Investigation into spiral phase plate contrast in optical and electron microscopy

TL;DR

This paper analyzes the use of spiral phase plates in optical and electron microscopy, providing theoretical insights into image contrast mechanisms and demonstrating phase and amplitude reconstruction from multiple images.

Contribution

It offers a theoretical framework for interpreting spiral phase plate images and introduces a method to reconstruct exit wave phase and amplitude using multiple images.

Findings

Average of ±1 SPP images relates to exit wave gradient squared

Conditions identified where SPP image differences vanish

Method demonstrated for phase and amplitude reconstruction

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 $e^{i\ell\phi}$ 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.