Abstract : The ideal perfect vortex beam is a special beam whose light intensity distribution does not change with the change of the topological charge. Compared with the ordinary vortex beam, it can greatly improve the application efficiency in particle manipulation and optical fiber transmission. In order to explore the free space propagation characteristics of the perfect vortex beam, the influence of the topological charge, the radius of the initial surface halo, and the width of the ring on its diffraction characteristics is calculated and analyzed in detail by using the Hank transformation, and it is found that the perfect vortex beam does not have no diffraction characteristics, As the diffraction distance increases, the halo widens and gradually transforms into a Bessel function. The diffraction effect is enhanced when the halo radius of the initial surface increases or the halo width decreases, and the effect of the halo width is greater than that of the halo radius. Compared with the former two, the topological charge has less influence on the diffraction effect. The research results provide a useful theoretical reference for the further application of perfect vortex light.

The vortex beam has a helical wavefront phase, and its important feature is that there will be a phase singularity. This is because the orbital angular momentum (OAM) carried by the photon will change the phase structure of the beam. The topological charge is an important parameter of the vortex beam, and the OAM of the beam is proportional to the magnitude of the topological charge. The vortex beam carrying OAM has great potential application value in the fields of quantum information encoding, particle rotation and manipulation, optical super-resolution imaging, etc., and is one of the current research hotspots in the field of optics.

Applying the beam carrying OAM to the optical tweezers system can make the suspended particles rotate. Recently, the researchers realized the ultra-high-speed rotation of the particles trapped in vacuum. In addition to light waves, sound waves propagating in free space can also carry corresponding OAM. When the sound wave carries a large enough OAM, it will cause the suspended macroscopic object to rotate. Through the interaction between OAM and nanostructured materials, silver nanowires with a length of more than 10 μm can be induced to generate sinusoidal optical moments on the dielectric surface, so that they can obtain angular acceleration during motion. The OAM-based communication system can realize high-density data communication, and the data rate in free space has reached 200Tbit/s. Existing studies have shown that it can be used in free space and optical fibers, and its working wavelength extends from optical band to millimeter wave. Combining the OAM-carrying beam with beam replication and phase correction technology improves the fringe definition after diffraction, thereby reducing the cost of an OAM-based communication system. Combining OAM with microscopy, helical spectrum analysis is used to identify gradients and dislocations, while Fourier methods are used for image reconstruction. However, for Laguerre-Gaussian or Bessel-Gaussian light that can generally carry vortex phase, the change of its topological charge will cause the change of the radius and width of the bright ring of the beam, thus weakening its performance in image processing and optical fiber transmission. and other aspects of application efficiency. In order to improve this situation, in 2013, Ostrovsky et al. first proposed a perfect vortex beam (PVB) concept in which the radius and width of the bright ring do not depend on the topological charge, and used a spatial light modulator to generate PVB, but The initially generated PVB has a low beam quality due to the influence of the surrounding secondary bright ring. The proposal of PVB has aroused the interest of researchers, and a lot of research work has focused on the generation method of ideal PVB.

Chen et al. generated PVB with the help of an axicon lens. This method greatly reduces the impact of the secondary bright ring on the outer ring and realizes the operation of fine particles. Based on the Fourier transform of the Bessel-Gaussian function, Vaity et al. used a phase mask instead of the axicon lens in the literature, which improved the quality of the beam generated in the experiment and realized the real-time controllability of PVB parameters. Chen et al. used a digital micromirror device to generate PVB with large topological charges. This method expanded the application range of PVB in high-dimensional quantum entanglement, optical transmission and spanning, and material processing. Zhao Jianlin et al. cleverly used the Sagnac interference optical path to generate vector PVB, whose lateral light intensity distribution does not depend on the polarization degree and topological charge value, and the beam has good stability in the longitudinal propagation space. In 2021, Zhou et al. used the dielectric TiO2 nanopillar metasurface to generate PVB with different parameters in different wavelength bands in the visible light range, and could flexibly control the shape of the light intensity distribution of the beam, breaking the limitation that the light intensity distribution is generally circularly symmetric. . In 2019, Forbes et al. proved through theoretical analysis that ideal PVB does not exist, and the influence of topological charges on light intensity distribution cannot be eliminated, and pointed out that the larger the ratio of the radius of the halo to the width of the halo, the closer it is to the ideal PVB.

Figure 1. Experimental light path diagram for producing PVB

As mentioned above, the current research on PVB mainly focuses on the optimization and improvement of the generation method, and there are few reports on its spatial propagation characteristics. For vortex beams, the propagation characteristics of free space are also an important factor affecting its application value. Judgment criteria. Based on this, this paper first studies the influencing factors of the degree of coincidence between the actual PVB and the ideal PVB; then studies the free space propagation characteristics of the actual PVB, and uses the diffraction theory to simulate and analyze in detail the topological charge of the initial surface of the beam, the radius of the halo and The effect of the ring width on its diffraction characteristics is expected to provide a useful theoretical reference for the further application of PVB.

Figure 2. The parameters R=3mm, T=0.04mm remain unchanged, the light intensity and phase distribution of PVB with different topological charge l The closeness not only depends on the value, but also is affected by the topological charge, the larger the topological charge, the stricter the approximation condition. Through the numerical calculation of the diffraction integral, it is found that PVB does not have the non-diffraction characteristics similar to Bessel beams, the width of the halo will expand with the diffraction distance, and as the diffraction distance further increases, the shape of the halo distribution will gradually approach its spatial spectrum form , that is, the Bessel function transformation. In the Fresnel diffraction region, the diffraction effect will be enhanced when the ring radius of the initial surface increases or the ring width decreases, but the effect of the ring width is significantly stronger than that of the ring radius. When the initial surface halo radius and halo width are constant, the widening speed of the halo is almost linear with the propagation distance. At this time, the increase of the topological charge will slightly reduce the widening speed of the halo width with distance, but the effect is very small.