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Spin-controlled vortex generation is a manifestation of the spin-orbit interaction (SOI) of light, which has been
extensively studied in Pancharatnam-Berry geometric phase elements in recent years. The SOI under the normal
incidence of a light beam at a sharp interface, also shown by a spin-dependent vortex, has attracted little attention, except
for a few exceptions. Here, we establish a Fresnel Jones matrix to fully describe the dynamics of beam reflection and
refraction at sharp interfaces under normal incidence. It is pointed out that the vortex phase originates from the
topological structure of the beam itself and is essentially a spin-redirection Berry phase. Although the geometric phase in
Pancharatnam-Berry elements comes from the anisotropy of the external material, which shows a fundamental difference,
they are the same in form. We then give a comparative study of the two kinds of SOI, and reveal the intrinsic connection
and difference between them. Our research not only establishes a unified framework to describe the two SOIs, but also
offers a new perspective for studying the SOIs in other physics.
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