physics of high intensity laser plasma · pdf fileshort-wavelength radiation sources. ... a...

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Physics of High IntensityLaser Plasma InteractionsVarenna Summer School onLaser-Plasma Acceleration

20–25 June 2011 Paul Gibbon

Course outline

Lecture 1: Introduction – Definitions and Thresholds

Lecture 2: Interaction with Underdense Plasmas

Lecture 3: Interaction with Solids

Lecture 4: Numerical Simulation of Laser-Plasma Interactions

Lectures 5 & 6: Tutorial on Particle-in-Cell Simulation

2 132

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Physics of High IntensityLaser Plasma InteractionsPart I: Definitions and thresholds


Lecture 1: Definitions and thresholds

Introduction

Field ionization

Relativistic threshold

Plasma Debye length

Plasma frequency

Further reading

Definitions and thresholds 4 132

Laser technology progress: chirped pulseamplification

Electronenergy

2

1970 1980 1990 2000 2010

1010

1510

10

2010

23

Year

1 keV

1 MeV

1 GeV

1 meV

CPA

1 eVMultiphoton physicsLaser medicine

Particle accelerationFusion schemes

ray sourcesγ−

Hot dense matter

Hard X−ray flash lamps

Field ionisation of hydrogen

Intensity (W/cm )

1960

Relativistic ions

Laser−nuclear physics

Relativistic optics: v ~ cosc

Figure 1: Progress in peak intensity since the invention of the laser.Definitions and thresholds Introduction 5 132

Extreme conditions: nonlinear, strong-field science

Ordinary matter — solid, liquid or gas — rapidly ionized whensubjected to high intensity irradiation.

Electrons released are then immediately caught in the laserfield

Oscillate with a characteristic energy which dictates thesubsequent interaction physics.

Basis for laser-based particle accelerator schemes andshort-wavelength radiation sources.

Definitions and thresholds Introduction 6 132

Electrons in intense electromagnetic fieldsprehistory

Volkov (1935): electron ‘dressed’ by field – relativistic massincrease

Schwinger (1949): cyclotron radiation

Invention of laser (1960): theoretical works on electrondynamics

Figure of merit q:

q =eEL

mωc, (1)

e = electron charge, m = electron mass, c = speed of light;EL = laser electric field strength; ω = light frequency.

Ostriker & Gunn (1969) – electron dynamics in vicinity ofpulsars: q ∼ 100

Definitions and thresholds Introduction 7 132

Field ionizationAt the Bohr radius

aB =~2

me2 = 5.3× 10−9 cm,

the electric field strength is:

Ea =e

4πε0a2B

' 5.1× 109 Vm−1. (2)

This leads to the atomic intensity :

Ia =ε0cE2

a

2' 3.51× 1016 Wcm−2. (3)

A laser intensity of IL > Ia will guarantee ionization for any targetmaterial, though in fact this can occur well below this thresholdvalue via multiphoton effects.

Definitions and thresholds Field ionization 8 132

Tunneling ionization

Keldysh (1965) and Perelomov (1966): introduced aparameter γ separating the multiphoton and tunnelingregimes, given by:

γ = ωL

√2Eion

IL∼

√Eion

Φpond

. (4)

where

Φpond =e2E2

L

4mω2L

(5)

is the ponderomotive potential of the laser field.

Regimes

γ < 1⇒ tunneling – strong fields, long wavelengthsγ > 1⇒ MPI – short wavelengths


Tunnelling: barrier suppression model

��

��

��

��

V(x)

0

−E

x

xmax

ε x

ion

−e

Figure 2: a) Schematic picture of tunneling or barrier-suppressionionization by a strong external electric field.


Barrier suppression model II

Coulomb potential modified by a stationary, homogeneouselectric field, see Fig. 13:

V (x) = −Ze2

x− eεx .

⇒ suppressed on RHS of the atom, and for x � xmax is lowerthan the binding energy of the electron.

If the barrier falls below Eion, the electron will escapespontaneously⇒ barrier suppression (BS) ionization.

Set dV (x)/dx = 0 to determine the position of the barrier:

xmax = (Ze/ε),


Barrier suppression model III

Set V (xmax) = Eion to get the threshold field strength for BS:

εc =E2ion

4Ze3 . (6)

Equate critical field to the peak electric field of the laser –appearance intensity for ions created with charge Z :

Iapp =c

8πε2

c =cE4

ion

128πZ 2e6 , (7)

Appearance intensity

Iapp ' 4× 109(

Eion

eV

)4

Z−2 Wcm−2 (8)

Eion is the ionization potential of the ion or atom with charge(Z − 1).


Appearance intensities of selected ions

Ion Eion Iapp

(eV) ( Wcm−2)

H+ 13.61 1.4× 1014

He+ 24.59 1.4× 1015

He2+ 54.42 8.8× 1015

C+ 11.2 6.4× 1013

C4+ 64.5 4.3× 1015

Ne+ 21.6 8.6× 1014

Ne7+ 207.3 1.5× 1017

Ar8+ 143.5 2.6× 1016

Xe+ 12.13 8.6× 1013

Xe8+ 105.9 7.8× 1015

Table 1: BS ionization model –Eq. (8).

Figure 3: Auguste et al., J. Phys. B(1992)


Relativistic field strengths

Classical equation of motion for an electron exposed to a linearlypolarized laser field E = yE0 sinωt :

dvdt

' −eE0

mesinωt

→ v =eE0

meωcosωt = vos cosωt (9)

Dimensionless oscillation amplitude, or ’quiver’ velocity:

a0 ≡vosc≡ pos

mec≡ eE0

meωc(10)

Definitions and thresholds Relativistic threshold 14 132

Relativistic intensity

The laser intensity IL and wavelength λL are related to E0 and ωby:

IL =12ε0cE2

0 ; λL =2πcω

Substituting these into (10) we find (Exercise):

a0 ' 0.85(I18λ2µ)1/2, (11)

where

I18 =IL

1018 Wcm−2 ; λµ =λL

µm.

Implies that for IL ≥ 1018 Wcm−2, λL ' 1 µm, we will haverelativistic electron velocities, or a0 ∼ 1.

Definitions and thresholds Relativistic threshold 15 132

Plasma definitions: the Debye lengthA plasma created by field-ionization of a gas or solid will initiallybe quasi-neutral. This means that the number densities ofelectrons and ions with charge state Z are locally balanced:

ne ' Zni .

Any local disturbance in the charge distribution will be rapidlyneutralised by the lighter, faster electrons – Debye shielding:shields potential around exposed charge: φD = e−r/λD

r .

Debye length

λD =

(ε0kBTe

e2ne

)1/2

= 743(

Te

eV

)1/2( ne

cm−3

)−1/2

cm (12)

Definitions and thresholds Plasma Debye length 16 132

Plasma classificationAn ideal plasma has many particles per Debye sphere:

ND ≡ ne4π3λ3

D � 1. (13)

10 104 107 1010

Temperature (K)

102

106

1010

1014

1018

1024

1028

1032

Ele

ctro

nde

nsity

(cm

-3)

gaseous nebulaionosphere(F layer)

magneticfusion

electron gasin a metal

laser-plasmaslaser-plasmasJovian interior

stellar interiorstellar interior

solaratmosphere

solar corona

white dwarf

non-ideal p

lasmas

Figure 4: Ideal and non-ideal plasmas

Definitions and thresholds Plasma Debye length 17 132

Plasma classificationAn ideal plasma has many particles per Debye sphere:

ND ≡ ne4π3λ3

D � 1. (13)

10 104 107 1010

Temperature (K)

102

106

1010

1014

1018

1024

1028

1032

Ele

ctro

nde

nsity

(cm

-3)


magneticfusion




solaratmosphere

solar corona

white dwarf

non-ideal p

lasmas

Figure 4: Ideal and non-ideal plasmasDefinitions and thresholds Plasma Debye length 17 132

Plasma frequency

For a thermal plasma with temperature Te one can also define acharacteristic reponse time (eg: to disturbances from externallaser fields or particle beams):

tD 'λD

vt=

(ε0kBTe

e2ne· m

kBTe

)1/2

=

(e2ne

ε0me

)−1/2

.

Electron plasma frequency

ωp ≡(

e2ne

ε0me

)1/2

' 5.6× 104(

ne

cm−3

)1/2

s−1. (14)

Definitions and thresholds Plasma frequency 18 132

Underdense and overdense plasmas

If the plasma response time is shorter than the period of aexternal electromagnetic field (such as a laser), then thisradiation will be shielded out.

Figure 5: Underdense, ω > ωp:plasma acts as nonlinearrefractive medium

Figure 6: Overdense, ω < ωp:plasma acts like mirror


The critical density

To make this more quantitative, consider ratio:

ω2p

ω2 =e2ne

ε0me· λ2

4π2c2 .

Setting this to unity defines the wavelength for which ne = nc , orthe

Critical density

nc ' 1021λ−2µ cm−3 (15)

above which radiation with wavelengths λ > λµ will be reflected.cf: radio waves in ionosphere.


Laser-generated plasmas

Target material Electron density ne ne/nc ( 800nm)( cm−3)

Capilliary discharge 1016 10−5

Gas jet 1018 10−3

Foam/aerogel 1021 0.1− 5Frozen H 1022 36CH foil 5× 1023 600

Table 2: Electron densities in typical laser-produced plasmas


Further reading

IC Press, London (2005)

Definitions and thresholds Further reading 22 132

Summary of Lecture 1

Introduction

Field ionization

Relativistic threshold

Plasma Debye length

Plasma frequency

Further reading

Definitions and thresholds Further reading 23 132

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Physics of High IntensityLaser Plasma InteractionsPart II: Interaction with UnderdensePlasmas


Lecture 2: Interaction with Underdense PlasmasPlasma responseCold fluid equations

EM wavesPlasma waves

EM wave propagationNonlinear refractionSelf focussingSF power thresholdPonderomotive channel formation

Plasma wave propagationDispersionNumerical solutionsWave breaking

Wakefield excitationElectron accelerationBench-Top Particle Accelerators

Interaction with Underdense Plasmas 25 132

Ionized gases: when is plasma response important?

Simultaneous field ionization of many atoms produces a plasmawith electron density ne, temperature Te ∼ 1− 10 eV. Collectiveeffects important if

ωpτL > 1

Example (Gas jet)

τL = 100 fs, ne = 1017 cm−3 → ωpτL = 1.8Typical gas jets: P ∼ 1bar; ne = 1018 − 1019 cm−3

Recall that from Eq.15, critical density for glass lasernc(1µ) = 1021 cm−3. Gas-jet plasmas are thereforeunderdense, since ω2/ω2

p = ne/nc � 1.

Exploit plasma effects for: short-wavelength radiation; nonlinearrefractive properties; high electric/magnetic fields.

Interaction with Underdense Plasmas Plasma response 26 132

Wave propagation

The starting point for most analyses of nonlinear wavepropagation phenomena is the Lorentz equation of motion forthe electrons in a cold (Te = 0), unmagnetized plasma, togetherwith Maxwell’s equations.

We also make two assumptions:

1 The ions are initially assumed to be singly charged (Z = 1)and are treated as a immobile (vi = 0), homogeneousbackground with n0 = Zni .

2 Thermal motion is neglected – justified for underdenseplasmas because the temperature remains small comparedto the typical oscillation energy in the laser field.

Interaction with Underdense Plasmas Cold fluid equations 27 132

Lorentz-Maxwell equations

Starting equations (SI units) are as follows

∂p∂t

+ (v · ∇)p = −e(E + v × B), (16)

∇·E =eε0

(n0 − ne), (17)

∇×E = −∂B∂t, (18)

c2∇×B = − eε0

nev +∂E∂t, (19)

∇·B = 0, (20)

where p = γmev and γ = (1 + p2/m2ec2)1/2.

Interaction with Underdense Plasmas Cold fluid equations 28 132

Electromagnetic waves

To simplify matters we first assume a plane-wave geometry likethat above. A laser pulse can thus be described by theelectromagnetic fields EL = (0,Ey , 0); BL = (0, 0,Bz).

Exercise: From Eq. (16) one can show that the transverseelectron momentum is then simply given by:

py = eAy , (21)

where Ey = ∂Ay/∂t . This relation expresses conservation ofcanonical momentum.

Interaction with Underdense Plasmas Cold fluid equations EM waves 29 132

Electromagnetic waves

To simplify matters we first assume a plane-wave geometry likethat above. A laser pulse can thus be described by theelectromagnetic fields EL = (0,Ey , 0); BL = (0, 0,Bz).Exercise: From Eq. (16) one can show that the transverseelectron momentum is then simply given by:

py = eAy , (21)

where Ey = ∂Ay/∂t . This relation expresses conservation ofcanonical momentum.


The EM wave equation I

Substitute E = −∇φ− ∂A/∂t ; B = ∇× A into Ampère Eq.(19):

c2∇× (∇× A) +∂2A∂t2 =

Jε0−∇∂φ

∂t,

where the current J = −enev .

Now we use a bit of vectorial magic, splitting the current intorotational (solenoidal) and irrotational (longitudinal) parts:

J = J⊥ + J || = ∇×Π +∇Ψ

from which we can deduce (see Jackson!):

J || −1c2∇

∂φ

∂t= 0.


The EM wave equation I

Substitute E = −∇φ− ∂A/∂t ; B = ∇× A into Ampère Eq.(19):

c2∇× (∇× A) +∂2A∂t2 =

Jε0−∇∂φ

∂t,

where the current J = −enev .Now we use a bit of vectorial magic, splitting the current intorotational (solenoidal) and irrotational (longitudinal) parts:

J = J⊥ + J || = ∇×Π +∇Ψ

from which we can deduce (see Jackson!):

J || −1c2∇

∂φ

∂t= 0.


The EM wave equation IINow apply Coulomb gauge ∇ · A = 0 and vy = eAy/γ from (21),to finally get:

EM wave

∂2Ay

∂t2 − c2∇2Ay = µ0Jy = − e2ne

ε0meγAy . (22)

The nonlinear source term on the RHS contains two importantbits of physics:

ne = n0 + δn → Coupling to plasma waves

γ =√

1 + p2/m2ec2 → Relativistic effects


Electrostatic (Langmuir) waves I

Taking the longitudinal (x)-component of the momentumequation (16) gives:

ddt

(γmevx ) = −eEx −e2

2meγ

∂A2y

∂x

We can eliminate vx using Ampère’s law (19)x :

0 = − eε0

nevx +∂Ex

∂t,

while the electron density can be determined via Poisson’sequation (17):

ne = n0 −ε0

e∂Ex

∂x.

Interaction with Underdense Plasmas Cold fluid equations Plasma waves 32 132

Electrostatic (Langmuir) waves IIThe above (closed) set of equations can in principle be solvednumerically. For the moment, we simplify things by linearizingthe plasma quantities:

ne ' n0 + n1 + ...

vx ' v1 + v2 + ...

and neglect products like n1v1 etc. This finally leads to:

Driven plasma wave

(∂2

∂t2 +ω2

p

γ0

)Ex = −

ω2pe

2meγ20

∂

∂xA2

y (23)

The driving term on the RHS is the relativistic ponderomotiveforce, with γ0 = (1 + a2

0/2)1/2.


Cold plasma wave equations: recap

Electromagnetic wave

∂2Ay

∂t2 − c2∇2Ay = µ0Jy = − e2ne

ε0meγAy

Electrostatic (Langmuir) wave

(∂2

∂t2 +ω2

p

γ0

)Ex = −

ω2pe

2meγ20

∂

∂xA2

y

These coupled fluid equations describe a vast range of nonlinearlaser-plasma interaction phenomena: parametric instabilities,self-focussing, channelling, wakefield excitation, harmonicgeneration, ...


Dispersion properties: EM waves

This time we switch the plasma oscillations off (ne = n0) inEq.(22) and look for solutions:

Ay = A0 sin(ωt − kx),

to obtain

ω2 =ω2

p

γ0+ c2k2. (24)

From this relation we can derive a

Nonlinear refractive index

η =

√c2k2

ω2 =

(1−

ω2p

γ0ω2

)1/2

(25)

Interaction with Underdense Plasmas EM wave propagation 35 132

Nonlinear refraction effectsHave so far assumed plane wave approximation for laser pulse –’photon bullet’Real laser pulses are created with focusing optics & are subjectto:

1 diffraction due to finite focal spot σL:

ZR = 2πσ2L/λ

2 ionization effects: refraction due to radial density gradients

3 relativistic self-focusing. Power threshold:

Pc ' 17(ω0

ωp

)2

GW, (26)

4 ponderomotive channelling

All nonlinear effects important for PL > 2TW

Interaction with Underdense Plasmas EM wave propagation Nonlinear refraction 36 132

Relativistic self-focussing: Geometric opticsConsider laser beam with a radial profile

a(r) = a0 exp(−r2/2σ2L),

spot size σ0 just inside a region of uniform, underdense plasma,see Fig. 7.

plasma

θ α

α

∆L

σ

R

laser

a) b)

ZZ

Figure 7: a) diffraction, b) self-focusing

Interaction with Underdense Plasmas EM wave propagation Self focussing 37 132

Geometric optics picture: power threshold

Beam spreading due to diffraction will be cancelled byself-focusing effects if θ = α, (Exercise!),

a20

(ωpσL

c

)2≥ 8. (27)

This represents a power threshold, since the laser powerPL ∝ a2

0σ2L . In numbers:

PL > 9(ω

ωp

)2

GW

This is ∼ ×2 too low because we didn’t take beam profile intoaccount.

Interaction with Underdense Plasmas EM wave propagation SF power threshold 38 132

Focussing threshold – practical unitsLitvak, 1970; Max et al.1974, Sprangle et al.1988

Relation between laser power and critical power:

PL =(mωc

e

)2(

cωp

)2 cε0

2

∫ ∞0

2πra2(r)dr

=12

(me

)2c5ε0

(ω

ωp

)2

P

' 0.35(ω

ωp

)2

P GW.

The critical power Pc = 16π thus corresponds to:

Power threshold for relativistic self-focussing

Pc ' 17.5(ω

ωp

)2

GW, (28)


Focussing threshold – example

Critical power

Pc ' 17.5(ω

ωp

)2

GW, (29)

Example

λL = 0.8µm, ne = 1019 cm−3

⇒ ne

nc=(ωp

ω

)2=

1019

1.6× 1021 = 6× 10−3

⇒ Pc = 2.6TW


Transverse plasma responseCigar-shaped pulse: take ∇ = ∇⊥, and apply Poisson’sequation (φ and n normalized as before)

∇2⊥φ = k2

p (n − 1),

to obtain density perturbation:

n = 1 + k−2p ∇2

⊥γ. (30)

0 1 2 3 4 5kpr

0.0

0.3

0.6

0.9

1.2

1.5

Ele

ctro

nde

nsity

n(r)

a0=2.0a0=1.0a0=0.5a0=0.2

Interaction with Underdense Plasmas EM wave propagation Ponderomotive channel formation 41 132

Cavitation condition

Consider Gaussian pulse profile a(r) = a0 exp(−r2/2σ2). Aftertime-averaging over the laser period, the density depressionterm in cylindrical coordinates can be written:

∇2⊥γ =

14γ∇2⊥a2 =

14γ

4a20

σ2

(r2

σ2 − 1)

exp(−r2/σ2).

The deepest depression is on the laser axis at r = 0, giving

Cavitation condition, (n = 0):

I18λ2µ >

120

n18σ2µ, (31)

– quite easily fulfilled with TW lasers.


Relativistic beam propagationNumerical solution of NLSE with an initial radial Gaussian beamprofile with σ0 = 7.5 µm and pump strengths a0.Beam powers P/Pc : 0.5, 1.0 and 3.0 respectively.

0 1 2 3 4

z (mm)

0

5

10

15

20

25

30

Bea

mra

dius

r(

m)

vacuumP=Pc/2P=PcP=3Pc


Plasma (Langmuir) wave propagation

Without the laser driving term (Ay = 0), Eq.(23) describes linearplasma oscillations with solutions

Ex = Ex0 sin(ωt),

giving the dispersion relation:

−ω2 + ω2p = 0. (32)

The linear eigenmode of a plasma has ω = ωp.

Including finite temperature Te > 0 yields the Bohm-Grossrelation:

ω2 = ω2p + 3v2

t k2. (33)

Interaction with Underdense Plasmas Plasma wave propagation Dispersion 44 132

Numerical solutions – linear Langmuir wave

Numerical integration of the electrostatic wave equation on slide1 for vmax/c = 0.2

0.0 3.1 6.2

p

-0.5

0.0

0.5

1.0

1.5

u,E

,ne/

n 0

uEne/n0

NB: electric field and density 90o out of phase

Interaction with Underdense Plasmas Plasma wave propagation Numerical solutions 45 132

Numerical solutions – nonlinear Langmuir waves

Parameters here: a) phase velocity vp/c ' 1 and vm/c = 0.9; b)vp/c = 0.6, vm/c = 0.55

0 2 4 6 8

p

-2

0

2

4

6

8

u,E

,ne/

n 0

a)

0.0 3.1 6.2

p

-2

0

2

4

6

8

u,E

,ne/

n 0

b)uEne/n0

Typical features : i) sawtooth electric field; ii) spiked density; iii)lengthening of the oscillation period by factor γ.

Interaction with Underdense Plasmas Plasma wave propagation Numerical solutions 46 132

Maximum field amplitude - wave-breaking limitFor relativistic phase velocities, find

Emax ∼ mωpc/e

– wave-breaking limit – Dawson (1962), Katsouleas (1988).

Example

me = 9.1× 10−28g

c = 3× 1010cms−1

ωp = 5.6× 104(ne/cm−3)1/2

e = 4.8× 10−10statcoulomb

Ep ∼ 4× 108(

ne

1018 cm−3

)1/2

V m−1

Interaction with Underdense Plasmas Plasma wave propagation Wave breaking 47 132

Wakefield excitation

fpond

+

+

+

t1

fpond

eEx

t2 +

+

+

eEx

t3

fpond

+

+

+

Interaction with Underdense Plasmas Wakefield excitation 48 132

Resonance conditionThe amplitude of the longitudinal oscillation will be enhanced ifthe pulse length is roughly matched to the plasma period:

τL ' ω−1p .

Example

What plasma density do we need to match a 100 fs pulse?

ωp ' 5× 104n1/2e s−1

Matching condition:

ne ' 4× 1014τ−2ps cm−3

For 100 fs, need ne = 4× 1016 cm−3.


Numerical solution: small laser amplitude

0 5 10 15 20 25

kp

-0.02

0.0

0.02

0.04

0.06w

akefi

eld

Enpump


Numerical solution: resonance condition (smallamplitudes)

0 5 10 15 20kp L

0.0

0.001

0.002

0.003

0.004

0.005E

max


Numerical solution: large laser amplitude

0 2 4 6 8 10 12 14 16 18 20

kp

-1

0

1

2

3

wak

efiel

d

pump

En


Wake amplitude scaling in nonlinear regimeMurusidze & Berzhiani, 1990

Analytical solution possible for a square pump in the limit βg → 1⇒ Scaling of the wake-variable maxima:

φmax ∼ γ2⊥ − 1

Emax ∼γ2⊥ − 1γ⊥

(34)

pmax ∼ (γu)max =γ4⊥ − 12γ2⊥

where γ⊥ = (1 + a2)1/2 as on p. ??.


2D wakefield excitation


Electron acceleration by wakefields

Conventional synchrotrons and LINACS operate with fieldgradients limited to around 100 MVm−1.

Plasma is already ionized; can theoretically sustain a field 104

times larger, given by:

Ep =mecωp

eε

' n1/218 ε GV cm−1, (35)

where n18 is the electron density in units of 1018 cm−3.

Interaction with Underdense Plasmas Wakefield excitation Electron acceleration 55 132

Laser-electron acceleratorTajima & Dawson, 1979

Laser-driven wakefields must propagate with velocitiesapproaching the speed of light (vp = vg < c).Plasma wave has a phase velocity:

vp = c

(1−

ω2p

ω2o

) 12

' c(

1− 12γ2

p

), (36)

where γp = ω20/ω

2p .


Acceleration length

A relativistic electron (v ' c) trapped in such a wave will beaccelerated over at most half a wavelength in the wave-frame,after which it starts to be decelerated.Effective acceleration length:

La =λpc

2(c − vp)' λpγ

2p

=ω2

ω2pλp

' 3.2 n−3/218 λ−2

µm cm. (37)


Maximum energy gain

Combine Eq. (35) and Eq. (37) to obtain the maximum energygain:

∆U = eEp.La

= e(mωpc

e

)εω2

ω2p

2πcωp

= 2π(ω

ωp

)2

εmc2

' 3.2 n−118 λ

−2µm GeV. (38)


Limiting factors

In principle, TW laser is capable of accelerating an electron to5 GeV in a distance of 5 cm through a plasma with density1018 cm−3.Spoiling factors:

Diffraction: typically have ZR � La, so some means ofguiding the laser beam over the dephasing length is essential

Propagation instabilities – beam break-up: modulation;hosing; Raman


Bench-Top Particle Accelerators

Standard electron acceleration in fast plasma wave (recap):

Acceleration length

La ' 3.2 n−3/218 λ−2

µm cm

Energy gain∆U ' 3.2 n−1

18 λ−2µm GeV

Interaction with Underdense Plasmas Wakefield excitation Bench-Top Particle Accelerators 60 132

Acceleration mechanisms

Large variety of attributed acceleration mechanisms inexperiments (long and short pulse):

self-modulated

forced-wave

wave-breaking

guided

bubble-regime

GeV milestone reached September 2006 (LBL). More to comeon Wed–Fri ....


What mechanisms are at work here?


Livingstone chart for laser-plasma electronaccelerators

1980 1985 1990 1995 2000 2005 2010

Year

1

2

5

10

2

5

102

2

5

103

2

Max

.ele

ctro

nen

ergy

(MeV

)

IC/RALUCLA

INRS

ILE

UCLA

LULI

RAL

ILECUOS

NRL

RAL

CUOS

MPQ

LOA

RAL

RAL (Astra)

LBL

LOA

LBL

Figure 8: Open circles represent early beat-wave experiments; filledcircles single pulse wakefield experiments; trianglesquasi-monoenergetic electron beams



Plasma response

Cold fluid equations

EM wave propagation

Plasma wave propagation

Wakefield excitation


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chaf

t

Physics of High IntensityLaser Plasma InteractionsPart III: Interaction with Solids


Lecture 3: Interaction with SolidsShort pulse interaction scenarios

Collisional AbsorptionNormal skin effect

Collisionless AbsorptionResonance absorptionBrunel model

Hot Electron GenerationScalingMeasurements of hot electron temperature

Ion accelerationMechanismsSheath modelHole boringLight sail

Interaction with Solids 66 132

Short pulse vs. long pulse interactions

Long-pulse interaction physics (ICF – ns lasers):

Collisional heating and creation of long scale-length plasmas

Laser reflected at critical density surface

Fast (keV) particles produced at ’high’ intensities(1016 Wcm−2)

Femtosecond pulses

Pulse length typically < ion motion timescale(hydrodynamics)

Huge intensity range 107

No single interaction model possible

Interaction with Solids Short pulse interaction scenarios 67 132

Typical interaction scenario: ICreation of critical surface

Comination of field and collisional ionization over the first fewlaser cycles rapidly creates a surface plasma layer with a densitymany times the critical density nc .

ω2 =4πe2nc

me, (39)

where e and me are the electron charge and mass respectively.

In practical units (see Eq. 15):

nc ' 1.1× 1021(

λ

µm

)−2

cm−3. (40)


Interaction scenario: IIIonization degree

Example

Al has 3 valence electrons; 6 more can be released for a fewhundred eV. The electron density is given by:

ne = Z ∗ni =Z ∗NAρ

A. (41)

effective ion charge: Z ∗ = 9atomic number: A = 26Avogadro number: NA = 6.02× 1023

mass density: ρ = ρsolid = 1.9 g cm−3

electron density: ne = 4× 1023 cm−3

density contrast (1 µm): ne/nc ' 400


Interaction scenario: IIIHeating

Target is heated via electron-ion collisions to 10s or 100s of eVdepending on the laser intensity.The plasma pressure created during heating causes ion blow-off(ablation) at the sound speed:

cs =

(Z ∗kBTe

mi

)1/2

' 3.1× 107(

Te

keV

)1/2(Z ∗

A

)1/2

cm s−1, (42)

where kB is the Boltzmann constant, Te the electron temperatureand mi the ion mass.


Interaction scenario: IVExpansion

Because of ion ablation, density profile formed is exponentialwith scale-length:

L = csτL

' 3(

Te

keV

)1/2(Z ∗

A

)1/2

τfsÅ. (43)

Example

100 fs Ti:sapphire pulse on Al foil heats the target to a fewhundred eV→ plasma with scale-length L/λ= 0.01–0.1. (cf:100-1000 for ICF plasmas).


Collisional absorption

Modelled using Helmholtz wave equations: standard method forelectromagnetic wave propagation in an inhomogeneous plasma– see books by Ginzburg, Kruer.

Start from Maxwell’s equations with small field amplitudes and anon-relativistic fluid response including collisional damping:

m∂v∂t

= −e(E +vc×B)−mνeiv , (44)

where νei is the electron-ion collision frequency.

Physically arises from binary collisions, resulting in a frictionaldrag on the electron motion.

Interaction with Solids Collisional Absorption 72 132

Electron-ion collisional frequencySpitzer-Härm

Collision rate:

νei =4(2π)1/2

3neZe4

m2v3te

ln Λ

' 2.91× 10−6ZneT−3/2e ln Λ s−1. (45)

Z = number of free electrons per atomne = electron density in cm−3

Te = temperature in eVln Λ is the Coulomb logarithm, with usual limits, bmin and bmax,of the electron-ion scattering cross-section.


Coulomb logarithm

Limits are determined by the classical distance of closestapproach and the Debye length respectively, so that:

Λ =bmax

bmin

= λD.kBTe

Ze2 =9ND

Z, (46)

where

λD =

(kBTe

4πnee2

)1/2

=vte

ωp, (47)

andND =

4π3λ3

Dne

is the number of particles in a Debye sphere.


Absorption in steep density profiles: skin effect

Density profile can be approximated by a Heaviside stepfunction:

n0(x) = n0Θ(x),

giving a dielectric constant (cf: dispersion relation Eq. 24):

ε(x) ≡ c2k2

ω2 = 1−ω2

p

ω2(1 + iνei/ω)Θ(x). (48)

Interaction with Solids Collisional Absorption Normal skin effect 75 132

Solution for laser field I

For normally incident light, the transverse electric field has thesolution

Ez =

{2E0 sin(kx cos θ + φ), x < 0E(0) exp(−x/ls), x ≥ 0

(49)

where ls ' c/ωp is the collisionless skin-depth, k = ω/c, E0 isthe amplitude of the laser field and φ a phase factor.

Matching vacuum and solid solutions at the boundary x = 0gives:

E(0) = 2E0ω

ωpcos θ

tanφ = −lsω

ccos θ.


Solution for laser field I

-4 -3 -2 -1 0 1 2kx

0

1

2

3

4

5

6F

ield

inte

nsity

Ez2

By2

E2(0) = 2E02nc/n0

n0/nc


Reflectivity: Drude model

Example

Al: Z ∗ = 3, ne ' 2× 1023 cm−3, λL = 0.8 µm , ne/nc ' 100.

0 15 30 45 60 75 90

(o)

0.0

0.2

0.4

0.6

0.8

1.0a

=1-

R

FresnelHelmholtz

Figure 9: Angular absorption for a step-profile with ne/nc = 100 andν/ω = 5 calculated analytically from the Fresnel equations (solid) andnumerically from the Helmholtz wave equations (dotted).


Collisional frequency turn-offquiver velocity correction

Effective collision frequency reduced by quiver motion in laserfield

νe� ' νeiv3

te

(v2os + v2

te)3/2. (50)

A temperature of 1 keV corresponds to a thermal velocityvte' 0.05, so collisional absorption starts to turn off forirradiances Iλ2≥ 1015 Wcm−2µm2.

Interaction with Solids Collisionless Absorption 79 132

Collisionless absorption mechanisms

What other absorption mechanisms can couple laser energy to ahot, solid-density target?

1 Resonance absorption (L/λL � vos/ω) – Denisov (1957)

2 Anomalous (collisionless) skin effect – Weibel (1967)

3 ’Vacuum heating’ – Brunel (1987), Gibbon (1992)

4 Relativistic j × B heating – Kruer (1988), Wilks (1992)

All of these mechanisms will generate fast electrons withenergies Th ∼ keV–MeV.

Interaction with Solids Collisionless Absorption 80 132

Collisionless resonance absorption

nc

n cosc2

x

θ

en (x)

yθ

B

E p

kE

Bs

Figure 10: Standard picture of resonance absorption: a p-polarizedlight wave tunnels through to the critical surface (ne = nc) and drivesup a plasma wave. This is damped by particle trapping and wavebreaking at high intensities.

Interaction with Solids Collisionless Absorption Resonance absorption 81 132

Resonance absorption: Denisov function

Self-similar dependence on the parameter ξ = (kL)1/3 sin θ,kL� 1.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

=(kL)1/3 sin

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4(

)


Absorption rate for long scalelengths

To a good approximation,

φ(ξ) ' 2.3ξ exp(−2ξ3/3), (51)

and the fractional absorption is given by:

ηra =12

Φ2(ξ).

Behavior nearly independent of the damping mechanism.


Kinetic simulation of resonance absorptionParticle-in-Cell

0 2 4 6 8 10x/

0.0

0.5

1.0

1.5

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c)

Figure 11: PIC simulation of resonance absorption for parametersθ = 9o, vos/c = 0.07, L/λ = 5 (kL = 10π), and nmaxe /nc = 1.5 : a)Density profile and electric field normal to the target, b) electric fieldparallel to the target, c) particle momenta, d) laser magnetic field.


Vacuum heating: Brunel modelResonance absorption not possible if oscillation amplitudeexceeds the density scale length L, i.e. if vos/ω > L.

Capacitor approximation: magnetic field of the wave is ignored;laser electric field EL has some component Ed normal to thetarget surface.

��

��

E

E n

v

kθ

d

d

L

e

∆x0 x

Figure 12: Capacitor model of the Brunel heating mechanism.

Interaction with Solids Collisionless Absorption Brunel model 85 132

Vacuum heating: Brunel modelResonance absorption not possible if oscillation amplitudeexceeds the density scale length L, i.e. if vos/ω > L.Capacitor approximation: magnetic field of the wave is ignored;laser electric field EL has some component Ed normal to thetarget surface.

��

��

E

E n

v

kθ

d

d

L

e

∆x0 x

Figure 12: Capacitor model of the Brunel heating mechanism.


Brunel model II

Driving electric fieldEd = 2EL sin θ. (52)

Field pulls a sheet of electrons out to a distance ∆x from itsinitial position. Surface number density of this sheet isΣ = ne∆x , so the electric field created between x = −∆x andx = 0 is

∆E = 4πeΣ.

Equating this to the driving field and solve for Σ:

Σ =2EL sin θ

4πe. (53)


Brunel model III

When the charge sheet returns to its original position, it acquiresvelocity vd ' 2vos sin θ, where vos is the usual electron quivervelocity.Assuming these electrons are all ‘lost’ to the solid, then theaverage energy density absorbed per laser cycle is given by:

Pa =Σ

τ

mv2d

2

' 116π2

emω

E3d .


Brunel model IV

Compare to the incoming laser power:

PL = cE2L cos θ/8π.

Substituting Eq. (52), we obtain the

fractional absorption rate for the Brunel mechanism:

ηa ≡Pa

PL=

4π

a0sin3 θ

cos θ, (54)

where a0 = vos/c.

Expect more absorption at large angles of incidence and higherlaser irradiance, Iλ2 ∝ a2

0.


Brunel model: refinements

Two corrections to improve the model:

1 Account for the reduced driver field amplitude due toabsorption, replacing Eq. (52) by

Ed = [1 + (1− ηa)1/2]EL sin θ. (55)

2 Relativistic return velocities: replace kinetic energy with

Uk = (γ − 1)mc2


Vacuum heating


Fully relativistic modelGibbon, Andreev & Platanov (2011)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0 10 20 30 40 50 60 70 80 90

(deg)

a0=0.027a0=0.085a0=0.27a0=0.85a0=2.7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

10172 5 1018

2 5 10192 5 1020

2 5 1021

I 2 [Wcm-2 m2]

=0=6=45=60=85

Figure 13: Absorption fraction vs. angle and intensity


Hot electron generation

Typical signature of collisionless laser heating – bi-Maxwellianelectron spectrum with characteristic temperatures Tc and Th.

0 100 200 300 400 500U (keV)

1

10

102

103

104

105f(

U)d

U

Te ~ 5 keV

Th ~ 75 keV

Interaction with Solids Hot Electron Generation 92 132

Short pulse Th scaling

Simple Brunel model:

TBh =

mv2d

2' 3.7I16λ

2µ (56)

EM PIC simulations:

TGBh ' 7

(I16λ

2)1/3, (57)

Relativistic j × B model (ponderomotive scaling):

TWh ' mc2(γ − 1)

' 511[(

1 + 0.73I18λ2µ

)1/2 − 1]

keV. (58)

Interaction with Solids Hot Electron Generation Scaling 93 132

Hot electron temperature: short pulses

1015 1016 1017 1018 1019 1020

I 2 (Wcm-2 m2)

2

5

102

5

1022

5

1032

5

Tho

t(k

eV)

LLE-93

IOQ-99 LULI-94

IOQ-96

LULI-97

INRS-99

MBI-97

MBI-95

RAL-96

IOQ-97

IOQ-00

RAL-99

CELV-96

CELV-96

MBI-00

STA-92 LLNL-99

LLNL-00

Th (B)Th (GB)Th (W)Th (FKL)Experiments

Figure 14: Hot electron temperature measurements in femtosecondlaser-solid experiments (squares) compared with various models: longpulse — Eq. (??); Brunel — Eq. (56); Gibbon & Bell — Eq. (57) andWilks — Eq. (58).

Interaction with Solids Hot Electron Generation Measurements of hot electron temperature 94 132

Ion acceleration

Direct acceleration in laser field inefficient, since

vi

c' eEL

miωc=

mi

mea0 ≤

a0

1836

Get relativistic ion energies for

a0 ∼ 2000

or Iλ2L ≥ 5× 1024 Wcm−2µm2

Therefore need means of transmitting laser energy to ions overmany cycles→ exploit electrostatic field in plasma.

Interaction with Solids Ion acceleration 95 132

Mechanisms

1 Coulomb explosion: clusters; ponderomotive channelling ingas jets

2 Electrostatic sheath formed by hot electron cloud (TNSA)

3 Collisionless shock formation: hole boring

4 Light sail: radiation pressure on mass-limited target

Interaction with Solids Ion acceleration Mechanisms 96 132

Sheath modelElectrostatic plasma expansion into vacuum: ions initially at rest(ni = ni0), hot electrons described by Boltzmann distribution:

ne = ne0 exp(eφ/Th)

where ne0 = Zni0, and φ satisfies Poisson’s equation:

∂2φ

∂x2 =eε0

(ne − Zni)

Ion expansion is described by continuity and momentumequations: (

∂

∂t+ vi

∂

∂x

)ni = −ni

∂vi

∂x

mi

(∂

∂t+ vi

∂

∂x

)vi = −Ze

∂φ

∂x(59)

Interaction with Solids Ion acceleration Sheath model 97 132

Self-similar solutionIf the plasma stays quasineutral everywhere (ne ' Zni ), thenEqs. (59) have a self-similar solution in x/t :

Zni = ne0 exp(−x/cst − 1)

vi = cs + x/t (60)

eφ = −Te(x

cst+ 1)

where cs =√

ZTe/mi is the ion sound speed.

Max ion velocity is

vf = 2cs log(τ +√τ2 + 1) (61)

where

τ =ωpi t√

2e; ωpi =

(Z 2e2ni0

ε0mi

)1/2


Self-similar solutionIf the plasma stays quasineutral everywhere (ne ' Zni ), thenEqs. (59) have a self-similar solution in x/t :

Zni = ne0 exp(−x/cst − 1)

vi = cs + x/t (60)

eφ = −Te(x

cst+ 1)

where cs =√

ZTe/mi is the ion sound speed.Max ion velocity is

vf = 2cs log(τ +√τ2 + 1) (61)

where

τ =ωpi t√

2e; ωpi =

(Z 2e2ni0

ε0mi

)1/2


Energy spectrumIon energy spectrum is given by:

dNdU

=ni0cst

(2UU0)1/2e−(2UU0)1/2

(62)

where U0 = ZkBTh.

Figure 15: Ion energy spectrum from Mora expansion model


Hole boring

On the front side of an overdense plasma, the ponderomotiveforce will displace and compress electrons into the target,creating an electrostatic field acting on the ions.

Momentum balance across the collisionless, electrostatic shockthus formed implies:

ui = 2us = 2(

ILminic

)1/2

= 2

√ZA

me

mp

nc

ne(63)

where us is the velocity of the shock front. This leads to aquasi-monoenergetic component in the ion spectrum. See:Denavit, PRL (1992); Wilks, PRL (1992); Macchi, PRL (2005)Relativistic HB formula: Robinson, PPCF (2009).

Interaction with Solids Ion acceleration Hole boring 100 132

Hole boring



ui = 2us = 2(

ILminic

)1/2

= 2

√ZA

me

mp

nc

ne(63)

where us is the velocity of the shock front.

This leads to aquasi-monoenergetic component in the ion spectrum. See:Denavit, PRL (1992); Wilks, PRL (1992); Macchi, PRL (2005)Relativistic HB formula: Robinson, PPCF (2009).


Hole boring



ui = 2us = 2(

ILminic

)1/2

= 2

√ZA

me

mp

nc

ne(63)

where us is the velocity of the shock front. This leads to aquasi-monoenergetic component in the ion spectrum. See:Denavit, PRL (1992); Wilks, PRL (1992); Macchi, PRL (2005)Relativistic HB formula: Robinson, PPCF (2009).


Light sail accelerationA mass-limited target, such as a nm-thick foil, allows nearlycomplete displacement of electrons, thereby maximizing the ESfield. A simple capacitor model suffices to determine thethreshold intensity for this scenario.

The charge separation field of a foil with thickness d is:

∆E =eε0

ned

This is balanced by the net laser field at the surface,2EL = 2meωca0/e, leading to the matching condition:

a0 ' πne

nc

dλL

(64)

Under these conditions, find max ion energy Ui ∼ t1/3 – seeEsirkepov, PRL (2004).

Interaction with Solids Ion acceleration Light sail 101 132

Light sail accelerationA mass-limited target, such as a nm-thick foil, allows nearlycomplete displacement of electrons, thereby maximizing the ESfield. A simple capacitor model suffices to determine thethreshold intensity for this scenario.The charge separation field of a foil with thickness d is:

∆E =eε0

ned

This is balanced by the net laser field at the surface,2EL = 2meωca0/e, leading to the matching condition:

a0 ' πne

nc

dλL

(64)

Under these conditions, find max ion energy Ui ∼ t1/3 – seeEsirkepov, PRL (2004).



Short pulse interaction scenarios

Collisional Absorption

Collisionless Absorption

Hot Electron Generation

Ion acceleration


Mitg

lied

derH

elm

holtz

-Gem

eins

chaf

t

Physics of High IntensityLaser Plasma InteractionsPart IV: Numerical Simulation ofLaser-Plasma Interactions


Lecture 4: Numerical Simulation of Laser-PlasmaInteractions

Plasma modelsClassificationHierarchy

HydrodynamicsThermal transportSingle fluid modelNumerical scheme

Particle-in-Cell CodesVlasov equationParticle pusherField solverAlgorithm3D codes

Numerical Simulation of Laser-Plasma Interactions 104 132

Plasma classification

10 104 107 1010

Temperature (K)

102

106

1010

1014

1018

1024

1028

1032

Ele

ctro

nde

nsity

(cm

-3)


magneticfusion




solaratmosphere

solar corona

white dwarf

non-ideal p

lasmas

Figure 16: Plasma classification in the density-temperature plane.

Numerical Simulation of Laser-Plasma Interactions Plasma models Classification 105 132

Plasma model hierarchy

Hydrodynamics

Kinetic theory:Boltzmann equation

Fokker−Planck

Vlasov

Particle−in−Cell

Macroscopic:fluid equations

Molecular dynamics

Monte−Carlo

MHD

Steady−state

average

velocitymoments

Microscopic

collisionless

collisional

statistical

equilibrium

non−eqm

d/dt = 0

+ E, B

Figure 17: Physical basis of common plasma models andcorresponding numerical approaches.

Numerical Simulation of Laser-Plasma Interactions Plasma models Hierarchy 106 132

Numerical modelling of laser-plasma interactions

Two types of modelling dominate LPI:

1 Hydrodynamic modelling to follow the macroscopic, dynamicbehavior of the plasma, including external electric andmagnetic fields, and heating by laser or particle beams: ps-nstimescale. Good for prepulse modelling.

2 Kinetic modelling for situations in which ’non-Maxwellian’particle distributions fα(v) have to be determinedself-consistently – this is the method of choice forlaser-particle accelerator schemes!

Numerical Simulation of Laser-Plasma Interactions Plasma models Hierarchy 107 132

Hydrodynamic plasma simulation

Figure 18: Building blocks of a laser-plasma hydro-code.

Numerical Simulation of Laser-Plasma Interactions Hydrodynamics 108 132

Heating rate of plasma slab

If penetration depth of the heat wave lh < ls = c/ωp (skin depth),then the thermal transport can be neglected.

Volume heated simultaneously: V ' lsπσ2L .

Setting ε = 32nekBTe and ∇.Φa ∼ Φa/ls, have

dTe

dt' Φa

nels, (65)

heating rate:

ddt

(kBTe) ' 4Φa

Wcm−2

(ne

cm−3

)−1( lscm

)−1

keV fs−1. (66)

Numerical Simulation of Laser-Plasma Interactions Hydrodynamics Thermal transport 109 132


If penetration depth of the heat wave lh < ls = c/ωp (skin depth),then the thermal transport can be neglected.Volume heated simultaneously: V ' lsπσ2

L .

Setting ε = 32nekBTe and ∇.Φa ∼ Φa/ls, have

dTe

dt' Φa

nels, (65)

heating rate:

ddt

(kBTe) ' 4Φa

Wcm−2

(ne

cm−3

)−1( lscm

)−1

keV fs−1. (66)



If penetration depth of the heat wave lh < ls = c/ωp (skin depth),then the thermal transport can be neglected.Volume heated simultaneously: V ' lsπσ2

L .Setting ε = 3

2nekBTe and ∇.Φa ∼ Φa/ls, have

dTe

dt' Φa

nels, (65)

heating rate:

ddt

(kBTe) ' 4Φa

Wcm−2

(ne

cm−3

)−1( lscm

)−1

keV fs−1. (66)


Thermal transportEnergy transport equation for a collisional plasma:

∂ε

∂t+∇.(q + Φa) = 0, (67)

where ε is the energy density, q is the heat flow and Φa = ηaΦL

is the absorbed laser flux.

Huge temperature gradients are generated after a few fs: heat iscarried away from the surface into the colder target materialaccording to Eq. (67). For ideal plasmas, we write

ε =32

nekBTe

as before, and

q(x) = −κe∂Te

∂x, (68)

which is the usual Spitzer-Härm heat-flow.


Thermal transportEnergy transport equation for a collisional plasma:

∂ε

∂t+∇.(q + Φa) = 0, (67)

where ε is the energy density, q is the heat flow and Φa = ηaΦL

is the absorbed laser flux.Huge temperature gradients are generated after a few fs: heat iscarried away from the surface into the colder target materialaccording to Eq. (67). For ideal plasmas, we write

ε =32

nekBTe

as before, and

q(x) = −κe∂Te

∂x, (68)

which is the usual Spitzer-Härm heat-flow.


Spitzer-Härm heat-flow

Substituting for ε and q in Eq. (67) and restricting ourselves to1D by letting ∇ = (∂/∂x , 0, 0), gives a diffusion equation for Te:

32

nekB∂Te

∂t=

∂

∂x

(κe∂Te

∂x

)+∂ΦL

∂x. (69)

κe is known as the Spitzer thermal conductivity and is given by:

κe = 32(

2π

)1/2 ne

ν0m5/2T 5/2

e , (70)

where

ν0 =2πneZe4 log Λ

m2 .


Nonlinear heat wave

0 40 80 120 160 200x (nm)

0

100

200

300

400

Te

(eV

) t = 400 fst = 300 fst = 200 fst = 100 fs

Figure 19: Nonlinear heat-wave advancing into a semi-infinite,solid-density plasma. The curves are obtained from the numericalsolution of the Spitzer heat flow equation for constant laser absorptionat the target surface (left boundary).


Single-fluid approximation

Fluid model

∂ρ

∂t+∇·(ρu) = 0 (71)

∂ρu∂t

+∇·(ρuu) +∇P − f p = 0 (72)

∂εe

∂t+∇·

[u(εe + Pe)− κe∇Te −

Qei

γe − 1−Φa

]= 0 (73)

∂εi

∂t+∇·

[u(εi + Pi)− κi∇Ti +

Qei

γi − 1

]= 0,(74)

Numerical Simulation of Laser-Plasma Interactions Hydrodynamics Single fluid model 113 132

Notes

Average fluid density ρ and velocity u defined thus:

ρ ≡ niM + nem ' niM,

u ≡ 1ρ

(niMui + nemue) ' ui .

The energy density εα is the sum of internal and kinetic fluidenergies:

εα =Pα

γα − 1+

12ρu2,

with γα defined as the number of degrees of freedom.Qei is the electron-ion equipartition rate

Qei =2mM

nekB(Te − Ti)

τei(75)

τei = ν−1ei is the electron-ion collision time.

Coupling to the laser Φa and f p.

Numerical Simulation of Laser-Plasma Interactions Hydrodynamics Single fluid model 114 132

Example: laser prepulse heating solid Al target

Figure 20: Density (solid line) and temperature (dashed line) profilesfrom two interactions with pulse durations of a) 100 fs and b) 100ps.The initially cold Al target extends to x = 1 µm. a) Snapshots 0 fs, 300fs and 5 ps after the peak of the pulse; in b) snapshots are at -150 ps,-100 ps and 0 ps.

Numerical Simulation of Laser-Plasma Interactions Hydrodynamics Numerical scheme 115 132

Kinetic plasma simulation

Simplest kinetic description of a plasma uses single-particlevelocity distribution function f (r, v), evolving according to theVlasov equation:

∂f∂t

+ v· ∂f∂x

+ q(E +vc× B)· ∂f

∂p= 0. (76)

Distribution function f (r, v) is 6-dimensional – direct solutionof Eq. (76) generally impractical.

Even 1D spatial geometry still needs 2 or 3 velocitycomponents to couple to Maxwell’s equations, ie: 3- or4-dimensional code.

Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes Vlasov equation 116 132

Particle-in-Cell simulationImportant method developed in the 1960s is the so-calledParticle-in-Cell (PIC) technique.Distribution function is represented instead by a large number ofdiscrete macro-particles, each carrying a fixed charge qi andmass mi .

x

p

x

pf(x,p)

i iq(x ,p )

Figure 21: Relationship between a) Vlasov and b) PIC approaches.

Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes Vlasov equation 117 132

Particle pusherGeometry for simplified Lorentz equation (see Eq. 16):E = (0,Ey , 0),B = (0, 0,Bz):1/2-acceleration:

p−x = pn− 1

2x ; p−y = p

n− 12

y +∆t2

Eny

rotation:

γn = (1 + (p−x )2 + (p−y )2)1/2 ; t =∆t2

Bnz

γn ; s =2t

1 + t2

p′x = p−x + p−y t

p+y = p−y − p′xs (77)

p+x = p′x + p+

x t

1/2-acceleration:

pn+ 1

2x = p+

x ; pn+ 1

2y = p+

y +∆t2

Eny .

Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes Particle pusher 118 132

Density & current gather

The density and current sources needed to integrate Maxwell’sequations are obtained by mapping the local particle positionsand velocities onto a grid as follows:

ρ(r) =∑

j

qjS(rj − r),

J(r) =∑

j

qjv jS(rj − r), j = 1...Ncell (78)

where S(rj − r) is a function describing the effective shape ofthe particles. Usually it is sufficient to use a linear weighting forS — the ‘Cloud-in-Cell’ scheme — although other more accuratemethods (eg: cubic spline) are also possible.

Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes Field solver 119 132

Field solver: FDTD

Once ρ(r) and J(r) are defined at the grid points, we canproceed to solve Maxwell’s equations to obtain the new electricand magnetic fields. In 2D geometry, can define:

F + =12

(Ey + Bz)

F− =12

(Ey − Bz)

Then Faraday and Ampere laws reduce to (normalised units):

F +i+1(n + 1) = Fi(n)− ∆t

2Jn+1/2

y ,i+1/2 (forward)

F−i (n + 1) = Fi+1(n)− ∆t2

Jn+1/2y ,i+1/2 (backward) (79)

Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes Field solver 120 132

PIC algorithm summary

The fields are then interpolated back to the particle positions sothat we can go back to the particle push step Eq. (??) andcomplete the cycle, see Fig. 22.

i

i i jgρ

g g

i

3. Mesh −> Particles 2. Maxwell equations

4. Push particles 1. Particles −> Mesh r v

E B g

E B

Figure 22: Iteration steps of the particle-in-cell algorithm.

Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes Algorithm 121 132

PIC codes for laser-plasma interaction studiesBecause of its simplicity and ease of implementation, thePIC-scheme sketched above is currently one of the mostimportant plasma simulation methods.

Name Authors Group(s)

OSIRIS Mori, Silva UCLA, IST LisbonVLPL Pukhov U. DüsseldorfREMP Esirkepov Kyoto, MoscowVPIC Lin, Bowers LANLEPOCH Arber, Bennett U. Warwick (CCPP)CALDER Lefebvre CEASPSC Ruhl, Brömmel LMU, FZ Jülich

Table 3: Parallel 3D, electromagnetic PIC codes

Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes 3D codes 122 132


Plasma models

Hydrodynamics

Particle-in-Cell Codes

Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes 3D codes 123 132

Mitg

lied

derH

elm

holtz

-Gem

eins

chaf

t

Physics of High IntensityLaser Plasma InteractionsPart V: Tutorial on Particle-in-CellSimulation


Lecture 5: Tutorial on Particle-in-Cell Simulation

The PIC code BOPS

Prerequisites

Installation

Running BOPS

Project I: Laser wakefield accelerator

Project II: Ion acceleration – TNSA vs. RPA

Tutorial on Particle-in-Cell Simulation 125 132

The PIC code BOPS

BOPS is a one-and-three-halves (1 spatial, 3 velocitycoordinates: 1D3V) particle-in-cell code originally created byPaul Gibbon and Tony Bell in the Plasma Physics Group ofImperial College, London.

Based on standard algorithm for a 1D electromagnetic PICcode from Birdsall & Langdon.

Optionally employs a Lorentz ‘boost’ along the target surfaceto mimic 2D, periodic-in-y geometry, with big savings incompute time.

Applications: absorption, electron heating, high harmonicgeneration, ion acceleration

Tutorial on Particle-in-Cell Simulation The PIC code BOPS 126 132

Prerequisites

Current version: BOPS 3.4

Download site:https://trac.version.fz-juelich.de/bops

Compiler: sequential code written in Fortran 90 (Intel ifort orGNU’s gfortran)

Run scripts provided are designed for generic Unix systems:Ubuntu, SuSE, RedHat and also MacOS.

Linux and MAC users:

Unpack the tar file (name may differ) with:

tar xvfz bops3.4.tar.gz

and cd to the installation directory bops.

Tutorial on Particle-in-Cell Simulation Prerequisites 127 132

https://trac.version.fz-juelich.de/bops

Windows users1 First install CYGWIN (www.cygwin.com). This creates a

fully-fledged Unix environment emulator under Windows. Inaddition to the default tools/packages offered during theinstallation you will also need:

Gnu compilers gcc, g77, g95 etc. (devel package)make (devel package)vi, emacs (editors package)X11 libraries if you want to generate graphics directly undercygwin (eg using gnuplot)

If you forget anything first time, just click on the Cygwininstaller icon to locate/update extra packages.

2 Download and unpack the bops3.4.tar.gz file with an archivingtool (eg PowerArchiver: www.powerarchiver.com). Put this inyour ’home’ directory under CYGWIN, e.g.:

C:\cygwin\home\<username>\

3 Open a Cygwin terminal/shell and ‘cd’ to the bops directory.

Tutorial on Particle-in-Cell Simulation Prerequisites 128 132

Installation

The directory structure resulting from unpacking the tar filesshould look like this:src/ ... containing the fortran90 source codedoc/ ... some documentation in html and pstutorial/ ... scripts and files for current tutorialbenchmarks/ ... additional sample scripts for running the codetools/ ... postprocessing tools

To compile:

Go to the source directory src, and edit the Makefile: adjustand tune the flags to match your machine type (FC=ifort orgfortran etc). Then do:

make

Tutorial on Particle-in-Cell Simulation Installation 129 132

Running BOPSOnce the code has compiled, go back up to the base (or top)directory and enter the tutorial directory. Here you will find therun scripts (suffix .sh) which launch the example simulations.These can be edited directly or copied as needed.

To run from this directory, just type eg:

./wake.sh

Or if you prefer to stay in the top directory – adjust paths inwake.sh first:

tutorial/wake.sh

Further examples can be found in benchmarks

Tutorial on Particle-in-Cell Simulation Running BOPS 130 132

Project I: Laser wakefield accelerator

1 Edit and run the script: ./wake.sh. Note that the inputparameters are normalised to laser wavelength andwavenumber.

2 Look at the printed output to check the actual laser andplasma parameters (are they what you intended?)

Plot filesThese are in run directory (foil_tnsa1) in ASCII formatand have a suffix NN.xy, where NN is the snapshotnumber. See bops.oddata for complete list

ezsi EM field Ez (S-polarized wave)nenc Electron densityexsi Electrostatic fieldpxxe Electron momenta (x − px phase space)

3 Examine the field and particle phase space plots atsuccessive time intervals (e.g.: t=100, t=200). Thegnuplot script wake.gp should help you get started.

4 How can the plasma wave amplitude | Ex |max beoptimised? Hint: try changing the electron density and/orpulse length.

5 Increase the laser amplitude to (a0 = 3) and observe thedifference in the electron phase space and plasma wave.How does the matching condition need to be altered in thisregime? Why is the amplitude of the latter reduced towardsthe back of the wake?

6 (advanced) Set up a plasma equivalent to twice thedephasing length (Eq. 37) and determine the maximumenergy of electrons trapped in the plasma wave. Can youbeat the scaling predicted by Eq. 38?

Project II: Ion acceleration – TNSA vs. RPA

1 TNSA regime.a) The script foil.sh sets up a 2 µm ‘foil’ out of frozenhydrogen (Z = A = 1), with density ne/nc = 36. This isirradiated by a linearly polarized 50 fs pulse with intensity1019 Wcm−2 .Run the script and inspect the following plots at thesnapshot times (0, 50, 100, 150) fs:

Plot files

nenc, ninc Electron & Ion densitiesexdc Cycle-averaged electrostatic fieldpxxe, pxxi Electron & ion momentafuep, fuip Electron, ion energy spectrauinc, ubac Incoming and outgoing wave energy

b) Try varying, eg: pulse amplitude or duration andcompare the scaling of the maximum ion energy againstthe theoretical prediction of Eq. (15).

2 HB regime.a) Copy the script to a new file (eg: hb.sh), switch thepolarization from linear to circular (set cpolzn='C'),change the run directory – eg: RUN=hb1 and rerun. (Whatphase space variables can you check the laser polarizationwith?) Compare the results to the TNSA case.

b) To compare with HB theory, it is convient to specify a‘square’ pulse profile. This can be done with, eg: ilas=1,trise=5, tpulse=150. Another important constraint inthis case is to avoid numerical heating arising from anunderresolved (cold) electron Debye length. Adjust the gridsize/resolution to ensure that ∆x < 0.5λD . Rerun andcompare the ion shock velocity with Eq. (63).

3 Light sail regime.a) Use the matching condition Eq. (64) to determine the foilwidth (using the same laser and target material) for whichthe light sail regime is reached. Set ilas=4 withtpulse=50, tfall=5, and ensure that the new foil isresolved by the grid. Create a new script with theseparameters and rerun. NB: you may have to increase theparameter uimax for this case!

b) Investigate the scaling behaviour of the monoenergeticion feature with intensity, pulse length etc.

c) Now repeat with a Gaussian or sin2 pulse shape.Compare the hot electron spectrum and phase space withthe flat-top case at early times: what causes thisdifference? What could you change to improve the far-fieldion sail stability for realistic pulse forms?

d) Now try the same thing with a carbon foil. Here, you willneed higher (solid) number densities and multiply chargedions (see Eq. 41). What happens to the max. ion energy?How does the energy/nucleon compare with the hydrogencase?

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