The propagation dynamics of an azimuthally polarized dark hollow laser beam described by a first-order Bessel–Gauss laser beam in a parabolic plasma channel is investigated by adopting the weakly relativistic limit. By using the variational method, the evolution equation of the ring-beam radius is derived and the ring-beam width is proportional to and synchronous with the radius. It is found that the azimuthal polarization can weaken the vacuum diffraction effect and the propagation dynamics of the dark hollow laser beam may be classified into three types, i.e., propagation with a constant ring-beam radius and width, or synchronous periodic defocusing oscillation, or synchronous periodic focusing oscillation. Their corresponding critical conditions and characteristic quantities, such as the amplitudes and spatial wavelengths, are obtained. Further investigation indicates that, with the increase in the initial laser power or the ratio of initial ring-beam radius to channel radius, the dark hollow beam may experience a process from synchronous periodic defocusing oscillation to constant propagation and then to synchronous periodic focusing oscillation, in which the corresponding amplitudes decrease sharply to zero (constant propagation) and then increase gradually, while the spatial wavelength decreases continuously. The evolution type of this kind of dark hollow beam also depends on its initial amplitude but is insensitive to the initial laser profile which, however, has a large influence on the spatial wavelength. These results are well confirmed by the numerical simulation of the wave equation. A two-dimensional particle-in-cell simulation of an azimuthally polarized laser beam is performed finally and also reveals the main results.

Considering the relativistic self-focusing, the ponderomotive self-channel, and the preformed channel focusing, the effect of a density hump on the laser propagation in a preformed plasma channel is studied. The evolution equation of the laser spot size is derived by using the source-dependent expansion technique. It is found that the laser behavior after the hump strongly depends on the hump position and width and is also related to the hump altitude. For the incident laser with a constant spot size, the laser after the hump may oscillate or not change, only depending on the hump width under a certain hump altitude. For the incident laser with oscillation, the laser oscillation can be enlarged, decreased, unchanged, according to the hump width, position, and altitude. So, the density hump can play the role like a filter, or like an oscillator, or be ineffective by adjusting its width, position, and altitude. These results are well confirmed by the final numerical simulations.

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We report a generation of energetic protons by the interaction of a high-energy electron driving beam with an underdense plasma slab. After an interaction period of approximately 4000 fs, a proton beam with maximum energy greater than 250 MeV can be achieved by applying a driving beam with energy 1.0 GeV to a 200 $\mathrm {\mu }$ m plasma slab. Our two-dimensional particle-in-cell simulations also show that the proton acceleration process can be divided into two stages. In the first stage, a strong positive longitudinal electric field appears near the rear boundary of the plasma slab after the driving beam has passed through it. This acceleration process is similar to the target normal sheath acceleration scheme by the interaction between intense pulsed laser with overdense plasma targets. In the second stage, the accelerated protons experience a long-range acceleration process with a two-stream instability between the high-energy driving beam and the proton beam. Further analyses show that this accelerated proton beam is equipped with the property of good collimation and high energy. This scheme presents a new way for proton or ion acceleration on some special occasions.