0.5$: the focal plane size needs to be sufficiently large so that light going through the mask can be balanced with light going outside the mask.

The same pupil apodization technique is used in the Apodized Pupil Lyot Coronagraph (APLC) to optimize the pupil entrance complex amplitude to the hard edged opaque focal plane mask [@2003AA...397.1161S]. In the APLC, $t=0$ in equation \[equ:PsiC\], and the coronagraphic extinction is therefore not total for an on-axis point source. Equation \[equ:PsiC\] shows that the on-axis PSF in the final focal plane mask is an exact copy of the non-coronagraphic PSF, scaled by $(1-\Lambda_a)^2$ in intensity. For large focal plane masks diameter $a$, $\Lambda_a$ is close to 1, and the coronagraphic extinction is satisfactory. The APLC concept has been adopted for the Palomar Observatory high contrast imaging program [@2011PASP..123...74H] and the Gemini Planet Imager [@2008SPIE.7015E..31M] and has been validated in laboratory demonstrations [@2011AJ....142..119T]. The APCMLC concept is very similar to the APLC, the only fundamental difference being that its focal plane mask transmission is allowed to be non-zero, therefore allowing full coronagraphic extinction for any focal plane mask size $a$ for which $\Lambda_a > 0.5$.

APCMLC designs for segmented apertures {#ssec:APCMLCdesigns}
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Apodized pupil 