Apodized phase mask coronagraphs for arbitrary apertures

Context. Phase-mask coronagraphs can be seen as linear systems that spatially redistribute in the pupil plane the energy collected by the telescope. Most of the on-axis light must ideally be rejected outside the aperture so as to be blocked with a Lyot stop, while almost all of the off-axis light must go through it. The unobstructed circular apertures of off-axis telescopes make this possible but all of the major telescopes are on-axis, however, and the performance of these coronagraphs is dramatically reduced by their central obstructions.