  Second harmonic generation (SHG) microscopy is a valuable tool for optical
microscopy. SHG microscopy is normally performed as a point scanning imaging
method, which lacks phase information and is limited in spatial resolution by
the spatial frequency support of the illumination optics. In addition,
aberrations in the illumination are difficult to remove. We propose and
demonstrate SHG holographic synthetic aperture holographic imaging in both the
forward (transmission) and backward (epi) imaging geometries. By taking a set
of holograms with varying incident angle plane wave illumination, the spatial
frequency support is increased and the input and output pupil phase aberrations
are estimated and corrected -- producing diffraction limited SHG imaging that
combines the spatial frequency support of the input and output optics. The
phase correction algorithm is computationally efficient and robust and can be
applied to any set of measured field imaging data.
