Schematic of a vortex coronagraph. The first pupil plane (PP1) is typically an image of the telescope aperture. Lens L1 forms focal plane FP1 with field F(ρ,φ). A phase-only element known as a vortex phase mask (VPM) is located at FP1, with transmission t = exp(ilφ), where l is a nonzero, even integer. Lens L2 forms the output pupil plane (PP2) with field E(r,θ). The on-axis starlight (α = 0, red rays) is diffracted outside of the Lyot stop (LS). Lens L3 forms the subsequent focal plane (FP2) with the on-axis starlight removed, while light from off-axis sources (α ≠ 0, blue rays) propagates through the LS. A Lyot-plane phase mask (LPM) may be introduced at PP2 to improve the contrast between off-axis and on-axis sources in the image at FP2. Lenses L1 and L2 have focal length f.