Generalized vectorial Rayleigh-Sommerfeld diffraction integrals are developed for the cross-spectral-density matrices of spatially partially coherent beams. Using the Gaussian Schell-model (GSM) beam as an example, we derive the expressions for the propagation of cross-spectral-density matrices and intensity of partially coherent vectorial nonparaxial beams, and the corresponding far-field asymptotic forms, beyond the paraxial approximation. The propagation of the vectorial nonparaxial GSM beams are evaluated and analyzed. It is shown that a 3×3 cross-spectral-density matrix or a vector theory is required for the exact description of nonparaxial GSM beams.
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