Fig. 1. Schematics and principle.

a Schematic of the WS₂ monolayer and optical selection rules at the K and K’ valleys. σ₊ (σ₋) excitation corresponds to interband optical transition at the K (K’) valley. b The normalized Poincaré sphere. Different azimuthal positions on the equator correspond to different linearly polarized states. Two poles correspond to two types of circularly polarized states. The centre of the Poincaré sphere corresponds to vortex singularity. c PhC slabs with C₄ symmetry and without in-plane inversion symmetry. d Illustration of the photoluminescence of the WS₂ monolayer on the PhC slab without in-plane inversion symmetry. Valley photons with different chiralities separate and radiate directionally in the far field