  Light possesses both spin and orbital angular momentum (AM). While spin AM is
determined by helicity of circular-polarization, orbital AM is characterized by
topological charge of vortex beam. Interaction of AM with optical beam orbit
leads to optical spin Hall or orbital Hall effect, exhibited as spin-dependent
or topological charge-dependent transverse shift of optical beam. Conservation
of AM enables spin-to-orbital AM conversion, where circular-polarized Gaussian
beam is converted to opposite-helicity circular-polarized vortex beam with
topological charge $\pm 2$, an example of controlling spatial beam profiling by
spin flip. However, the resultant vortex beam has the beam center of gravity
unchanged, the same as that of incident Gaussian beam, meaning a null
transverse shift. Here we introduce a cyclic group symmetric metasurface to
demonstrate generation of vortex beam exhibiting spin-dependent transverse
shift, namely, spin- and orbital-Hall effect, attributed to an alteration of
dynamical phase of scattered beam according to the order $n$ of cyclic group
while keeping geometric phase constant. Capability of spin-controlled spatial
beam profiling with a transverse shift via spin- and orbital-Hall effect has
important implications for spatial demultiplexing in optical communication
utilizing orbital AM mode division multiplexing as well as for optical vortex
tweezer and signal processing involving vortex beams.
