In a recent publication [Opt. Lett.42, 1512 (2017)OPLEDP0146-959210.1364/OL.42.001512], a novel class of partially coherent sources with circular coherence was introduced. In this paper, we examine the propagation behavior of the spectral density and the spectral degree of spatial coherence of a beam generated by such a source in free space and in oceanic turbulent media. It is found that the beam exhibits self-focusing, which is dependent on the initial coherence and the parameters of oceanic turbulence. The self-focusing phenomenon disappears when the initial coherence is high enough or the oceanic turbulence is strong. The area of high coherence appears in the center and along two diagonal lines. With increasing turbulence, the coherence area reduces gradually along one diagonal line and is retained along the other one. A physical interpretation of the self-focusing phenomenon is presented, and potential applications in optical underwater communication and beam shaping are considered.
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