Photonic spin-Hall effect at the strong gain points in parity-time symmetric bilayer systems

Photonic spin-Hall effect (PSHE) at the strong gain points (SGPs) in parity-time (PT) symmetric systems exhibits intriguing features, yet its underlying physics and connection to the PT-symmetric phase are unclear. Here, we clarify this using rigorous full-wave theory calculations. At the SGPs, the significant phase gradient of the s- or p-wave reflected Fresnel coefficients ($r_s$ or $r_p$) causes a large longitudinal shift while suppressing the longitudinal PSHE. In the mixed PT-symmetric phase, the transverse PSHE in the reflected beam is enhanced at the SGPs. However, in the broken PT-symmetric phase, the transverse PSHE is suppressed at the SGPs but enhanced in its vicinity when the SGPs of $r_s$ coincides with that of $r_p$, with no topological phase transition unlike at the Brewster angle. The modulation mechanism is that PT-symmetric phase transitions at the SGP enable control over the intrinsic/extrinsic orbital angular momentum distributions, thereby regulating the transverse PSHE. These findings reveal the underlying mechanism and provide a new perspective for understanding the PSHE at the SGPs.