Torsional selection rule for the spin--orbit conversion of light

Abstract

Standard Pancharatnam-Berry and linear-birefringent media convert optical spin into orbital angular momentum (OAM) through an anisotropy \emph{director}, a rank-two, headless field, and therefore obey the selection rule $Δ\ell=2q$ per unit texture charge $q$. We show that a medium with geometric \emph{torsion}, the continuum limit of a screw-dislocation array, can convert spin to OAM through the \emph{contortion} of its material connection, which enters the effective paraxial dynamics as a rank-one vector field. The resulting selection rule is $Δ\ell=q$. Its winding is fixed by geometry and symmetry, not by a Pancharatnam--Berry director, and the process conserves the screw charge $\tilde J_z=L_z+(q/2)σ_z$ while exchanging $(2-q)\hbar$ of angular momentum per converted photon with the defect lattice. Paraxial simulations confirm the rule: a circular Gaussian input develops a stable, topologically quantized $\ell=+q$ vortex in the reversed helicity, with $83\%$ conversion over three Rayleigh ranges and no fine-tuning. We propose a polarization-resolved photonic-lattice discriminator in which the slope of the measured OAM versus the independently written texture charge, one for torsion, two for birefringence, separates the two mechanisms.