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The analytical expressions of hypergeometric-Gaussian type-II (HyGG-II) beams propagating in the quadratic-index medium are deduced based on the ABCD optical transformation matrix method. The propagation properties of HyGG-II beams in the quadratic-index medium are numerically and theoretically analyzed in detail. The intensity distribution and the second-order moment-based beam width vary periodically when the HyGG-II beams pass through the quadratic-index medium. The interactions of two HyGG-II beams are also discussed graphically.

Based on the generalized diffraction integral formula, analytical propagation expressions for a partially coherent anomalous hollow beam (AHB) propagating through a quadratic-index medium are derived. The distributions of intensity and Poynting vector for a partially coherent AHB through the quadratic-index medium are numerically and theoretically studied. It is found that the intensity and Poynting vector of the partially coherent AHB take on a periodical change. The evolution properties of the partially coherent AHB in a quadratic-index medium are closely relevant to its transverse coherence width and the quadratic-index parameter of media.

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