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 φ 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 = ±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.

Theory and applications of free-electron vortex states

“…Lacking such an -varying phase plate one, could resort to only measuring the image intensities with two opposite topological charges: e.g., = ±1. In this case, the difference and sum of the image intensities have been demonstrated to yield [237]:…”

“…17(a). Applying such a spiral phase filter results in an image where the intensity in each point is directly proportional to the weight of the th OAM component measured with respect to the same point [237]. Recording a series of such images ρ with covering a range of values provides a point-by-point OAM spectrum of the conventional image wave ψ 0 .…”

Extension of Friedel's law to vortex-beam diffraction

“…Since their first experimental demonstration in optics [6], they have been studied extensively, both theoretically and experimentally, which has led to numerous applications in fields such as nano-manipulation [7][8][9], astrophysics [10][11][12][13] and telecommunications [14][15][16]. Although slightly more complicated, a vast amount of methods have been developed to produce vortex beams in an electron microscope [17][18][19][20] as well and several studies suggest their use to probe magnetism [18,21], nano-manipulation [22], spin-polarization devices [23] and measuring the chirality of crystals [24]. Additionally, the expression in eq.…”

Near-4D STEM with an Orbital Angular Momentum Sorter: Advantages and Challenges

“…An OAM sorter permits OAM and radial momentum to be measured for each scan position. It can provide edge enhancement, as well as measurements of vorticity and, in principle, full phase information [5]. We have coupled an OAM sorter to a scanning convergent probe and a system of de-scan coils, in order to be able to record position-dependent OAM spectra.…”

“…CNR -Istituto Nanoscienze, Modena, Emilia-Romagna, Italy, 2 Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons, Jülich, Nordrhein-Westfalen, Germany, 3 Università di Modena e Reggio Emilia, Modena, Emilia-Romagna, Italy, 4 CNR-IMM, Bologna, Emilia-Romagna, Italy, 5 ThermoFisher, Eindoven, Noord-Brabant, Netherlands…”