Revisiting physical mechanism of longitudinal spin splittings and Goos-Hänchen shift

Abstract The intrinsic connection between the transverse photonic spin Hall effect (PSHE) and the Imbert-Fedorov shift has been well characterized. However, physical insights into the longitudinal spin splittings associated with the Goos-Hänchen (GH) shift remain elusive. This paper aims to expand on the theory of the PSHE generation mechanism by examining the reflection of each spin component from an arbitrarily linearly polarized incident Gaussian beam on the air-dielectric interface in the longitudinal case. Unlike the transverse case, both spin-maintained and spin-flipped modes exhibit non-zero longitudinal displacements, with the latter being affected by the second-order expansion term derived from the in-plane wave-vector component. The polarization angle plays a crucial role in determining the longitudinal PSHE since each reflected total spin component is a coherent superposition of these two corresponding modes. Remarkably, the imaginary part of the relative permittivity of the dielectric significantly affects the symmetry of the longitudinal PSHE. Furthermore, the GH shift results from a superposition of individual spin states' longitudinal displacements, taking into account their energy weights. By incorporating the corresponding extrinsic orbital angular momentum, this paper explores the physical mechanism of the longitudinal PSHE. This unified physical framework provides a comprehensive understanding of the fundamental origin of spin separations and beam shifts, paving the way for potential applications in spin-controlled nanophotonics.