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.
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