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, which adds an angularly dependent phase of the form $e^{i\ell {\varphi }_k}$ to the exit wave in Fourier space. In the limit of large collection angles, we analytically calculate that the average of a pair of $\ell =\pm 1$ SPP filtered images is directly proportional to the gradient squared of the exit wave, explaining the edge contrast previously seen in optical SPP work. We discuss the difference between a clockwise-anticlockwise pair of SPP filtered images and derive conditions under which the modulus of the wave's gradient can be seen directly from one SPP filtered image. This work provides the theoretical background to interpret images obtained with a SPP, thereby opening new perspectives for new experiments to study, for example, magnetic materials in an electron microscope.

• Received 25 May 2016

DOI:https://doi.org/10.1103/PhysRevA.94.023838