Fig. 1. (a) The phase differences of the plasma wave after 20T₀ passage in a transversely tailored plasma (red solid line) and a uniform plasma (black solid line). The spatial distribution of the transverse magnetic field B_z (b) and the backward electron density (c) at 20T₀. (d) A few typical trajectories of the re-injected electrons.

Fig. 2. The spatial distribution of the longitudinal electrostatic field (a) and the transverse field E_ys − cB_zs in the (x, y) plane (b) at 20T₀. The black dashed box marks the position of the bubble.

Fig. 3. (a) The Hamiltonian and (b) the electron density distribution in phase space (ψ, γ). The terms in (6) are normalized with the simulation results for the fields and the electron motion. It follows that eE_L/m_eω₀c = 50, eE_Sx/m_eω₀c = 9.32, kE=eESy/meω02=1, kB=eBSzc/meω02=1.47, v_y = 0.2, and v_x = 0.98.

Fig. 4. (a) The spatial distribution of the plasma electron density at 32T₀. (b) and (c) are plots of the on-axis longitudinal electric field E_x and the transverse field E_y − cB_z, respectively. (d) The trajectory of a sample electron and the electromagnetic field that the electron experiences. (e) The evolution of the transverse and longitudinal momenta, normalized by m_ec, for the same electron. (f) The work done by the longitudinal and transverse fields. The two vertical dashed black lines in (a)–(c) mark the acceleration zone. The first and second vertical dashed lines in (d)–(f) show the start of the betatron resonance acceleration, while the third dashed line marks the end of the wake-field acceleration. The horizontal dashed line in (b)–(f) gives the zero value of the corresponding quantity.

Fig. 5. (a) The phase space x–p_x of electrons inside the bubble at 32T₀. (b) The longitudinal distribution of the electron density at the position with peak electron density. (c) The angular energy of the accelerated electrons. (d) The spectrum of the first electron microbunch.

Fig. 6. (a) The emitted photon density at 32T₀. (b) The evolutions of the local electron oscillation period (black solid line) and the local strength parameter K(red solid line). (c) The angle energy of the generated photons. (d) The spectrum of photons emitted by the first electron bunch.