Abstract
1 Introduction
The muon
[
Laser wakefield acceleration (LWFA), which promises the next generation compact high-energy electron beam source
[
This new all-optical ‘Generator and Booster’ scheme can supply a prompt, compact, low cost and controllable muon source which would have potential applications in muon collider, neutrino physics and Higgs Factory
[
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In the scheme, the Generator would produce a muon bunch with short pulse duration, small source emittance and continuous energy distribution
[
In this paper, we investigate the trapping and acceleration of muons with continuous energy distribution from the ‘Generator’. The motion of muons is analyzed by one-dimensional analytic model and verified by two-dimensional particle-in-cell (PIC) simulation of a typical laser wakefield. It is shown that muons can be trapped in a broad energy range and accelerated to higher energy than that of electrons for longer dephasing length. We further extrapolate the muon acceleration to anticipate a muon energy up to 15.2 GeV on the existent short pulse laser facilities, which is exciting for the application in the laser laboratories.
2 Muon motion in one-dimensional analytic model
We first illustrate a typical laser wakefield in Figure
We analyze the motion of muons in such a laser wakefield in a one-dimensional analytic model. Similar as electrons, only muons locate in
in the electrostatic field shown in Figure
For convenience, we define forward phase (
) and backward phase (
) denoting the initial direction of the muon in the rest frame of the bubble as shown in Figure
Then the trajectories of muons giving
and
can be calculated with this analytic model. Giving
, we choose the
randomly from 0.2 GeV to 2.0 GeV. When muons drop into
or
regions, the final acceleration energy is recorded as
. We calculate the trajectories for
which are solid lines in Figure
For
case,
muons would drop out of the bubble as
when falling back to
and
muons change to forward phase before falling back to
resulting in insufficient acceleration, which illustrate the decline from the maximum acceleration energy. For higher
, muons dephase directly. Thus the lowest boundary of the solid line in Figure
For case, muons drop out of the bubble without trapping. It is worth to mention that muons with higher energy ( ) would dephase more quickly resulting in less energy gain than the case. As a result, the energy spread of muons would be narrowed.
For
case, lines stand above the
boundary (the lowest boundary in Figure
3 Muon motion in the two-dimensional PIC simulation
With the estimation of the one-dimensional analytic model, we choose a flat energy distribution range from 0.7 GeV to 2.2 GeV with initial position
denoting the forward muons and energy range from 0.2 GeV to 1.2 GeV with initial position
denoting the backward muons in the two-dimensional PIC simulation. The muons are located in
in the transverse direction with a density of
(roughly
muons located in a
plate), which is lower enough to avoid disturbing the bubble’s plasma structure. In the simulation, we trigger the movement of muons in
direction when the bubble structure is formed. The snapshots in Figure
We see in Figure
The
and
of muons at
are plotted in Figure
4 Extrapolation of muon acceleration in laser wakefield
The good agreement of the one- and two-dimensional simulations gives us more confidence to extrapolate the estimation of muon acceleration. Obviously, to accelerate muons to higher energies, longer dephasing time is needed. Therefore the relativistic factor of the bubble
would be the most important parameter in the extrapolation. The initial energy of muons
is another important parameter for the finite muon energy from the ‘Generator’. Considering the status of LWFA electrons up to now
[
We have also shown the extrapolation of electron acceleration in the same parameters for comparison as the dashed line in Figure
5 Summary
Therefore, compared to electron or proton laser plasma accelerations, this all-optical muon acceleration scheme has particular characteristics. For the massive invariant mass, muons could be injected into the whole bubble acceleration region with a broader energy spread. Furthermore, higher energy gain compared to electrons could be achieved from the longer dephasing length. On the other hand, the light invariant mass decreases the trapping energy threshold which makes muons easier to catch up the bubble. Considering the crucial requirement of laser intensity for proton acceleration in the laser wakefield discussed in Ref. [
In conclusion, we propose a new all-optical ‘Generator and Booster’ scheme to accelerate muons from the Bethe–Heitler dimuon production process by the laser wakefield to supply a prompt, compact, low cost and controllable muon source in the laser laboratories. To our knowledge, it is the first research on muon acceleration in the laser wakefield. By applying a one-dimensional analytic model, the muon trapping energy threshold depending on the phase space of the bubble region is discussed in detail. A two-dimensional PIC simulation is carried out to validate the acceleration picture. The forward and backward muons in the bubble region are simulated and well agreement with the one-dimensional estimation is presented. We also extrapolate the estimation to higher energy muon acceleration. It is shown that a maximum energy up to 15.2 GeV could be achieved with an initial energy
by accelerating muons for 300 ps with a bubble of relativistic factor
. This fact seems quite promising on existing short pulse laser facilities
[
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