Average intensity and spreading of partially coherent dark hollow beam through the atmospheric turbulence along a slant path

The propagations of a partially coherent dark hollow beam (PC-DHB) through the atmospheric turbulence along a slant path (i.e., inhomogeneous atmospheric turbulence) are investigated. Analytical propagation formulae for the average intensity and spreading of PC-DHB in inhomogeneous turbulence are derived based on the extended Huygens-Fresnel integral. It is shown that the dark area of PC-DHB in inhomogeneous turbulence will drop more rapidly with the increase of propagation distance, and the evolution from a DHB to Gaussian beam becomes quicker for a smaller waist width, lower beam order, smaller inner scale of the turbulence and transverse coherence length, and larger zenith angle. Furthermore, PC-DHB spreads in inhomogeneous turbulence more rapidly than the free space, and the saturation propagation distance (SPD) of relative spatial spreading for uplink slant paths with zenith angles of 45° or less is about 5 km. The relative spreading becomes quicker for the larger waist width and zenith angle, lower beam order, smaller parameter p and inner scale of the turbulence.