Fig. 2

Spin decomposition of non-paraxial beams. The beams, each of waist \(1.5\lambda\) and separated in columns, are (a) a linearly polarised Gaussian beam (\({{\mathbf{E}}}^{T}{||}\hat{{\mathbf{x}}}\)), (b) an azimuthally polarised (\({\mathbf{E}}{||}\hat{{\boldsymbol{\phi}}}\)), \(l=0\) doughnut beam, and (c) a linearly polarised (\({{\mathbf{E}}}^{T}{||}\hat{{\mathbf{x}}}\)) vortex beam with topological charge \(l=1\). The top row of (isometric) plots across each sub figure column shows the beam energy density in colour, as well as the Poynting vector and electric (blue) and magnetic (green) polarisation ellipses, which are elliptical due to a significant \(z\) field component. Subsequent rows are vector plots of each beam’s Poynting vector \(\mathbf{P}\), total spin \(\mathbf{S}\) Eq. (3), canonical spin \({\mathbf{s}}_{\text{c}}\) and Poynting spin \({\mathbf{s}}_{\text{p}}\) from Eq. (4), respectively. White arrows are projections of the corresponding vector into the \({xy}\) plane, while the red arrows are projections of the vector onto longitudinal \({yz}\) and \({xz}\) cut planes. Within each beam, arrows in the three spin decomposition plots are drawn to a consistent scale. Each non-paraxial beam is generated using an angular spectrum integration method⁴⁶. Defining the 3D vector vortex beam (c) is a difficult problem and the method we used produces a small and physical longitudinal spin (third row of c), contributed by the magnetic field, which would not be present in a (non-physical) perfect paraxial beam