Presentation is loading. Please wait.

Presentation is loading. Please wait.

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,

Similar presentations


Presentation on theme: "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,"— Presentation transcript:

1 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 T hot. 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 Beg’s empirical scaling of T h (keV)=215(I 18 2  m ) 1/3 for 70 < T h < 400keV & 0.03 < I 18 < 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 n h 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. Transform to the axial rest-frame of the beam: Equate E 0 to m e  0 c 2 ;  0 indicates the thermal energy in the rest frame of the beam; because transverse momenta are unaffected by the transformation

7 In dimensionless parameters, t h = eT h /m e c 2 and a 0, t h = (1+2 1/2 a 0 ) 1/2 - 1 (2) This contrasts with the ponderomotive scaling: t h = (1+a 0 2 ) 1/2 - 1 S.C.Wilks et al PRL(1992)69,1383 Simple model of Beg scaling, Eq.1, gives t h = 0.5 a 0 2/3 (3) Eqs (2) and (3) agree to within 12% over the range 0.3

8 Various scaling laws; Beg’s empirical law is almost identical to Haines- classical and relativistic up to I = 5  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 T hot is reduced. The accelerating electrons will form a moving mirror, but the return cold electrons ensure that the net J z, 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 = n c p z (  h -1)c 2 (4) while momentum flux conservation is I/c + (1-  abs )I/c = n c p z 2 /m e (5) Define I r = (1-  abs )I; (5)  c+(4) gives 2I = n c p z c 2 [p z /m e c + (  h - 1)], while (5)  c-(4) gives 2I r = n c p z c 2 [p z /m e c - (  h -1)], or dimensionlessly i i = 2I/n c p z c 2 = p z ' +  h - 1 (6) i r = 2I r /n c p z c 2 = p z ' -  h + 1 (7) where p z ' = p z /m e c

11 As before, transform the energy to the beam rest-frame E 0 2 = E 2 - p z 2 c 2 = (  h m e c 2 ) 2 - p z 2 c 2 = m e 2 c 4 (  h 2 -p z ' 2 ) = m e 2 c 4  0 2 Hence T h as measured in the beam rest frame is t h = eT h /m e c 2 =  = [(  h +p z ')(  h - p z ')] 1/2 - 1 = [(1+i i )(1- i r )] 1/2 - 1 Use (6) and (7) to eliminate p z ' to give i i +i r =2p z '. Define r = i r /i i ; then i i = 2 1/2 a o (1+r) -1/2 and t h = [{ /2 a 0 /(1+r) 1/2 }{ /2 a 0 r/(1+r) 1/2 }] 1/2 - 1 (8) This becomes Eq (2) for r = 0, and for r > 0, t h is reduced. The condition t h > 0 becomes f (r )  (1 - r 2 )(1 - r)/(2r 2 ) > a 0 2 and df/dr<0 for 0

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

13 Table of f(r) and t h (  ) versus r r = f(r) = t h (1.1) t h (1.2) t h (2) For a given value of (intensity) f( r) must be larger than this, leading to a restriction on r (reflectivity). t h 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  Wcm -2. The low T hot 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 (E x,B y ) of arbitrary form in vacuum an electron starting from rest at E x =0 will satisfy p z =p x 2 /2mc A wave E 0 sin(  t-kz) and proper time gives  x/c = a 0 (s - sin s)  z/c = a 0 2 ( 3s/4 - sin s sin 2s)  t = s + a 0 2 ( 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 = 3a 0 2 /4.

16 But in an overdense plasma c/  pe < /2 .  for a 0 ≥ ~ 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 T hot 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 = n pr exp(-z/z 0 ) with energy content 1.5n pr eTz 0 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 / (cn pr m i )] 1/2 For I = Wm -2, n pr = m -3, m i = 27m p, this gives 2.7  10 6 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 E z 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, B 2 /2  0 = n h eT h =  h n c m e c 2 [(1+2 1/2 a 0 ) 1/2 -1] and  h = 1+a 0 /2 1/2. E.g. I = 9  Wcm -2, a o = 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 T hot. A fully relativistic model including photon momentum extends this to higher intensities where T hot  (I 2 ) 1/4. Electrons leave the collisionless skin depth in less than a quarter-period for a o 2 > 1. Including reflected light deposits more photon momentum, lowers T hot, and restricts the reflectivity at high intensity. Precursor plasma can change the scaling law. More data, more physics (e.g. inclusion of E z to drive the return current, time-dependent resistivity) are needed.


Download ppt "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,"

Similar presentations


Ads by Google