Lyot coronagraphs diffract light within the geometrical aperture, making it impossible to achieve very high contrast on segmented apertures. Moderate-contrast band-limited Lyot coronagraphs have been designed for the NIRCAM instrument [@2009SPIE.7440E..28K] on the James Webb Space Telescope, but the aggressive Lyot stop designs, which remove much of the residual diffraction from the secondary structure and the segmented primary, come at the cost of significant companion throughput.]{}

-   [[**Phase mask coronagraphs.**]{} Coronagraphs using phase focal plane masks are also affected by central obscuration and spiders/gaps. [@2003SPIE.4860..171L] showed that the 4-quadrant phase mask can only achieve full coronagraphic suppression on unobstructed pupils free of gaps or spiders, as any obscuration diffracts light outward in the Lyot plane. The optical vortex coronagraph is similarly affected by obscurations, although [@2011OptL...36.1506M] showed that central obstruction can be mitigated by a dual-stage approach, where the second stage rejects most of the light diffracted by the central obstruction. For both the vortex and the 4 quadrant phase mask coronagraphs, no solution has been found to eliminate the light diffracted by spiders and gaps. ]{}

-   [[**Shaped Apertures.**]{} For moderate contrast level and relatively large IWA, shaped apertures can be designed for centrally obscured and segmented pupils. [@2006PASJ...58..627T] 