focal plane mask size is increased - while mitigation of undesired chromatic effects at the focal plane mask may require a larger and more opaque mask.

Phase-induced Amplitude Apodization (PIAA) uses aspheric mirrors to achieve a lossless beam apodization [@2003AA...404..379G], and can therefore produce a highly apodized beam suitable for high contrast imaging without the angular resolution loss and throughput loss of a conventional apodizer. PIAA can also be used to replace the entrance apodization in the APCMLC design described in Section \[sec:APCMLC\], as previously proposed for unobstructed circular pupils [@2010ApJS..190..220G]. The resulting coronagraph, denoted Phase-induced Amplitude Apodization Complex mask coronagraph (PIAACMC), offers simultaneously full throughput, small inner working angle and total on-axis extinction.

An example PIAACMC design is shown in Figure \[fig:PIAACMCprinc\] for a segmented centrally obscured pupil. The entrance pupil P (image shown in the lower left of the figure) is apodized with lossless aspheric PIAA optics. Because the PIAA optics perform apodization by remapping instead of selective transmission, the resulting pupil P1 shape is modified. A conventional apodizing mask may be used to fine-tune the apodization if the PIAA optics do not exactly produce the required amplitude distribution (this will be addressed in the following section). The 