Lecture 2: Basic plasma equations, self-focusing,
direct laser acceleration
Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996)
plasma box (ne/nc=0.6)
B ~ mcwp/e ~ 108 Gauss
1
Laser Interaction with Dense Matter
Plasma approximation:
Laser field at a > 1 so large that atoms ionize within less than laser cycle
Free classical electrons (no bound states, no Dirac equation)
Non-neutral plasma (
nelectron  nion
, usually fixed ion background)
2
Single electron
plasma (ncrit = 1021cm-3)
In plasma, laser interaction generates additional
•
E-fields (due to separation of electrons from ions)
•
B-fields (due to laser-driven electron currents)
They are quasi-stationary and of same order as laser fields:
EL  31012 V/m  a0
BL  108 Gauss  a0
Plasma is governed by collective oscillatory electron motion.
3
The Virtual Laser Plasma Laboratory
Three-dimensional electromagnetic fully-relativistic Particle-Cell-Code
A. Pukhov, J. Plas. Phys. 61, 425 (1999)
Fields
Particles
 
1  E 4
rot B 

j
c t
c
dp
q
 qE 
p B
dt
m
1 B
rot E  
c t
p2
  1 2 2
mc
divE  4
divB  0
109 particles in 108 grid cells
are treated on 512 Processors
of parallel computer
4
Theoretical description of plasma dynamics
Distribution function:
f (r , p, t )
Kinetic (Vlasov) equation (
p   mv ,   1  ( p / mc) 2
):


 
  v   e E  (v / c)  B   f (r , p, t )  0 (collisions ?)
r
p 
 t


Fluid description:
Approximate equations for density, momentum, ect. functions:
N ( r , t )   f ( r , p, t ) d 3 p
P ( r , t )   p f ( r , p, t ) d 3 p
5
Problem: Light waves in plasma
Starting from Maxwell equations
E 4
 B 

J
ct c
B
 E  
ct
 B  0
B   A ,  A  0
  E  4 e( N 0  N e )
E
A
 
ct
and assuming that only electrons with density Ne contribute to the plasma current
J  eN e P /  m with electron momentum P   mu and   1  ( P / mc) 2 ,
while immobile ions with uniform density Ni =N0/Z form a neutralizing background.
using normalized quantities and plasma frequency
eA
e
P
Ne
4 e2 N0
2
a ( r , t )  2 ,  ( r , t )  2 , p( r , t ) 
, n(r , t ) 
, wp 
,
mc
mc
mc
N0
m
derive
 2 1 2 
 w p np
 2
,
  2 2 a 
c t 
ct
c 

2
2  (wp2 / c2 )(n 1)
6
Problem: Derive cold plasma electron fluid equation
In this approximation, electrons are described as cold fluid elements which have
relativistic momentum P   mu and satisfy the equation of motion
dP(r , t ) / dt  e  E  (u / c)  B 
where pressure terms proportional to plasma temperature have been neglected.
Using again the potentials A and  and replacing the total time derivative by
by partial derivatives, find
 A

u


    ( A) 
  u   P(r , t )  e 
c
 t

 ct

and show that this leads to the equation of motion of a cold electron fluid
1 
( p  a )  u    ( p  a )  (   ) ,
c t
written again in normalized quantities (see previous problem). Here, make use of
relations    1  p 2  p 2 / 2 and u  (  p)  c  (u  ) p .
7
Basic solution of
1 
( p  a )  u    ( p  a )   (   )
c t
Solution for electron fluid initially at rest, before hit by laser pulse,
p  a and  =
implying balance between the electrostatic force   and the
ponderomotive force
   1  p 2   1  a 2  a 2 / 2
This force is equivalent to the dimensional force density
2
2
w


E
p
2
F  N0 mc   2  

w  8 
It describes how plasma electrons are pushed in front of a laser pulse
and the radial pressure equilibrium in laser plasma channels, in which
light pressure expels electrons building up radial electric fields.
8
Propagation of laser light in plasma
For low laser intensities ( a 1 ), the p  a solution implies
The wave equation for laser propagation in plasma
  1 and n  1 .
w p na w p
 2 1 2 
 2 a,
  2 2 a  2
c t 
c 
c

2
2
then leads to the plasma dispersion relation
w2  wp2  c2k 2
For increasing light intensity, the plasma frequency is modified
w
2
p , rel
4 e2 Ne (r , t )
w


m  (r , t )
2
p
n
by changes of electron density and relativistic  – factor, giving rise to effects of
relativistic non-linear optics.
9
Relativistic Non-Linear Optics
Induced transparency:
w2 = wp2 + c2k2
wp2= 4e2 ne /(m<>
 =(1- v2/c2)-1/2
nR = (1 - wp2/ w2)1/2
Self-focussing: vph= c/nR
Profile steepening: vg = cnR
10
Problem:
Derive phase and group velocity of laser wave in plasma
Starting from the plasma dispersion relation
w2  wp2  c2k 2 ,
show that the phase velocity of laser light in plasma is
vphase  w / k  c / nR
and the group velocity
vgroup  dw / dk  cnR ,
where nR is the plasma index of refraction
nR  1  w p2 / w 2 .
11
3D-PIC simulation of laser beam selffocussing in plasma
Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996)
plasma box (ne/nc=0.6)
12
Problem: Derive envelope equation
Consider circularly polarized light beam
a  Re (ey  iez )a0 (r , z , t ) exp(ikz  iwt )
Confirm that the squared amplitude depends only on the slowly varying
envelope function a0(r,z,t), but not on the rapidly oscillating phase function
a  a0 (r , z , t ) 2 , a0 / t
2
w a0
a0 / z
ka0
Derive under these conditions the envelope equation for propagation in
vacuum (use comoving coordinate =z-ct, neglect second derivatives):
 2 1 2 
  2 2 a  0
c t 


 2

   2ik



 a0 (r ,  )  0

13
Problem: Verify Gaussian focus solution
Show that the Gaussian envelope ansatz
a0 (r, z)  exp( P( z)  Q( z)(r / r0 )2 )
inserted into the envelope equation
 
1  
r

2
ik

 a0 (r , z )  0
z 
 r r r
leads to
a0 (r , z ) 
where
e
 r 2 /[ r02 (1 z 2 / L2R )]
1  z 2 / L2R

 z
exp  i arctan 

 LR

2
  r  z / LR
i 
2
2
r
1

z
/
L
R
  0




LR  kr02 / 2 is the Rayleigh length giving the length of the focal region.
14
Relativistic self-focusing
For increasing light intensity, non-linear effects in light propagation first show up
In the relativistic factor 1/   1/ 1  a 2 1  a 2 2 giving
w p2 na
w p2  a 
 2 1  
 a,
 2 1 
  2 2 a  2
c t 
c 
c 
2 

and leads to the envelope equation (using w2  wp2  c2k 2 !)
2
2
w p a02

 2
a0
   2ik  a0 (r, z)   2
z 
c 2

2
While  a0 is defocusing the beam (diffraction), the term (wp2 / c2 )(a02 / 2)a0
is focusing the beam. Beyond the threshold power
2
Pcrit  2Po (w / wp )2  17.4 GW (ncrit / ne )
the beam undergoes relativistic self-focusing.
15
2D versus 3D relativistic self-focusing
Relativistic self-focusing develops differently in 2D and 3D geometry.
Scaling with beam radius R :
diffraction
2
relativistic non-linearity
1/ R2
(wp2 / c2 )(a02 / 2)
P / R (for 2D: P Ra02 )
(wp2 / c2 )(a02 / 2)
P / R2 (for 3D: P  R2a02 )
2D leads to a finite beam radius (R~1/P), while 3D leads to beam collapse (R->0).
For a Gaussian beam with radius r0:
power:
beam radius evolution
(Shvets, priv.comm.):
critical power:
P   R2 I0 / 2  P0 (w 2 /16c2 ) a02 R2
2
dR 2 ( z )
4 
1 wp 2 2 
 2 3 1 
a0 R 
2
2


dz
k R  32 c

Pcrit  2Po (w / wp )2  17.4 GW (ncrit / ne )
16
3D-PIC simulation of laser beam selffocussing in plasma
Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996)
plasma box (ne/nc=0.6)
B ~ mcwp/e ~ 108 Gauss
17
Relativistic self-focussing of laser channels
wp2= 4 e2 ne / m eff
w p2

ne
nR  1  w p2 wL2
radius
relativistic electrons
Relativistic mass increase ( )
and electron density depletion (ne )
increases index of refraction in the
channel region, leading to
selffocussing
B-field
laser
18
Relativistic Laser Plasma Channel
Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996)
I
B
L

jx
Intensity
80 fs
ne
ne/<>
B-field
Intensity
330 fs
Ion density
19
Plasma channels and electron beams observed
C. Gahn et al. PRL 83, 4772 (1999)
laser
gas jet
6×1019 W/cm2
plasma 1- 4 × 1020 cm-3
electron spectrum
observed channel
20
Scaling of Electron Spectra
Pukhov, Sheng, MtV, Phys. Plasm. 6, 2847 (1999)
Teff =1.8 (Il2/13.7GW)1/2
electrons
21
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)22
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

laser pulse length
20
2
40
eEx/w
mc
0
What drives electrons to  ~ 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
23
Channel fields and direct laser acceleration
eEr  (1  f )mwp2 R / 2
eB  f  mwp2 R / 2
d 2R
m 2  eEr  eB  mw p2 R / 2
dt
B
E
space charge
n = e(1-f)n0
j = efn0c
Radial electron oscillations
  w p / 2
2wL
electron
momenta
wL
wp/c)
24
How do the electrons gain energy?
2x103
Long pulses (> lp)
G
dt p = e E + e
c vB
Direct Laser
Acceleration
(long pulses)
0
dt p2/2 = e E  p = e E|| p|| + e E p
Gain due to transverse (laser) field:
-2x103
0
G||
103
Short pulses (< lp)
0
G|| =  2 e E|| p|| dt
Laser Wakefield
Acceleration
(short pulses)
G
Gain due to longitudinal (plasma) field:
104
G =  2 e E pdt

0
G||
4
1025
Selected papers:
J. Meyer-ter-Vehn, A. Pukhov, Z.M. Sheng,
in Atoms, Solids, and Plasmas In Super-Intense Laser Fields
(eds. D.Batani, C.J.Joachain, S. Martelucci, A.N.Chester), Kluwer, Dordrecht, 2001.
A. Pukhov, J. Meyer-ter-Vehn, Phys. Rev. Lett. 76, 3975 (1996).
C. Gahn, et al. Phys.Rev.Lett. 83, 4772 (1999).
A. Pukhov, Z.M. Sheng, Meyer-ter-Vehn, Phys. Plasmas 6, 2847 (1999)
26
Problem: Derive envelope equation
Consider circularly polarized light beam
a  Re (ey  iez )a0 (r , z , t ) exp(ikz  iwt )
Confirm that the squared amplitude depends only on the slowly varying
envelope function a0(r,z,t), but not on the rapidly oscillating phase function
a  a0 (r , z , t ) 2 , a0 / t
2
w a0
a0 / z
ka0
Derive under these conditions the envelope equation for propagation in
vacuum (use comoving coordinate =z-ct, neglect second derivatives):
 2 1 2 
  2 2 a  0
c t 


 2

   2ik



 a0 (r ,  )  0

27
Problem: Verify Gaussian focus solution
Show that the Gaussian envelope ansatz
a0 (r, z)  exp( P( z)  Q( z)(r / r0 )2 )
inserted into the envelope equation
 
1  
r

2
ik

 a0 (r , z )  0
z 
 r r r
leads to
a0 (r , z ) 
Where
e
 r 2 /[ r02 (1 z 2 / L2R )]
1  z 2 / L2R

 z
exp  i arctan 

 LR

2
  r  z / LR
i 
2
2
r
1

z
/
L
R
  0




LR  kr02 / 2 is the Rayleigh length giving the length of the focal region.
28
Problem: Derive channel fields
B
E
space charge
n = e(1-f)n0
j = efn0c
Consider an idealized laser plasma channel with uniform charge density
N = e(1-f)N0c , i.e. only a fraction f of electrons is left in the channel after
Expulsion by the laser ponderomotive pressure, and this rest is moving
With velocity c in laser direction forming the current j = efN0c. Show that
the quasi-stationary channel fields are
eEr  (1 f )mwp2 R / 2,
eB  f  mwp2 R / 2
and that elctrons trapped in the channel l perform transverse oscillations
at the betatron frequency, independent of f,
  w p / 2
29