Small angle expansion, a solution to the phase-retrieval problem using generalised phase diversity in pupil space

Phase diversity is a phase-retrieval algorithm that uses a pair of defocused intensity images taken symmetrically about the wavefront to be determined. Generalised phase diversity is a phase-retrieval algorithm that uses diversity functions other than defocus. The approach adopted assumes that unknown phase changes satisfy the small-angle approximation over spatial regions that can be selected by choice of the diversity function. For smooth functions, and for discontinuous functions with only small discontinuities, this leads to a very simple analytic solution. Computer simulations were used to validate this method for the retrieved phase.