Spin-orbit interaction of light in inhomogeneous-anisotropic medium: higher-order effects

Light propagating along an arbitrary path can carry intrinsic spin (SI) and orbital (OI) and extrinsic orbital (OE) angular momenta (AM) associated respectively with its polarization, helical phase structure and the off-axis trajectory. The coupling of any two of the AM leads to first-order effects which includes spin-to-orbital AM conversion, spin-Hall effect (SHE) and orbital-Hall effect (OHE) of light, widely investigated in different systems and configurations. Moving forward, it is essential to study and understand the interaction and influence of any two AM on the third, leading to second-order effects. Here we investigate the influence of SI and OI on OE by measuring the SHE for different off-axis propagation of linearly polarized optical vortex beam through an inhomogeneous, weakly-anisotropic graded-index rod. The spin-Hall shifts are amplified and measured via optical weak measurement and Stokes parameter. The influence of the orbital Berry phase modifies the spin-Hall shift into a general beam shift, which also significantly modifies the output beam's spatial structure and orientation. In addition, the influence of the OHE on the SHE results in asymmetric transverse shift. The appearance of X-type beam shift is attributed to orbital Berry phase and the beam shift curve flips with the sign change of topological charge. Interestingly, the measured shift is asymmetric for +/- charge values and right- / left- circular polarization due to orbital-Berry phase