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Laser-plasma interactions
Reference material:
“Physics of laser-plasma interactions” by W.L.Kruer
“Scottish Universities Summer School Series on Laser Plasma Interactions”
number 1 edited by R. A. Cairns and J.J.Sanderson
number 2 edited by R. A. Cairns
number 3 - 5 edited by M.B. Hooper
Peter Norreys
Central Laser Facility
Outline
1. Introduction
Ionisation processes
1. Laser pulse propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
Outline
1. Introduction
Ionisation processes
1. Laser pulse propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
Structure of a laser produced plasma
Laser systems at the CLF
• Laser systems for laser plasma
research
Vulcan: Nd: glass system
•
long pulse (100ps-2ns), 2.5
kJ, 8 beams
•
100TW pulse (1ps) and 10TW
pulse (10ps) synchronised with
long pulse
•
Petawatt (500fs, 500J)
•
Upgrading to 10 PW peak
power
UK Flagship 2002-2008: Vulcan facility
8
3
3
1
Beam CPA Laser
Target Areas
kJ Energy
PW Power
Increased complexity, accuracy & rep-rate
UK Flagship 2008-2013: Astra-Gemini
facility
Dual Beam Petawatt Upgrade of Astra (factor 40 power upgrade)
1022 W/cm2 on target irradiation
1 shot every 20 seconds
Opened by the UK Science Minister (Ian Pearson MP) Dec 07
Significantly over-subscribed. Assessing need for 2nd target area.
Ian Pearson, MP
Minister of State for Science and
Innovation , Dec 2007
The next step: Vulcan 10 PW (>1023 W/cm2)
100 fold intensity enhancement
OPCPA design
Coupled to existing PW and long pulse beams
Introduction
SOLIDS
GASES
Non-linear optics (2nd harmonic
generation)
104
Dielectric breakdown (collisional ionisation)
109
1011 Avalanche breakdown in air
Collisional absorption
1013 Non-linear optics regime
1014 Tunnel ionisation regime for hydrogen
Resonance absorption
Profile steepening
1015 Scattering instabilities (SRS, SBS etc)
become important
1016
3 1016
1017 Self focusing
Hole boring
Brunel heating
2 1018 Relativistic effects particle acceleration
1020
1021
Power and Intensity
1PW = 1015W
Power =
Energy
pulse duration
=
500 J
= 11015 W
500 fs
To maximise the intensity on target, the beam must be focused to a small spot.
The focal spot diameter is 7 mm and is focused with an f/3.1 off-axis parabolic
mirror. 30% or the laser energy is within the focal spot which means
Intensity =
Power
=
Focused area
1x 1021 W cm-2
Vulcan
diffraction gratings
Units
In these lecture notes, the units are defined in cgs. Conversions to SI units are given
here in the hand-outs.
But note that in the literature
Energy – Joules
Pulse duration  - ns nanoseconds 10-9 sec (ICF)
ps picoseconds 10-12 sec
fs femtoseconds 10-15 sec (limit of optical pulses)
as attoseconds 10-18 sec (X-ray regime)
Distance – microns (mm)
Power – Gigawatts 109 Watts
Terawatts 1012 Watts
Petawatts 1015 Watts
Exawatts 1018 Watts
Intensity – W cm-2
Irradiance – W cm-2 mm2
Pressure – Mbar – 1011 Pa
Magnetic field – MG – 100 T
http://wwppd.nrl.navy.mil/formulary/NRL_FORMULARY_07.pdf
Critical density and the quiver velocity
• The laser can propagate in a plasma until what is known as the critical
density surface. The dispersion relationship for electromagnetic waves in a
2
2
2 2
plasma is    pe  k c so that when  = pe the wave cannot propagate
Critical density
ncritical = 1.1  1021 cm-3
lmm is the laser wavelength in mm
lmm2
• An electron oscillates in a plane electromagnetic wave according to the Lorentz
forceF  e( E  c 1 v  B). Neglecting the vB term, then the quiver velocity vosc is
vosc 
eEL
m
• The dimensionless vector potential is defined as a 
vosc
c
Note that a can be > 1. This is because a = g b = g velectron / a and


Il2
a
18
2 
1
.
4

10
Wcm


1/ 2
~ 1 if relativity is important and g electron
 [ 1  a 2 ]1/ 2
Laser-plasmas can access the high energy
density regime
• The ratio of the quiver energy to
the thermal energy of the plasma is
given by
2
mvosc
1.8 10 18 Il2

2kT
T eV 
< 1 for ICF plasmas
>> 1 for relativistic intensities
• The laser electric field is given
by
E0 
3.2a0
l
10 
10
Vcm 1
• The Debye length lD for a
laser-produced plasma is ~ mm
for gas targets and ~ nm for
solids. n lD - 50 -100
• Strongly coupled plasma
regime can be accessed
Outline
1. Introduction
Ionisation processes
2. Laser pulse propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
Ionisation processes
• Multi-photon ionisation
Non linear process
Very low rate
Scales as In n = number of photons
• Collisional ionisation (avalanche ionisation)
Seeded by multiphoton ionisation
Collisions of electrons and neutrals  plasma

ne 2Te3 / 2
2
•Tunnel and field ionisation
Laser electric field “lowers” the Coloumb barrier which
confines the electrons in the atom
Tunnelling
Isolated atom
Atom in an electric field
Atom in an oscillating field
Keldysh parameter g = time to tunnel out / period of laser field

2I p m
eE
I
6 p

  2.3  10 
 Il
1/ 2



1
l
i.e.
if g    1  TUNNELLING
t
Ip

g  1  MULTIPHOTON
2 (EZe3)
if
2I p m 
[g = 1, I =61013 Wcm-2, l 1mm, xenon]
eE
IONISATION
Laser pulse
Solid target
Heated region
skin layer
vapour expansion
into vacuum
Outline
1. Introduction
Ionisation processes
2. Laser propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
EM wave propagation in plasma
Start with an electric field of the form
E  E( k ) exp( i t )
(1)
The current density can be written j   E where the conductivity  is  
i pe2
4
2
and  pe
 4 e 2n0 / me
1 B
i
gives   E  
B
c
c t
4
1 E
Substituting (1) and j into Ampere’s law   B 
gives
J
c
c t
4
i
i
(3)
 B 
 E E   E
c
c
c
2


 pe
Where we have defined the dielectric function of the plasma   1  2 
 

Substituting (1) into Faraday’s law   E  
Taking the curl of Faraday’s law (2) and substituting Ampere’s law (3) with a
standard vector identity gives
 2 E  .E 
2
c
2
E  0
i.e.
2 E 
2
c
2
E  0
(4)
(2)
EM wave propagation in a plasma with a
linear density gradient
Consider plane waves at normal incidence. Assume the density gradient is
in the propagation direction Z. This
n0  n0 (z)
   ( , z)
E( x)  E( z) exp( it)
Substituting (5) into (4) gives
(5)
d2
2
Ex , y  2  Ex , y  0
2
dz
c
(6)
 Ez  0
 pe
n( x)
z
 1 2  1
We can rewrite the dielectric function as   1 
nc

L
2
m 2
z
when n( x)  nc and the critical density is nc 
L
4 e 2
2
2
d
E

1  z E  0
Equation (6) can then be written as



dz 2 c 2  L 
(8)
(7)
EM wave propagation in a plasma with a linear
density gradient continued (1)
1/ 3
Change variables
Gives
 2 
   2  ( z  L)
c L
d2 E
 E  0
2
d
whose solution is an Airy function
E( )  Ai ( )  bBi ( )
(9)
 and b are constants determined by the boundary condition matching.
Physically, E should represent a standing wave for  < 0
and to decay at  . Since Bi()  as  , b is
chosen to b = 0.  is chosen by matching the E-fields at
the interface between the vacuum and the plasma at
z=0. This gives  = (L/c)2/3. If we assume that L/c >>1
and represent Ai() by the expression
Ai (  ) 
1
  1/ 4
2

cos   2 / 3  
4
3
(10)
EM wave propagation in a plasma with a linear
density gradient continued (2)
Then E( z  0) 

2  ( L / c)1/ 6
2 L  
2 L  



exp i 3 c  4   exp  i 3 c  4 


(11)
Now E(z=0) can be represented as the sum of the incident wave with amplitude
EFS with a reflected wave with the same amplitude but shifted in phase
4 L  
E( z  0)  EFS 1  exp  i(
 )
3 c
2 

(12)
L 
i
Provided that
  2  
 EFS e
 c 
EFS is the free space value of the incident light and  is an arbitrary phase that does
not affect E
1/ 6
 L 
1/ 6
 E   2  

 c 
EFS e i Ai ( )
(13)
This function is maximised at  = 1 corresponding to (z - L)= -(c2 L / 2)1/3
E
 max
EFS
2
L 
 3.6


 c 
1/ 3
2L 
 3.6


 l 
1/ 3
(14)(valid for L/l>>1)
Outline
1. Introduction
Ionisation processes
2. Laser propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
Oblique incidence
z
nc
sin2

y
x
kx  0

k y    sin 
c

k z    cos 
c
s-polarised

E  Ex x

0
x
Eqn (4) becomes
 2 Ex  2 Ex  2
 2  2  ( z)Ez  0
2
y
z
c
(15)
Since  is a function only of z, ky must be conserved and hence k y  ( / c) sin  and
 iy sin  
(16)

c


2
2


d
E
z

2
Substituting (16) into (15) gives



(
z
)

sin
 E( z)  0
2
2
dz
c

Ex  E z  exp 
(17)
Reflection of the light occurs when (z) = sin2 . Remember that  = (1 – pe2(z) /  2)
and therefore reflection takes place when pe=  cos  at a density lower than
critical given by ne = nc cos2  .
Oblique incidence – resonance absorption
p-polarised
z
In this case, there is a component of the electric field that oscillates along the
density gradient direction, i.e. E. ne  0. Part of the incident light is
transferred to an electrostatic interaction (electron plasma wave) and is no
longer purely electromagnetic.
nc
sin2

E  Ey y  Ez z
. E  0

y
x

(18)
Remember the vector identity . E  .E   .E
then
.E  
1 
Ez
 z
(19)
This means that there is a resonant response when  = 0, i.e. when  = pe
We seen previously that the light does not penetrate to nc. Instead,the field
penetrates through to  = pe to excite the resonance.
Oblique incidence – resonance absorption
continued

Note that the magnetic field B  Bx x
 
and use the conservation of ky    sin 
c

iy


The B-field can be expressed as B  B z  exp   it 
sin   x
c


Substituting into Ampere’s law (equation (3))   B  
Ed
sin 
Ez 
Bz 
 ( z)
 ( z)
(20)
i
 E gives
c
(21)
This implies that Ez is strongly peaked at the critical density surface and the
resonantly driven field Ed is evaluated there by calculating the B-field at nc.
ic E
 ic    E 
B

B


(22)
From Faraday’s law
or
  


z


  
 z
L 
E   2  

c


1/ 6
1/ 6
EFS e i Ai ( )
c 
B   i 2  

 L 
B decreases as E gets larger near the reflection point
EFS e i Ai ( )
(23)
1/ 6
At the reflection point,
c 
B(  0)  0.9
 EFS
 L 
The B field decays like exp (-b) where b 

1
b   2  p2   2 cos 2 
L cos  c
L
 2L  sin 3 

 3c 
Solution: b  

1/ 2
1/ 3
Defining
sin 
L cos 2 
k( z)dz .
 pe2 z

2 L
(25)
(26)
 c 
 B( z  L)  0.9 EFS 


L


L
   
 c 
dz

L
(24)
1/ 6
 2L sin 3  

exp  
3c


and remember that Ez 
(27)
Ed
B sin 

 ( z)
 ( z)
 Ed 
EFS
 ( )
( 2L / c )
 ( )  2.3 exp   3 
2
 3

(28)
The driver field Ed vanishes as   0, as the component along z varies as sin.
Also, Ed becomes very small when   , because the incident wave has to tunnel
through too large a distance to reach nc
The simple estimate for  () from (28) is
plotted here against an numerical
solution of the wave equation.
Note that the maximum value of  () is
~1.2 (needed to estimate the absorption
fraction in the next section)
Resonance absorption – energy transfer
Start from the electrostatic component of the electric field.
Now include a damping term  in the dielectric
function of the plasma that can arise form collisions,
wave particle interactions etc.
Ez2

I abs   
dz 
0
8
8



0
Ez 
Ed
 (z )
 p2
 ( z)  1 
 (  i )
Ed 2 ( z)
dz
2

 can be considered as the rate of energy dissipation and Ez2/8 the incident
energy density.
z     z2

  1      2
 L   L
2
For a linear density profile, (ne = ncr z / L)
2
2
Substituting and approximating that Ed
2

E
dz
is constant over a narrow width of the I  d z  L 
abs
0 (1  z / L)2   /  2
8
resonance function gives
The integral gives    ,
 I abs  LE / 8
2
d
2
cEFS
I abs  f A
8
f A   2   / 2
Iabs peaks at ~0.5
The absorption is maximised at an angle given by
 max  sin 1 0.8(c / L)1/ 3 
This is the characteristic signature of resonance absorption – the dependence
on both angle of incidence and density scale-length.
Outline
1. Introduction
Ionisation processes
2. Laser propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
Collisional absorption
Principal energy source for direct drive inertial fusion plasmas
From the Vlasov equation, the linearized force equation for the electron fluid is
ue
t

where ei is the damping term.
e
E  ei ue
m
Since the field varies harmonically in time
ue 
 ieE
m  i ei 
The plasma current density is J  ne eue 
2
i pe
4 (  i ei )
E
 pe2
The conductivity  (J=E)  
and dielectric function   1 
 (  i ei )
4 (  i ei )
i pe2
Collisional absorption
i
B
c
i
 B    E
c
 E  
Faraday’s law (cf (eqn (2))
Ampere’s law (cf eqn (3))
Combining them gives
 E
2
2
c
2
E  0
To derive the dispersion relation for EM waves in a spatially uniform plasma, take
E( x) ~ e i k.x and substitute for  and assume that ei/ << 1 gives
 2  k 2 c 2   pe2 (1 
Expressing   r  i / 2
gives
where  is the energy damping term.
i ei

)
 r    k c
2
pe
 pe2
  2  ei
r

2 2 1/ 2
collisional damping in linear density gradient
The rate of energy loss from the EM wave [ = (E2/8  must balance the oscillatory
velocity of the electrons is randomised by electron- ion scattering. Assume again
that ne = ncr z / L and write down equation (4) again
2E  2

 ( z) E  0
z 2 c 2
z
The dielectric function  (z) is now   1 
L1  i e*1 /  
where *ei is its value at ncr

d2 E  2 
z

E  0

1

2
2 
*
dz
c  L1  i ei /   
d2 E
 E  0
Changing variables again, gives Airy’s equation
2
d
1/ 3

 
 i ei  
2
  z  L1 
 
   2

 

 c L1  i ei /    
E( )   Ai ( )
Again, match the incident light wave to the vacuum plasma boundary interface
1
at z=0
 2  2/3   
Ai (  ) 
cos


4
  1/ 4
3
Energy absorption
Thus at z=0, E can be represented as an incident and reflected wave whose
amplitude is multiplied by ei
4

3
2
As  is now complex, there is both a phase shift and a damping term for the
reflected wave
2/3
  L  i ei  
 
 (0)    1 
 
  c 
   (z  0) 
For ei/ <<1
4L 
real 

3c
2
4 ei L
imaginary 
3c
 8  ei L 
Energy ~ ( amplitude)2 ~ exp  

 3 c 
 8 ei L 

 f A  1 exp 
3c 

 ei  3  10 6 ln 
ne Z
1
sec
TeV 3 / 2
Experiment shows
fA
Z3 / 2 L0.6

I L0.4l2L
Outline
1. Introduction
Ionisation processes
2. Laser propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
Collisionless “Brunel” absorption
E
•Dominant mechanism is L/l < 1
vacuum
plasma
•Quiver motion of the electrons in
the field of a p-polarised laser pulse
• Laser energy is absorbed on each
cycle
• Can result in higher absorption
than classical resonance absorption
(up to 0.8)
•Not as dependent on angle
From P.Gibbon Phys. Rev. Lett. 76, 50 (1996)
Harmonic generation: the moving mirror
model
z
Incident
light
Reflected
light from
mirror at
0t/2 = 0
Reflected light
from mirror at
0t/2 = 0.5
see references:
D. von der Linde and K. Rzazewski: Applied Physics B 63, 499 (1996).
R. Lichters, J. Meyer-ter-Vehn and A. Pukhov Phys. Plasmas 3, 3425 (1996).
The moving mirror model
Ignoring retardation effects, the phase shift of the reflected wave resulting from a
sinusoidal displacement of the reflecting surface in the z direction s(t)=s0 sinwmt is
 (t )  (20 s0 ) cos sin mt
 is the angle of incidence, m is the modulation frequency. The electric field of the
reflected wave is
n 
i 0t i ( t )
 i 0 t
in m t
R
n
n  
E e
e
e
J n (  ) is the Bessel function of order n and
J
( )e
  (20 s0 / c) cos 
where we’ve made use of the Jacobi expansion
e iz sin mt 
n 
in m t
J
(

)
e

n  
Conversion efficiency dependence on
intensity on target
5x1017
5x1018
1x1019
Energy conversion
efficiencies of 10-6 in X-UV
harmonics up to the 68th
order were measured and
scaled as E() = EL(/0)-q
Good agreement was found
with particle in cell simulations
P.A.Norreys et al. Physical Review Letters 76, 1832 (1996)
Complex dynamics of nc motion
For Il2 = 1.21019 Wcm-2mm2
and n/nc = 3.2 only 0
oscillations are observed. The
power spectrum extends to ~35
and still shows modulations.
Over dense regions of plasma
are observed at the peak of the
pulse.
Il2 = 1.2  1019 Wcm-2mm2
and n/nc = 6.0, both 0 and
20 oscillations appear.
Additional order results in the modulation of the harmonic spectrum
I.Watts et al., Physical Review Letters 88, 155001 (2002).
• Harmonic efficiency changes
with increasing density scalelength and mirrors the change
in the absorption process
• As the density scale-length
increases, the absorption
process changes from Bruneltype to classical resonance
absorption.
• Maximum absorption at L/l
~0.2
M. Zepf et al., Physical Review E 58, R5253 (1998).
Intense laser-plasma interaction physics
High harmonic generation – towards attosecond interactions
B.Dromey, M.Zepf….P.A.Norreys Nature Physics 2 546 (2006)
Outline
1. Introduction
Ionisation processes
2. Laser propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
Ponderomotive force
Neglecting electron pressure, the force equation on the electron fluid is
ue
t
To a first approximation
 ue  ue  
e
E( x) sin t
m
ue  uh where uh 
eE
cos t
m
u h e
 E( x) sin t
t m
Averaging the force equation over the rapid oscillations of the electric field,
u s
m
 e Es  m  uh .uh  t
t
< > denotes the average of the high frequency oscillation, u  ue  t
s
u s
1 e2
s
2
m
 e E 

E
(x)
2
t
4 m
E s  E  t
Ponderomotive force
• An electron is accelerated from its initial position x0
to the left and stops at position x1.
• It is accelerated to the right until its has passed its
initial position x0.
• From that moment, it is decelerated by the reversed
electric field and stops at x2
• If x0’ is the position at which the electric field is
reversed, then the deceleration interval x2- x0’ is larger
than that of the acceleration since the E field is weaker
and a longer distance is needed to take away the
energy gained in the former ¼ period.
x1 x3
x0
• On its way back, the electron is stopped at position x3
The result is a drift in the direction of decreasing wave amplitude.
x2
Outline
1. Introduction
Ionisation processes
2. Laser propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
Outline
1. Introduction
Ionisation processes
2. Laser propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
Summary of laser-plasma instabilities
Stimulated Brillouin Scatter - SBS. Scattered light is downshifted by ia ~ L
Source of large losses for ICF experiments
Very dangerous for laser systems
Stimulated Raman Scatter – SRS.
Two Plasmon Decay
Filamentation
- TPD.
Scattered light downshifted by p
Produces large amplitude plasma waves
Hot electrons (Landau damping)
Preheating of fuel
Decays into two plasma waves
Not as serious
Localised near ncr / 4.
Can cause beam non-uniformities
problematic for direct drive ICF
Regimes of applicability
n
Dispersion relation
  c k   pe  ncr 
2
2 2
2
21
10 cm
lmm 2
3
SBS – back and side
Self focusing and filamentation
ncrit
SRS
   sc   p
SBS
  sc  ia
TPD
  p  p
TPD
ncrit / 4
Raman back
and side scatter
x
Parametric instability growth
Large amplitude
light wave couples
to a density
fluctuation
ELn
Enhances
the density
perturbation
n
Ep
Drives an electric
field associated
with an electron
plasma wave
EL Ep
The electric fields
combine to generate
a fluctuation in the
field pressure
The threshold for the instabilities is determined by spatial inhomogeneities and
damping of plasma waves – this reduces the regions where the waves can
resonantly interact
Parametric decay

k
plasmons
photons
ion acoustic
phonons
Outline
1. Introduction
Ionisation processes
2. Laser propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
Stimulated Raman Scatter
0   s   pe
k 0  k s  k pe
0  2 pe
The instability requires
i.e
n
ncr
4
10 % in ICF plasmas. The instability causes heating of the plasma, due
to damping
Can have high phase velocity – produces high energy electrons
Stimulated Raman scattering

k
plasmons
photons
General instability analysis
Derive an expression that relates the creation of the scattered EM waves (by
coupling the density perturbations generated by the laser field) to the transverse
current that produces the scattered light wave.
Start from Ampere’s law
  B  4 J 
Remember that   A  0
E
1 A
 
c t
1 E
c t
(1)
B   A
  (  A)  (.A)   2 A
 1 2

4
1 
  2 2   2  A 
j

c
c t
 c t

(2)
General instability analysis continued…1
Separate J into a EM part jt and a longitundinal part jl
Poisson’s equation
Conservation of charge
 2  4
t
l
(3)
 is the charge density
(4)

 . j  0
t
jj j
(5)
Taking /t of (4) and substituting for /t from (5)
 (

  4 j )  0
t

   4 j
l
t
Equation (2) becomes
(6)
because
 1 2

4
 2 2   2  A 
j
c t
 c t

. j  0
t
(7)
(8)
If we restrict the problem to A. ne = 0, the transverse current becomes jt = - ne e ut
where ut is vosc. For ut <<c, ut =eA/mc because
 ut
e
e A
  Et 
(9)
t
m
mc t
This gives eqn for propagation of a light wave in a plasma
 2
4e 2
2 2
 2  c   A  
ne A
m
 t

(10)
Substitute for A  A L  A1 and ne = n0 + n1
 2
4e 2
2 2
2 
 2  c    pe  A1  
n1 A L
m
 t

(11)
The right hand side is simply the transverse current ( n1vL ) that produces the
light wave.
Now we need an equation for the density perturbation
Continuity equation
Force equation
ne
   ne ue   0
t
(12)
u e
ue  B  pe
e 
 ue  ue 
E



t
m
c  ne m
Here the ions are fixed and provide a neutralising background
(13)
General instability analysis continued…2
eA
ue  u L 
mc
transverse
(14)
longitudinal
pe e 

eA  
eA  
eA 
1
eA  
u


u



u




E

u


 L
  L
  L

 L
  B
t 
mc  
mc  
mc 
ne m m 
c
mc  
substituting
1 A
 
c t
1
2
B   A
E
gives
u L e
1 
eA
    uL 
t
m
2 
mc
Electric
field
 A   A  A    A   A2
2
 pe
 
 ne m
Ponderomotive Plasma
force
pressure
(15)
General instability analysis continued…3
pe/ne3 = constant
Use adiabatic equation of state
(16)
Linearise (12) and (15). Do this by taking uL  u, ne  no  n1 , A  AL  A1
(
pe
)0
3
ne
p e 
3p e
n1
ne
(17)
pe 3vth2

n1
ne me
n0
3vth 2
u e
1 
e

A L  A1  
    u1 
n1
t m
2 
mc
n0

2
(18)
General instability analysis finalised
To 1st order ignore A12, u12 and u1  A  0
3vth 2
u e
e2
   2 2  A L  A1  
n1
t m
mc
n0
n1
 n0.u1  0
t
(19)
(20)
Combining by taking time derivative of (20), a divergence of (19) gives
n0 e 2 2
2
2
2 2
( 2   pe  3ve  )n1  2 2  A L  A1 
t
mc
(21)
2
Where    4n1e
This is now the equation for the density perturbations generated by
fluctuations in the intensity of the EM wave
Dispersion relationship for SRS
Start with AL  A0 cos( k 0  x  0t)
Fourier analyse

2
 k c 
2 2
2
pe
4 e 2
A1( k , ) 
A0 n1( k  k0 ,  0 )  n1 k  k0 ,  0 
2m

k 2 e 2n0
   n1( k , )  2 2 A0  A1( k  k0 ,  0 )  A1 k  k0 ,  0 
2m c
2
2
ek

(22)
(23)

Where  ek   pe2  3k 2 ve2
is the Bohm-Gross frequency and 0 and k0 are the
frequency and wave number of the laser wave. Also
1/ 2
A1  A1 exp( ikx  iwt)
n1  n1 exp( ikx  it)
A 0  A0 cos( k0 x  w0t )
A 0  A0 [exp i( k0 x  w0t )  exp  i( k0 x  w0t )]
(24)
Growth rates
Use (19) to eliminate A1

2
  ek2  
 p2 k 2 vos2 

1
1







D



,
k

k
D



,
k

k
0
0
0
0


4
D , k    2  k 2c 2   pe2
(25)
vos is the oscillatory velocity (eA0/mc)
Note: in deriving (25), it was assumed that ~pe and terms n1 (k-2k0,  -20) and
n1 (k+2k0,  +20) were neglected as non-resonant.
From this dispersion relation, the growth rates can be found. Neglecting the upshifted light wave as non-resonant gives

2


  ek2    0   k  k 0 2 c 2   pe2 
2
 pe2 k 2 vos2
4
(26)
Maximum growth rate
Take ek   where   ek . Note that maximum growth rate
occurs when the scattered light is resonant, i.e. when
ek  0 2  k  k 0 2 c 2  pe2  0
Then take ig where

 pe2
kvos 
g


4  ek  0   ek 
(27)
1/ 2
(28)
The wave number k is given by (27). For backscattered light the growth rate
maximises for
1/ 2
 0  2 pe 
k  k0 
1

c 
 0 
(29)
The wave numbers start at k = 2k0 for n << nc/ 4 and goes to k = k0 for n~ncr/4
Growth rate in low density plasmas
For forward scattered light at low density k<<0/c, both the up-shifted
and down-shifted light waves can be nearly resonant
(30)
D   , k  k   2    
0
0

pe
0

Substitute into (25), the maximum growth rate ig
 pe2 vos
g 
2 2 0 c
(31)
Outline
1. Introduction
Ionisation processes
2. Laser propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
Stimulated Brillouin Scatter

k
photons
ion acoustic
phonons
Stimulated Brillouin Scattering (SBS)
• density fluctuations associated with high frequency ion acoustic waves causes
scattering of light
•> 60 % has been observed in experiment
0  s  ia
k 0  k s  k ia
• Similar analysis to SRS, except n1 is density fluctuation associated with an ion
acoustic wave. We write
4 e 2
 2
2 2
2 
 2  c    pe  A1  
n1 A L
m
 t

2
2
Zn
e

n
2
2
0
1
• Force equation gives
AL  A1 

c

n


s
1
2
2
t
mMc
c s  (ZkTe / M )1/ 2 ion acoustic velocity
(32)
(33)
Growth rate
Fourier analyse (32) and (33)
  kcs  ig
g  kcs
gives
k  2k 0 
g
for backscattered light
1
2 0 c c
c2
(34)
k0 vos pi
2 2  0 k0c s 1/ 2
(35)
reflectivity
average ion
energy
Holhraum irradiation uniformity by SBS
S. Glenzer et al., Science 327, 1228 (2010)
P. Michel et al., Phys. Rev. Lett. 102, 025004 (2009)
Outline
1. Introduction
Ionisation processes
2. Laser propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
Two plasmon decay instability
• Occurs in a narrow density region at ncr / 4
   pe   pe
k  k1  k 2
• Similar analysis as SRS except the equations describe the coupling of
the electron plasma waves with wave number k and k  k 0 with the
large light wave
k0 vos
for plasma waves at 45 to both k and vos
4
• Threshold is much less than SRS at ncr / 4 unless the plasma is hot
•For k>>k0, g 
Two plasmon decay

k
plasmons
photons
Outline
1. Introduction
Ionisation processes
2. Laser propagation in plasmas
Resonance absorption
Inverse bremsstrahlung
Not-so-resonant resonance absorption (Brunel effect)
3. Ponderomotive force
4. Instabilities in laser-produced plasmas
Stimulated Raman scatter (SRS) instability
Stimulated Brillouin scatter (SBS) instability
Two plasmon decay
Filamentation
Filamentation instability
• Same dispersion relation as SBS also describes filamentation
• Zero frequency density perturbation corresponding to modulations of
the light intensity
• Occurs in a plane orthogonal to propagation vector of light wave
k  ko  0
  ig  0
2
1  vos   p
g   
8  vt   0
2
•
Filamentation can also be caused by thermal or relativistic effects:
1. enhanced temperature changes refractive index locally
2. relativistic electron mass increase has same effect as reducing the density
2
Pcritical  20 2 GW for self-focusing
 pe
Efficient Raman amplification into the
petawatt regime
R. Trines , F. Fiuza, R. Bingham, R.A. Fonseca,
L.O.Silva, R.A. Cairns and P.A. Norreys
Nature Physics 7, 87 (2011)
Raman amplification of laser
pulses
•A long laser pulse
(pump) in plasma will
spontaneously scatter
off Langmuir waves:
Raman scattering
Stimulate this
scattering by sending in
a short, counter
propagating pulse at
the frequency of the
scattered light (probe
pulse)
Because scattering
happens mainly at the
location of the probe,
most of the energy of the
long pump will go into
the short probe: efficient
pulse compression
Miniature pulse compressor
Solid state compressor (Vulcan)
Volume of a plasmabased compressor
Image: STFC Media Services
Simulations versus theory
RBS growth increases with pump amplitude and plasma density,
but so do pump RFS and probe filamentation
Optimal simulation regime corresponds to at most 10 e-foldings
for pump RFS and probe filamentation
Growth versus plasma density
Growth versus pump amplitude
A bad result
For a 2*1015 W/cm2 pump and ω0/ωp = 10, the probe is still
amplified, but also destroyed by filamentation
A good result
For a 2*1015 W/cm2 pump and ω0/ωp = 20, the probe is amplified
to 8*1017 W/cm2 after 4 mm of propagation, with limited
filamentation
10 TW → 2 PW and transversely extensible!