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With a clear circular aperture, the vortex coronagraph perfectly cancels anon-axis point source and offers a 0.9 or 1.75 lambda/D inner working angle fortopological charge 2 or 4, respectively. Current and near-future largetelescopes are on-axis, however, and the diffraction effects of the centralobscuration, and the secondary supports are strong enough to prevent thedetection of companions 1e-3 - 1e-5 as bright as, or fainter than, their hoststar. Recent advances show that a ring apodizer can restore the performance ofthis coronagraph by compensating for the diffraction effects of a circularcentral obscuration in a 1D modeling of the pupil. We extend this work andoptimize apodizers for arbitrary apertures in 2D in order to tackle thediffraction effects of the spiders and other noncircular artefacts in thepupil. We use a numerical optimization scheme to compute hybrid coronagraphdesigns that combine the advantages of the vortex coronagraph (small in IWA)and of shaped pupils coronagraphs (robustness to central obscuration and pupilasymmetric structures). We maximize the apodizer transmission, whileconstraints are set on the extremum values of the electric field that iscomputed in chosen regions of the Lyot plane through closed form expressions.Optimal apodizers are computed for topological charges 2 and 4 vortexcoronagraphs and for telescope apertures with 10-30% central obscurations and0-1% thick spiders. We characterize the impacts of the obscuration ratio andthe thickness of the spiders on the throughput and the IWA for the twotopological charges.

Apodized phase mask coronagraphs for arbitrary apertures. II. Comprehensive review of solutions for the vortex coronagraph