Vortex beam is a beam that carries orbital angular momentum (OAM). The perfect vortex beam (PVB) is a new type of beam that has emerged in recent years. Compared with other traditional vortex beams, the PVB has the property that the radius of the optical ring does not increase with the increase in OAM mode, which has attracted much attention in the field of free-space optical communication. Moreover, the different OAM modes of the vortex beam are orthogonal to each other and can be used to expand the channel capacity of optical communication systems. The OAM dimension of the vortex beam can also be used for signal coding, and since the number of modes of OAM modes is not limited (it can be any integer), it is theoretically possible to carry an infinite amount of bits of information in a single code element. However, vortex beam transmission in the atmosphere will be affected by atmospheric turbulence and produce distortion, and atmospheric turbulence makes its light intensity distribution uneven. Spiral phase distortion can result in the expansion of the spiral spectrum, cause crosstalk between different modes of the vortex beam, and reduce the signal-to-noise ratio of the communication system, thus leading to the degradation of the communication quality in practical applications. In this study, based on the Rytov approximation, the analytical expression of the spiral phase spectrum of the PVB at the receiving aperture is derived, and the probability of detection and crosstalk probabilistic models of the OAM mode of the PVB is established. The effects of different parameters on the PVB in a turbulent atmosphere are analyzed in the context of the light intensity distribution characteristics of the PVB in free space transmission. These results are expected to provide a reference for the application of PVB in free-space optical communication.

In this paper, an analytical expression for the spiral phase spectrum of the PVB is derived theoretically. First, the complex amplitudes of PVBs transmitted in atmospheric turbulence in the weakly turbulent region are obtained using the Rytov approximation based on the optical field distribution of PVBs in the source plane and in free space. Then, in order to describe the OAM mode of the PVB more clearly, the expression of the vortex beam is decomposed into the form of a spiral harmonic function. After that, the non-Kolmogorov probability spectrum is used to describe the effect of atmospheric turbulence on the OAM of the PVB. Then, by using the quadratic approximation of the wave structure function, the analytical expression of the OAM mode probability density of the PVB is obtained. In the next step, the spiral phase spectrum is defined, and the detection probability and the crosstalk probability of the OAM mode of the PVB are modeled. In addition, the effect of each beam element on the beam transmission in atmospheric turbulence and the light intensity characteristics of PVB transmission are analyzed using MATLAB software.

The PVB has the property that the radius of the beam does not increase with the increase in the OAM mode. As the OAM mode at the transmitter changes, the detection probability and crosstalk probability curves corresponding to different initial OAM modes at the transmitter almost coincide when transmitting to the near field, and the difference between the detection probability and crosstalk probability curves corresponding to different initial OAM modes at the transmitter increases significantly when transmitting to the far field (Fig. 3). In addition, when the beam is transmitted to the far field, and the quantum number difference is 1, larger OAM mode at the emission indicates higher crosstalk probability. The crosstalk probability occurs mainly between two neighboring OAM states (Fig. 3). Furthermore, the variation of PVB light intensity with distance in atmospheric turbulence can reveal the evolution of the PVB light field (Fig. 4). In addition, the crosstalk probability curve of PVB has a significant feature in the state with a large refractive index structure constant near the ground: it first increases to a maximum with the transmission distance and then slowly decreases (Fig. 5).

In this paper, an analytical expression for the spiral phase spectrum of a PVB under non-Kolmogorov turbulence is derived theoretically. Theoretical models of OAM modal detection probability and crosstalk probability are developed. The results show that the atmospheric turbulence significantly causes the spiral phase spectrum expansion of the PVB. The detection probability curve of the PVB in the near field hardly varies with the OAM mode at the transmitter, while it varies significantly with the OAM mode at the transmitter when the beam is transmitted to the far field. This is because the PVB transmitted to the far field becomes a Bessel-like beam, and its beam radius varies significantly with the OAM mode at the transmitter. Moreover, a long transmission distance of the PVB beam indicates a more severe negative impact of atmospheric turbulence. The detection probability of the beam after atmospheric turbulent transport decreases as the number of OAM modes at the transmitter, beam radius, refractive index structure constant near the ground, and turbulence coefficient increase. As the beam wavelength increases, the detection probability of the PVB after atmospheric turbulent transport increases. These results provide a certain reference value for the implementation of PVBs in atmospheric turbulence for optical communication links in free space.