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Theoretical study on the spectral shifts of partially coherent array beams passing through ABCD optical systems

According to the extended Huygens–Fresnel integral, the expressions for the on-axis spectrum of partially coherent Gaussian Schell-model (GSM) rectangular array beams passing through ABCD optical systems have been derived. The generalized Fresnel number of the system, the spatial coherent parameter of array beamlets and the array beam parameters including the number of beamlets and the separation distance between beamlets have been taken as the characteristic parameters to compare the spectral shifts of GSM array beams for the two types of the superposition, i.e. the correlated superposition and the uncorrelated superposition. In particular, the effect of characteristic parameters on the on-axis relative spectral shifts has been discussed in detail. The results show that the spectral intensity of GSM array beams for the two types of the superposition passing through ABCD systems depends on the source spectral density S 0(ω), the spatial coherent parameter of array beamlets β, the generalized Fresnel number of the system F and the array beam parameters. Furthermore, for the uncorrelated superposition, the spectrum of GSM array beams only exhibit the blue-shift, whereas for the correlated superposition, GSM array beams exhibit the spectral switch and the number of spectral switches increases with the increase of array beam parameters in the near field due to the interference between beamlets. In particular, the effect of the array beam parameters on the on-axis relative spectral shift is more obvious.