  With a clear circular aperture, the vortex coronagraph perfectly cancels an
on-axis point source and offers a 0.9 or 1.75 lambda/D inner working angle for
topological charge 2 or 4, respectively. Current and near-future large
telescopes are on-axis, however, and the diffraction effects of the central
obscuration, and the secondary supports are strong enough to prevent the
detection of companions 1e-3 - 1e-5 as bright as, or fainter than, their host
star. Recent advances show that a ring apodizer can restore the performance of
this coronagraph by compensating for the diffraction effects of a circular
central obscuration in a 1D modeling of the pupil. We extend this work and
optimize apodizers for arbitrary apertures in 2D in order to tackle the
diffraction effects of the spiders and other noncircular artefacts in the
pupil. We use a numerical optimization scheme to compute hybrid coronagraph
designs that combine the advantages of the vortex coronagraph (small in IWA)
and of shaped pupils coronagraphs (robustness to central obscuration and pupil
asymmetric structures). We maximize the apodizer transmission, while
constraints are set on the extremum values of the electric field that is
computed in chosen regions of the Lyot plane through closed form expressions.
Optimal apodizers are computed for topological charges 2 and 4 vortex
coronagraphs and for telescope apertures with 10-30% central obscurations and
0-1% thick spiders. We characterize the impacts of the obscuration ratio and
the thickness of the spiders on the throughput and the IWA for the two
topological charges.
