Investigation into spiral phase plate contrast in optical and electron microscopy

Abstract

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 $e^{i\ell \varphi }$ 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 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 $\ell =\pm 1$ 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.

7 pages, 4 figures

References in corpus (3)

• Using electron vortex beams to determine chirality of crystals in transmission electron microscopy
• Orbital angular momentum in electron diffraction and its use to determine chiral crystal symmetries
• Local orbital angular momentum revealed by spiral phase plate imaging in transmission electron microscopy

Cited by in corpus (5)

• Theory and applications of free-electron vortex states
• Extension of Friedel's law to Vortex Beam Diffraction
• Edge-Enhanced Microscopy of Comlplex Object using Scalar and Vectorial Vortex Filtering
• Focused Ion Beam fabrication of Janus bimetallic cylinders acting as drift~tube Zernike phase plates
• Demonstration of a 2x2 programmable phase plate for electrons