challenges listed above (circular design problem due to effect of PIAA on aperture shape, and complexity/impossibility of designing PIAA optics for non-circular symmetric remapping), a hybrid PIAACMC design is adopted, which includes a conventional apodizer after the remapping to produce the required prolate function. Thanks to this post-apodizer, the output of the PIAA apodization does not need to exactly match the generalized prolate function, allowing radial remapping functions to be used on non-circular symmetric apertures. The goal of the design optimization is to bring the PIAA apodization and generalized prolate functions close, in order to minimize the strength of the post-apodizer and thus maintain a high system throughput.

The proposed steps for designing a PIAACMC for complex shaped apertures are :

1.  [Choose radial remapping function $r_1=f_b(r_0)$, where $r_0$ and $r_1$ are the radial coordinate in the input (before remapping) and output(after remapping) pupils respectively. For convenience, the remapping function is selected among a pre-computed set of functions used to produce prolate spheroidal apodizations on circular apertures. The focal plane mask diameter corresponding to the prolate spheroidal function is denoted $b$, and the corresponding remapping function and apodization intensity profile are respectively $f_b$ and $I_b$.]{}

2.  [Apply the remapping function 