Transcript Folie 1 - University of Arizona

Lecture 3:
Laser Wake Field Acceleration (LWFA)
1D-Analytics:
1. Nonlinear Plasma Waves
2. 1D Wave Breaking
3. Wake Field Acceleration
Bubble Regime (lecture 4):
1.
2.
3.
4.
5.
3D Wave Breaking and Self-Trapping
Bubble Movie (3D PIC)
Experimental Observation
Bubble Fields
Scaling Relations
1
Direct Laser Acceleration versus Wakefield Acceleration
DLA
electron
B
LWFA
laser
Non-linear plasma wave
E
plasma channel
acceleration by
transverse laser field
Free Electron Laser (FEL) physics
Pukhov, MtV, Sheng,
Phys. Plas. 6, 2847 (1999)
acceleration by
longitudinal wakefield
Tajima, Dawson, PRL43, 267 (1979)
2
3
Laser pulse excites plasma wave of length lp= c/wp
eEz/wpmc
eEz/wpmc
lp
0.2
0.2
wakefield breaks
after few oscillations
-0.2
40
-0.2
g
laser pulse length
20
2
40
eEx/w
mc
g0
What drives electrons to g ~ 40
in zone behind wavebreaking?
-2
20
20
px/mc
-20
a
eEx/w0mc
px/mc
p /mc
3
3
-3-3
2020
z
zoom
zoom
Laser amplitude
a0 = 3
l
00
-20
-20
270
270
Transverse momentum
p/mc >> 3
Z /l
280
Z /l
280
4
0
dt p2/2 = e E  p = e E|| p|| + e E p
G
dt p = e E + e
c vB
2x103
How do the electrons gain energy?
Gain due to longitudinal (plasma) field:
G|| =  2 e E|| p|| dt
0
103
104
G||
G
G =  2 e E pdt
-2x103
0
Gain due to transverse (laser) field:

0
G||
104
5
Phase velocity and gph of Laser Wakefield
L
laser
density
lp
Short laser pulse
( L  lp  2 c / w p )
excites plasma wave with
large amplitude.
Light in plasma (linear approximation)
2
2
wLaser
 wp2  c2kLaser
laser
vgroup

d wL
dk L
 c 1  w p2 /wL2  vpplasma
hase
g ph  1/ 1  v 2ph /c 2  wL /w p  ncrit /ne
6
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:
N 
 ( Nu )  0
t x
 
1 p

  u  (g mu )  eE 
x 
N x
 t
g  1/ 1   2
cold plasma
E
 4 e( N 0  N )
x
7
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
N1 ( x, t ), u1 ( x, t ), E1 ( x, t ) exp(ikx  iwt )
Show that the 1D plasma equations, keeping only terms linear in the perturbed
quantities, have the form
iwmN1  ikN0u1  0,  iwmu1  eE1 , ikE1  4 eN1
giving the dispersion relation
4 e2 No
w 
 w p2
m
2
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
laser
plasma waves driven by a laser pulse at its group velocity (  ph  vph / c   group),
one has
2
g ph  1/ 1   ph
 wL / w p  Ncrit / N0
8
10. Problem: Normalized non-linear 1D plasma equations
We now look for full non-linear propagating wave solutions of the form
N ( ), u( ), E( ), with   wp (t  x / vph )
Using the dimensionless quantities
n( )  N / N0 ,  ( )  u / c, Eˆ ( )  E / E0 , E0  mcwp /e
show that the the 1D plasma equations reduce to
nˆ  1/(1   /  ph )
d
(1   /  ph ) ( g )   Eˆ
d
dEˆ / d   /(1   /  ph )
9
Nonlinear 1D Relativistic Plasma Wave
nˆ  1/(1   /  ph )
1. integral: energy conservation
(1   /  ph )
dEˆ
d ˆ2
d
dg
ˆ
E

( E /2)  
( g ) 
d d
d
d
(use g 2 (1   2 )  1 )
dEˆ / d   /(1   /  ph )
 ( )
umax
(g max )
2
Eˆ ( )2 /2 + g ( )  Eˆmax
/2 + 1  g max
Eˆ (g )= 2(g max -g )
d
( g )   Eˆ
d
u0
(g  1)
Eˆ ( )
E0
Emax
Eˆ max = 2(g max -1)
10
Wave Breaking
n( ) 
n0
density spikes diverge
(1  u ( ) / v ph )
for u  v ph
v ph
u

Maximum E-field at wave breaking (Achiezer and Polovin, 1956)
EWB  E0 2(g ph  1)
Non-relativistic limit (Dawson 1959)
EWB  E0  ph  mvphwp / e
11
11. Problem: Derive non-linear wave shapes
Show that the non-linear velocity  ( )
can be obtained analytically in non-relativistic
2
approximation  ph 1, g ph  1   ph
/ 2, from
 d 
(1   /  ph )d ( g )
2(g m  g )

 ( )
(1   /  ph ) d (  /  m )
1  ( / m )2
with the implicit solution
n( )
(   0 )  arcsin(  /  m )  (  m /  ph ) 1  (  /  m ) 2
Notice that this reproduces the linear plasma
wave for small wave amplitude m. Then
discuss 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.
E ( )
 /
12
Wakefield amplitude
The wake amplitude is given between laser ponderomotive and electrostatic force
E / E0 
c
wp
 
c
wp
g
Using g  1  a0 and ( a ) max  2k p a0 with k p  w p / c for circular polarization,
one finds
2
2
2
2
c  a 
Emax / E0 



w p  2g 

max
a02
1  a02
laser
density
For linear polarization,
replace a02  a02 / 2 .
13
14
Dephasing length
Acceleration phase
  w p (t  x/vph )  
E-field
Td  Ld / c
vph  ve  c
Emax
Time between injection
and dephasing
v ph
ve


lp
Ld
Ld 
 /w p
(1/v ph  1/c)

Dephasing
length
 /w p  2c
1  v ph /c
2
2
2
 lpg ph
Estimate of maximum particle energy
2
Wmax  Emax Ld  g ph
(Emaxlp )
15
PHASE-SPACE ANALYSIS
FLUID VS. TRAPPED ORBITS
trapped orbit
(e- “kicked” from
fluid orbit)
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
1D case:
Trapped electrons require a
sufficiently high momentum
to reside inside 1D separatrix
cold fluid orbit
(e- initially at rest)
16
Maximum electron energy gain Wmax in wakefield
Electron acceleration (norm. quantities)
dP / dt  Eˆ ( )
1



2
Pmax
P0
( d / dt )dP
1
2g
(dP / dt )d
2
ph
( Pmax  P0 )
p  g
Pmax  g max
  t  x(t ) /  ph
2
d / dt  1 1/  ph  1/ 2g ph

1

Eˆ ( )d
2
P0  (g ) ph
pm
0

 
  1 
d ( g )
  ph 
 pm 

 2 pm 2g m Em2
pm
2
W  (Pmax  P0 )c  2g ph
Em2
 pm 
1
2
acceleration
range
For maximum wave amplitude pm  (g )m  g ph
Emax  Pmaxc  4mc2g 3ph
(in units, first obtained by Esarey, Piloff 1995)
17
1
Wave Breaking
single electron motion
injected at phase velocity
p/mc = g
g max
Emax Ld
Wave-Breaking at
(g)ph
p/mc = 
0
EWB / E0  2(g ph  1)
collective
motion of
plasma
electrons
E/E0
Longitudinal
E-field
18
Example
Plasma: N0  1019 cm3 , lp  2 c / wp  10 m,
Laser:
E0  mcwp / e  3 1011 V/m
l  1 m, Ncrit  1021cm3,  pulse  lp / 2c  15 fs
g ph  Ncrit / N0  10,
E-field at wave-breaking:
Wmax  4mc2g 3ph  2 GeV,
EWB  E0 2(g ph  1)  1012 V/m
2
Ld  lpg ph
 1 mm
Dephasing length:
Required laser power:
EWB / E0  2(g ph  1) 
a02 / 2
1  a02 / 2
 a02  36, I  5 1019 W/cm 2
P  I lp2  50 TW, WLas  P Las  80 mJ
19
Nature Physics 2, 456 (2006)
L=3.3 cm, f=312 m
Laser
1.5 J, 38 TW,
40 fs, a = 1.5
1 GeV electrons
Plasma filled capillary
Density: 4x1018/cm3
Divergence(rms): 2.0 mrad
Energy spread (rms): 2.5%
Charge: > 30.0 pC
20
GeV: channeling over cm-scale
• Increasing beam energy requires increased dephasing length and power:
W[GeV] ~ I[W/cm2 ] n[cm-3 ]
Capillary
• 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

Plasma channel technology: Capillary
1 GeV
e- beam
Laser: 40-100 TW,
40 fs 10 Hz
21
3 cm
0.5 GeV Beam Generation
225 m 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%
22
Charge: ~50 pC
1.0 GeV Beam Generation
312 m 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
23
Laser power fluctuation, discharge timing, pointing stability
Wake Evolution and Dephasing
Longitudinal
Momentum
200
WAKE FORMING
0
Propagation Distance
200
Longitudinal
Momentum
INJECTION
0
Propagation Distance
Longitudinal
Momentum
200
DEPHASING

3
2
3 / 2
DEPHASING
L
dph  (lp / l ) ne
0
Propagation Distance
24
Geddes et al., Nature (2004) & Phys. Plasmas (2005)
Bubble regime: Ultra-relativistic laser, I=1020 W/cm2
A.Pukhov & J.Meyer-ter-Vehn, Appl. Phys. B, 74, p.355 (2002)
trapped e-
laser
12J, 33 fs
Time evolution of electron spectrum
Ne / MeV
1 109
t=750
t=650
20
t=850
t=550
X/l
5 108
-20
t=450
t=350
-50
cavity
Z/l 0
0
200
400
E, MeV
25