Near-field and far-field spreading of partially coherent annular beams propagating through atmospheric turbulence

Based on the extended Huygens–Fresnel principle, closed-form expressions for the Rayleigh range and the far-field divergence angle of partially coherent annular beams propagating through atmospheric turbulence are derived by using the Wigner distribution function (WDF). Taking the Rayleigh range and the far-field divergence angle as the characteristic parameters of near-field and far-field spreading, respectively, the spreading of partially coherent annular beams both in free space and in turbulence is studied in detail. It is found that the effect of the strength of turbulence and beam parameters (e.g., the spatial correlation length, the waist width, and the wave length) on the beam spreading in the near field is in agreement with that in the far field. However, in turbulence, the effect of the obscure ratio of annular beams on the spreading is different between in the near field and in the far field. Namely, in turbulence the beam spreading in the near field becomes smaller and the beam spreading in the far field becomes larger as the obscure ratio increases. In particular, the effect of turbulence on the Rayleigh range and the far-field divergence angle is nearly unchanged versus the obscure ratio when the spatial correlation length is small. The main results obtained in this paper are explained physically.