Spectral shifts of partially coherent radial array beams passing through ABCD optical systems

The expressions for the spectral intensity of partially coherent Gaussian Schell-model (GSM) radial array beams for both the correlated and uncorrelated superpositions passing through ABCD optical systems have been derived by using the extended Huygens-Fresnel diffraction integral. The effects of the normalized radius R, the number of beamlets N, the spatial coherent parameter of array beamlets β and the optical system parameters on the on-axis and off-axis relative spectral shifts for the two types of superposition have been discussed in detail. The results show that for the correlated superposition, the on-axis spectral intensity in free space and the off-axis spectral intensity on the geometrical focal plane depends on the source spectral density S0(ω), the spatial coherent parameter of array beamlets β, the generalized Fresnel number of the system F, the normalized radius R and the number of beamlets N, whereas the spectral intensity for the uncorrelated superposition is independent of the number of beamlets N. Furthermore, as for on the actual focal plane, the off-axis spectral intensity for the two types of superposition is closely related to N.