Fig. 1. All-optical ‘Generator and Booster’ scheme of muon source. Muons are first generated by the Bethe–Heitler process, the ‘Generator’, via high-energy photons from Bremsstrahlung radiation of laser wakefield accelerated electrons interacting with high materials. After a proper collection and focusing system, muons are boosted by another laser wakefield, the ‘Booster’.

Fig. 2. (a) The bubble’s electron density and (b) electrostatic field in the simulation box were obtained from a two-dimensional particle-in-cell simulation code Opic2D for a plasma density . The laser pulse of wavelength , normalized peak amplitude , pulse duration 33 fs and FWHM spot size entered from the left of the simulation box in the direction. A muon with critical trapping energy entered into the simulation box following the trajectory shown in (c) and the energy increasing process in (d).

Fig. 3. (a) The estimated acceleration energy of muons depending on the initial energy with different positions in the rest frame of bubble from the one-dimensional analytical model denoted by the solid lines. Clearly trapping energy thresholds presented from the sharp peaks of the lines. The two-dimensional PIC simulations of the forward (red dots) and backward (blue dots) muons at show well agreement with the one-dimensional estimation. (b) The inner plot shows the spectra of the forward (red lines) and backward (blue lines) muons, where dashed lines denote the initialized energy spectra and the solid lines denote the acceleration energy spectra at .

Fig. 4. The snapshots of the acceleration processes of forward (red dots) and backward (blue dots) muons from the two-dimensional PIC simulations in Figure  2 at (a) ps, (b) 1.65 ps, (c) 16.5 ps and (d) 33 ps. (e)–(h) The and (i)–(l) phase spaces of forward (red dots) and backward (blue dots) muons at the four snapshots are also shown, respectively.

Fig. 5. The extrapolated relationships of the maximum acceleration energy of muons depending on (a) the bubble’s relativistic factor and (b) the relevant trapping energy threshold from the one-dimensional analytic model. The extrapolation of electrons (open circles) in the same parameters is also shown for comparison.