Shaping the electron wavefronts in (scanning) transmission electron microscopy

Dr Laura Clark1
1. School of Physics and Astronomy, Monash University, Australia
Laura Clark works at the School of Physics and Astronomy at Monash University. Since her first foray into electron microscopy in a BSc project entitled “Electron beams with a twist” at the University of York, she has been developing new imaging techniques for TEM – firstly during her PhD in the EMAT lab at the University of Antwerp (awarded 2016) and now as a postdoctoral researcher at Monash. Following 6 years of electron vortices, she is now working on quantitative differential phase contrast STEM. Her work is based on advanced optical modelling of (S)TEM imaging – and seeing what happens when the usual limits are broken.
The shape and intensity profile of the electron beam are critical in determining the final image quality that can be achieved in (scanning) transmission electron microscopy. Classic imaging theory would have us believe that optimal transmission electron microscopy images are formed with a wavefront that is as close to planar as possible, and that perfect scanning transmission electron microscopy images are made with probes formed from crisp circular apertures, with no aberrations. Neither of these need be strictly true – and as has been emphasised by developments in recent years with demonstrations of vortex beams, phase plates and apodisation – perhaps breaking free from these simple imaging regimes will allow for images with better contrast, resolution and use of electron dose. Perhaps we can form (scanning) transmission electron microscopy images in ways that are both more effective, and more efficient.
Corresponding author
A workman is only as good as his tools –and an electron microscope is only as good as the electron beam it can form.  The fundamental goal of imaging tools is to obtain information about a specimen and transfer this information to a detector of some form. In (scanning) transmission electron microscopy, the electron beam is the information-carrier, and the imaging mode we select determines what information we can collect. Diffraction patterns reveal crystal structure, while bright-field transmission electron microscopy displays the direct image of the specimen in real space (without spatial frequency information).
From this perspective, it then seems natural to survey what imaging modes can be created within a typical transmission electron microscope, and thus what specimen information we can potentially be sensitive to when different optical arrangements are implemented. This idea is at the heart of the study of “beam shaping”– the development of different methods to shape the electron beam (in phase and/or amplitude) in order to obtain new specimen information. In this article I hope to give an overview of the many fields in which these methods can be useful. 
Beam shaping includes the use of carefully designed phase plates (whether a physical object, shifting the phase of the beam using the mean inner potential of the material, or controlled electric or magnetic fields used to tune the wavefront) and unusual aperture geometries (such as the forked masks used to make vortex beams, described herein). These change how the electron beam moves through the microscope, and ultimately determines what specimen information can be collected by the detector at the far end.
The need for careful tuning of the wavefront in transmission electron microscopes has been known for many decades – this requirement was at the heart of the push for the development of aberration correctors. However, the breadth of optical flexibility available on a normal scanning transmission electron microscopy instrument for wavefront shaping was not widely appreciated until the recent experimental demonstration of electron vortex beams.1
Vortex beams are a rather odd family of wavefronts. Rather than the planar wavefronts of transmission electron microscopy, or the parabolic wavefronts of a focussed scanning transmission electron microscope probe, vortex wavefronts form long twisted spirals in 3D, very much like nanoscale fusilli pasta. These beams rotate around their central axis as they propagate – and this imbues the beams with the interesting property of quantised orbital angular momentum.
While lenses in standard configurations can warp and bend the electron wavefront, much like stretching a rubber sheet, one cannot transform from a normal scanning transmission electron microscope probe to a vortex scanning transmission electron microscopy probe without cutting and restitching the wavefronts. This is because vortex beams have a different ‘topology’ to planar wavefronts – and the vortex core is ‘topologically protected’.
With special apertures (an example of which is illustrated in figure 1) or phase plates with a spiralling thickness profile, it is possible to cut and restitch the wavefronts in order to generate vortex beams. The fork aperture shown in Figure 1 is one of the simplest methods to implement experimentally. If this shape is cut out using a focussed ion beam (with a diameter of a few tens of micrometres) and put in the condenser aperture plane –a set of vortex probes will be formed with doughnut intensity profiles,  in place of the usual probe intensity profile.

Figure 1
.Example of a “vortex forked mask” - an aperture design able to produce electron vortex beams
The orbital angular momentum possessed by vortex beams is one of the major reasons for their popularity – an atomically sized electron vortex probe ought to couple well to magnetic states within a crystal.1 There is hope that this behaviour can be controlled in order to allow for a high-resolution magnetically-sensitive imaging mode. Initial demonstrations have shown promise, but the signal-to-noise ratio is challenging to increase.
A slightly simpler route to eke out the magnetic signal is possible for some categories of material. If we know that the magnetic structure of the material has a particular symmetry, a probe matched to this symmetry ought to boost the contrast of the signal. Jan Rusz and colleagues showed an example of this method by mis-using the aberration corrector to introduce four-fold stigmation to their probe.2
Introducing phase profiles to the probe to bring out particular specimen symmetries and contrast has also found application in allowing both light and heavy elements to be visualised simultaneously in a method dubbed “matched illumination and detector interferometry” (MIDI) STEM by Colin Ophus and colleagues. Building on ideas of Harald S Rose in the 1970s, MIDI STEM combines a phase plate with a sequence of phase-shifting rings, with a detector of the same geometry in order to tune the image contrast into a desired range – showing both gold and amorphous carbon simultaneously.3
It’s not only the shape of the wavefront in the sample and detector planes that can be controlled with apertures and phase plates, but also how the beam diverges and reshapes in the third dimension as it propagates through the electron column.
Classic scanning transmission electron microscopy probes, made from a simple circular aperture, form an Airy disc intensity profile in the sample plane – a sharply peaked central lobe, surrounded by many, low intensity rings. The depth of focus of these probes is inversely proportional to the aperture radius. A broader aperture (creating a narrower probe) leads to a smaller depth of focus, than a narrower aperture/broader probe arrangement. It is this characteristic that is used to enable optical sectioning imaging methods in scanning transmission electron microscopy. But – we need not be limited to just these simple circular apertures.
A small step in complexity from the standard aperture design is to create an aperture that is a narrow circular ring. A probe formed from such an aperture is an approximation to a Bessel beam. A mathematically perfect Bessel beam will have an infinite depth of field. An experimentally realistic Bessel beam will have a depth of field that is much longer than the standard Airy probe, when propagating in free space – this could be useful for tomographic-type imaging.
While Bessel beams may be useful for applications in need of long depths of field, they are broader than comparable Airy probes (a higher proportion of the electron intensity will be in the tails surrounding the central lobe). In some imaging modes, the presence of the tails around the central lobe can make image interpretation more complex.  It has recently been shown that differential phase contrast scanning transmission electron microscopy (used to measure electric and magnetic field gradients in a sample), using quadrant detectors can be made more quantitative by reshaping the probe to remove the additional tails using an additional aperture in the microscope.4
With applications in magnetic imaging, 3D probe design and electric field imaging, probe shaping in scanning transmission electron microscopy seems here to stay – but tuning and shaping the wavefront in conventional transmission electron microscopy also holds potential for across many different aspects of microscopy.
Shaping the electron beam wavefronts in conventional transmission electron microscopy has been being exploited for many decades – but with recent developments, far more applications are becoming feasible.
One of the earliest discussions of wavefront shaping in transmission electron microscopy was the development of Scherzer defocus, where the spherical aberration (Cs) inherent to circularly symmetric magnetic lenses can be partially balanced by the tuneable defocus aberration in order to increase the contrast of the resultant image – which can be framed in the optical theory of Zernike.
From classical imaging theory, if the sample does not absorb a significant proportion of the transmitted wave intensity (such samples are known as phase objects), an aberration free, bright field image will have minimal contrast. Image contrast can be generated either by use of small objective apertures, or through applying phase shifts in the back focal plane. This method was developed in the early 20th century by Fritz Zernike, and ultimately led to his being awarded the Nobel Prize. If the phase shifts induced by the specimen are small, then a shift of the unscattered wave by 90°, will lead to the resultant image having strong contrast, which is approximately linearly proportional to the phase shifts induced by the specimen – a very intuitive image to understand.
The goal behind balancing the Cs with defocus is not to reach a flat contrast transfer function, but one which shifts the phase of the central, unscattered part of the electron wave, relative to the scattered rays through manipulation of the optical back focal plane (usually in the objective aperture plane). If the specimen only weakly shifts the transmitted electron wave, then an image formed in this manner will have contrast linearly proportional to the phase shift imposed by the specimen, as prescribed by Zernike.
With the advent of aberration correctors, the contrast transfer function can be tuned more freely, and the ideal Zernike image contrast can be approximated increasingly well5. However, even the most modern aberration corrector has only a finite number of orders of correction –and so using the corrector alone to tune the image contrast towards linearity has a finite range of flexibility. More adjustable methods to control image contrast can be made using physical phase plates – many different implementations of which have been demonstrated.
Phase plates are additional optical elements, added in to the microscope column – most commonly in the optical back focal plane, in place of one of the standard objective aperture options. These phase plates reshape the transmitted wavefront, using a range of techniques, including microelectronic circuits built into the aperture strip, applied nanoscale magnets (used carefully, to avoid issues with the objective lens) or using a thickness-profiled thin film (which relies on the phase shift induced by the projected mean inner potential of the film being proportional to the thickness). Figure 2 illustrates some of the methods demonstrated by different groups used to enable a Zernike-type linear phase contrast.

Figure 2
. Illustrations of the different forms of Zernike phase plate: (a) Hole in carbon film, (b) Boersch ring electrode, (c) Zach electrode phase plate, (d) Volta phase plate. References to original demonstrations and details available within9
However, much as in scanning transmission electron microscopy imaging, while linear image contrast is most simple to interpret, linear image contrast is not the most dose efficient method. For many important samples (such as batteries, photovoltaics, or biological specimens), the achievable image resolution can be limited by the sample deforming under the electron beam illumination – such materials are described as beam-sensitive, and the beam-induced damage can prevent clear, high-resolution images from being obtained. More dose efficient imaging modes can be applied by breaking free from the constraints of simple image interpretability that restrict us to linear image contrast regimes. The flexibility of beam shaping using thickness-profile phase plates has been comprehensively demonstrated by Roy Shiloh et al 6 where a highly structured logo-design was imprinted onto the beam experimentally.
One method of low-dose transmission electron microscopy imaging using wavefront shaping that has seen wide application in cryo-TEM is applying high values of defocus, in order to increase the image contrast. The detail in these images is not directly interpretable (requiring some computational processing), but contrast above the noise level is achievable at lower dose levels.
Images formed with applied high levels of defocus no longer have contrast that is linearly proportional to the specimen mass-thickness, but instead to the second derivative of this – leading to the well-known strong edge-contrast. For some types of specimen, the recorded intensity values can be computationally propagated back to the focal plane (using a variety of different algorithms and approximations) – but one particular algorithm has seen a huge variety of applications in the x-ray imaging literature and has been found to enable a boost in the signal-to-noise ratio both theoretically and experimentally – allowing for a dose reduction on the order of 10,000 times.7 Of more interest to the electron microscopy community is the recent demonstration of the same method in the transmission electron microscopy. The Paganin algorithm has been found to be applicable in some transmission electron microscopy imaging regimes and experimental tests of the signal-boosting properties in transmission electron microscopy are promising8.
An alternative imaging method to generate the dose-efficient, Laplacian-type contrast is by using a spiral phase plate (in a reciprocal arrangement to that used to generate vortex beams).9,10
A conventional transmission electron microscopy image of a typical phase object, taken in-focus, but with a spiral profile phase plate carefully centred in the objective aperture will reveal an image with strong-edge contrast: the vortex phase masks acts as an approximate Laplacian, similarly to the defocus method discussed above. However – if a pair of such images are taken: one using a left-handed spiral phase plate and the other with a right-handed spiral phase plate, the image of the difference between the two recorded images (assuming sufficient sample and microscope stability) will also reveal how the magnetic field varies in the specimen plane.10
Currently, experimental application of this technique is limited by the precision demanded in the manufacture of the spiral phase plates to ensure the central point of the phase plate is sharply defined. Phase plate manufacturing techniques are rapidly advancing, so robust implementation of this method is perhaps not so far away.
The shaping of the electron wavefronts in (scanning) transmission electron microscopy has a long and wide-reaching history and has played a role in enabling many interesting materials to be imaged. However, following recent advances in the implementation of aberration correctors (in both number of tuneable order of aberration, and the precision with which they can be adjusted), and developments in the design and control of experimental phase plates are having a huge influence on the ease with which new or modified imaging techniques can be implemented, and the range of materials for which phase-contrast manipulation can be usefully applied.
While this article has given an overview of many ways in which beam shaping has been applied to overcome imaging challenges – two of the emerging applications are particularly promising: using azimuthally-structured wavefronts to image magnetic fields, and combining structured wavefronts, with judiciously designed reconstruction algorithms in order to enable lower-dose imaging. The broader adoption of these techniques holds great promise to rapidly advance the imaging capabilities of existing (scanning) transmission electron microscopes with only small tweaks needed.

1. Uchida, Masaya, and Akira Tonomura, Nature 464.7289 (2010): 737, Verbeeck, Johan, He Tian, and Peter Schattschneider, Nature 467.7313 (2010): 301, McMorran, Benjamin J., et al. Science 331.6014 (2011): 192-195.

2. Rusz, Ján, Juan-Carlos Idrobo, and Somnath Bhowmick, Physical review letters 113.14 (2014): 145501.

3. Ophus, Colin, et al., Nature communications 7 (2016): 10719, Rose, H., Ultramicroscopy 2.0 (1976): 251-267.

4. Clark, L., et al., Physical Review A 97.4 (2018): 043843.

5. Jia, C. L., et al., Ultramicroscopy 110.5 (2010): 500-505.

6. Shiloh, Roy, et al., Ultramicroscopy 144 (2014): 26-31.

7. Paganin, David, et al., Journal of microscopy 206.1 (2002): 33-40, Gureyev, Timur E., et al., JOSA A 34.12 (2017): 2251-2260.

8. Liu, A. C. Y., et al., Ultramicroscopy 111.8 (2011): 959-968, Clark L., et al., EMAG conference, July 2018, University of Warwick.

9. Clark, L., Creation and quantification of electron vortex beams, towards their application, PhD thesis, 2016.

10. Juchtmans, Roeland, et al., Physical Review A 94.2 (2016): 023838.

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