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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
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
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
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