A decomposition for transverse spins in structured vector fields

Classical vector waves can possess dedicated spin angular momenta (SAM), which are \emph{perpendicular} to the propagation direction, as surprisingly revealed by the recent recognition of transverse SAM in electromagnetic (EM) fields. In this paper, we adopt the Hertz potential method to define structured vector fields and derive analytically the SAM of the wave fields in closed form. Our calculations not only confirm that transverse SAM may originate from the first-order spatial inhomogeneity of the momentum of EM waves, but also point out that for \emph{non-planar vector waves with near fields}, an extraordinary spin appears as a distinct part out of transverse spin. We further demonstrate that the proposed transverse spins prevail universally in both propagating and evanescent waves. This work renews our fundamental understanding of the decomposition of SAM for classical vector waves.