1

Lecture 3:Laser Wake Field Acceleration (LWFA)

1D-Analytics:

1. Nonlinear Plasma Waves2. 1D Wave Breaking 3. Wake Field Acceleration

Bubble Regime (lecture 4):

1. 3D Wave Breaking and Self-Trapping2. Bubble Movie (3D PIC)3. Experimental Observation4. Bubble Fields5. Scaling Relations

2

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

3

4

Laser pulse excites plasma wave of length p= c/p

-0.2-0.2

0.20.2eEz/pmc

22

-2-2eEx/0mc

-20-20

2020

px/mc

4040

2020

eEx/0mc

Z / 270 280

33

--332020

-20-20

00px/mc

zoom

-0.2-0.2

0.2

eEz/pmc wakefield breaksafter few oscillations

4040

2020 What drives electrons to ~ 40

in zone behind wavebreaking?

Laser amplitude a0 = 3

Transverse momentum p/mc >> 3

p /mc

zoom3

-3a

20

-20

0

Z / Z / 270 280

z

laser pulse length

p

5

How do the electrons gain energy?

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

dt p = e E + v Bec

|| = 2 e E|| p||

dt

Gain due to longitudinal (plasma) field:

= 2 e E pdt

Gain due to transverse (laser) field:

||

0 104

0

10

4

-2x103 0 103

0

2x1

03

||

6

Phase velocity and ph of Laser Wakefield

2 2crit1/ 1 / / /ph ph L p ev c n n

2 2 2 2Laser p Laserc k

2 2 plasmap

lasergroup hase1 / L

p L

L

d

dkv c v

Light in plasma (linear approximation)

L

p

density

lase

r

Short laser pulse ( )excites plasma wave withlarge amplitude.

2 /p pL c

7

1D Relativistic Plasma Equations (without laser)

( ) 0N

Nut x

1( )

pu mu eE

t x N x

04 ( )E

e N Nx

cold plasma

21/ 1

Consider an electron plasma with density N(x,t), velocity u(x,t), and electric field E(x,t), all depending on one spatial coordinate x and time t.Ions with density N0 are modelled as a uniform, immobile, neutralizingbackground. This plasma is described by the 1D equations:

8

Problem: Linear plasma waves

Consider a uniform plasma with small density perturbation N(x,t)=N0+N1(x,t), producing velocity and electric field perturbations u1(x,t) and E1(x,t) ,respectively.Look for a propagating wave solution

1 1 1( , ), ( , ), ( , ) exp( )N x t u x t E x t ikx i tShow that the 1D plasma equations, keeping only terms linear in the perturbedquantities, have the form

1 0 1 1 1 1 10, , 4i mN ikN u i mu eE ikE eN giving the dispersion relation 2

2 24 op

e N

m

Apparently, plasma waves oscillate with plasma frequency for any k, in thislowest order approximation, and have phase velocity vph=p/k. Show that for

plasma waves driven by a laser pulse at its group velocity ( ), one has

201/ 1 / /ph ph L p critN N

/ laserph ph groupv c

9

10. Problem: Normalized non-linear 1D plasma equations

show that the the 1D plasma equations reduce to

We now look for full non-linear propagating wave solutions of the form

with ( ), ( ), ( ), ( / )p phN u E t x v Using the dimensionless quantities

0( ) / ,n N N ( ) / ,u c 0ˆ ( ) / ,E E E

0 /pE mc e

ˆ 1/(1 / )phn

ˆ(1 / ) ( )ph

dE

d

ˆ / /(1 / )phdE d

10

Nonlinear 1D Relativistic Plasma Wave

ˆ 1/(1 / )phn

ˆ(1 / ) ( )ph

dE

d

ˆ / /(1 / )phdE d

1. integral: energy conservation

2ˆ

ˆ ˆ( /2) ( )dE d d d

E Ed d d d

2 2(1 ) 1 (use )

0u

maxE

( 1)

maxumax( )

0E

( )

ˆ ( )E

2 2max max( ) 2 ( ) 2 ˆ ˆ/ + / + 1 E E

max( )=ˆ 2( - ) E

max max=ˆ 2( -1) E

11

Wave Breaking

0( )( )(1 / )ph

nn

u v

Maximum E-field at wave breaking (Achiezer and Polovin, 1956)

WB 0 2( 1)phE E

WB 0 /ph ph pE E mv e Non-relativistic limit (Dawson 1959)

density spikes diverge

for phu v

phvu

12

11. Problem: Derive non-linear wave shapes

Show that the non-linear velocity can be obtained analytically in non-relativisticapproximation from

2

(1 / ) ( ) (1 / ) ( / )

2( ) 1 ( / )

ph ph m

m m

d dd

21, 1 / 2,ph ph ph

( )

with the implicit solution2

0( ) arcsin( / ) ( / ) 1 ( / )m m ph m

Notice that this reproduces the linear plasmawave for small wave amplitude m. Thendiscuss the non-linear shapes qualitatively:Verify that the extrema of , n(), and the zeros of E() do not shift in when increasing m, while the zeros of (), n(), and the extrema of E() are shifted such that velocity and density develop sharp crests, while the E-field acquires a sawtooth shape.

( )

( )n

( )E

/

13

Wakefield amplitude

density

laser

0/p p

c cE E

2 20

max 0 20max

/2 1p

ac aE E

a

The wake amplitude is given between laser ponderomotive and electrostatic force

For linear polarization,replace .2 2

0 0 / 2a a

Using with for circular polarization, one finds

22 20 max 01 (an 2d ) pa a k a /p pk c

14

15

Dephasing length

22 21 1 1

/ / 2 ( / /c) /c

p pd p ph

ph ph

cL

v v

ph( / )p t x v Acceleration phase

Estimate of maximum particle energy2

max max max( )d ph pW E L E

/d dT L c dLTime between injection

and dephasingDephasing

length

E-field

p

Emaxphv

ev

cph ev v

16

1D case:

Trapped electrons require a sufficiently high momentum to reside inside 1D separatrix

cold fluid orbit

(e- initially at rest)

trapped orbit

(e- “kicked” from fluid orbit)

1D separatrix

PHASE-SPACE ANALYSISFLUID VS. TRAPPED ORBITS

Viewgraph taken from E. EsareyTalk at Dream Beam Symposiumwww.map.uni-muenchen.de/events.en.htmlUID: symposium PWD: dream beams

17

Maximum electron energy gain Wmax in wakefield

p

mp

mp0

0 ( ) phP

max max 1P

1 2

accelerationrange

ˆ/ ( )dP dt E ( ) / pht x t 2/ 1 1/ 1/ 2ph phd dt

Electron acceleration (norm. quantities)

For maximum wave amplitude ( )m m php

(in units, first obtained by Esarey, Piloff 1995)2 3

max max 4 phE P c mc

2

1 ( )

2 2

m

m

p

php

m m m

d

p E

max

0

max 02

( / )

1( )

2

P

P

ph

d dt dP

P P

1 1

2 2

ˆ( / ) ( )dP dt d E d

2 2max 0( ) 2 ph mW P P c E

18

0

()ph

p/mc =

collectivemotion ofplasma

electrons

single electron motioninjected at phase velocity

E/E0

LongitudinalE-field

Wave Breaking

p/mc =

Wave-Breaking at

0/ 2( 1)WB phE E

max max dE L

19

Example

E-field at wave-breaking:12

0 2( 1) 10 V/mWB phE E

Plasma: 19 3 110 010 cm , 2 / 10 m, / 3 10 V/mp p pN c E mc e

Laser: 21 3pulse1 m, 10 cm , / 2 15 fs crit pN c

0/ 10,ph critN N 2 3max 4 2 GeV,phW mc

Dephasing length: 2 1 mmd p phL

Required laser power:2

2 19 200 02

0

/ 2/ 2( 1) 36, 5 10 W/cm

1 / 2WB ph

aE E a I

a

2

Las Las50 TW, 80 mJpP I W P

20

Nature Physics 2, 456 (2006)

L=3.3 cm, =312 mLaser

1.5 J, 38 TW, 40 fs, a = 1.5

Plasma filled capillary

Density: 4x1018/cm3

Divergence(rms): 2.0 mradEnergy spread (rms): 2.5%Charge: > 30.0 pC

1 GeV electrons

21

GeV: channeling over cm-scale

• Increasing beam energy requires increased dephasing length and power:

• Scalings indicate cm-scale channel at ~ 1018 cm-3 and ~50 TW laser for GeV

• Laser heated plasma channel formation is inefficient at low density

• Use capillary plasma channels for cm-scale, low density plasma channels

Capillary

W[GeV] ~ I[W/cm2] n[cm-3]

3 cm

e- beam

1 GeV

Laser: 40-100 TW, 40 fs 10 Hz

Plasma channel technology: Capillary

22

0.5 GeV Beam Generation

Density: 3.2-3.8x1018/cm3

Laser: 950(15%) mJ/pulse (compression scan)

Injection threshold: a0 ~ 0.65 (~9TW, 105fs)

Less injection at higher power

-Relativistic effects

-Self modulation

500 MeV Mono-energetic beams:

a0 ~ 0.75 (11 TW, 75 fs)

Peak energy: 490 MeVDivergence(rms): 1.6 mradEnergy spread (rms): 5.6%Resolution: 1.1%Charge: ~50 pC

Stable operation

a0

225 m diameter and 33 mm length capillary

23

1.0 GeV Beam Generation

Laser: 1500(15%) mJ/pulse

Density: 4x1018/cm3

Injection threshold: a0 ~ 1.35 (~35TW, 38fs)

Less injection at higher power

Relativistic effect, self-modulation

1 GeV beam: a0 ~ 1.46 (40 TW, 37 fs)

Peak energy: 1000 MeVDivergence(rms): 2.0 mradEnergy spread (rms): 2.5%Resolution: 2.4%Charge: > 30.0 pC

Less stable operation

312 m diameter and 33 mm length capillary

Laser power fluctuation, discharge timing, pointing stability

24

Wake Evolution and Dephasing

WAKE FORMING

INJECTION

DEPHASINGDEPHASING

Propagation Distance

Lon

gitu

dina

l M

omen

tum

200

0

Propagation DistanceL

ongi

tudi

nal

Mom

entu

m

200

0

Propagation Distance

Lon

gitu

dina

l M

omen

tum

200

0

Geddes et al., Nature (2004) & Phys. Plasmas (2005)

Ldph p3 /2 ne

3 / 2

25

Bubble regime: Ultra-relativistic laser, I=1020 W/cm2

A.Pukhov & J.Meyer-ter-Vehn, Appl. Phys. B, 74, p.355 (2002)

laser12J, 33 fstrapped e-

2 0

-2 0

X /

0Z/-50 cavity0 200 400

E, MeV

t=350

t=450

t=550

t=650

t=750

t=850

5 108

1 109Ne / MeV

Time evolution of electron spectrum