Fig. 1

Component parts of light’s total spin angular momentum density visualised at a single point (red circle) in a 3D electromagnetic field. The field is generated by interfering randomly polarised plane waves with random \(\mathbf{k}\) vectors, bearing minimal spatial symmetry and allowing the spin decomposition to be depicted in its most generic form. An \({x\text{'}y\text{'}}\) cut plane of the field is provided, on which two elements are plotted: the field’s non-uniform energy density, indicated by the colour, and streamlines of the Poynting vector’s \({x\text{'}y\text{'}}\) projection. The co-ordinate axes \({x\text{'}}\), \({y\text{'}}\) and \({z\text{'}}\) are specially chosen so that \({z\text{'}}\) aligns with the Poynting spin vector, purely to allow the curling of the Poynting vector streamlines to be visualised on the \({x\text{'}y\text{'}}\) plane. The blue and green ellipses represent the electric and magnetic polarisation ellipses respectively, and are oriented freely in 3D (partially submerged by the cut plane as a 3D visual aid). Total spin (red arrow) is equal to the sum of the electric \({\mathbf{S}}_{\text{e}}\) and magnetic \({\mathbf{S}}_{\text{m}}\) spins and the sum of canonical spin (yellow arrow) and Poynting spin (black arrow) which have no orientation restrictions and all point in different 3D directions. Only \({\mathbf{S}}_{\text{e}}\) and \({\mathbf{S}}_{\text{m}}\) hold a geometric significance in relation to polarisation by being orthogonal to the plane of the \(\mathbf{E}\) and \(\mathbf{H}\) ellipses (according to the right-hand rule) as illustrated separately below the main combined diagram. Dotted lines within the ellipses represent the ellipse major and minor axes (note that the ellipses are distorted as the 3D field is projected into a 2D image)