1. Lecture 3: Laser Wake Field Acceleration (LWFA)
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

2. Direct Laser Acceleration versus Wakefield Acceleration
Pukhov, MtV, Sheng,
Phys. Plas. 6, 2847 (1999)
plasma channel E B
laser
electron
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

4. Laser pulse excites plasma wave of length lp= c/wp
-0.2
0.2
eEz/wpmc 2 -2
eEx/w0mc
-20 20
px/mc 40 g
Z / l
270
280 3 -3
zoom
-0.2
0.2
eEz/wpmc
wakefield breaks
after few oscillations 40 20 g
What drives electrons to g ~ 40
in zone behind wavebreaking? lp
laser pulse length
Laser amplitude
a0 = 3
Transverse momentum
p/mc >> 3
p /mc
zoom 3 -3 a 20
-20
Z / l
270
280 l z

5. How do the electrons gain energy?
-2x
x103
G|| G
dt p2/2 = e E  p = e E|| p|| + e E p
dt p = e E v  B e c
G = 2 e E pdt  
Gain due to transverse (laser) field:
G|| G
G|| = 2 e E|| p|| dt  
Gain due to longitudinal (plasma) field:

6. Phase velocity and gph of Laser Wakefieldlp
density
laser
Short laser pulse
( )
excites plasma wave with
large amplitude.
Light in plasma (linear approximation)

7. 1D Relativistic Plasma Equations (without laser)
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, neutralizing
background. This plasma is described by the 1D equations:
cold plasma

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
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

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
Using the dimensionless quantities

10. Nonlinear 1D Relativistic Plasma Wave
1. integral: energy conservation

11. Wave Breaking density spikes diverge t
Maximum E-field at wave breaking (Achiezer and Polovin, 1956)
Non-relativistic limit (Dawson 1959)

12. 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 in t when 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.

13. Wakefield amplitude
The wake amplitude is given between laser ponderomotive and electrostatic force
Using with for circular polarization,
one finds
density
laser
For linear polarization,
replace

15. Time between injection
Dephasing length
Acceleration phase
E-field t lp
Emax Dt
Time between injection
and dephasing
Dephasing
length
Estimate of maximum particle energy

16. FLUID VS. TRAPPED ORBITS
Viewgraph taken from E. Esarey
Talk at Dream Beam Symposium
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
1D separatrix
cold fluid orbit
(e- initially at rest)

17. Maximum electron energy gain Wmax in wakefield
Electron acceleration (norm. quantities)
acceleration
range
For maximum wave amplitude
(in units, first obtained by Esarey, Piloff 1995)

18. Wave Breaking single electron motion injected at phase velocity
(bg)ph
p/mc = bg
collective
motion of
plasma
electrons
single electron motion
injected at phase velocity
E/E0
Longitudinal
E-field
Wave-Breaking at
p/mc = b

19. Example Plasma: Laser: E-field at wave-breaking: Dephasing length:
Required laser power:

20. Plasma filled capillary
Nature Physics 2, 456 (2006)
L=3.3 cm, f=312 mm
Laser
1.5 J, 38 TW,
40 fs, a = 1.5
Plasma filled capillary
Density: 4x1018/cm3
Divergence(rms): 2.0 mrad
Energy 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
Plasma channel technology: Capillary
1 GeV
e- beam
Laser:
TW, 40 fs 10 Hz
3 cm

22. 225 mm diameter and 33 mm length capillary
0.5 GeV Beam Generation
225 mm diameter and 33 mm length capillary
Density: x1018/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)
With 225 micron diameter and 33 mm length capillary,
stable generation of half GeV mono-energetic beams was found.
This is a typical image from the spectrometer.
Horizontal axis is Energy in MeV, and vertical axis is space.
Input laser parameter was about 11 TW 75 fs.
For this particular shot, peak energy was 490 MeV, divergence was 1.6 mrad,
energy spread was 5.6 %, and charge was around 50 pC.
Peak energy: 490 MeV
Divergence(rms): 1.6 mrad
Energy spread (rms): 5.6%
Resolution: 1.1%
Charge: ~50 pC

23. 312 mm diameter and 33 mm length capillary
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
With 312 micron diameter, 33 mm length capillary,
1 GeV electron beam was generated.
This is a spectrum. Input laser was 40 TW - 37 fs. Peak energy was 1 GeV, divergence was 2.0 mrad, energy spread was 2.5 %.
Here, the resolution of the magnetic spectrometer was 2.4 %.
This spread was clearly resolution limited, means might have been smaller.
Charge was 30 pC.
This results is published very recently from Nature Physics.
Less stable operation
Laser power fluctuation, discharge timing, pointing stability

24. Wake Evolution and Dephasing
WAKE FORMING
200
Longitudinal
Momentum
Propagation Distance
INJECTION
200
Longitudinal
Momentum
Propagation Distance
200
DEPHASING
DEPHASING
Longitudinal
Momentum
Propagation Distance
Geddes et al., Nature (2004) & Phys. Plasmas (2005)

25. Bubble regime: Ultra-relativistic laser, I=1020 W/cm2
A.Pukhov & J.Meyer-ter-Vehn, Appl. Phys. B, 74, p.355 (2002)
laser
12J, 33 fs
trapped e- Z/ l
-50
cavity
E, MeV
t=350
t=450
t=550
t=650
t=750
t=850
5 10 8
1 10 9 N e
/ MeV
Time evolution of electron spectrum