Range of turbulence-independent propagation and Rayleigh range of partially coherent beams in atmospheric turbulence.

Two characteristic distances for partially coherent beams propagating in atmospheric turbulence have been proposed. The turbulent Rayleigh range is used for characterizing the range over which the beams propagate in turbulence without spreading appreciably; i.e., the concept of the well-known Rayleigh range in free space is extended to the case of turbulence. In this paper the range of turbulence-independent propagation of the beams, in contrast to similar characteristic distances in previous published works, is based on the formula of the beam propagation factor (M(2) factor) and is used for describing the range over which the spatial and angular spreading and the M(2) factor increase due to turbulence are sufficiently small and negligible. Several simple formulas used for calculating the approximate values of these distances are given, and the formulas are applied to Gaussian Schell-model (GSM) beams and illustrated by examples. Furthermore, as a typical example, the effect of the angular spread of GSM beams in turbulence on a thin-lens optical system is also discussed. We show that the turbulent Rayleigh range depends on the Rayleigh range in free space, the waist width, and the spatial power spectrum of the refractive-index fluctuations of the turbulent atmosphere, and that the range of turbulence-independent propagation depends on the waist width, the initial angular spread in the waist plane, and the spatial power spectrum.