  Quantitative phase imaging has become a topic of considerable interest in the
microscopy community. We have recently described one such technique based on
the use of a partitioned detection aperture, which can be operated in a single
shot with an extended source [Opt. Lett. 37, 4062 (2012)]. We follow up on this
work by providing a rigorous theory of our technique using paraxial wave
optics, where we derive fully three-dimensional spread functions for both phase
and intensity. Using these functions we discuss methods of phase reconstruction
for in- and out-of-focus samples, insensitive to weak attenuations of light.
Our approach provides a strategy for detection-limited lateral resolution with
an extended depth of field, and is applicable to imaging smooth and rough
samples.
