The kurtosis parameter of partially coherent controllable dark hollow beams in free space

Based on the extended Huygens–Fresnel principle and the definition of second-order moments of the Wigner distribution function, the analytical expression for the kurtosis parameter of partially coherent controllable dark hollow beams in free space is derived, and used to study the influence of beam parameters on the kurtosis parameter. It is shown that the kurtosis parameter increases with increasing the wavelength and the waist width, or decreasing the scaling factor, however, the effect of the beam order and the spatial correlation length on the kurtosis parameter depends on the propagation distance. The results can be interpreted physically.