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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  ( mu )  eE 
x 
N x
 t
E
 4 e( N 0  N )
x
  1/ 1   2
cold plasma
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    p (t  x / v p )
Using the dimensionless quantities
n( )  N / N0 ,  ( )  u / c, Eˆ ( )  E / E0 , E0  mc p /e
show that the the 1D plasma equations reduce to
nˆ  1/(1   /  p )
d
(1   /  p )
(  )   Eˆ
d
dEˆ / d   /(1   /  p )
11. Problem: Derive non-linear wave shapes
Show that the non-linear velocity  ( )
can be obtained analytically in non-relativistic
approximation  p 1,  p  1   p2 / 2, from
 d 
(1   /  p )d (  )
2( m   )

 ( )
(1   /  p ) d (  /  m )
1  ( / m )2
with the implicit solution
n( )
(   0 )  arcsin( / m )  (m /  p ) 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 ( )
 /
13. Problem: Maximum electron energy gain Wmax
Verify energy gain for electron injected at phase
Velocity according to drawing:
dP / dt  Eˆ ( )
p  
Pmax   max
  t  x(t ) /  ph
2
d / dt  1 1/  ph  1/ 2 ph
1

(dP / dt )d
2

1

P0  (  ) ph
Eˆ ( )d
pm
2
0

Pmax
P0

(d / dt )dP
1
2
2
ph
( Pmax  P0 )

 
  1 
d (  )
  ph 
 pm 

 2 pm 2 m Em2
pm
2
W  ( Pmax  P0 )c  2 ph
Em2
pm  (  ) m   ph
Show that maximum energy for wavebreaking is
Emax  Pmaxc  4mc2 3ph
 pm 
1
2
acceleration
range
1
14. Problem: Verfy the Example
Plasma: N0  1019 cm3 , p  2 c /  p  10 m,
Laser:
E0  mc p / e  3 1011 V/m
  1 m, Ncrit  1021 cm3 ,  pulse  p / 2c  15 fs
 ph  Ncrit / N0  10,
E-field at wave-breaking:
Wmax  4mc2 3ph  2 GeV,
EWB  E0 2( ph  1)  1012 V/m
2
Ld  p ph
 1 mm
Dephasing length:
Required laser power:
EWB / E0  2( ph  1) 
a02 / 2
1  a02 / 2
 a02  36, I  5 1019 W/cm 2
P  I p2  50 TW, WLas  P Las  800 mJ
Electron Trapping in the Broken-Wave Regime
Pukhov, MtV, Appl.Phys. B74, 355 ( 2002)
Laser pulse
Plasma density:
3.5×1019 /cm3
Laser pulse :
6.6 fs
20 mJ
3 TW
a0=1.7
Trapped electrons:
colour ~ pz/mc
2000
Bubble regime:
Ultra-relativistic laser, I=1020 W/cm2:
VLPL
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/
5 108
-20
t=450
t=350
-50
cavity
Z/ 0
0
200
400
E, MeV
The Bubble: Details
VLPL
A.Pukhov & J.Meyer-ter-Vehn, Appl. Phys. B, 74, p.355 (2002)
laser
12J, 33 fs
T=500
20
X/
(d)
600
a
2
Nonlinear laser
compression
down to a
one-cycle pulse
400
trapped e-
-20
200
(g) 
500
0
T=700
0.2
20
(e) n e / n c
650
cavity wall
X/
cavity
-20
e- beam
0
( f) eE z /mc  0
0.5
20
accelerating
field
0
R/
-50
Z/ 
0
Z/ 
700
-factor of electrons
0.1
0
1000
650
Z/ 700
The trapping cross-section is
str~ 3 m2
Observation of monoenergetic electron beams
Nature 431 (2004):
Mangles et al, Rutherford:
70 MeV beam
Geddes et al, LBNL:
85 MeV beam
Faure et al, LOA:
170 MeV beam
First observation of bubble acceleration
J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gordienko, E. Lefebvre, J. Rousseau, F. Burgy,
V. Malka, Nature 431, 541 (2004)
energy: 170 ± 20 MeV
charge: 0.5 nC
divergence: 10 mrad
10% laser->electron conv.
B=0
He-gas
ne= 61018 cm-3
B>0
Laser pulse:
1J, 30 fs
ne= 21019 cm-3
The experimental spectrum is peaked in energy
in good agreement with 3D-PIC simulation
J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gordienko, E. Lefebvre, J. Rousseau, F. Burgy,
V. Malka, Nature 431, 541 (2004)
Electron spectrum
peaked in energy
Simulation
(Pukhov)
Experiment
Bubble Acceleration (3D-PIC)
M.Geissler, J. Schreiber, J. Meyer-ter-Vehn, New J. Phys. 8, 186 (2006).(http://www.njp.org/)
(a0=5)
electron path
180
z (m)
1
-1
210
Ez (TV/m)
0
ne (1020 cm-3)
1
Laser
Laser
5fs, 115 mJ
2
2x109 electrons
at 70 ± 10 MeV
15 % of
laser energy
Electrons/MeV
plasma: 2 x1019 cm-3
0
40
80
E (MeV)
3D wakefield, transverse electron motion,
single bubble formation
p = 2c/p
electric
potential
peak at rear
bubble vertex
void of
electrons
ions
only
driving
laser
pulse
electron trajectories
seen in frame comoving with bubble)
for analytic description see: Kostyukov, Pukhov, Kiselev, Phys. Plasmas 11, 5256 (2004)
S. Gordienko, A. Pukhov, Phys. Plasmas 12, 043109 (2005)
3D-PIC movie (5 fs, 115 mJ,a0=5) M. Geissler (2005)
M.Geissler, J. Schreiber, J. Meyer-ter-Vehn, New J. Phys. 8, 186 (2006).
Intensity threshold for electron trapping in bubble
M.Geissler, J. Schreiber, J. Meyer-ter-Vehn, NJP submitted (2006).
a0 = 3
a0 =5
Number and total energy of 70 MeV electrons
extract beam
Number of electrons
in peak > 60 MeV
Laser energy
In bubble volume initially
Energy in peak
Particle trajectories: 5fs, a=5, 60-80MeV e-
“Monoenergetic” Energy Spectra
from Regular Structures
Regular accelerating structure 
dN dN  dE 



dE dx0  dx0 
1
E =E(x0)
“Monoenergetic” spikes at stationary points
At a first-order maximum
dE
0
dx0
E  Em   ( x0  x0 m ) 2
Universal spectrum for continuous particle trapping
dN
 2s tr ne Lacc
dE
1
Em ( Em  E )
“Monoenergetic” Energy Spectra
from Stationary Points
1st order maximum
E
dN
~
dE
Em
1
Em  E
edN/dE
1 109
t=750
t=650
t=850
t=550
5 108
t=450
t=350
N
x0m
x0
0
200
400
E
E, MeV m
E
The bubble fields
Kostyukov, Pukhov, Kiselev, Phys. Plasmas 11, 5256 (2004)
spherical bubble:
Maxwell
potentials
nion  n0 , ne  0
  E  1 n
E np
 B 

t 
E
E ?
B?
c
4 e2 n0
plasma  
m
2
p
laboratory frame
   (  x  t , y, z )
ax  ax (  x  t , y, z )
a  0
choose gauge
  ax  0
a
 
t
B  a
  3 / 4
solution: harmonic oscillator potential
  ax  ( R 2   2  y 2  z 2 ) 8
normalized quantities: eE / m p c, eB / m p c
n / n0 , e / mc 2 , ea / mc 2 ,  pt , (c /  p ), etc.
The bubble fields
potentials:
  ax  ( R 2   2  y 2  z 2 ) 8
  x t
E field:
E    (  / 2, y / 4, z / 4)
B field:
B    (ax ex )  ( 0, z / 4, - y / 4)
Force on electron with velocity
v  cex
:
F   E  v  B  ( Ex ,  E y  Bz ,  Ez  By )
= (   / 2,  y / 2,  z / 2)
Force on electron with velocity
F  (   / 2, 0, 0)
v  cex:
12. Problem: Bubble fields
(Kostyukov, Pukhov, Kiselev, Phys. Plasmas 11, 5256 (2004))
Consider a spherical bubble (radius R, ne= 0, ni=n0 for r < R) moving in the lab frame
in x-direction at velocity c. Show that the electric potential corresponds to a harmonic
oscillator
 ( x, y, z, t )  ( R 2  ( x  ct )2  y 2  z 2 ) / 8R 2
Start from Maxwell equations (no current inside bubble: ions static, no electrons!)
 E  1 n
 B  E / t
E  a / t  
B   a
with normalized quantities (confirm!)
eE / m p c, eB / m p c, n / n0 , e / mc 2 , ea / mc2 ,  pt , (c /  p ), etc.
Use gauge ax   (  x  ct , y, z ), a  0 for vector potential (why allowed ?).
Derive the electric and the magnetic field inside the bubble. Show that an electron
comoving with the bubble at velocity c experiences a force linear in bubble radius,
while no transverse force acts on an electron moving in opposite direction (v=-cex).
13. Problem: Ultra-relativistic laser plasma scaling
S. Gordienko, A. Pukhov, Phys. Plasmas 12, 043109 (2005)
rel. laser plasma equations
(  v  
t
r
)
 e( E  v  B / c)   p f (r , p, t )  0, p   mv ,   1  ( p / mc) 2 ,
cr  B  t E  4 j , cr  E  t B, r  E  4 e( N0  Ne ), r  B  0,
Inc.Pulse: A(t  0)  a(r 2 / R 2 , x / c )cos(k L x), N e   f d 3 p, j  e  v f d 3 p,
4 parameters: a0 , focus radius kL R, pulse duration L , plasma density Ne /Ncrit ,
scaled variables: S  Ne /(a0 Nc ),
1/ 2
1/ 2
eA
eS
E
eS
B
1/
2
1/
2
ˆ
ˆ
ˆt  S Lt , rˆ  S k L r , pˆ  p / mca0 , Aˆ 
,E 
,B 
,
2
2
2
mc a0
mc L a0
mc L a0
Show that, in ultra-relativistic limit (a0>>1), scaled. laser plasma equations
(   vˆ 
)
2
2
ˆ  0,
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
v

p
/
p

a

e
(
E

v

B
/
c
)

f
rˆ
pˆ
0 ,
tˆ
ˆ   Eˆ  4 (1  nˆ),  ˆ  Bˆ  0,
crˆ  Bˆ  tˆ Eˆ  4 ˆj, crˆ  Eˆ  tˆB,
r
r
3
3
Aˆ (tˆ  0)  aˆ (rˆ 2 / Rˆ 2 , xˆ / ˆ)cos( S 1/ 2 xˆ ), max aˆ  1, nˆ   fˆ d pˆ , ˆj  e  vˆ fˆ d pˆ ,
depend only on three parameters:
kL R, L , and S !
Geometric Similarity, S=10-3=const
Gordienko, Pukhov, Phys. Plasmas 12, 043109 (2005)
70
(i)
70
0.2
(ii)
ne/nc
Y/
a0=10
ne= 0.01nc
0.1
a2 200
0
0
680
0
600
70
ne/nc
0
0
680
0
600
0.4 70
(iii)
a2 800
a0=20
ne= 0.02nc
0.8
(iv)
ne/nc
ne/nc
a0=40
ne= 0.04nc
a0=80
ne= 0.08nc
0
0
600
a2 3200
X/
0
680
0 a2 12800
0
600
X/
0
680
bubble regime scales with
optimal focus
peak electron energy
number of acc. electrons
laser-to-electron efficiency
P0  m2c5 / e2  8.7 GW
S  (
Rel
p
L )  ne ancrit
2
k p R  a0
Emax  0.65mec P
c pulse
2
P0
L
1.8 P
N acc 
P0
k L re
  Emax Nacc / P  20%
re  e 2 / mc 2
Conventional accelerators
50 GeV LINAC at Stanford
3 km
International Liner Collider:
500 GeV, 31 km, $6.7 bln
30 km
The accelerating field is limited to a few 10 MeV/m
Present day accelerators are tens of kilometers long
and reach their limits in size.
The hope is that laser accelerators can be built
with much smaller dimensions
reaching higher energies
Accelerating fields
Conventional RF accelerators
E < 100 MV/m
limit set by electrical breakdown
Space charge fields, laser-generated
for solid-density plasma
Laser
++++++++
E  mcp /e < 100 TV/m
1023
W/cm2
-E -
5 GeV protons at 1023 W/cm2
1 kJ , 15 fs laser pulse
focussed on 10 m spot
of 1022/cm3 plasma
1012 protons
4 - 5 GeV
Selected Publications:
Recent Survey and Collection of papers on:
Laser-driven particle accelerators: new sources of energetic particles and radiation.
Ed. K.Burnett, D. Jaroszynski, S. Hooker,
Phil. Trans. Royal Soc. A 364, 551 – 778 (2006)
Exp. Observation of laser-driven mono-energetic electron beams:
Nature 431 (2004) Dream Beam Issue,
1. S.P.Mangles, C.D.Murphy, Z.Najmudin et al.,
2. C.G. Geddes, E.Esarey, W.P.Leemans et al.
3. J. Faure, Y. Glinec, A. Pukhov, V. Malka, et al.
T. Tajima, J.M. Dawson, Phys.Rev.Lett. 43, 267 (1979)
E.Easarey, P.Sprangle, J.Krall, A.Ting, IEEE Trans. Plas.Science 24, 252 (1966).
Pukhov & Meyer-ter-Vehn, Appl. Phys. B74, 355 (2002).
Kostyukov, Pukhov, Kiselev, Phys. Plasmas 11, 5256 (2004).
S. Gordienko, A. Pukhov, Phys. Plasmas 12, 043109 (2005).
F.S.Tsung, W. Lu, M.Tsoulas, W.B.Mori, C. Joshi, J.M. Vieira, L.O.Silva, R.A.Fonseca,
Phys. Plasmas 13, 056708 (2006).
M.Geissler, J. Schreiber, J. Meyer-ter-Vehn, New J. Phys. 8, 186 (2006).