Advanced modeling tools for laser- plasma accelerators (LPAs) 1/3 Carlo Benedetti LBNL, Berkeley, CA, USA (with contributions from R. Lehe, J.-L. Vay, T. Mehrling) Work supported by Office of Science, Office of HEP, US DOE Contract DE-AC02-05CH11231
Course overview ● 3 lectures: Monday, Tuesday, Thursday ● Topics: - [1] The Particle-In-Cell (PIC) method as a tool to study laser- plasma interaction in LPAs; - [2] Limits/challenges of conventional PIC codes; - [3] Tools to speed-up the modeling of LPAs (Lorentz-boosted frame, quasi-static approximation, Fourier-mode decomposition, ponderomotive guiding center description, etc.); 2
Overview of lecture 1 ● Basic physics of laser-plasma accelerators (LPAs); ● The Vlasov-Maxwell (V-M) equations system; ● The PIC approach to solve V-M equations system: – Numerical particles; – The PIC loop; – Force interpolation and current deposition; – Pushing “numerical” particles; – Solving Maxwell's equations on a grid; 3
LPA as compact accelerators Plasma wavelength: Short and intense laser propagating in a plasma: λ p ~ n 0 -1/2 ≈ 10-100 μm, - short → T 0 =L 0 /c ~ λ p /c (tens of fs) for n 0 ≈10 19 -10 17 cm -3 - intense → a 0 =eA 0 /mc 2 ~ 1 (λ 0 =0.8 um, I 0 >10 18 W/cm 2 ) Plasma density electron plasma waves (v phase ~ v laser ) laser ▲ = ponderomotive force: pulse Transverse, k p x F p ~ -grad[I laser ] w 0 ~ λ p → F p displaces electrons v laser ~ c (but not the ions) creating λ p charge separation from which EM fields arise Accel. Decel. T 0 ~ λ p /c Longitudinal wake Longitudinal, k p (z-ct) 4
LPAs produce 1-100 GV/m accelerating gradients + confining forces plasma waves laser wakefield, E z comoving coordinate, ζ E z ~mcω p /e~100 [V/m] x (n 0 [cm -3 ]) 1/2 e.g., n 0 ~10 17 cm -3 , a 0 ~ 1 → E z ~30 GV/m, ~ 10 2 -10 3 larger than conventional RF accelerators 5
Electron bunches to be accelerated in an LPA can be obtained from background plasma → external injection (bunch from a conventional accelerator) Requires: - short (~ fs) bunch generation - precise bunch-laser synchronization Electron bunch to be → trapping of background plasma electrons accelerated * self-injection (requires high-intensity, high plasma density) → limited control Bunch Transverse direction * controlled injection → use laser(s) and/or tailored plasma to manipulate the plasma wave properties and capture background electrons laser - laser-triggered injection (e.g., colliding pulse) 6 - ionization-induced injection - density gradient injection Longitudinal direction 6
Limits to energy gain in a (single stage) LPA ● laser diffraction (~ Rayleigh range) → mitigated by guiding: wakefield plasma channel and/or Z rayleigh = laser self-focusing/self-guiding πw 0 2 /λ 0 e-bunch ● beam-wave dephasing: β bunch ≈ 1, β wave ≈ 1 - λ 0 2 /(2λ p 2 ) plasma waves → slippage L d ≈ (λ p /4)/ (β bunch - β wave ) ~ n 0 -3/2 → mitigated by longitudinal density tailoring laser wakefield, E z ● laser energy depletion → energy loss into v wave plasma wave excitation, L pd ~n 0 -3/2 (L pd ≈ 1 cm for n 0 =10 18 cm -3 ) v bunch comoving coordinate, ζ → Energy gain ~ n 0 -1 7
Schematic of a “typical” LPA experiment + modeling needs Laser pulse [“known”] Diagnostics : Plasma target (gas-jet, gas cell, - laser (e.g., laser capillary, etc..): mode, spectrum, etc.) - Gas dynamics (gas target - bunch (charge, formation; ~ms scalr) spectrum, divergence, etc.) - Plasma formation (discharge, - radiation (betatron, MHD; 1 ns - 100 ns scale) etc.) - Laser-plasma interaction Bunch transport (laser evolution in the plasma, wake (transport optics, formation and evolution, [self-]injection, etc.) bunch dynamics; ~fs → ~ps scale) 8
Schematic of a “typical” LPA experiment + modeling needs Laser pulse [“known”] Diagnostics : Plasma target (gas-jet, gas cell, - laser (e.g., laser capillary, etc..): mode, spectrum, etc.) - Gas dynamics (gas target - bunch (charge, formation; ~ms scalr) spectrum, divergence, etc.) - Plasma formation (discharge, - radiation (betatron, MHD; 1 ns - 100 ns scale) etc.) - Laser-plasma interaction Bunch transport (laser evolution in the plasma, wake (transport optics, formation and evolution, [self-]injection, etc.) ← Computationally bunch dynamics; ~fs → ~ps scale) expensive part! 9
Laser-plasma interaction physics in LPAs described via Maxwell-Vlasov equations ● Statistical* description for the plasma in the 6D ( r , p ) phase-space → phase-space distribution function f s ( r , p ,t)d r d p = # particles (s=electron, ion) located between r and r +d r with a momentum between p and p +d p at time t ● Evolution of the distribution → Vlasov equation (collisionless plasma) Plasma dynamics ● Evolution of the fields E ( r , t), B ( r , t) → Maxwell equations Laser + Wakefield dynamics ● Coupling between Vlasov ↔ Maxwell n( r, t)=∫f s ( r , p ,t)d 3 p 10
Numerical solution of M-V equations requires using grids (spatial, phase-space) to represent physical quantities ● E.g., 3D grid for density L y L x Δx Δy Δz Grid points: N x ≈L x / Δx L z N y ≈L y /Δy N z ≈L z /Δz → N 3D =N x *N y *N z 11 *OSIRIS simulation
Use of a moving computational box greatly reduces memory requirements for LPA simulations Simulation box (moving window) v laser ≈ c v window ≈ c laser wakefield, E z v bunch ≈ c ← 10's-100's um → ← plasma (~mm to ~m scale) → 0 0 1 1 2 N Fixed grid Grid is shifted to follow the laser 0 M M << N Moving grid (window) 12
Numerical solution of the Maxwell-Vlasov equations: direct solution ● Direct numerical solution of MV equations is unfeasible E( r ), B( r ), J( r ), ... → discretized on a 3D spatial grid f s ( r, p, t) → discretized on a 6D = 3D (space) x 3D (momentum) phase-space grid Ex: Plasma: n 0 ~ 10 18 cm -3 → λ p ~ 30 um Laser: λ 0 ~ 1 um, L 0 ~ 10 um, w 0 ~ 30 um 3D spatial grid: 3D momentum grid: L x ~ L y ~ L z ~ 100 um [a few plasma lengths] L px ~ L py ~ 10 mc [transverse] Δx~Δy~λ p /60 [transverse] L pz ~ 2000 mc [longitudinal] Δz~λ 0 /30 [longitudinal] Δpx~Δpy~Δpz~mc/10 [transverse] N px ~N py ~100 , N pz ~20000 N x ~N y ~200 , N z ~3000 → Representing 1 double precision quantity N 3D =N x *N y *N z ~1.2x10 8 points (f s ) on a grid with N 6D points requires > 200 N 6D =N 3D *N px *N py *N pz ~ 2x10 16 points PBytes of memory ==> UNFEASIBLE !!! 13
Numerical solution of the Maxwell-Vlasov equations: particle method (PIC)/1 ● Vlasov equation solved using a particle method (+ 3D spatial grid for the fields) g → “compact support” function ∫g( r )d r =1 f s ( r , p ,t) = (1/N s )∑ k g[ r - r k (t)]δ[ p - p k (t)] δ → Dirac function N s → # “numerical” particles r k (t), p k (t) → phase-space orbit of the k-th Particle “shape” “numerical” particle (Vlasov characteristic) (finite spatial extent) f s ( r , p ,t) p p Sampling with r k (t), p k (t) “numerical” particles r r 14
Numerical solution of the Maxwell-Vlasov equations: particle method (PIC)/2 ● Equation for the characteristics of the Vlasov equation d r k /dt= v k = p k /m γ k d p k /dt= q s [ E k +( v k /c) B k ] where E k = ∫ E (r)g( r - r k )d r , f s ( r , p ,t) = (1/N s )∑ k g[ r - r k (t)]δ[ p - p k (t)] B k = ∫ B (r)g( r - r k )d r ==> evolution of f s described via the motion of a “swarm” of numerical particles ● Expressing the current density using numerical particles J =∑ s (q s /N s )∑ k=0, Ns v k g[ r - r k (t)] 15
Numerical solution of the Maxwell-Vlasov equations: particle method (PIC)/3 ● Example of particle shapes: g( r )=g x (x)g y (y)g z (z) Numerical particles on the spatial grid (“clouds” of charge) Spatial extension = Δx g 0 (x) g 1 (x) Spatial extension = 2Δx g 2 (x) Spatial extension = 3Δx g(x,y)=g 1 (x)g 1 (y) Δy x/Δx [Lapenta] Δx → describes interaction particles ↔ grid → finite spatial extension (to limit number of calculations) → type of shape controls noise in simulation (higher order reduces noise) 16
Memory requirements for the solution of Maxwell- Vlasov equations using the PIC technique Plasma: n 0 ~ 10 18 cm -3 → λ p ~ 30 um Laser: λ 0 ~ 1 um, L 0 ~ 10 um, w 0 ~ 30 um 3D spatial grid: L x ~ L y ~ L z ~ 100 um [a few plasma lengths] Δx~Δy~λ p /60 [transverse] Δz~λ 0 /30 [longitudinal] N 3D =N x *N y *N z ~1.2x10 8 points N x ~N y ~200 , N z ~3000 Particles: N tot =N 3D *N ppc ~10 8 -10 10 particles N ppc =1-100 Grid → (9 fields) x (8 bytes) x N 3D ~ 7 GBytes Particles → (6 coordinates) x (8 bytes) x N tot ~ (5-500) GBytes Memory requirements OK!!! Edison @ NERSC (10 5 CPUs) = 360 TBytes, 2.6 Pflops/s 17
Resolution in momentum space depends on number of “numerical” particles per cell many PPC few PPC p p structures ??? x x Δ x Δ x 18
The PIC loop: self-consistent solution of Maxwell-Vlasov equations Load initial EM Load initial Initial fields on the grid particle distribution condition → Force interpolation (E, B) i,j F k Push particle Evolve E, B (solution of Maxwell's equations) Δt Current deposition (r k ,p k ) J i,j 19
The PIC loop: self-consistent solution of Maxwell-Vlasov equations Load initial EM Load initial Initial fields on the grid particle distribution condition → Force interpolation (E, B) i,j F k Push particle Evolve E, B (solution of Maxwell's equations) Δt Current deposition (r k ,p k ) J i,j 20
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