Structure of transverse spin in focused random light

We analyze the spin structure of weakly and tightly focused random light beams in view of the characteristic decomposition of the $3\ifmmode\times\else\texttimes\fi{}3$ polarization matrix. We take the incident beam to be a mixture of mutually uncorrelated, orthogonally linearly polarized parts which entails zero spin for the beam but leads to a nonzero purely transverse spin in focusing via the generation of a longitudinal electric-field component. We show that the spin ensuing from the so-called discriminating component of the polarization matrix displays a vortex structure for any degree of polarization and level of focusing, while the spin distribution of the pure component may vary from linear to circular as the incident beam changes from fully polarized to completely unpolarized. The spin of the discriminating component is related to the recently introduced concept of polarimetric nonregularity and emerges only when the degree of nonregularity is different from zero. Our results demonstrate the relationship between the rich polarimetric structure of random three-component light and the composition of the optical electric-field spin.