Spectral shift of partially coherent rectangular array beams in free space

According to the extended Huygens-Fresnel integral, propagation formulas of the spectrum of partially coherent Gaussian-Schell model(GSM) rectangular array beams have been derived. The changes of on-axis and off-axis spectral shifts in free space have been analyzed quantitatively. The effects of array beam parameters and the spatial coherent parameter of array beamlets on the spectral shifts have been discussed in detail. The results show that the spectral shifts of GSM array beams in free space depend on the source spectral density, the spatial coherent parameter of array beamlets, as well as the array beam parameters, i.e., the number and the separation of beamlets. Especially, the effect of the array beam parameters on the relative spectral shifts is more obvious.