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 18.11.014 pukhov@tp1.uni-duesseldorf.de 64 3
Hybrid PIC/hydro codes 18.11.014 pukhov@tp1.uni-duesseldorf.de 65 Tuckmantel et al., IEEE TPS 38 (01) Hybrid PIC/hydro codes 18.11.014 pukhov@tp1.uni-duesseldorf.de 66 33
Tuckmantel et al., IEEE TPS 38 (01) Hybrid PIC/hydro codes 18.11.014 pukhov@tp1.uni-duesseldorf.de 67 Hybrid PIC/hydro codes 18.11.014 pukhov@tp1.uni-duesseldorf.de 68 34
Hybrid PIC/hydro codes www.cern.ch/awake 18.11.014 pukhov@tp1.uni-duesseldorf.de 69 Hybrid PIC/hydro codes T Tückmantel, 18.11.014 A Pukhov Journal pukhov@tp1.uni-duesseldorf.de of Computational Physics 69, 168-180 (014) 70 35
Hybrid PIC/hydro codes 18.11.014 pukhov@tp1.uni-duesseldorf.de 71 T Tückmantel, A Pukhov Journal of Computational Physics 69, 168-180 (014) Hybrid PIC/hydro codes 18.11.014 pukhov@tp1.uni-duesseldorf.de 7 36
Hybrid PIC/hydro codes 18.11.014 pukhov@tp1.uni-duesseldorf.de 73 T Tückmantel, A Pukhov Journal of Computational Physics 69, 168-180 (014) Hybrid PIC/hydro codes 18.11.014 pukhov@tp1.uni-duesseldorf.de 74 T Tückmantel, A Pukhov Journal of Computational Physics 69, 168-180 (014) 37
Hybrid PIC/hydro codes 18.11.014 pukhov@tp1.uni-duesseldorf.de 75 T Tückmantel, A Pukhov Journal of Computational Physics 69, 168-180 (014) Hybrid PIC/hydro codes 18.11.014 pukhov@tp1.uni-duesseldorf.de 76 38
Summary 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 11/18/014 pukhov@tp1.uni-duesseldorf.de 77 39