1

Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996)

Laser pulse1019 W/cm2

plasma box (ne/nc=0.6)

B ~ mcωp/e ~ 108 Gauss

Relativistic electron beamj ~ en

cc ~ 1012 A/cm2

10 kA of 1-20 MeV electrons

Lecture 2: Basic plasma equations, self-focusing, direct laser acceleration

2

Laser Interaction with Dense Matter

Plasma approximation:

Laser field at a > 1 so large that atoms ionize within less than laser cycle

Free classical electrons (no bound states, no Dirac equation)

Non-neutral plasma ( , usually fixed ion background) electron ionn n

3

Single electron plasma (ncrit = 1021cm-3)

In plasma, laser interaction generates additional

• E-fields (due to separation of electrons from ions)

• B-fields (due to laser-driven electron currents)

They are quasi-stationary and of same order as laser fields:

12L 03 10 V/m E a

8L 010 Gauss B a

Plasma is governed by collective oscillatory electron motion.

4

The Virtual Laser Plasma Laboratory

Fields

( )

22

2

1cm

p

Bpm

qEq

dt

pd

+=

×+=

γ

γ

Particles

109 particles in 108 grid cells are treated on 512 Processors

of parallel computer

A. Pukhov, J. Plas. Phys. 61, 425 (1999)

Three-dimensional electromagnetic fully-relativistic Particle-Cell-Code

0

4

1

41

=

=∂∂−=

+∂∂=

Bdiv

Ediv

t

B

cErot

jct

E

cBrot

πρ

π

5

Theoretical description of plasma dynamics

Distribution function: ( , , )f r p tr r

( ) (collisions ?( / ) ( , , ) 0 ) v e E v c B f r p tt r p

+ − + ≅r rr r r r

r r

Kinetic (Vlasov) equation ( ):2, 1 ( / )p mv p mcγ γ= = +r r

Fluid description:Approximate equations for density, momentum, ect. functions:

3( , ) ( , , ) N r t f r p t d p=r r r

3( , ) ( , , ) P r t p f r p t d p=r r r r r

6

Starting from Maxwell equations

0

4 0 , 0

4 ( ) e

EB B B A A

c t c

B AE E e N E

c t c

J

tN

π

π φ

=+ == =

=− =−=−−

r r rr r r

rr r r

r

r

Problem: Light waves in plasma

2 with electron momentu/ m and 1 / ) ,(eJ eN P m P mu P mcγ γ γ= − = = +r r r r

and assuming that only electrons with density Ne contribute to the plasma current

while immobile ions with uniform density Ni =N0/Z form a neutralizing background.

22 0

2 20

4( , ) , ( , ) , ( , ) , ( , ) , ,e

p

eA e P N e Na r t r t p r t n r t

mc mc mc N m

φ πϕ ω= = = = =r

r r r r r

using normalized quantities and plasma frequency

222

2 2 2

1 ,p np

ac t c t c

ωϕγ

− = +r 2 2 2( / )( 1)p c nϕ ω= −

derive

7

In this approximation, electrons are described as cold fluid elements which have relativistic momentum and satisfy the equation of motion

where pressure terms proportional to plasma temperature have been neglected.Using again the potentials A and φ and replacing the total time derivative by by partial derivatives, find

and show that this leads to the equation of motion of a cold electron fluid

written again in normalized quantities (see previous problem). Here, make use of

( , ) / ( / )dP r t dt e E u c B= − +r r rr r

( , ) ( )A u

u P r t e At c t c

φ+ =−−− +r r rrr r

1( ) ( ) ( ) ,p a u p a

c tϕ γ−− −= −r r r r r

P muγ=r r

2 2relations and 1 / 2 ( ) ( ) . p p u p c u pγ γ γ= += = − r r r r

Problem: Derive cold plasma electron fluid equation

8

1( ) ( ) ( )p a u p a

c tϕ γ−− −=−r r r r r

Basic solution of

Solution for electron fluid initially at rest, before hit by laser pulse,

implying balance between the electrostatic force and theponderomotive force

This force is equivalent to the dimensional force density

and = p a ϕ γr r

2 2 21 1 / 2p a aγ γ= + = + =

ϕ

2 22

0 2 8p E

F N mcω

γγω π

= =r

It describes how plasma electrons are pushed in front of a laser pulseand the radial pressure equilibrium in laser plasma channels, in whichlight pressure expels electrons building up radial electric fields.

9

For low laser intensities ( ), the solution implies and .The wave equation for laser propagation in plasma

p a=r r1a= 1γ ≅ 1n ≅

then leads to the plasma dispersion relation 2 2 2 2

p c kω ω= +For increasing light intensity, the plasma frequency is modified

22 2

,

( , )4

( , )e

p rel p

N r tn e

m r t

πω ωγ γ

= =rr

by changes of electron density and relativistic γ – factor, giving rise to effects ofrelativistic non-linear optics.

Propagation of laser light in plasma

2 222

2 2 2 2

1 ,p pna

a ac t c c

ω ωγ

− = ≅rr r

10

Relativistic Non-Linear Optics

Self-focussing: vph= c/nR Profile steepening: vg = cnR

ωp2= 4πe2 ne /(m<γ>)

nR = (1 - ωp2/ ω2)1/2

ω2 = ωp2 + c2k2

γ =(1- v2/c2)-1/2

Induced transparency:

11

Problem: Derive phase and group velocity of laser wave in plasma

Starting from the plasma dispersion relation

show that the phase velocity of laser light in plasma is

and the group velocity

where nR is the plasma index of refraction

/ /phase Rv k c nω= =

/ ,group Rv d dk cnω= =

2 21 / .R pn ω ω= −

2 2 2 2 ,p c kω ω= +

12

3D-PIC simulation of laser beam selffocussing in plasma

Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996)

Laser pulse1019 W/cm2

plasma box (ne/nc=0.6)

13

0Re ( ) ( , , ) exp( )y za e ie a r z t ikz i tω= −r r r

2 20 0 0 0 0( , , ) , / / a a r z t a t a a z kaω=r = =

22

022

2

10 2 ( , ) 0ik a

c tra ζ

ζ⊥+ =− =r

Problem: Derive envelope equation

Consider circularly polarized light beam

Confirm that the squared amplitude depends only on the slowly varyingenvelope function a0(r,z,t), but not on the rapidly oscillating phase function

Derive under these conditions the envelope equation for propagation invacuum (use comoving coordinate ζ=z-ct, neglect second derivatives):

14

0

1 2 ( , ) 0r ik a r z

r r r z+ =

Problem: Verify Gaussian focus solution

Show that the Gaussian envelope ansatz2

0 0 ( , ) exp( ( ) ( )( / ) )a r z P z Q z r r= +

2 2 2 20

2/[ (1 / )]

0 2 22 20

/ ( , ) exp arctan

1 /1 /

Rr r z LR

R RR

z Le z ra r z i i

L r z Lz L

− +

= − +++

inserted into the envelope equation

leads to

where is the Rayleigh length giving the length of the focal region.20 / 2RL kr=

15

Relativistic self-focusing

2 21/ 1/ 1 1 2a aγ = + −;

22 222

2 2 2 2

1 1 ,

2p p ana

a ac t c c

ω ωγ

− = ≅ −rr r

2 22 0

0 022 ( , )

2p a

ik a r z az c

ω⊥+ = −

For increasing light intensity, non-linear effects in light propagation first show up In the relativistic factor giving

and leads to the envelope equation (using !)2 2 2 2p c kω ω= +

While is defocusing the beam (diffraction), the termis focusing the beam. Beyond the threshold power

2 2 20 0( / )( / 2)p c a aω−2

0a⊥

22 ( / ) 17.4 GW ( / )crit o p crit eP P n nω ω≅ =the beam undergoes relativistic self-focusing.

16

2D versus 3D relativistic self-focusing

Relativistic self-focusing develops differently in 2D and 3D geometry.Scaling with beam radius R :

2 21/ R⊥ :

diffraction

2 2 2 2 2 20 0( / )( / 2) / (for 3 D: )p c a P R P R aω π− : :

2 2 2 20 0( / )( / 2) / (for 2D: )p c a P R P Raω− : :

relativistic non-linearity

2D leads to a finite beam radius (R~1/P), while 3D leads to beam collapse (R->0).For a Gaussian beam with radius r0:

22 ( / ) 17.4 GW ( / )crit o p crit eP P n nω ω≅ =

2 2

0

2 2 20 0( /16 )/ 2 P cP R I a Rωπ= =

2

2

22 202 2 3

1

32

( ) 41 p

c

dR za R

dz k R

ω= −

power:

beam radius evolution (Shvets, priv.comm.):

critical power:

17

3D-PIC simulation of laser beam selffocussing in plasma

Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996)

Laser pulse1019 W/cm2

plasma box (ne/nc=0.6)

B ~ mcωp/e ~ 108 Gauss

Relativistic electron beamj ~ en

cc ~ 1012 A/cm2

10 kA of 1-20 MeV electrons

18

Relativistic self-focussing of laser channels

relativistic electrons

laser

B-field

ω p2

radius

γ

ne

ωp2= 4π e2 ne / mγ eff

2 21R p Ln ω ω= −

Relativistic mass increase (γ ) and electron density depletion (ne )increases index of refraction in the

channel region, leading to selffocussing

19

Relativistic Laser Plasma ChannelPukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996)

ne/<γ>ne

Intensity

B-field

Intensity

Ion density

80 fs

330 fs

B⊥ jxIL

20

Plasma channels and electron beams observedC. Gahn et al. PRL 83, 4772 (1999)

gas jet laser

6×1019 W/cm2

observed channel

electron spectrum plasma 1- 4 × 1020 cm-3

21

Scaling of Electron Spectra

Pukhov, Sheng, MtV, Phys. Plasm. 6, 2847 (1999)

electrons

Teff =1.8 (Iλ2/13.7GW)1/2

22

Direct Laser Acceleration versus Wakefield Acceleration

Pukhov, MtV, Sheng, Phys. Plas. 6, 2847 (1999)

plasma channel

EB

laserelectron

Free Electron Laser (FEL) physics

DLA

acceleration by transverse laser field

Non-linear plasma wave

LWFA

Tajima, Dawson, PRL43, 267 (1979)

acceleration by longitudinal wakefield

23

Laser pulse excites plasma wave of length λp= c/ωp

-0.2-0.2

0.20.2eEz/ωpmc

22

-2-2eEx/ω0mc

-20-20

2020

px/mc

4040

2020γ

eEx/ω0mc

Z / λ270 280

33

--332020

-20-20

00px/mc

zoom

-0.2-0.2

0.2

eEz/ωpmc wakefield breaksafter few oscillations

4040

2020γ What drives electrons to γ ~ 40

in zone behind wavebreaking?

Laser amplitude a0 = 3

Transverse momentum p⊥/mc >> 3

p⊥ /mc

zoom3

-3a

20

-20

0

Z / Z / λ270 280

λ

z

laser pulse length

λp

24

Channel fields and direct laser acceleration

EB

j = efn0cspace chargen = e(1-f)n0

2(1 ) / 2r peE f m Rω= −2 / 2peB f m Rϕ ω=

22

2/ 2r p

d Rm eE eB m R

dt ϕγ ω= − − = − Radial electron oscillations

2/p γωΩ =

electronmomenta

Lω

2 Lω

(ωp/c)

25

How do the electrons gain energy?

dt p2/2 = e E • p = e E|| p|| + e E⊥ p⊥

dt p = e E + v × Bec

Γ|| = 2 e E|| p|| dt

Gain due to longitudinal (plasma) field:

Γ⊥ = 2 e E⊥ p⊥dt

Gain due to transverse (laser) field: -2x103 0 103

Γ||

Γ

⊥

0

2

x103 Direct Laser

Acceleration(long pulses)

Long pulses (> λp)

0 104

Γ||

Γ

⊥

0

104

Laser WakefieldAcceleration

(short pulses)

Short pulses (< λp)

26

Selected papers:

C. Gahn, et al. Phys.Rev.Lett. 83, 4772 (1999).

J. Meyer-ter-Vehn, A. Pukhov, Z.M. Sheng, in Atoms, Solids, and Plasmas In Super-Intense Laser Fields (eds. D.Batani, C.J.Joachain, S. Martelucci, A.N.Chester), Kluwer, Dordrecht, 2001.

A. Pukhov, J. Meyer-ter-Vehn, Phys. Rev. Lett. 76, 3975 (1996).

A. Pukhov, Z.M. Sheng, Meyer-ter-Vehn, Phys. Plasmas 6, 2847 (1999)

27

0Re ( ) ( , , ) exp( )y za e ie a r z t ikz i tω= −r r r

2 20 0 0 0 0( , , ) , / / a a r z t a t a a z kaω=r = =

22

022

2

10 2 ( , ) 0ik a

c tra ζ

ζ⊥+ =− =r

Problem: Derive envelope equation

Consider circularly polarized light beam

Confirm that the squared amplitude depends only on the slowly varyingenvelope function a0(r,z,t), but not on the rapidly oscillating phase function

Derive under these conditions the envelope equation for propagation invacuum (use comoving coordinate ζ=z-ct, neglect second derivatives):

28

0

1 2 ( , ) 0r ik a r z

r r r z+ =

Problem: Verify Gaussian focus solution

Show that the Gaussian envelope ansatz2

0 0 ( , ) exp( ( ) ( )( / ) )a r z P z Q z r r= +

2 2 2 20

2/[ (1 / )]

0 2 22 20

/ ( , ) exp arctan

1 /1 /

Rr r z LR

R RR

z Le z ra r z i i

L r z Lz L

− +

= − +++

inserted into the envelope equation

leads to

Where is the Rayleigh length giving the length of the focal region.20 / 2RL kr=

29

Problem: Derive channel fields

EB

j = efn0cspace chargen = e(1-f)n0

2(1 ) / 2,r peE f m Rω= − 2 / 2peB f m Rϕ ω=

2/p γωΩ =

Consider an idealized laser plasma channel with uniform charge densityN = e(1-f)N0c , i.e. only a fraction f of electrons is left in the channel afterExpulsion by the laser ponderomotive pressure, and this rest is movingWith velocity c in laser direction forming the current j = efN0c. Show thatthe quasi-stationary channel fields are

and that elctrons trapped in the channel l perform transverse oscillations at the betatron frequency, independent of f,