Spin-orbital conversion of the light field immediately behind an ideal spherical lens

The Richards-Wolf equations not only adequately describe a light field distribution at the sharp focus, but are also able to describe a light field distribution just behind an ideal spherical lens, i.e. on a converging spherical wavefront. Knowing all projections of light field strength vectors behind the lens, longitudinal components of the spin angular momentum and orbital angular momentum (SAM and OAM) can be derived. In this case, the longitudinal projection of the SAM just behind the lens either remains zero or decreases. This means that the spin-orbital conversion (SOC), where part of the “spin transfers orbit”, occurs just behind the ideal spherical lens. Notably, the sum of the longitudinal projections of SAM and OAM is conserved. Regarding the spin Hall effect, it is revealed that rather than forming just behind the lens, it appears as focusing occurs. Thus, we find that while just behind the lens there is no Hall effect, it becomes maximally pronounced in the focal plane. It is because just behind the ideal spherical lens, two optical vortices with topological charges (TCs) –2 and 2 and opposite-sign spins (with right and left circular polarization) are generated. However, the total spin is equal to zero because the two vortices have the same amplitudes. The amplitudes of the optical vortices become different in the course of focusing and in the focal plane and, therefore, areas with opposite-sign spins (Hall effect) are formed. We also present a general form of the incident light fields whose longitudinal component is zero in the focal plane. In this case, the SAM vector can only have the longitudinal non-zero component. The notion of the SAM vector elongated only along the optical axis in the focal plane is applied for solving magnetization problems.