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### Lecture 3:Laser Wake Field Acceleration (LWFA)

### Nonlinear 1D Relativistic Plasma Wave

1D-Analytics:

- Nonlinear Plasma Waves
- 1D Wave Breaking
- Wake Field Acceleration

Bubble Regime (lecture 4):

- 3D Wave Breaking and Self-Trapping
- Bubble Movie (3D PIC)
- Experimental Observation
- Bubble Fields
- Scaling Relations

DLA

LWFA

Non-linear plasma wave

electron

B

E

acceleration by

transverse laser field

plasma channel

laser

Free Electron Laser (FEL) physics

acceleration by

longitudinal wakefield

Pukhov, MtV, Sheng,

Phys. Plas. 6, 2847 (1999)

Tajima, Dawson, PRL43, 267 (1979)

Direct Laser Acceleration versus Wakefield Acceleration

0.2

0.2

eEz/wpmc

wakefield breaks

after few oscillations

-0.2

eEz/wpmc

-0.2

40

g

20

40

What drives electrons to g ~ 40

in zone behind wavebreaking?

2

g

eEx/w0mc

-2

20

20

px/mc

laser pulse length

-20

zoom

3

zoom

Laser amplitude

a0 = 3

a

3

-3

eEx/w0mc

20

-3

l

20

Transverse momentum

p/mc >> 3

0

p/mc

0

px/mc

-20

-20

270

280

Z / l

280

270

Z / l

Laser pulse excites plasma wave of length lp= c/wplp

z

dt p = e E + v B

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

G

0 2x103

Gain due to transverse (laser) field:

-2x103 0 103

G||

e

G = 2 e E pdt

c

Gain due to longitudinal (plasma) field:

G

0104

G|| = 2 e E|| p|| dt

0 104

G||

How do the electrons gain energy?L

density

Short laser pulse

( )

excites plasma wave with

large amplitude.

laser

lp

Light in plasma (linear approximation)

Phase velocity and gph of Laser Wakefieldcold plasma### 1D Relativistic Plasma Equations (without laser)

Consider an electron plasma with density N(x,t), velocity u(x,t), and

electric fieldE(x,t), all depending on one spatial coordinate x and timet.

Ions with densityN0 are modelled as a uniform, immobile, neutralizing

background. This plasma is described by the 1D equations:

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

Show that the 1D plasma equations, keeping only terms linear in the perturbed

quantities, have the form

giving the dispersion relation

Apparently, plasma waves oscillate with plasma frequency for any k, in this

lowest order approximation, and have phase velocity vph=wp/k. Show that for

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

one has

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

Using the dimensionless quantities

show that the the 1D plasma equations reduce to

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

1. integral: energy conservation

density spikes diverge ### Wave Breaking

t

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

Non-relativistic limit (Dawson 1959)

11. Problem: Derive non-linear wave shapes

Show that the non-linear velocity

can be obtained analytically in non-relativistic

approximation from

with the implicit solution

Notice that this reproduces the linear plasma

wave for small wave amplitude bm. Then

discuss the non-linear shapes qualitatively:

Verify that the extrema of b(t), n(t), and the

zeros of E(t)do not shift intwhen increasing bm,

while the zeros of b(t), n(t), and the extrema

of E(t)are shifted such that velocity and density

develop sharp crests, while the E-field acquires

a sawtooth shape.

Using with for circular polarization,

one finds

For linear polarization,

replace .

density

laser

Wakefield amplitude

The wake amplitude is given between laser ponderomotive and electrostatic force

E-field

Emax

t

Estimate of maximum particle energy

Dt

lp

Dephasing lengthAcceleration phase

Time between injection

and dephasing

Dephasing

length

1D separatrix

Viewgraph taken from E. Esarey

Talk at Dream Beam Symposium

www.map.uni-muenchen.de/events.en.html

UID: symposium PWD: dream beams

PHASE-SPACE ANALYSIS

FLUID VS. TRAPPED ORBITS

trapped orbit

(e- “kicked” from fluid orbit)

1D case:

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

cold fluid orbit

(e- initially at rest)

0

acceleration

range

For maximum wave amplitude

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

Maximum electron energy gain Wmaxin wakefield

Electron acceleration (norm. quantities)

single electron motion

injected at phase velocity

p/mc = bg

Wave-Breaking at

(bg)ph

collective

motion of

plasma

electrons

0

E/E0

Longitudinal

E-field

Wave Breaking

p/mc = b

Nature Physics 2, 456 (2006)

L=3.3 cm, f=312 mm

Laser

1 GeV electrons

Divergence(rms): 2.0 mrad

Energy spread (rms): 2.5%

Charge: > 30.0 pC

Plasma filled capillary

Density: 4x1018/cm3

1.5 J, 38 TW,

40 fs, a = 1.5

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

Plasma channel technology: Capillary

1 GeV

e- beam

40-100 TW, 40 fs 10 Hz

Laser:

3 cm

0.5 GeV Beam Generation

225 mm diameter and 33 mm length capillary

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

a0

Stable operation

500 MeV Mono-energetic beams:

a0 ~ 0.75 (11 TW, 75 fs)

Peak energy: 490 MeV

Divergence(rms): 1.6 mrad

Energy spread (rms): 5.6%

Resolution: 1.1%

Charge: ~50 pC

1.0 GeV Beam Generation

312 mm diameter and 33 mm length capillary

- 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 MeV

Divergence(rms): 2.0 mrad

Energy spread (rms): 2.5%

Resolution: 2.4%

Charge: > 30.0 pC

Less stable operation

Laser power fluctuation, discharge timing, pointing stability

Wake Evolution and Dephasing

200

WAKE FORMING

Longitudinal

Momentum

0

Propagation Distance

200

INJECTION

Longitudinal

Momentum

0

Propagation Distance

200

DEPHASING

DEPHASING

Longitudinal

Momentum

0

Propagation Distance

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

N

/ MeV

laser

12J, 33 fs

e

Time evolutionof electron spectrum

trapped e-

9

1 10

t=750

t=650

t=850

t=550

8

t=450

5 10

t=350

0 200 400

-50

0

Z/

l

E, MeV

cavity

Bubble regime: Ultra-relativistic laser, I=1020 W/cm2A.Pukhov & J.Meyer-ter-Vehn, Appl. Phys. B, 74, p.355 (2002)

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