  The vortex coronagraph is an optical instrument that precisely removes
on-axis starlight allowing for high contrast imaging at small angular
separation from the star, thereby providing a crucial capability for direct
detection and characterization of exoplanets and circumstellar disks.
Telescopes with aperture obstructions, such as secondary mirrors and spider
support structures, require advanced coronagraph designs to provide adequate
starlight suppression. We introduce a phase-only Lyot-plane optic to the vortex
coronagraph that offers improved contrast performance on telescopes with
complicated apertures. Potential solutions for the European Extremely Large
Telescope (E-ELT) are described and compared. Adding a Lyot-plane phase mask
relocates residual starlight away from a region of the image plane thereby
reducing stellar noise and improving sensitivity to off-axis companions. The
phase mask is calculated using an iterative phase retrieval algorithm.
Numerically, we achieve a contrast on the order of $10^{-6}$ for a companion
with angular displacement as small as $4~\lambda/D$ with an E-ELT type
aperture. Even in the presence of aberrations, improved performance is expected
compared to either a conventional vortex coronagraph or optimized pupil plane
phase element alone.
