With the development of high-power laser technology, laser plasma acceleration has developed rapidly due to its excellent acceleration structure. Nearly one-hundred-MeV proton beams and several GeV electron outputs are obtained. The laser-driven proton beams have excellent quality of μm-scale sizes and ps-scale pulse lengths. Owing to the existence of the accelerating laser field, direct application is difficult, so the proton beams need to be transmitted to the application terminal through the beamline. However, the wide energy spectrum and large divergence angle bring difficulties in transmitting the beam. The weak focusing in the constant gradient magnetic field is neglected in the transmission of laser-driven particle beams because of the relatively weak focusing force. But weak focusing has special advantages: simultaneous focusing in the horizontal direction and the vertical direction, energy analysis in the horizontal direction, focusing force in the horizontal and vertical direction distributed by the field index n, and smaller influence of chromatic aberration effect.

In this paper, we propose the beam transmission with weak-focusing magnet. The requirements for the focusing of proton beams with the same energy and different divergence angles in the X direction and Y direction in the weak-focusing magnetic field are explored by studying the linear beam dynamics of the beams. Then the conditions of precise energy analysis for particle beams with large divergence angle can be determined. For beams with 2% energy spread, the lengths of the drift space before and after the weak-focusing magnet and deflection radius are scanned to find out the minimum beam size and the shortest pulse length after transmission. It is found that a certain combination of drift space and deflection radius can minimize the beam size or the pulse length. Focusing and energy selection can be achieved while compressing the pulse length and effectively reducing the size of the beamline, which has significant advantages. When the deflection radius is 0.65 m, the proton beam with 20 MeV energy, 2% energy spread, and an initial divergence angle of ±50 mrad has the root-mean-square size of 108 μm in both the X direction and the Y direction, and a pulse length of 154 ps at the application terminal.

Comparing with common beam transmission elements such as quadrupole lenses and deflection magnets, the laser-accelerated ion beam benefits from the integration of focusing and energy analysis of weak-focusing magnetic fields (focusing and energy analysis exist at the same time and continuously change with deflection angle), as well as the horizontal and vertical focusing forces can be distributed by the magnetic field index n (the larger the n, the stronger the focusing force in the vertical direction is and the weaker the focusing force in the horizontal direction). When the proton beam is transmitted in a weak-focusing magnetic field, the advantages of the focusing element and the energy selection element are combined, so the influence of the chromatic aberration effect can be reduced, the pulse length can be compressed, and the beamline size can be effectively reduced.