4th December 2014, Praha - cvut.czkfe.fjfi.cvut.cz/~horny/LPA.pdf · 4th December 2014, Praha yech...

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


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


ianvisits.co.uk

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


Alternative possibility: Electron acceleration in laser plasma


Electron acceleration in laser plasmaPrehistory

Breakthrough article (1979, before CPA discovered).Based on simulations, Dawson was a pioneer in PIC computing.


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


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)


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


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


http://1t2src2grpd01c037d42usfb.wpengine.netdna-cdn.com/wp-content/uploads/sites/2/2011/08/laser-wakefield1-300x110.jpg

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).


Generation of the wake wavesWake wave in my own simulation


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).


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


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.


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.


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.


Laser wake�eld electron acceleration

From www.vacet.org


www.vacet.org

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


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.


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


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


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, . . . ).


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.


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


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)


http://www.particleincell.com/

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


img/video_n.mp4

img/video_px.mp4

Example of own simulation: Electron density plot2D PIC simulation for Ti:sapphire system at PALS

Generation of the bubble



Self-injection



Divergence of accelerated beam in vacuum



Electric �eld Ex during propagation


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.


Thanks for your attention

• Vojt�ech Horn�y

• �UFP AV �CR

• KFE FJFI �CVUT v Praze

• [email protected]

• kfe.fjfi.cvut.cz/~horny

Presentation available at kfe.fjfi.cvut.cz/~horny


kfe.fjfi.cvut.cz/~horny

kfe.fjfi.cvut.cz/~horny

4th December 2014, Praha - cvut.czkfe.fjfi.cvut.cz/~horny/LPA.pdf · 4th December 2014, Praha yech...

Documents

Transcript of 4th December 2014, Praha - cvut.czkfe.fjfi.cvut.cz/~horny/LPA.pdf · 4th December 2014, Praha yech...