Longitudinal Spin Hall Effect of Polarization Vortices with a Carrier Frequency

We derive the axial projection of the spin angular momentum (SAM) vector for a vector Gaussian beam with 1D periodic amplitude modulation. Such a beam has zero SAM in the source plane and its polarization is inhomogeneous linear. We demonstrate that when such a beam propagates, its periodic modulation effectively splits the beam into two, having elliptic polarization of opposite handedness. Therefore, the spin Hall effect is shown to arise, since the cross section of the beam contains alternating domains with nonzero spin of opposite sign. For generating such a field, a metasurface can be employed, whose transmittance is changing with one coordinate periodically.