Simulation of the laser plasma interaction with the PIC code ALaDyn - PowerPoint PPT Presentation

Simulation of the laser plasma interaction with the PIC code ALaDyn Carlo Benedetti Department of Physics, University of Bologna & INFN/Bologna, ITALY Oxford, November 20, 2008 – p.1/58

Overview of the presentation 1. Presentation of ALaDyn 2. Relevant features of ALaDyn 3. Benchmarks of the code 4. Application I : ALaDyn @ AO-FEL 5. Application II : ALaDyn @ PLASMONX 6. Conclusions and outlooks Oxford, November 20, 2008 – p.2/58

1. Presentation of ALaDyn Oxford, November 20, 2008 – p.3/58

1. Presentation of ALaDyn : general features = A cceleration by La ser and Dyn amics of charged particles ALaDyn born in 2007 fully self-consistent, relativistic EM-PIC code “virtual-lab”: laser pulse(s) + injected bunch(es) + plasma ⇒ defined by the user written in C / F 90 , parallelized with MPI, organized as a LIBRARY the (same) code works in 1D, 2D and 3D Cartesian geometry relevant features: low/high order schemes in space/time + moving window + stretched grid + boosted Lorentz frame + hierarchical particle sampling devel. & maintain. @ Dep. of Phys. - UniBo for the INFN-CNR PlasmonX collaboration ” ALaDyn -philosophy” : IMPROVE algorithms/numerical schemes to REDUCE computational requirements ⇒ run 2D/3D simulations in few hours/days on SMALL CLUSTERS ( < 100 CPUs) with an ACCEPTABLE accuracy Oxford, November 20, 2008 – p.4/58

1. Presentation of ALaDyn : basic equations Maxwell Equations [ME] Vlasov Equation [VE], f s ( s = e, i, · · · ) ∂ B 8 ∂t = − c ∇ × E > > ∂f s ∂t + v · ∂f s · ∂f s E + v < “ ” ⇔ ∂ r + q s c × B ∂ p = 0 > ∂ E > R ∂t = c ∇ × B − 4 π P s q s v f s d p : ∇ · B ( t ) = 0 if ∇ · B (0) = 0 ⇒ ∇ · E ( t ) = 4 πρ ( t ) if ∇ · E (0) = 4 πρ (0) and ∂ρ ⇒ ∂t + ∇ · J = 0 • fields E , B , J → discretized on a grid with N x × N y × N z = 10 7 − 8 points • num. particles ( r i , p i ) → sample the phase space distribution ( ∼ 10 8 − 9 particles): N ( s ) p q ( s ) δ ( r − r ( s ) ( t )) δ ( p − p ( s ) X q s f s ( r , p , t ) → C Np ( t )) i i i i 8 d r ( s ) = v ( s ) > i > > dt i > > < i = 1 , 2 , · · · , N ( s ) V E [ f s ] ⇒ p d p ( s ) > v ( s ) „ « > = q ( s ) E ( r ( s ) × B ( r ( s ) > i i ) + ) > > dt i i i c : Oxford, November 20, 2008 – p.5/58

2. Relevant features of ALaDyn Oxford, November 20, 2008 – p.6/58

2. Relevant features of ALaDyn : high order schemes • Spatial derivatives in the ME ⇒ (compact) high order schemes † Denoting by f i /f ′ i the function/derivative on the i − th grid point i +1 = a f i +1 − f i − 1 + b f i +2 − f i − 2 + c f i +3 − f i − 3 αf ′ i − 1 + f ′ i + αf ′ ( ∗ ) 2 h 4 h 6 h ⇒ relation between a, b, c and α by matching the Taylor expansion of (*) ⇒ if α � = 0 , f ′ i obtained by solving a tri-diagonal linear system ⇒ “classical” 2 nd order: α = b = c = 0 , a = 1 Numerical dispersion relation 1.00 theory 0.75 1. improvement in the spectral accuracy 8comp ω = ω ( k ) ⇒ ( ω /c) h / π � even with few (10-12) points/wavelength 6comp 0.50 6expl the wave phase velocity is well reproduced 4expl 0.25 2expl 0.00 0.00 0.25 0.50 0.75 1.00 k h / π † S.K. Lele, JCP 103 , 16 (1992) Oxford, November 20, 2008 – p.7/58

2. Relevant features of ALaDyn : high order schemes 2. improvement in the isotropy Compact scheme 8 o Explicit scheme 4 o 3.14 3.14 2.4 2.4 0.6 0.6 k y h y� k y h y� 1.6 1.6 0.7 0.7 0.8 0.8 0.8 0.8 0.9 0.9 0.99 0.99 0.00 0.00 0.00 0.8 1.6 2.4 3.14 0.00 0.8 1.6 2.4 3.14 k x h x k x h x • Time integration in the ME & particle Eq. of motion: high accuracy in the spatial derivatives requires high order time integration ⇒ 4 th order Runge-Kutta scheme With high order schemes we can adopt, for a given accuracy, a coarser computational grid allowing to use a higher particles per cell number and a ⇒ ⇐ larger time step compared to standard PIC codes (factor 3-10 gain). Oxford, November 20, 2008 – p.8/58

2. Relevant features of ALaDyn : stretched grid • Stretched grid: high accuracy in the centre (sub- µ m resolution in transv. plane) VS low accuracy in the borders (not interesting!) x i → “physical” transv. coordinate / ξ i → “rescaled” transv. coordinate ξ i unif. distributed x i = α x tan ξ i , α x → “stretching parameter” ( α x → ∞ unif. grid, α x → 0 super-stretched grid) ⇒ Adopting a transverse stretched grid we (considerably) reduce the number of grid points allowing to save memory (keeping fixed the accuracy) com- ⇒ ⇐ pared to an uniform grid (max. gain ∼ 100). Oxford, November 20, 2008 – p.9/58

2. Relevant features of ALaDyn : hierarchical particle sampling • A given particle species ( e.g. electrons) can be sampled by a family of macroparticles with different charge putting more macroparticles in the physically interesting zones (center/high energy tails) and less in the borders... We can reduce the total number of particles involved in the simulation (es- pecially when the stretched grid is enabled) AND decrease the statistical ⇒ ⇐ noise ( i.e. increase the reliability of the results). Oxford, November 20, 2008 – p.10/58

2. Relevant features of ALaDyn : the Boosted Lorentz Frame • The space/time scales spanned by a system are not invariant under Lorentz transform. † ⇒ the “computational complexity” can be reduced changing the reference system Laboratory Frame Boosted Lorentz Frame ( β ∗ ) IMPULSO LASER (P=300 TW) λ 0 → laser wavelength λ ′ 0 = γ ∗ (1 + β ∗ ) λ 0 > λ 0 ℓ → laser length 40 fs ℓ ′ = γ ∗ (1 + β ∗ ) ℓ > ℓ PLASMA L p → plasma length 1.2 mm L ′ p = L p /γ ∗ < L p c ∆ t < ∆ z ≪ λ 0 , λ 0 < ℓ ≪ L p ⇒ t ′ simul ∼ ( L ′ p + ℓ ′ ) / ( c (1 + β ∗ )) ⇒ t simul ∼ ( L p + ℓ ) /c # steps ′ = t ′ L p L p t simul # steps = simul ∝ ≫ 1 ∝ λ 0 γ 2 ∗ (1+ β ∗ ) 2 ∆ t λ 0 ∆ t ′ large # of steps # of steps reduced (1 /γ 2 ∗ ) ⇒ diagnostics is more difficult ( t = cost in the LF � t ′ = cost in the BLF) We can reduce the simulation length changing the reference system (useful ⇒ ⇐ for parameter scan). † J.L. Vay, PRL 98 , 130405 (2007) Oxford, November 20, 2008 – p.11/58

2. Relevant features of ALaDyn : the Boosted Lorentz Frame without BLF [t=46.3 h] with BLF , β ∗ = 0 . 9 [t=8.1 h] 9e+11 500 -Whithout_BLF -Whithout_BLF -With_BLF 6e+11 -With_BLF -Pukhov_Theory 400 3e+11 300 Ez [V/m]� p z /mc� 0 200 -3e+11 100 -6e+11 -9e+11 0 525 550 575 600 625 525 550 575 600 625 z [um] z [um] Oxford, November 20, 2008 – p.12/58

3. Benchmarks of the code Oxford, November 20, 2008 – p.13/58

3. Benchmarks of the code: analytic solutions has been benchmarked against “standard” plasma physics problems • ALaDyn 1.50 Linear Landau damping -theory *simulation 1.00 f e = (1 + 0 . 02 sin( kx )) × √ × exp( − v 2 / 2) / 0.50 2 π (max)� - grid: 16 points 0.00 E x (t)/E x - 10 4 − 10 6 particles/cell -0.50 agreement with -1.00 Vlasov-fluid (512 × 1024) -1.50 0 2 4 6 8 10 ⇓ t’ ⇑ 0 Plasma oscillation - δn/n 0 ∼ 3% -2 - grid: 19 points log (E 1 (t)/E 1 (0))� - 200 particles/cell -4 - ∆ t = T plasma / 15 P = 2 . 52 · 10 14 rad/s ω th -7 -theory P = 2 . 51 · 10 14 rad/s -10^4ppc ω si -10^5ppc error < 0.4 % -10^6ppc -9 0 10 20 30 40 Oxford, November 20, 2008 – p.14/58 t

3. Benchmarks of the code: analytic solutions • 1D EM Solitons in a e + /e − overdense plasma + trapped radiation with CP a 1.2 10.8 1.0 10.7 0.8 10.6 2 ) 1/2� density � 2 +E’ z 0.6 10.5 (E’ y 0.4 10.4 0.2 electron(t=0) 10.3 electron(t=1000) 0 10.2 -30 -15 0 15 30 0 250 500 750 1000 x t’ ⇒ Stationary solution of the VE: f e + = f e − = exp( − β γ ( x,u x )) 1 + | a | 2 + u 2 where γ = x , p 2 K 1 ( β ) a ( x, t ) = a y ( x, t ) + i a z ( x, t ) = a 0 ( x ) exp( iωt ) . The vector potential satisfies q ! q q 1+ a 2 1+ A 2 K 0 ( β 0 ) K 1 ( β 0 ) d 2 a 0 1 0 + 2 dz 2 + ω 2 a 0 = 2 a 0 2 ω 2 A 2 1 + A 2 , − 1 = 0 0 K 1 ( β ) β K 1 ( β ) ⇒ Simulation: grid with 150 points + 10 4 particles/cell the soliton is stable a M. Lontano, et. al , Phys. Plas. 9 /6, 2562 (2002) Oxford, November 20, 2008 – p.15/58

3. Benchmarks of the code: HO vs LO schemes • Test based on the nonlinear LWFA regime: Plasma: DENSITY LASER - first plateau: L 1 = 30 µm , density 10 19 e/cm 3 DENSITY - accelerating plateau: L 2 = 220 µm Laser: - λ 0 = 0 . 8 µ m, P = 60 TW, τ F W HM = 17 fs, w 0 = 16 µ m 300 um High Order (3.8h on 4 CPUs)) • ALaDyn - domain: (60 × 80) µ m 2 , grid: (750 × 200) points ⇒ (10 × 2) points/ λ - plasma sampled with: 20 electrons/cell - derivatives: compact h.o. schemes (8 th order), time evolution: 4 th -order Runge-Kutta Low Order (14h on 4 CPUs) • ALaDyn - domain: (50 × 80) µ m 2 , grid: (1200 × 320) points ⇒ (20 × 3.2) points/ λ - plasma sampled with: 20 electrons/cell - derivatives: 2 nd -order accurate, time evolution: 2 nd -order accurate (leap-frog) N ( grid ) N ( particles ) ∆ t HO HO = HO = 0 . 4 ∆ t LO = 1 . 6 N ( grid ) N ( particles ) LO LO Oxford, November 20, 2008 – p.16/58