  Cylindrical vector (CV) beams are a set of transverse spatial modes that
exhibit a cylindrically symmetric intensity profile and a variable polarization
about the beam axis. They are composed of a non-separable superposition of
orbital and spin angular momentum. Critically, CV beams are also the eigenmodes
of optical fiber and, as such, are of wide-spread practical importance in
photonics and have the potential to increase communications bandwidth through
spatial multiplexing. Here, we derive the coupled amplitude equations that
describe the four-wave mixing (FWM) of CV beams in optical fibers. These
equations allow us to determine the selection rules that govern the
interconversion of CV modes in FWM processes. With these selection rules, we
show that FWM conserves the total angular momentum, the sum of orbital and spin
angular momentum, in the conversion of two input photons to two output photons.
When applied to spontaneous four-wave mixing, the selection rules show that
photon pairs can be generated in CV modes directly and can be entangled in
those modes. Such quantum states of light in CV modes could benefit
technologies such as quantum key distribution with satellites.
