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A set of pupil apodization functions for use with a vortex coronagraph ontelescopes with obscured apertures is presented. We show analytically thatpupil amplitudes given by real-valued Zernike polynomials offer ideal on-axisstarlight cancellation when applied to unobscured circular apertures. Thecharge of the vortex phase element must be a nonzero even integer, greater thanthe sum of the degree and the absolute value of its azimuthal order of theZernike polynomial. Zero-valued lines and points of Zernike polynomials, orlinear combinations thereof, can be matched to obstructions in the pupils ofground-based telescopes to improve the contrast achieved by a vortexcoronagraph. This approach works well in the presence of a central obscurationand radial support structures. We analyze the contrast, off-axis throughput,and post-coronagraph point spread functions of an apodized vortex coronagraphdesigned for the European Extremely Large Telescope (E-ELT). This techniqueoffers very good performance on apertures with large obscuring supportstructures similar to those on future 30-40m class ground-based telescopes.

Zernike amplitude pupil apodization for vortex coronagraphy with obscured apertures