A decomposition for transverse spins in structured vector fields

Classical vector waves can possess intricate spin angular momenta (SAM), which are \emph{perpendicular} to the propagation direction, as revealed by the recent recognition of surprisingly transverse SAM in electromagnetic (EM) fields. In this paper, we employ the Hertz potential method to define structured vector fields and analytically decompose the SAM of the wave fields in two parts. Our novel approach of decomposition not only confirms that transverse SAM may originate from the first-order spatial inhomogeneity of the Poynting momentum, but also points out that for \emph{non-planar vector waves with near fields}, an extraordinary spin appears as a distinct part out of transverse spin. By four examples of vector beams, 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.