A vortex beam has a helical phase, and each photon carries orbital angular momentum (OAM). It has extensive applications in fields such as particle manipulation, free-space optical communication, and micro-displacement measurement. Research on its transmission characteristics has received widespread attention. The beam propagating in free space inevitably encounters obstacles. When the beam passes through an obstacle, its intensity is disrupted due to the presence of the obstacle, and the degree of disruption is directly related to the size of the obstacle. However, the beam exhibits self-healing characteristics after encountering obstacles. Understanding the self-healing characteristics of vortex beams is conducive to its applications in optical communication and particle manipulation.

Using numerical analysis, based on Babinet's principle, a Gaussian absorption function was employed to describe the obstacle. The transmission expression of vortex beams passing through the obstacle was derived, the intensity distribution of vortex beams propagating through the obstruction was theoretically calculated, and the influence of obstacle radius on self-healing properties was analyzed.

As the radius of the obstacle increases, the missing part of the vortex beams became larger. As the transmission distance increased, the missing part of the vortex beams gradually disappeared. When the transmission distance was long enough, the intensity of the vortex beams basically returned to that without the obstacle; that was, the intensity distribution of light became circular, indicating that the vortex beams had self-healing characteristics(Fig.1).It could be observed that the energy flow of the vortex beams moves counterclockwise. The main reason was that the topological charge of the vortex beams was 1 during the propagation process. When the topological charge of the vortex beams was set to -1, the energy flow of the beams changed from counterclockwise to clockwise. It was not difficult to find that when the vortex beams passed through the obstacle, the direction of the Poynting vector around the intensity void pointed toward the center of the void. It was inferred that the energy flow around the intensity void would continuously move towards the break, which essentially explained the self-healing property of the vortex beam as it passed through the obstacle. Due to the presence of the obstacle, there was no Poynting vector in the intensity void part, which meant that the beam energy was completely absorbed by the obstacle. The numerical analysis results were consistent with the actual situation. As the transmission distance increased, due to the self-healing characteristics of the vortex beams, the intensity was restored, and at the same time, the Poynting vector also appeared successively(Fig.3 and Fig.4).Energy flow distribution of the vortex beams on the far-field plane of 0.1 m when the radii of the obstacles were 0.1 mm, 0.2 mm and 0.3 mm respectively(Fig.5). It could be seen from the figure that at the source plane, there was no energy flow at the defect position, which was consistent with the result of the intensity distribution in Fig.1. It could be seen that the energy flow distribution presented a spiral shape, which was consistent with the counterclockwise rotation of the Poynting vector direction in Fig.4. As the transmission distance increased, the energy flow also showed a circular distribution. The self-healing characteristics of vortex beams could also be observed from the evolution of energy flow.Intensity distribution at the far-field plane of 0.1 m after the vortex beams passed through the obstacle with 0.1 mm diameter for topological charges of 1, −1, −3, −3, 5, and −5, respectively are show in Fig.7. By comparing the intensity distribution of different topological charges, it could be observed that when the topological charge of the vortex beams was a positive integer, the intensity rotated counterclockwise; when the topological charge was a negative integer, the intensity of the vortex beams rotated clockwise. The obvious phenomenon that could also be observed was that the intensity distribution of different topological charges was uneven, with multiple bright spots, and the number of bright spots was consistent with the number of topological charges. This phenomenon could also provide a method for detecting the topological charge number of vortex beam. By comparing the intensity distribution of the theory and the experiment, it could be seen that the theoretical and experimental results were highly consistent, further verifying that the vortex beams had self-healing characteristics.

The transmission expression of vortex beams after passing through an obstacle is derived, and the intensity distribution of the vortex beams after passing through the obstacle is theoretically calculated. In addition, the Poynting vector of the vortex beams after passing through the obstacle is also calculated. It can be seen that as the radius of the obstacle increases, the missing part of the vortex beams intensity becomes larger. Theoretical results confirm that the vortex beams has self-healing characteristics. When a femtosecond laser is used to focus onto a coated plane mirror, defect spots appearing on the plane mirror can be used as obstacles in experiments. The self-healing characteristics of vortex beams are verified experimentally. The experimental results and theoretical results are consistent, further confirming the self-healing characteristics of the vortex beam. The research results have important significance for understanding the propagation characteristics of vortex beam.