Induced Berry Connection and Photonic Spin Hall Effect in Optical Dirac Theory

Within the framework of optical Dirac theory, we present a field-theoretical model of spin-orbit interaction and photonic spin/orbit Hall effects. Our approach reformulates light propagation along helical paths as solving the Maxwell equations in a ray-based curvilinear coordinate system. This system can alternatively be interpreted as a spin-degenerate medium with antisymmetric elements in the dielectric tensor, corresponding to the spin-1 excitation mode characterized by a rotational dipole vector. We show that, at leading order, the resulting effective Hamiltonian is equivalent to the Maxwell theory in a uniformly rotating frame, incorporating both spin-rotation and orbit-rotation coupling. This rotation arises from the torsion of helical paths, manifesting as extrinsic orbital angular momentum (EOAM) of photons. Notably, the spin angular momentum (SAM) of the photon and its intrinsic orbital angular momentum (IOAM) contribute jointly to the geometric phase. In the Heisenberg picture, the spin and orbital Hall effects naturally emerge from interaction terms in the ray equations. Furthermore, we find that the transverse spin of evanescent waves couples with EOAM, revealing that the geometric phase of elliptically polarized light differs from that of circularly polarized light. This distinction underscores the role of spin-orbit coupling in modifying phase accumulation.