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A partially coherent off-axis double vortex beam (PCOADVB) composed of two off-axis vortices is theoretically presented. The analytical equations of a PCOADVB in oceanic turbulence are presented, and the intensity profiles and the number of coherence vortices of the PCOADVBs are investigated based on the derived expressions. The numerical results show that the intensity profiles of PCOADVBs are determined by the initial topological charges M1 and M2 and the positions of the off-axis vortices (x1d,y1d) and (x2d,y2d). The intensity profiles of PCOADVBs will lose the off-axis ring intensity profile and acquire a Gaussian-like profile as z increases, and stronger oceanic turbulence and a smaller σ can help the PCOADVB evolve into a spot with a Gaussian-like profile faster on propagation. The number of coherence vortices of a PCOADVB in oceanic turbulence will increase on propagation. The PCOADVB may have potential applications in underwater laser sensing and wireless communications.

The descriptions of a radially phased-locked Hermite–Gaussian correlated beam array are introduced, the equation of this beam array in oceanic turbulence is derived, and the intensity profiles of this beam array are shown and analyzed. The results imply that the evolutions of the sub-beam of this beam array in free space are the same as the Hermite–Gaussian correlated beam, while the intensity of this beam array can be adjusted by controlling the initial beam radius R and the coherence length. The intensity profiles of this beam array in free space have multiple spots during propagation, while the same beam array in oceanic turbulence can become a beam spot due to the influences of R and oceanic turbulence. The beam array with smaller coherence length in oceanic turbulence retains the splitting properties better during propagation.

The theoretical descriptions for a radial phase-locked multi-Gaussian Schell-model vortex (RPLMGSMV) beam array is first given. The normalized intensity and coherence distributions of a RPLMGSMV beam array propagating in free space and atmospheric turbulence are illustrated and analyzed. The results show that a RPLMGSMV beam array with larger total number N or smaller coherence length σ can evolve into a beam with better flatness when the beam array translating into the flat-topped profile at longer distance z and the flatness of the flat-topped intensity distribution can be destroyed by the atmospheric turbulence at longer distance z. The coherence distribution of a RPLMGSMV beam array in atmospheric turbulence at the longer distance will have Gaussian distribution. The research results will be useful in free space optical communication using a RPLMGSMV beam array.