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Electron Acceleration in Laser Plasma
Vojt�ech Horn�y
IPP AV CR
4th December 2014, Praha
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 1 / 34
List of Contents
1 Motivation
2 Physics of electron acceleration in laser plasma
3 Particle-in-cell
4 Own simulation
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 2 / 34
Conventional way of electron accelerationClassical electron accelerators
• betatrons (pioneers, 1940, up to 300 MeV)
• synchrotrons (GeV+), rather for X-ray radiation generationdE/dz ∼ E4/(m4R2)
• Linacs (SLAC, 3.2 km, 90 GeV electrons)
• Limit accelerating �eld < 100 MV/m
Example
Diamond Synchrotron
Lenght: 150 mElectron energy: 33 GeVCosts: 13 GK�c
Operating since 2007. Located in Oxfordshire, England.Image from ianvisits.co.uk
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 3 / 34
Advantages and disadvantages of classical accelerators
Advantages
• quasimonoenergetical resulting electron energy distribution
• understood, proven and estabilished technology
• high resulting electron energies
• relatively simple principle
Disadvantages
• huge facilities including several buildings
• high acquisition costs
• high running costs
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 4 / 34
Advantages and disadvantages of classical accelerators
Advantages
• quasimonoenergetical resulting electron energy distribution
• understood, proven and estabilished technology
• high resulting electron energies
• relatively simple principle
Disadvantages
• huge facilities including several buildings
• high acquisition costs
• high running costs
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 4 / 34
Alternative possibility: Electron acceleration in laser plasma
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 5 / 34
Electron acceleration in laser plasmaPrehistory
Breakthrough article (1979, before CPA discovered).Based on simulations, Dawson was a pioneer in PIC computing.
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 6 / 34
Laser Plasma AcceleratorsNew possibillities
• Laser Plasma Accelerators have electric �eld 100 GV/m, i.e. 1,000 ×higher than conventional accelerators
• Implies tens meter's to centimeters reduction in size for same electronenergy - attractive
• To date have always produced broad range of energies which severelylimited application
• Quasimonoenergetic electrons up to GeVs already produced
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 7 / 34
Ponderomotive force, Hora (1969)
Fp = − e2
4meω2∇E2 = −mec
2∇(a2
2
)(1)
• non-linear associated with the intensity gradients in the pulse
• pushes electron and ions out of high-intensity region
• ions are slow, i.e. relativistic plasma wave is formed in underdenseplasma
• its �eld can accelerate electrons
Ponderomotive force
Ponderomotive force drives wake�elds in laser plasma acceleration.
• breaks the quasineutrality of plasma
• generates longitudinal plasma wave
• compromises Lawson-Woodward theorem (EMW does not acceleratecharged particle in vaccuum)
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 8 / 34
Electron acceleration in laser plasmaConditions and expectations
Longitudinal accelerating electric �eld generated by the ponderomotiveforce of an ultrashort and ultraintense laser.
Parameter overview
• electron densities 1018 � 1019 cm−3
• laser intensities higher than 1017 W/cm2
• electric �eld amplitude up to several hundred GV/m
• size of plasma in order if milimeters
• electron energies 10 MeV � several GeV
• energy spread 5-10%
• charge in the electron bunch in order of hundreds of picocoulombs
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 9 / 34
Generation of the wake wavesGeneral idea
Electron oscillation is exited by a force traveling in the plasma at the forcefront. The phase velocity of the wake wave is to the velocity of the forceperturbation.
Simulation by Jean-Luc Vay and Cameron Geddes, Berkeley Lab. newscenter.lbl.gov
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 10 / 34
Generation of the wake waves
Ex =meωpu0
esin(ωpτ)Θ(τ) (2)
ne − n0 = n0u0vf
cos(ωpτ)Θ(τ) (3)
From Macchi, A. A Superintense Laser-Plasma Interaction Theory Primer. (2013).
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 11 / 34
Generation of the wake wavesWake wave in my own simulation
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 12 / 34
Wave-breaking
Electron in wake�eld
Ex = E0 cos(kpx− ωpt) (4)
If electron velocity v → vp, dens maximum diverges, i. e. wave-breaking.Non-relativistic wave-breaking limit
E0 =meωpvp
e(5)
Relativistic wave-breaking limit
E0 =meωpc
e
√2√γp − 1, (6)
where γp refers to phase velocity of wake wave (Gibbon, 2004).
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 13 / 34
Various electron acceleration regimes
1 Laser wake�eld accelerator
2 Plasma beat wave accelerator
3 Multiple laser pulses
4 Self-modulated laser wake�eldaccelerator
5 Blow-out regime
6 Other• "pseudo-resonance"• "forced" laser wake�eld
regimes
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 14 / 34
Laser wake�eld electron acceleration
• Acceleration by short intense laser pulse cτ = λp ( = tens fs).
• Ti:sapphire laser appropriate
• Accelerating distance
Lacc =λ
π
(ω
ωp
)3
(7)
on order of milimeters.
Example
Required electron energy: 100MeV
• ω/ωp = 10
• ne = 1019 cm−3
• λ = 1 µm
• Lacc = 300 µm
Similar to my own experiences gained by PIC simulations.
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 15 / 34
Bubble regime (cavitated wake�eld regime)
Ponderomotive force generated byintense laser pulse expels electronsand creates ion cavity.
Condition for bubble generation
• kpw0 = 2√a0
a
• cτ ∼ λp/2 b
• a0 > 2
aa0 = 0.855√I[1018 W/cm2]× λ2
L[µm]bλp[µm] = 3.34× 1010/
√ne[cm
−3]
Corde, S. et al. Femtosecond x rays from laser-plasmaaccelerators. Rev. Mod. Phys. 85, 1�48 (2013).
Size of the bubble
Size of bubble from the balance between ponderomotive expulsion andCoulomb repulsion R = w0.
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 16 / 34
Bubble regime (cavitated wake�eld regime)Electric �eld
In the rear part of thebubble electronsaccelerated.
In the front part ofthe bubble electronsdecelerated.
Observation
There is an optimum
plasma width
depending on plasmaand pulse parameters.
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 17 / 34
Laser wake�eld electron acceleration
From www.vacet.org
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 18 / 34
Useful scaling laws overviewAccording to Esarey, RMP, 2009
• dephasing length
Ld 'λ3p2λ2×{
1 for a20 � 1
(√
2/π)a0/Np for a20 � 1
• pump depletion length
Lpd 'λ3pλ2×{
2/a20 for a20 � 1
(√
2/π)a0 for a20 � 1
• energy gain if limited by dephasing
Wd(MeV) ' 630I(W/cm2)
n(cm−3)×{
1 for a20 � 1(2/π)/Np for a20 � 1
• energy gain if limited by depletion
Wpd(MeV) '{
3.4× 1021/[λ2(µm)n(cm−3)] for a20 � 1400I(W/cm2)/n(cm−3) for a20 � 1
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 19 / 34
Plasma beat-wave acceleration (PBWA)
Principle
• two longer pulses
• di�erent frequency
• beat equals toplasma frequency
Resonance condition
ω1 − ω2 = ωp
From Wiki From Malka et al., Nature Physics, 2008.
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 20 / 34
Injection of electrons into the bubble
Plasma wave only accelerates electrons; electrons have to be delivered intoa bubble to be accelerated.
Injection mechanisms
1 external injection of electron bunch into bubble• electron buch has to be preaccelerated to achieve e�ective acceleration
from plasma wave
2 self-injection of plasma electrons• ionisation by optical �eld inside the bubble• using second pulse• change of plasma density by bubble shape development
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 21 / 34
Particle-in-cellOverview
PIC method enables to simulate the development of relatively large amountof physical particles using a smart trick.
• Macroparticles representing up to million physical particles areintroduced.
• Advantages of grid and gridless computing connected.
Particle-in-cell cycle
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 22 / 34
Particle-in-cell methodPrinciple
PIC solves Vlasov equation:
∂fs∂t
+ v∂fs∂x
+qsE
ms
∂fs∂v
= 0. (8)
Probability density function of particle of spicies s is sum throughtmacroparticles
fs(x, v, t) =∑p
fp(x, v, t). (9)
Basic idea is to express fp as a certain function with a few free parameters
fp(x, v, t) = NpSx(x− xp(t))Sv(v − vp(t)), (10)
where shape of functions Si is simple (δ, saw, Bessel, . . . ).
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 23 / 34
Particle-in-cell: Equations of motion solver
1 Converting s Vlasov equations into p× s equations for everymacroparticle
∂fp∂t
+ v∂fp∂x
+qsE
ms
∂fp∂v
= 0. (11)
2 Solution should satisfy also several moments of (11)
dNp
dt= 0, (12)
dxpdt
= vp, (13)
dvpdt
=qsms
Ep. (14)
3 Boris scheme or scheme leap-frog used after discretisation.
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 24 / 34
Particle-in-cell: Maxwell's equations solver
Set of Maxwell's equations on the discrete grid
∇ · E =ρ
ε0(15)
∇×E = −∂B∂t
(16)
∇ · B = 0 (17)
∇×B = µ0j + µ0ε0∂E
∂t(18)
is usually solved using
• �nite di�erence method (advanced schemes)
• spectral methods.
• �nite elements method
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 25 / 34
Particle-in-cell: Interpolations
• Macroparticles can be foundanywhere in space.
• Macroscopic quantitiesreprezented only in the gridpoints.
• Interpolation functions are used,e.g. Charge of the particle (in gray) is distributed
among the surrounding nodes. Charge contributedto each node is based on the proximity of theparticle to that node.From http://www.particleincell.com/
W (xi − xp) =
∫dxSx(x− xp) b0
(x− xi
∆x
). (19)
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 26 / 34
Example of own simulation2D PIC simulation for Ti:sapphire system at PALS
Physical parameters
• Gaussian beam
• E = 500 mJ
• λ = 800 nm
• τ = 40 fs
• w0 = 7 µm
• ne = 7×1018 cm−3
• 100µm exponentionaldensity ramp
• L = 3.2 mm
Simulation Parameters
• 2D PIC code EPOCH used
Size of domain
• 100 µm×80µmNumber of particles per λ
• nx = 24
• ny = 8
Technical Details
• 11 hours on 24 CPUs
• computed at MetaCentrum
Video: density n Video: momentum px
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 27 / 34
Example of own simulation: Electron density plot2D PIC simulation for Ti:sapphire system at PALS
Generation of the bubble
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 28 / 34
Example of own simulation: Electron density plot2D PIC simulation for Ti:sapphire system at PALS
Self-injection
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 29 / 34
Example of own simulation: Electron density plot2D PIC simulation for Ti:sapphire system at PALS
Divergence of accelerated beam in vacuum
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 30 / 34
Example of own simulation: Electron density plot2D PIC simulation for Ti:sapphire system at PALS
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 31 / 34
Example of own simulation: Electron density plot2D PIC simulation for Ti:sapphire system at PALS
Electric �eld Ex during propagation
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 32 / 34
ConclusionsRepetitio est mater studiorum
1 Laser-plasma interaction o�ers a new possibility to generate highenergy electron beams cheaper and easier in comparison withconventional accelerators.
2 Electron can be accelerated as a consequence of laser-plasmainteration up to tens or hunderds of MeV even in our Ti:sapphire lasersystem.
3 Bubble regime of acceleration seems to be the most promising way.
4 There is an ideal length of plasma accelerator, limited by dephasing orlaser depletion.
5 Beam characteristics as monochromacity and beam divergence isquestionable.
6 Particle-in-cell method (PIC) o�ers reasonable simulation insight intothe topic.
7 Own simulations related to our Ti:sapphire laser system introduced.
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 33 / 34
Thanks for your attention
• Vojt�ech Horn�y
• �UFP AV �CR
• KFE FJFI �CVUT v Praze
• horny@pals.cas.cz
• kfe.fjfi.cvut.cz/~horny
Presentation available at kfe.fjfi.cvut.cz/~horny
Vojt�ech Horn�y (IPP AV CR) Electron Acceleration in Laser Plasma 4th December 2014, Praha 34 / 34
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