Beam propagation factor of partially coherent Hermite-Gaussian array beams

The analytical expression for the beam propagation factor (M-2-factor) of partially coherent Hermite-Gaussian (H-G) array beams is derived. It is shown that for the superposition of the intensity the M-2-factor increases monotonically with increasing the beam order, the beam number and the relative beam separation distance, and decreasing the beam coherence parameter. However, for the superposition of the cross-spectral density function there may appear a minimum of the M-2-factor as the beam number or the beam coherence parameter changes. On the other hand, a comparison of the M-2-factor between the two types of superposition is also given. It is found that the M-2-factor for the superposition of the cross-spectral density function may be smaller or larger than that for the superposition of the intensity depending on the beam order and the relative beam separation distance. However, the M-2-factor for the superposition of the cross-spectral density function is always smaller than that for the superposition of the intensity when the beam order is equal to zero or the relative beam separation distance is small enough. In particular, the M-2-factor is nearly the same for the two types of superposition when the beam coherence parameter is small enough or the relative beam separation distance is large enough.