The muon ^{[ 1 ]} plays a key role in particle physics and applied physics, such as the muon anomalous magnetic dipole moment measurement ^{[ 2 ]} , muon collider and neutrino physics ^{[ 3 ]} , muon catalyzed fusion ^{[ 4 ]} , muon probe of the microscopic magnetic properties of materials ^{[ 5 ]} , and muon radiography ^{[ 6 ]} . However, muon physics is limited by either the low flux cosmic-ray muons ^{[ 7 ]} or high cost accelerator muons ^{[ 8 – 11 ]} .

Laser wakefield acceleration (LWFA), which promises the next generation compact high-energy electron beam source ^{[ 12 , 13 ]} , could also have potential application in muon source researches. While electrons can be simply self-injected through wave-breaking, an external injection is needed for muons. Such external muons could be conveniently supplied by the Bethe–Heitler lepton pair production proposed by Titov  et al.^{[ 14 ]} , where the high-energy photons come from the LWFA electrons with energy up to several GeV ^{[ 15 – 20 ]} interacting with high  materials. dimuons could be produced by a 100 J petawatt laser facility ^{[ 14 ]} and diagnosed efficiently ^{[ 21 ]} . But those muons with lower mean energies and broad energy spread ranging from hundreds of MeV to GeV ^{[ 14 ]} limit the applications. On the other hand, those muons could be accelerated by the laser wakefield, called ‘bubble’, since the velocities are close or even higher than that of the bubble. Therefore for the first time we suggest an all-optical muon acceleration scheme as shown in Figure  1 , in which muons could be generated by the Bethe–Heitler process via high-energy photons from the Bremsstrahlung radiation of the LWFA electrons interacting with the high materials, called ‘Generator’, and boosted by another laser wakefield, called ‘Booster’.

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 ^{[ 3 ]} , muon acceleration in cosmic-ray sources ^{[ 22 ]} and other related muon physics. Peano et al. also proposed a scheme ^{[ 23 ]} to accelerate ‘cold’ muons by a slow ponderomotive beat-wave structure driven by two counterpropagating laser beams with variable frequencies which is quite different from the scheme proposed here.

In the scheme, the Generator would produce a muon bunch with short pulse duration, small source emittance and continuous energy distribution ^{[ 14 ]} . In principle, muons produced in the Generator could be collected and focused to adapt the bubble’s size before injecting into the Booster by the plasma lens ^{[ 24 , 25 ]} or magnetic quadrupoles lens ^{[ 26 ]} . The time synchronization between Generator and Booster is also guaranteed in such an all-optical system. However, the huge invariant mass difference by a factor of 207 ( ) would result in different behaviors of muons compared to electron and positron accelerations in the bubble ^{[ 27 , 28 ]} . Therefore, considering the continuous energy ^{[ 14 ]} , the acceleration performance and phase matching of muons in the accelerated field of the bubble need to be studied in detail. It is worth to mention that the  acceleration would be easier than that of  in a nonlinear (bubble/blow-out) regime since the accelerating/focusing wakefields for  determined by background plasma density are much challenging for  same as in the cases of positrons ^{[ 29 ]} . In this paper, we focus on the  acceleration. Hereafter in this paper muon denotes  .

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.

We first illustrate a typical laser wakefield in Figure  2 (a) using the two-dimensional PIC simulation code Opic2D. A linearly polarized pulse of wavelength  , normalized peak amplitude  , pulse duration 33 fs and full width at half-maximum (FWHM) spot size  enters from the left of the simulation box of size  . A plasma of electron density (0.002 ) is located in . There are 4000 600 cells in the simulation windows. The particle number per cell is 4 for both electron and muon. Open boundaries are applied in both longitudinal and transverse directions. A moving window is applied in the simulation with velocity which is consistent with the laser group velocity (normalized by the light speed ) reading from the simulation. The lineout on-axis electrostatic field in the moving window (i.e., in the rest frame of the bubble) is shown in Figure  2 (b).

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  2 (b) are accelerated. The two zero points of are defined as and with denoting the position in the rest frame of the bubble. Considering the initial energy of muons and position , the position of the muon in the rest frame of the bubble is

 (1)

 (2)

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  2 (c). Giving finite laser and plasma condition, the trapping condition of muons is at . Therefore, backward muons should be located at to achieve sufficient acceleration. As an example, a backward muon with critical trapping energy located at would fall back to in the rest frame of the bubble with velocity climbing up to . After that the muon turns into forward phase until flying over the bubble region. The whole process would follow the trajectory shown in Figure  2 (c) and the energy increasing in Figure  2 (d).

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  3 (a). Clearly energy thresholds for trapping are presented by the sharp peaks of the lines. Muons located at have the highest trapping energy threshold and the lowest . Here we take and cases to discuss the acceleration pictures of muons in such an electrostatic field.

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  3 (a) corresponds to .

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 (a)) which means in a broad energy range muons could be accelerated efficiently as our expectation. The reason is even with such a broad energy spread, the velocity of muons can still follow a narrow velocity distribution adapting to the bubble’s velocity.

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  4 show the acceleration processes of these two groups of muons at , 1.65 ps, 16.5 ps and 33 ps, respectively.

We see in Figure  4 (a), at , that a typical electron bubble structure is formed. The forward muons (red dots) and backward muons (blue dots) are located around the initial positions. At in Figure  4 (b), the backward muons are focused by the transversal field . Part of the forward and backward muons with lower energy have dropped out of the bubble from the left side ( ). At in Figure  4 (c), both forward and backward muons are accelerated in the bubble sufficiently. However, the bubble structure has become unstable for the perturbation of laser transmission in the plasma. Finally at in Figure  4 (d), the bubble structure is almost absent. Both of the forward and backward muons fly freely. The and phase spaces of forward (red dots) and backward (blue dots) muons at the four snapshots are also shown, respectively, in Figure  4 (e)– 4 (l). The oscillations of the two groups of muons in the phase space present effective constraints by the transversal field which would result in fine beam collimation.

The and of muons at are plotted in Figure  3 (a). Clearly consistency and same trends exhibit as the expectation from the one-dimensional analytic model especially the sharp peaks of the lines denoting the trapping energy thresholds for different initial positions. The trapping energy thresholds are higher than the one-dimensional model estimations because the bubble structure distortion is not considered in the analytic model. Furthermore, in Figure  3 (b), the energy spectra of both groups of muons are narrowed after the acceleration as expected.

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 ^{[ 15 – 20 ]} , we set the maximum . With the one-dimensional analytic model, we estimate the extrapolated relationships of the maximum acceleration energies of muons  depending on the bubble’s relativistic factor  in Figure  5 (a). For each setting of the bubble’s relativistic factor , the relevant trapping energy threshold  is different which is shown in Figure  5 (b). We can see that giving the relativistic factor of the bubble and the initial energy , the extrapolated maximum acceleration energy of muons could be up to 15.2 GeV. To obtain such high-energy muons, one needs to accelerate muons for 300 ps in the bubble’s plasma channel (plasma density is  and length 9 cm) which has been realized in the experiment ^{[ 20 ]} . To accelerate muons to more higher energy such as a Higgs factory, a collision energy of around 125 GeV (muon energy 62.5 GeV) is needed which requires lower plasma density (plasma density is  and length 46 cm) and higher relativistic factor ( ) than the plasma channel in the literature. This kind of plasma channel has not been actualized but it might be realized in the future.

We have also shown the extrapolation of electron acceleration in the same parameters for comparison as the dashed line in Figure  5 (a), in which we can find that the maximum acceleration of muons is larger than that of electrons by a factor of 2 to 3 for longer dephasing length. In fact, the dephasing length of trapped electrons reads ^{[ 30 ]}

 (3)

 (4)

 (5)

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. [ 31 ], muons ( and ) are the feasible choice besides the electron and positron in LWFA mechanism based on the short pulse laser facilities now. Additionally, as an unstable particle, muon’s lifetime multiplied by the relativistic factor is long enough compared to the production and acceleration processes proposed in our scheme. Thus the decay losses could be neglected unlike traditional accelerator based muon production and acceleration processes. It is worth to mention that traditional accelerator muon sources are also suitable for the discussion here, but high time synchronization precision is needed.

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 ^{[ 32 ]} . We expect such a new all-optical ‘Generator and Booster’ muon source to be realized in the near future.