Pair Production and Cascading in Counterpropagating Laser Beams

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Pair production and cascading in counterpropagating laser beams
I. Arka1, J. G. Kirk1, A. R. Bell2,3
1Max-Planck-Institut für Kernphysik, Heidelberg, Germany
2Clarendon Laboratory, University of Oxford, Oxford, UK
3STFC Central Laser Facility, RAL, Didcot, UK
Introduction
Pair production has been reported in several experiments that used intense laser beams either
as an accelerator (e.g. [3]) or as a target for electron beams [7]. Here we propose the use of laser
beams both as an accelerator and as a target. This could be achieved using counterpropagating
beams of intensity 1023 −1024Wcm−2in an underdense plasma. Electrons accelerated this way
radiate gamma rays, which can subsequently produce pairs via the multi-photon pair production
process. As shown in a companion paper (Kirk, Bell, Arka, these proceedings, referred to as
paper I in the following), these high energy photons reach energies of 775I3/2
24
λ
2
µ
mMeV, where
I24 is the single beam intensity in units of 1024Wcm−2and
λµ
mis the laser wavelength in
microns. The produced pairs are expected to be accelerated inside the interacting pulses, and to
lead to an electromagnetic cascade which could deplete much of the beam’s energy.
Pulse models and electron acceleration
-50
0
50
Φ
-0.2
-0.1
0.0
0.1
0.2
Ex
Incoming-Reflected
Figure 1: Shape of incoming and reflected
pulse. The pulses are not plotted to scale.
We have used realistic models of finite laser pulses
to investigate the acceleration and radiation of elec-
trons. The pulses are modeled choosing an envelope
function
f±(
φ
) = 1
41∓tanh
φ
∆1±tanh
φ
±L
∆
by which we multiply a monochromatic plane wave of
linear polarization. The phase is
φ
=z∓ct,Lis the
duration of the pulse, ∆adjusts the thickness of the
pulse’s edge and the upper(lower) sign refers to waves
propagating in the +z(−z) direction. We have chosen three different models: the beams have
aligned polarizations, crossed polarizations, or aligned with the beam propagating in the −z
direction simulating a wave reflected by a solid target. This wave includes high harmonics [4]:
E=ˆ
x2
π
s√3
2
nmax
∑
n=0sin[(2n+1)
φ
−]
2n+1−2cos[(2n+1)]
φ
−
π
(2n+1)2
36th EPS Conference on Plasma Phys. Sofia, June 29 - July 3, 2009 ECA Vol.33E, P-1.021 (2009)

The pulses are set to have a finite transverse size, and to occupy a cylinder with a radius of one
wavelength.
We calculate the electron trajectory numerically using the classical equations of motion [1]
dpi
d
τ
=eFij pj+gi
where piis the four-momentum of the particle, Fij the electromagnetic field tensor,
τ
the proper
time and githe radiation reaction force given by the Landau-Lifshitz prescription [1]. The im-
portance of quantum effects is determined by the Lorentz invariant parameter
η
, introduced in
paper I. When
η
>0.1 then quantum effects influence the electron motion, and pair production
becomes important when
η
∼1. We do not take into account the discrete emission of photons,
but we normalize the radiation emitted by the electron (and thus the radiation reaction force)
by the reduction in the total power radiated in the quantum case in comparison to the classical
case.
Pair creation and cascading
To calculate the pair creation rate by a single electron, the total optical depth
τγ
along a pho-
ton’s path from emission until escaping the laser pulse is computed. The photon pair production
rate is given by dN±
dt =Z
η
/2
0d
χ
dN
d
χ
dt 1−e−
τγ

where
χ
is a dimensionless photon frequency referred to in paper I, Ecr =1.3×1018V m−1is the
Aligned linear polarization
23 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 24
log(I)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
log(N)
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
Figure 2: Logarithm of pair creation probability per intensity bin
in the case of aligned linear polarizations of the pulses.
Schwinger field value and dN
d
χ
dt is
the electron’s instantaneous radi-
ation spectrum. The trajectories
of electrons in the beams are sen-
sitive to the initial position of
the particles. Electrons initially at
rest are picked up by one of the
pulses and accelerated along the
pulse propagation direction. They
emit strong radiation when they
reach the pulse interaction region.
However, in most cases they drift
out of the cylindrical volume of
the pulse before they reach this
36th EPS 2009; I.Arka et al. : Pair Production and Cascading in Counterpropagating Laser Beams 2 of 4

region. For this reason we have numerically computed trajectories with randomly chosen ini-
tial positions in the range −8
λ
<z<0 and randomly chosen x,yon a disc of radius
λ
. The
intensity of the beams is also randomly chosen in the interval 23<logI24 <24. We computed
105trajectories for each beam model discussed above. For the details of the calculation, see [6].
Crossed linear polarization
23 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 24
log(I)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
log(N)
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
Figure 3: Logarithm of pair creation probability in the case of
crossed linear polarizations of the pulses.
The results can be seen in
figures 2,3,4. Each figure corre-
sponds to a different beam model.
We have plotted the logarithm of
the probability that a certain num-
ber of pairs will be created if an
electron starts from rest with ran-
dom initial position in a given
logarithmic intensity interval. In
the cases of the reflected pulse
and aligned polarisation rougly
83% of the particles leave the
laser volume without having trig-
gered pair production, while in
the crossed polarisation case this percentage is close to 75%.
Reflected pulse
23 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 24
log(I)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
log(N)
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
Figure 4: Logarithm of pair creation probability per intensity bin
in the case of the interaction with a reflected beam.
Pair cascades should begin to
develop when each electron pro-
duces roughly one pair before it
escapes from the beams. From
figures 2,3 and 4 we see that
only for the highest intensities is
the probability of the creation of
one pair non negligible. For ex-
ample, in the aligned linear po-
larization case, the critical inten-
sity for pair creation is seen to be
logIcr
24 ∼ −0.14. The three differ-
ent beam configurations produce
similar results.
The above calculations show that a significant percentage of the electrons is going to emit
36th EPS 2009; I.Arka et al. : Pair Production and Cascading in Counterpropagating Laser Beams 3 of 4

radiation energetic enough to produce pairs on the beam photons if beams of intensity close to
1024Wcm−2are counterpropagating in an underdense plasma. Most of this radiation is emitted
at the region where the pulses interact. Because the pairs are produced in this region, they get
rapidly accelerated and thus can produce energetic radiation themselves. This should result in a
pair cascade at the focus of the beams, that could be energetic enough to deplete them of their
energy.
A more precise treatment of the problem should take into account the episodic emission of
high-energy photons by the electrons, rather than assume a smooth classical trajectory. The
pair yield would then be calculated by simulating the pair cascade that is initiated by the MeV
photons in the field of the laser pulses. This may enhance the amount of pairs produced, because
of the discrete energy loss of the electrons, which results in a higher energy spread [8].
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36th EPS 2009; I.Arka et al. : Pair Production and Cascading in Counterpropagating Laser Beams 4 of 4