With the advantages of high flexibility, high security, and large communication bandwidths, vortex beams play an important role in many fields, such as quantum entanglement, spatial optical communications, particle manipulation, and optical microscopy. In optical communication applications, the orbital angular momentums (OAMs) of vortex beams can be used as a new encoding method for high-dimensional information encoding. This method can not only achieve mode-division multiplexing and scale the capacity of optical communications but also improve the channel capacity and spectral efficiency of optical communications. It thus provides a potential solution for future high-speed, high-capacity, and high-spectral-efficiency optical communication technologies. This study proposes an encoding method based on OAM and radial modes for composite vortex beams. It uses a 5-bit binary sequence to encode the light intensity distributions of 32 different composite vortex beams generated by the coaxial superposition of two vortex beams. The topological charge and radial index of the incident vortex beam are detected by the proposed gradually-changing-period gratings for decoding purposes. The research results of this study provide a theoretical basis for extending the applications of the OAM modes of vortex beams in the encoding and decoding field.

To generate large topological charges and make demodulation easier, this study proposes an optical communication method and system featuring the encoding of composite vortex beams with spaced OAM modes. Specifically, a Laguerre-Gaussian (LG) vortex beam with fixed OAM and radial modes is coaxially superposed with an LG vortex beam with four radial modes (p=0, 1, 2, 3) and eight equally spaced OAM modes (l=±3, ±5, ±7, ±9) to generate and further encode the light intensity distributions of 32 different composite vortex beams with a 5-bit binary sequence. Then, Eq. (3) is used to convert the 32 composite vortex beams into 32 single LG vortex beams, which will irradiate the proposed gradually-changing-period gratings in the x-axis and y-axis directions. The p and l of the single LG vortex beams can be successfully detected by leveraging the far-field diffraction patterns of the gratings and then be used to derive the composite vortex beams. In this way, the information can be decoded correctly.

The light intensity distributions of the 32 composite vortex beams are shown in Fig. 1. The results reveal multi-ring patterns in the light intensity distributions. The radius of each ring increases as the OAM mode |l₂| rises, and the number of patterns in each ring is |l₂-l₁|. In addition, the number of rings in the light intensity distributions increases with the radial mode p₂, and the number of rings is p=max(p₁, p₂)+1. Figure 2 presents the encoding sequences for the composite vortex beams shown in Fig. 1. According to Fig. 2, the corresponding encoding sequences for the composite vortex beams LG02+LG03–LG06+LG3-9 are 00000-11111. Figure 3 illustrates the light intensity distributions of the 32 single LG vortex beams converted by Eq. (3) from the composite vortex beams shown in Fig. 1. In Fig. 3, the OAM value of a single LG vortex beam is the absolute value of the difference between the OAM values of the two superposed beams, and the radial mode of a single LG vortex beam is the maximum value of the radial modes of the two superposed beams. Figures 4(a) and 4(b) are the proposed gradually-changing-period gratings in the x-axis and y-axis directions, respectively. Figure 4 indicates that the period changes gradually in the x-axis and y-axis directions, respectively. A comparison between Figs. 5 and 7 suggests that when the beam passes through a gradually-changing-period grating, its adjacent far-field diffraction sub-pattern has dark nodal lines. The number of the nodal lines is related to the OAM value of the incident LG vortex beam and satisfies Eq. (6). Moreover, the direction of the nodal lines is determined by whether the OAM value of the incident LG vortex beam is positive or negative. Regarding the x-axis gradually-changing-period grating, the nodal lines of the diffraction pattern at the -1 diffraction order are from upper left to lower right while those of the diffraction pattern at the +1 diffraction order are from lower left to upper right when the l_m of the incident vortex beam is positive. In the case of the y-axis gradually-changing-period grating, the nodal lines of the diffraction pattern at the -1 diffraction order are horizontal while those of the diffraction pattern at the +1 diffraction order are vertical when the l_m of the incident vortex beam is positive. Figures 6 and 8 show the 32 far-field diffraction patterns produced after the 32 single LG vortex beams pass through the x-axis and y-axis gratings, respectively. The results suggest that the far-field diffraction patterns can be used to successfully detect the parameters of the single LG vortex beams for correct decoding without being affected by the increases in the OAM or radial modes.

This study derives the expression of the intensity of each composite vortex beam generated by the coaxial superposition of two LG vortex beams and uses a 5-bit binary sequence to encode the simulated light intensity distributions of 32 different composite vortex beams. The far-field diffraction patterns of the x-axis and y-axis gradually-changing-period gratings designed and proposed in this study can be used to successfully detect the parameters p and l of the single LG vortex beams. The results show that a multi-ring pattern can be observed in the light intensity distributions. The number of rings is p=max(p₁, p₂)+1, and the number of patterns in each ring is |l₂-l₁|. In addition, the proposed x-axis and y-axis gradually-changing-period gratings can be utilized to successfully detect the parameters of the incident beams for correct decoding without being affected by the radius of each ring or the number of rings in the light intensity distributions.