numerical simulation methods for laser target interactions
TRANSCRIPT
Numerical simulation methods for laser target interactions
Jiří Limpouch Czech Technical University in Prague
Faculty of Nuclear Sciences & Physical Engineering Brehova 7, 115 19 Praha 1, Czechia
Thanks to O. Klimo and M. Kucharik for preparation of slides
Courtesy: P. Gibbon, C. Ren
Numerical simulation is not theory! Approximate theoretical models important, even if analytical solution does not exist
What is numerical simulation? • The numerical simulation is: writing a program and using
the computer to perform a numerical experiment, which can show you the time evolution of the nonlinear phenomena.
• Linear problems: analytical solutions are available for most linear problems, so numerical simulation is not necessary. Linear theory provides theoretical basis for validating the simulation results.
• Unfortunately, most plasma-based systems are highly nonlinear. So the numerical simulation could provide a more solid theoretical basis for studying a highly nonlinear and highly relativistic system.
Numerical simulation methods • Plasma dynamics simulations
– Kinetic methods [⇒ distribution function f(r,p,t)] • Particle methods
– Test particle methods – Particle-particle methods (small particle number, inefficient) – Particle-mesh methods – PIC – Mesh-free methods – Tree codes
• Solution of equation for distribution function – Vlasov equation – Fokker-Planck equation (Vlasov-Boltzmann equation)
– Fluid (hydrodynamic) methods [ρ(r,t), u(r,t), T(r,t)] • Two-fluid hydrodynamics (rare) • One-fluid hydrodynamics (often 2 temperatures, radiation, …) • MHD
– Hybrid methods - part of plasma (e.g. electrons or fast electrons) treated e.g. in kinetic way and part (e.g. ions or ions and thermal electrons) e.g. via fluid methods
Numerical simulation methods (continued)
• Other simulations – Preprocessors
• Atomic physics codes – EOS, opacities, atomic physics rates (ionization, recombination, excitation etc)
– Integrated to plasma dynamics • Calculation of plasma ionization state • Populations of excitation states • Nuclear processes
– Post-processors • Simulations of plasma radiation or particle sources
– Line emission from plasma (including detailed atomic physics) – K-α emission (including fast electron transport)
• Simulation of diagnostics results – Synthetic shadowgrams, interferograms etc.
Introduction to particle simulations of plasmas
Particle ( )
Advantage of particle simulation approach
• Particle simulation techniques attempt to model many-body systems by solving the equations of motion of a set of particles.
• Tracking particle trajectories enables us to explore physical effects which are inaccessible to other modeling techniques.
• The method employs the fundamental equations without much approximation, allowing it to retain most of the physics.
• In comparison with Vlasov equation solvers, particle codes are: • Simpler to code up and analyze • More robust • Having a Lagrangian character, which lends them a certain economy • Applicable in 2 or 3D, where Vlasov solvers are generally impractical • However, they are generally noisier than Vlasov solvers
Particle-particle method • Algorithm consists of:
• Loading initial particle positions and velocities • Calculating force on each particle • Integrating of particle equations of motion
• Limitation is the number of arithmetic operations required in the force evaluation scales as N2.
• Therefore: • Particle-particle approach is proper for systems with N<106. • Otherwise, we need to reduce the scaling of the operation count in
the force evaluation below order N2.
Reducing operation count in force evaluation
• Tree code – replace particle-particle by particle-cluster interaction. • Particle-mesh – replace particle-particle by particle-mesh interaction.
Tree code principles
• The potential of a distant group of particles approximated by low-order multipole expansion.
• Tree codes are favored in systems with large density contrast. • Operation count scaling in the force evaluation is below order N×ln(N). • Tree code requires more CPU time than Particle-mesh code, but it does a
better job in resolving small-scale features of the solution. • It is more difficult to treat laser field in Tree code.
Mesh-free methods – Tree codes – less noise
Electromagnetic force calculations
Solving real 3D tasks possible – here laser interaction with
Particle-mesh technique (very wide-spread) • Numerical mesh added to more effectively compute the forces acting on
model particles.
• Force equation replaced with an evaluation based on continuum representa- tions of the charge density and electric field. (Particle-in-Cell - PIC)
• The number of floating point operations in ES1D with FFT for Poisson’s equation scales as αN+βNgln(Ng)+γNg (Ng - number of grid points).
Particle-in-Cell code • „Lagrangian Vlasov‟ with quasi-particles (3D + 3V)
Algorithm for integration of equations of motion
• Convergence – numerical solution converges to the exact solution of the differential equation in the limits as ∆t and ∆x tend to zero.
• Accuracy – truncation error associated with approximating derivatives with differences is minimized.
• Stability – total errors (including truncation error and round-off error) do not grows in time.
• Efficiency – critical consideration since whatever scheme we choose will be used for each particle at each time step.
• Dissipation – the dissipation of some physical quantities caused by the truncation error associated with approximating derivatives with differences should be minimized
• Conservation – the deviation of the conservation laws caused by the truncation error associated with approximating derivatives with differences should be minimized and should not grow in time.
Integration of equations of motion
Simple second order leapfrog achieves the best balance between accuracy, stability and efficiency.
Position (x) and force (F) are known in integer time steps, but not the velocity (v).
Stability of the Leapfrog scheme
For an electrostatic case, if ω0 Δt ≤ 2 (ω0=ωp), the leapfrog scheme is stable.
using finite differences and
Charge assignment and force interpolation
Once we introduce the grid we can no longer view the particles as point particles, this leads naturally to the idea of a finite sized particle.
For which systems does PIC work? • If the force on any particle is dominated by the contributions from its
nearest neighbors. Systems of this type are often called collisional or correlated. Then the force evaluation will be inaccurate unless we make Δ small compared with the typical distance of closest approach. PIC is expensive way to achieve even modestly accurate individual particle trajectories in collisional systems.
• Consider the opposite case, where close neighbors contribute very little to the force on a particle which is dominated by the sum of its interactions with distant particles.
• This type of system is called collisionless or uncorrelated.
• In these cases the important contributions to the force are accurately represented.
• PIC should be appropriate for collisionless, not collisional, systems.
• Many plasmas with collective behaviors (i.e. the Coulomb collisions could be ignored) fit this description.
Supported modes in collisionless plasmas
• For a plasma, the interaction force is a long range force. In principle and in practice the system supports long wavelength modes involving coherent collective motion of many particles.
• The highest characteristic frequency associated with collective modes is the electron plasma frequency ωp.
• The range of wavelengths of these coherent modes is bounded at the lower end by the Debye length λD. Electron thermal motions damp modes with λ<λD so strongly (Landau Damping), that we can say collective modes only exist for λ>λD.
• Particles separated by less than λD see each other as individuals; particles separated by more than λD interact with each other as participants in collective wave modes.
• To filter out the sub-λD scale components of the electric field, Δ=λD.
Categories of PIC models
• Field components: Electrostatic, Magnetostatic, Electromagnetic
• Geometry: 1/2/3D 1/2/3V, boost frame
• Equation of motion: Relativistic or Non-relativistic
• Boundary conditions: Absorbing, reflecting or periodic for both particles and fields
Flow scheme for a typical PIC-MCC code Collisional PIC
• A broad class of physical processes (resonance absorption, vacuum heating, j × B heating, anomalous skin effect) can be simulated in a reduced 1D2/3V geometry, in place of 2D periodic-in-y PIC simulation.
• Transforming to a simulation frame here called the boost frame moving at v0=c sin(θ) parallel to the target surface, where θ is the angle of incidence.
• The wave appears to be normally incident in the boost frame.
Boost frame – oblique incidence in 1D
PML splits the fields into subcomponents and intro-duces loss terms for the electric (σ) and magnetic (σ*) fields to cause the decay of propagating fields. The conductivities chosen to satisfy the matching condition
For particles: • Periodic • Reflecting • Absorbing (Thermal bath)
Boundary conditions
For fields: •Periodic •Conducting •Open, absorbing – Perfectly Matched Layer (PML)
• 1D3V PIC code evolved from LPIC++ code from Garching
• Electromagnetic + relativistic code, oblique incidence via boost frame
• Higher order particle shapes – cubic spline
• Elastic binary collisions approximate Coulomb collisions in real plasma
• Collisional and field ionization via tunneling approximation
• Massively parallel using MPI library
Applications
• Simulations of electron acceleration for K-α emission
• Transport of high-current fast electron beam in dielectric target
• Ion acceleration
• Laser plasma interaction for the conditions of Shock ignition
• Electron acceleration in underdense plasma (LWFA)
Our 1D3V PIC code and its applications
1D3V PIC + 3D time-resolved Monte Carlo simulations (J. Limpouch et al., LPB 22 (2004), 147–156)
I = 4×1016 W/cm2, θ = 30°, Aluminum
Impact of ASE and of ionization (O. Klimo et al., J. Physique IV 133 (2006), 1181-4)
Optical field and collisional ionization added into PIC code Electron density Ne and longitudinal electric field Ex (left), mean ion charge Z (right), Cu target. Maximum of sin2 pulse at t =27τ at x=7, angle of incidence 25° (Z = 11 is Ar-like copper). Conditions of exp. in Max-Born-Institute, Berlin, Ti-sapphire 45 fs, 5 mJ, 1 kHz, focal spot ∅ 6.7 µm, max. I = 1017 W/cm2, Cu foil target
• Ionization front and the processes taking place at the head of the beam
• Field ionization, acceleration of the return current, collisional ionization – avalanche, Ohmic heating, space-charge neutralization
• Dependence of the beam propagation on the beam density
Electron beam propagation in dielectrics O. Klimo et al., Phys. Rev E 75, 016403 (2007)
Phase space evolution: a) initial, b) n=1018 cm-3, c) n=1019 cm-3, d) n=1020 cm-3
Electron beam propagation in a dielectric
a) b) c) d)
Solution of equation for distribution function • Vlasov equation
– Computationally more demanding, originally 1D codes, but 2D simulations are performed at present
– Absence of noise, preferential when distribution tail matters – Trend to formation of small scale structures in configuration space,
small number of artificial collisions usually introduced
The phase space of relativistic KEEN wave in Vlasov simulations of stimulated Raman scattering (A. Ghizzo et al., Phys. Rev. E 74 (2006), 046407)
Phase-space contour plot of electron distribution function evolution at time (f) ωpet = 1100. (M. Masek, K. Rohlena, Eur. Phys. J. D (2009), 79-90.)
Collisional kinetic equations • Kinetic equations may use various collision term – Fokker-
Planck is the most frequent (Boltzmann; Lenard-Balescu) • Overdense plasma between critical and ablation surface is
usually highly collisional • Studied effects (most frequently)
– Non-local electron heat transport – Collisional (inverse bremstrahlung) laser absorption and its impact on
electron distribution – Impact of atomic processes (ionization, excitation,…)
• Types of codes – Vlasov-Fokker-Planck codes – fields calculated from field equations
(Poisson eq. or Maxwell’s eqn.) – high frequency processes included – Quasineutral Fokker-Planck codes – electric field calculated from
quasineutrality condition – only low frequency processes modelled
Fokker-Planck equation
where FP collisional term on the right hand side is 22 2
2 30
v v vln d
8 vab ij abi abja a b a b
ab b b abc ai ab aj bj
f q q f fp f ft p p p
δπε
−∂ ∂ ∂∂ = Λ − ∂ ∂ ∂ ∂ ∑ ∫
In non-relativistic case – velocity is usually used instead of momentum Distribution function is often written as series of spherical harmonics in velocity (eigenfunctions of FP collision operator)
( , , ) v ( , , ) ( , , ) a aa r a p a
bc cab
f ff r p t f r p t F f r p tt t t
∂ ∂∂ + ∇ + ∇ = = ∂ ∂ ∂ ∑
0 1 22
v vv 1( , v, ) ( , v, ) ( , v, ) ( , v, ) ....v v 3
i jij ijf r t f r t f r t f r tδ
= + + − +
In 1D case 0
( , v, ) ( , v, ) ( ) , where v / vk k xl
f x t f x t P µ µ∞
=
= =∑
In the simplest case only 2 first terms 0 1
v( , v, ) ( , v, ) ( , v, )v
xf x t f x t f x t= +
Numerical scheme • Discretization - method of alternating directions • Inner points
plus spatial and velocity boundaries plus edges
dotted – Spitzer-Harm
2D Fokker-Planck simulation of target heating by nonuniform laser irradiation
Thermal flux normalized on SH value for different
22oscv
vTe
ZZ
α
=
Fluid description • Variables are moments of the distribution function • Density
• Average velocity
• Thermal (internal) kinetic energy
• PDE in
• Less computationally demanding, can treat any state of matter, enables description of solid target
• Non-linear laser-target interaction treated only in phenomenological way, many processes have to be included using analytical models or experience from kinetic simulations
• Ofter 1-fluid (due to quasineutrality) 2-temperature HD
dva an f= ∫
1 vdva aa
w fn
= ∫
2(v ) dv2
aka a a
m f wε = −∫
( ), , not in vr t
movies\sedov\output\sedov.html
Lagrangian fluid simulation result (interaction of subnanosecond pulse with foil)
0.540 0.545 0.550 0.555 0.5600.0
0.5
1.0
1.5
2.0
wavelength (nm)
Emitt
ed e
nerg
y de
nsity
(arb
. uni
ts) Experiment
Simulation
⇓ LASER ⇓ LASER O. Renner, J. Limpouch et al., JQSRT 81 (2003), 385-394.
Ly-δ Ly-ε Ly-ζ
(a) Density profile, (b) Temperature profile – in time of maximum emission of studied lines - 300 ps after 400-ps laser pulse maximum. 2D Lagrangian fluid simulation in cylindrical geometry. Laser beam λ = 0.439 µm (3ω) of radius 40 µm normally incident on 5 µm-thick Al foil, laser energy 58 J, Imax= 3×1015 W/cm2
Time-integrated spectrum in region of lines of H-like Al emitted 50 µm behind original foil position tangentially to the s foil – measurement VJS and post-processor XEPAP
ρc≈0.02
movies\LagrxALE\impact_remeshed_11mum_240J_third_cyl_ALE.html
Detail of critical region at the impact edge
Conclusions • Extremely broad range of simulation types is
used for laser plasma interaction physics • Main types of plasma dynamic simulations are
– Kinetic models - simulate detailed non-linear physics of interaction, but cannot treat global evolution of dense targets, kinetic models are subdivided to
• Particle models • Solution of kinetic equations
– Fluid models – more simple, can treat solid targets globally, but interaction physics is described only phenomenologically
Thank you for attention