  We present methods for optimizing pupil and focal plane optical elements that
improve the performance of vortex coronagraphs on telescopes with obstructed or
segmented apertures. Phase-only and complex masks are designed for the entrance
pupil, focal plane, and the plane of the Lyot stop. Optimal masks are obtained
using both analytical and numerical methods. The latter makes use of an
iterative error reduction algorithm to calculate "correcting" optics that
mitigate unwanted diffraction from aperture obstructions. We analyze the
achieved performance in terms of starlight suppression, contrast, off-axis
image quality, and chromatic dependence. Manufacturing considerations and
sensitivity to aberrations are also discussed. This work provides a path to
joint optimization of multiple coronagraph planes to maximize sensitivity to
exoplanets and other faint companions.
