Fig. 2: Control of the 3D SAM vector in tightly focused fields.

a Controlling the direction of the SAM vector (\(\overset{\rightharpoonup }{{{\bf{S}}}_{{{\rm{all}}}}^{{\prime} }}\), yellow arrow) by tailoring the azimuthal distribution of the incident polarization helicity, \({S}_{\phi }\left(\phi \right)\), as indicated by Eq. (5). Focusing with a lens rotates the wavevector (\({\overset{\rightharpoonup}{{{\bf{k}}}}}\), red arrows) of a collimated beam (locally treated as geometrical rays) and yields a conical \(\overset{\rightharpoonup}{{{\bf{k}}}^{\prime}}\)-distribution in the focused field. The local spins (\(\overset{\rightharpoonup}{{{\bf{S}}}^{\prime}}\), purple arrows) that are attached to each \(\overset{\rightharpoonup}{{{\bf{k}}}^{\prime}}\) also rotates. As such, the total SAM vector will be the vector sum of all the local spins. Thus, the total SAM vector is determined by the transverse local polarization helicity of the incident field. (b–e) Typical 3D SAM vectors obtained by tailoring the designed polarization helicity. The desired 3D SAM vectors are indicated by the yellow arrows, and the local spins for \(\overset{\rightharpoonup}{{{\bf{S}}}^{\prime}_{{{\bf{x}}}}}\), \(\overset{\rightharpoonup}{{{\bf{S}}}^{\prime}_{{{\bf{y}}}}}\), \(\overset{\rightharpoonup}{{{\bf{S}}}^{\prime}_{{{\bf{z}}}}}\), and \({\overset{\rightharpoonup}{{{{\bf{S}}}^{\prime}}}}({60}^{\circ },\,{135}^{\circ })\) are represented by the purple, blue, green, and red vectors, respectively.