  The use of phase plates in the back focal plane of a microscope is a well
established technique in optical microscopy to increase the contrast of weakly
interacting samples and is gaining interest in electron microscopy as well. In
this paper we study the spiral phase plate (SPP), also called helical, vortex,
or two-dimensional Hilbert phase plate, that adds an angularly dependent phase
of the form $e^{i\ell\phi}$ to the exit wave in Fourier space. In the limit of
large collection angles, we analytically calculate that the average of a pair
of $\ell=\pm1$ SPP images is directly proportional to the gradient squared of
the exit wave, explaining the edge contrast previously seen in optical SPP
work. The difference between a clockwise-anticlockwise pair of SPP images and
conditions where this difference vanishes and the gradient of the exit wave can
be seen from one single SPP image, are discussed. Finally, we demonstrate how
with three images, one without and one with each of an $\ell=\pm1$ SPP, may
give enough information to reconstruct both the amplitude and the phase of the
exit wave. This work provides the theoretical background to interpret images
obtained with a SPP and can help enable new experiments to study for example
magnetic materials in an electron microscope.
