Abstract When the ring with the maximum intensity deviates from the central point, the dual autofocusing performance of the ring Pearcey edge dislocation (RPED) beams in free space is gradually destroyed. To address the degradation in the dual autofocusing ability, we investigate the propagation dynamics of the RPED beams in a system with fractional diffraction effect or parabolic potential. The simulation results show that there exists a critical value for the Lévy index, that results in the RPED beams exhibiting an obvious dual autofocusing phenomenon with equal focusing intensities. When the Lévy index is near the critical value, the RPED beams have dual autofocusing characteristics, and the focusing intensity and focal distance can be controlled by changing the Lévy index. The introducing of the parabolic potential leads to the periodic evolution of the RPED beams, and the dual autofocusing property of the RPED beams with smaller radius can be restored within one evolution cycle by changing the potential depth. Moreover, the positions of the edge dislocation affect the focusing intensity, but have no effect on the number of foci. Our research provides some inspiration for the control of dual autofocusing beams, and has potential applications in optical manipulation and optical trapping.
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