Scaling of the hot electron
temperature and laser absorption
in fast ignition
Malcolm Haines
Imperial College, London
Collaborators: M.S.Wei, F.N.Beg (UCSD, La Jolla) and
R.B.Stephens (General Atomics, San Diego)
Outline
• A simple energy flux model reproduces Beg’s
(I2)1/3 scaling for Thot.
• A fully relativistic “black-box” model including
momentum conservation extends this to higher
intensities.
• The effect of reflected laser light from the
electrons is added, leading to an upper limit on
reflectivity as a function of intensity.
• The relativistic motion of an electron in the laser
field confirms the importance of the skin-depth.
Beg’s empirical scaling of
Th(keV)=215(I182m)1/3
for 70 < Th < 400keV & 0.03 < I18 < 6 can be
found from a simple approximate model:
Assume that I is absorbed, resulting in a nonrelativistic inward energy flux of electrons:
1
1/2
3
and
I n m v
v  2eT / m
2
h
e h
h

h
e

Relativistic quiver motion gives
v osc
eE 0
 

 a0
c
me c
v osc
as
1
c
nh is the relativistic critical density
4 2 me
nh  nc  
 0 e2 2
2
2eTh 
2 m c a
1 
4

m
a

e
0
me 

I 
 
2
2 2
0 e 
2  0 e    me 
2
2
e
2
3
3/2
2
0
Taking the 2/3 power of this gives Eq.1
m ec 2 2/ 3
Th 
a0
2e
or
Th (keV)  230(I  )
2 1/ 3
18 m
Model 2: Fully relativistic with energy and
momentum balance
I  nh me ( h 1)vz c  nc pz ( h 1)c
2
2
Momentum conservation is
I
nc p2z where m  v  p
e h z
z
 nh pzvz 
c
me
consistent with electron motion in a plane wave
pz
 pˆ z   h  1
me c
h depends on the total velocity of an electron.
Transform to the axial rest-frame
of the beam:
2
2 

p

p
2 pz 
2
2
2 2
2 4 
2 4 
z 
z
E0  E  pz c  me c 1
 2 2  me c 1



m
c

m
c
m
c


e
e
e


Equate E0 to me0c2; 0 indicates the thermal
energy in the rest frame of the beam; because
transverse momenta are unaffected by the
transformation
1/2
 2 m I  
2
2
 e 
eTh  me c ( 0 1)  me c 
1
1

 mec nc c  
1/ 2
In dimensionless parameters, th = eTh/mec2 and a0,
th = (1+21/2a0)1/2 - 1
(2)
This contrasts with the ponderomotive scaling:
th = (1+a02)1/2 - 1 S.C.Wilks et al PRL(1992)69,1383
Simple model of Beg scaling, Eq.1, gives
th = 0.5 a02/3
(3)
Eqs (2) and (3) agree to within 12% over the range
0.3<a0<300, and intersect at a0 = 0.5685 and 112.55.
The total electron kinetic energy is (h - 1) = a0/21/2
Various scaling laws; Beg’s empirical law is almost identical to Hainesclassical and relativistic up to I = 51018 Wcm-2
Model 3: Addition of reflected or backscattered laser light
When light is reflected, twice the photon momentum is
deposited on the reflecting medium; thus the electrons
will be more beam-like, and we will find that Thot is
reduced.
The accelerating electrons will form a moving mirror,
but the return cold electrons ensure that the net Jz, and
thus the mean axial velocity of the interacting electrons
is zero.
If absorbed fraction is abs, energy conservation is
I - (1-abs)I = ncpz(h-1)c2
(4)
while momentum flux conservation is
I/c + (1-abs)I/c = ncpz2/me
(5)
Define Ir = (1-abs)I; (5)c+(4) gives
2I = ncpzc2[pz/mec + (h - 1)], while (5)c-(4) gives
2Ir = ncpzc2[pz/mec - (h-1)], or dimensionlessly
ii = 2I/ncpzc2 = pz' + h - 1
ir = 2Ir/ncpzc2 = pz' - h + 1
where pz' = pz/mec
(6)
(7)
As before, transform the energy to the beam rest-frame
E02 = E2 - pz2c2 = (hmec2)2 - pz2c2
= me2c4(h2-pz'2) = me2c402
Hence Th as measured in the beam rest frame is
th = eTh/mec2 = 0 - 1 = [(h+pz')(h- pz')]1/2 - 1
= [(1+ii)(1- ir)]1/2 - 1
Use (6) and (7) to eliminate pz' to give ii+ir=2pz'.
Define r = ir/ii ; then ii = 21/2ao(1+r)-1/2 and
th = [{1 + 21/2a0/(1+r)1/2}{1 - 21/2a0r/(1+r)1/2}]1/2 - 1 (8)
This becomes Eq (2) for r = 0, and for r > 0, th is reduced.
The condition th > 0 becomes
f (r )  (1 - r2)(1 - r)/(2r2) > a02 and df/dr<0 for 0<r<1
Defining  as f(r)  2a02 where  > 1, th becomes
th = {[1 +(1-r)/(r)][1 -(1-r)/]}1/2 - 1
Using r, (0<r<1), and , ( > 1) as parameters we can
also find a0 and the reflection coefficient, refl 1-abs = r
2
(1
r
)(1 r)
2
a0 
2 2
2r 
The condition refl ≤ 1 gives

(1 r 1/ 2 )2

  min  1
1/ 2
r
Table of f(r) and th() versus r
r = 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
f(r) = 44.6 9.6 3.54 1.58 .75 .356 .156 .0563 .0117 0
th(1.1) .265 .125 .065 .0365 .0204 .011 .0053.0021.0046 0
th(1.2) .44 .202 .108 .0607 .0341 .0184 .0089.0035.0077 0
th(2) .739 .342 .187 .107 .0607 .0328 .0159 .0062 .0014 0
For a given value of a 2 (intensity) f( r) must be larger than
0
this, leading to a restriction on r (reflectivity). th is
tabulated for 3 values of  where  > 1
Restriction of the fraction of laser light
reflected or back-scattered
2
a
For a given value of 0 (i.e. intensity) f(r) must be
larger than this which then leads to a restriction on
the fraction of light reflected.
2
0
For example we require r < 0.1 for a = 45,
i.e. I = 6  1019 Wcm-2.
The low Thot and low reflectivity are advantageous
to fast ignition, but require further experimental
verification, additional physics in the theory, and
simulations.
Relativistic motion of an electron in a plane
e.m. wave
In a plane polarized e.m.wave (Ex,By) of arbitrary form in
vacuum an electron starting from rest at Ex=0 will satisfy
pz=px2/2mc
A wave E0sin(t-kz) and proper time s   dt/  gives
x/c = a0 (s - sin s)
z/c = a02( 3s/4 - sin s + 8-1 sin 2s)
t = s + a02( 3s/4 - sin s + 8-1 sin 2s)
in a full period of the wave as seen by the moving
electron i.e. s=2, forward displacement is z = 3a02/4.
But in an overdense plasma c/pe < /2.
for a0 ≥ ~ 1 an electron will traverse a
distance greater than the skin depth without
seeing even a quarter of a wavelength, i.e. the
electron will not attain the full ponderomotive
potential, before leaving the interaction
region.
Thus it can be understood why the Thot scaling
leads to a lower temperature.
However if there is a significant laser prepulse
leading to an under-dense precursor plasma,
electrons here will experience the full field.
Relativistic collisionless skin-depth
1 Bˆ y me a0
J x  ncriteca0 (1 coss)  

sins
0 z
e 0 z
5
3s
a0
2
z  a0


5!
a0 / z
c
1/ 6
80 
 s   2 2 
a0  p 
2

2/ 3
c    1/ 3
  0.963   a0
 p  p 
Sweeping up the precursor plasma
Assuming a precursor density n = nprexp(-z/z0) with
energy content 1.5npreTz0 per unit area.
Using an equation of motion
dv/dt = - p + (I/c)
The velocity of the plasma during the high intensity
pulse I when p is negligible is
z/t ≈ [ I / (cnprmi)]1/2
For I = 1023 Wm-2, npr = 1027 m-3, mi = 27mp, this
gives 2.7 106 m/s, i.e. in 1ps plasma moves only
2.7m.
2D effect; Magnetic field generation
due to localised photon momentum deposition:
An Ez electric field propagates into the solid
accelerating the return current. It has a curl,
unlike the ponderomotive force which is the
gradient of a scalar.
At saturation there is pressure balance,
B2/20 = nheTh = hncmec2[(1+21/2a0)1/2 -1]
and h = 1+a0/21/2.
E.g. I = 91019Wcm-2, ao = 8.5 gives B = 620MG
(U.Wagner et al, Phys. Rev.E 70, 026401 (2004))
Summary
• A simple, approximate model has verified Beg’s empirical
scaling law for Thot.
• A fully relativistic model including photon momentum
extends this to higher intensities where Thot  (I2 )1/4.
• Electrons leave the collisionless skin depth in less than a
quarter-period for ao2 > 1.
• Including reflected light deposits more photon momentum,
lowers Thot, and restricts the reflectivity at high intensity.
• Precursor plasma can change the scaling law.
• More data, more physics (e.g. inclusion of Ez to drive the
return current, time-dependent resistivity) are needed.