equal in amplitude for every point within the pupil. Since this match does not naturally occur, all three concepts (APLC, Roddier phase mask coronagraph and APCMLC) require the entrance pupil to be amplitude apodized to enforce this match. Qualitatively, for small focal plane mask size, the apodization mostly changes the pupil light distribution for the “outside focal plane mask” component, while the light distribution for the “inside focal plane mask” component is mostly driven by the size of the focal plane mask. The entrance pupil apodization can therefore be iteratively derived to force the “outside focal plane mask” component to match the “inside focal plane mask component”, using the following steps:

1.  [Adopt a focal plane mask diameter $a$]{}

2.  [Compute the on-axis complex amplitude PSF for the aperture. This is the Fourier transform of the pupil complex amplitude P]{}

3.  [Clip the PSF: values outside the focal plane mask radius are forced to zero]{}

4.  [Inverse-Fourier transform the clipped PSF, and adopt this function as the apodized pupil plane amplitude function A, after multiplication by a factor $\Lambda_a$ so that its maximum value across the pupil is be equal to 1 (full transmission)]{}

5.  [Return to step (2), with 