  We develop for the first time a mathematical framework in which the class of
projection algorithms can be applied to high numerical aperture (NA) phase
retrieval. Within this framework, we first analyze the basic steps of solving
the high-NA phase retrieval problem by projection algorithms and establish the
closed forms of all the relevant prox-operators. We then study the geometry of
the high-NA phase retrieval problem and the obtained results are subsequently
used to establish convergence criteria of projection algorithms in the presence
of noise. Making use of the vectorial point-spread-function (PSF) is, on the
one hand, the key difference between this paper and the literature of phase
retrieval mathematics which deals with the scalar PSF. The results of this
paper, on the other hand, can be viewed as extensions of those concerning
projection methods for low-NA phase retrieval. Importantly, the improved
performance of projection methods over the other classes of phase retrieval
algorithms in the low-NA setting now also becomes applicable to the high-NA
case. This is demonstrated by the accompanying numerical results which show
that available solution approaches for high-NA phase retrieval are outperformed
by projection methods.
