1. Imperial College, London
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)

2. 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.

3. Relativistic quiver motion gives
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 non-relativistic inward energy flux of electrons:
and
Relativistic quiver motion gives

4. nh is the relativistic critical density Taking the 2/3 power of this gives Eq.1 or

5. Model 2: Fully relativistic with energy and momentum balance
Momentum conservation is
where
consistent with electron motion in a plane wave

6. h depends on the total velocity of an electron
h depends on the total velocity of an electron. Transform to the axial rest-frame of the beam: 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

7. In dimensionless parameters, th = eTh/mec2 and a0, th = (1+21/2a0)1/ (2) This contrasts with the ponderomotive scaling: th = (1+a02)1/ S.C.Wilks et al PRL(1992)69,1383 Simple model of Beg scaling, Eq.1, gives th = 0.5 a02/ (3) Eqs (2) and (3) agree to within 12% over the range 0.3<a0<300, and intersect at a0 = and The total electron kinetic energy is (h - 1) = a0/21/2

8. Various scaling laws; Beg’s empirical law is almost identical to Haines-classical and relativistic up to I = 51018 Wcm-2

9. Model 3: Addition of reflected or back-scattered 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.

10. If absorbed fraction is abs, energy conservation is I - (1-abs)I = ncpz(h-1)c (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 (6) ir = 2Ir/ncpzc2 = pz' - h (7) where pz' = pz/mec

11. 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/ (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

12. Defining  as f(r)  2a02 where  > 1, th becomes th = {[1 +(1-r)/(r)][1 -(1-r)/]}1/ Using r, (0<r<1), and , ( > 1) as parameters we can also find a0 and the reflection coefficient, refl 1-abs = r The condition refl ≤ 1 gives

13. Table of f(r) and th() versus r r = 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0
Table of f(r) and th() versus r r = f(r) = th(1.1) th(1.2) th(2) For a given value of (intensity) f( r) must be larger than this, leading to a restriction on r (reflectivity). th is tabulated for 3 values of  where  > 1

14. Restriction of the fraction of laser light reflected or back-scattered
For a given value of (i.e. intensity) f(r) must be larger than this which then leads to a restriction on the fraction of light reflected.
For example we require r < 0.1 for = 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.

15. 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 gives
x/c = a0 (s - sin s)
z/c = a02( 3s/4 - sin s sin 2s)
t = s + a02( 3s/4 - sin s 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.

16. But in an overdense plasma c/pe < /2
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.

17. Relativistic collisionless skin-depth

18. 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.

19. 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, (2004))

20. 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.