Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence

The closed-form expression for the angular spread of Gaussian Schell-model (GSM) array beams propagating through atmospheric turbulence is derived. It is shown that the angular spread theta (sp) of GSM array beams for the superposition of the cross-spectral density function is smaller than of those for the superposition of the intensity. However, the theta (sp) of GSM array beams for the superposition of the intensity is less sensitive to turbulence than that for the superposition of the cross-spectral density function. For the superposition of the cross-spectral density function, theta (sp) of GSM array beams with smaller coherence length sigma (0), smaller waist width w (0), smaller beam number N, and larger separation distance x (d) are less affected by turbulence than of those with larger sigma (0),w (0),N, and smaller x (d) ; while, for the superposition of the intensity, the effect of turbulence on theta (sp) is independent of N and x (d) . In addition, the angular spread is nearly the same for the two types of superposition when sigma (0) or w (0) is small enough, or x (d) is large enough. On the other hand, it is found that there exist equivalent GSM array beams for the two types of superposition which may have the same directionality as the corresponding fully coherent Gaussian beam in free space and also in turbulence.