Quantum Wavefront Shaping with a 48-element Programmable Phase Plate for
Electrons

C.P. Yu1,2, F. Vega Ibáñez1,2, A. Béché3 J. Verbeeck1,2*

1 University of Antwerp, EMAT, Groenenborgerlaan 171, 2020, Antwerp, Belgium

2 NANOlab Center of Excellence, University of Antwerp, Groenenborgerlaan 171,
2020 Antwerp, Belgium

3 AdaptEM, Interleuvenlaan 62, 3001 Heverlee, Belgium

*<EMAIL_ADDRESS>
## Abstract

We present a 48-element programmable phase plate for coherent electron waves
produced by a combination of photolithography and focused ion beam. This
brings the highly successful concept of wavefront shaping from light optics
into the realm of electron optics and provides an important new degree of
freedom to prepare electron quantum states. The phase plate chip is mounted on
an aperture rod placed in the C2 plane of a transmission electron microscope
operating in the 100-300 kV range. The phase plate’s behavior is characterized
by a Gerchberg-Saxton algorithm, showing a phase sensitivity of 0.075 rad/mV
at 300 kV, with a phase resolution of approximately 3$\cdot$10${}^{-3}~{}\pi$.
In addition, we provide a brief overview of possible use cases and support it
with both simulated and experimental results.

###### Contents

  1. 1 Introduction
  2. 2 Experimental considerations
    1. 2.1 Description of the Electrostatic Phase Plate
    2. 2.2 Characterization
  3. 3 Application Examples
    1. 3.1 Designer Electron Waveforms
    2. 3.2 Object Sampling with Different Wavefunctions
    3. 3.3 Adaptive Optics
    4. 3.4 Phase programmed Ptychography
  4. 4 Conclusion
  5. A Defocused Images of the Phase Plate
  6. B Pixel Index

##  1 Introduction

Wavefront shaping, or the spatial and time-dependent control over the phase in
coherent waves, has revolutionized many diverse scientific fields ranging from
radio and light astronomy [1, 2], radar [3], acoustics [4, 5, 6], seismology
[7], telecommunication [8, 9] and many more [10]. It requires a device that
can apply a position-dependent phase change and can be augmented by adding a
control loop to obtain adaptive optimization of the wave with respect to some
goal function. In optics, this can be realized with so-called spatial light
modulators, which can be based on moving arrays of mirrors or by liquid
crystal-based setups that can change refractive index when applying an
electric field [11, 12].

Matter waves, as introduced by de Broglie [13], are also amenable to this same
concept. Indeed, the working principle of an electron microscope is entirely
based on describing the free electrons as coherent quantum waves with
wavelengths of the order of picometers. The capability of manipulating these
electron waves is an indispensable part of a transmission electron microscope
(TEM). The most relevant addition to phase manipulating devices in recent
decades is, without a doubt, the spherical aberration corrector [14, 15, 16],
which flattens the phase front of the electron wave induced by (high-order)
geometric aberrations of the microscope lenses and allows the forming of a
sharper and more intense probe in scanning probe applications. Removing
unwanted phase aberrations has significantly increased the resolution and
current density of the scanning transmission electron microscope (STEM) with
many benefits in, e.g., spectroscopic applications.

Besides canceling geometric aberrations, the ability to arbitrarily shape the
electron wavefront is gradually gaining attention with the hope of improving
contrast or selectivity in electron microscopy setups. There has been a
renewed surge of such phase modulators and their applications in the past few
years. In soft material imaging, different phase plates such as Zernike [17,
18], Boersch[19, 20], Zach [21, 22], or Volta [23, 24, 25] have been
implemented in the TEM to imprint a constant phase shift to a (central) part
of the electron wave, to increase the contrast when imaging phase objects.
Some other designs with relatively higher complexity may modify both the
amplitude or phase configuration of the electron wave to create an electron
probe of specific shape [26], to increase contrast [27, 28], or to extract
specific information from the electron-sample interaction [29, 30], to name a
few. Some of these complex modulators even exhibit control over the parameters
or magnitude of the modulation. The electrostatic phase plate reported by
Verbeeck et al. [31] has demonstrated changes in interference between 4
partial waves by altering their mutual phase relation. Barwick and Batelaan
[32] showed that a pulsed laser beam could induce a phase shift in the
electron beam and that the contrast of the formed image can be optimized by
tuning these laser pulses. Different realizations of using the ponderomotive
force to change the phase of an electron beam appeared [33, 34, 35, 36]. The
electrostatic phase plate reported by Tavabi et al. [37] has demonstrated a
tuneable azimuthal phase by setting up specific electric field boundary
conditions, which was interpreted as adding orbital angular momentum to the
electron beam.

Here we report on an adaptive electrostatic phase plate based on the proof of
principle demonstration by Verbeeck et al. [31], but with significantly
increased complexity, performance, and practical usefulness. The phase plate
consists of 48 openings, or pixels, transparent to an incoming coherent
electron wave. The vertical walls of the pixels are made into electrodes so
that an electric potential can be established inside, changing the wavelength
of that part of the transmitted wave. Since separate voltage sources control
each of the 48 pixels, the phase of the entire transmitting coherent electron
wave can be programmed at will. This design and the electrostatic nature grant
the phase plate several advantages, such as short response time, the ability
to realize complex and arbitrary phase configurations, low power dissipation,
compactness, low weight, and high stability and repeatability.

The experimental part of the paper provides a concise summary of the reported
phase plate. The design of the phase plate is described first, as well as the
components and mechanism to create a phase shift on an electron wavelet. The
manufacturing design choices are briefly discussed in the scope of the
challenges faced. The device’s optical performance is then evaluated regarding
its phase sensitivity and response time.

We discuss the applications of the phase plate in the scope of electron
microscopy. Using the unique properties of a fast, hysteresis-free
programmable phase plate, we demonstrate how novel imaging setups can expand
or improve imaging modalities in TEM. We provide simulated examples and early
experimental attempts towards electron wave modulation, complex sampling
schemes, adaptive optics, and phase-coded ptychography to hint at what phase
plates could bring to the electron microscopy community.

##  2 Experimental considerations

###  2.1 Description of the Electrostatic Phase Plate

The basic working principle of the phase plate is sketched in Figure 1-a. A
coherent incoming electron wave is made to interact with an insulating
membrane that has several holes. The top and bottom surfaces of the membrane
are covered with a ground shield, while the inside of the holes is coated with
a conductive layer that can be put to a controlled electrostatic potential
($V_{1}$ and $V_{2}$ in the simplified sketch). The potential surrounding the
holes creates a potential landscape for the fast electrons that accelerates
the electron upon entering and decelerates upon leaving this area. This will
cause a phase change between the partial waves leaving these holes where one
could imagine them as coherent Huygens sources that will constitute a now
phase-programmed wave upon propagation in free space. The phase shift $\phi$
obtained is given by the electrostatic Aharonov-Bohm shift:

$\phi=\frac{\pi e}{\lambda E_{0}}\int_{\Gamma}V(\vec{r})dl$ (1)

For an electron wave with wavelength $\lambda$ and energy $E_{0}$ and crossing
a region of space with an electrostatic potential $V(\vec{r})$ along a
trajectory $\Gamma$. In the case of a weak perturbation, the electron’s
trajectory is not altered by this field, and the phase shift becomes directly
related to the projected electrostatic potential. The goal of a pixelated
phase plate is to create a potential profile that, in projection, leads to a
constant phase shift proportional to the voltage applied to each pixel
element. This occurs if the projected potential changes as little as possible
over the region of each hole, which can be obtained by choosing a high aspect
ratio (height/diameter$>1$).

From a practical perspective, the AdaptEM WaveCrafter phase plate [38]
comprises three main elements shown in Figure 1b-e: a dedicated condenser
aperture holder containing the phase plate chip, a 48-channel programmable
voltage source and a remote computer for control and user interface,
respectively. The phase plate used in this work is composed of 48 independent
active elements, or pixels, arranged in 4 concentric rings and 12 petals (see
Figure 1b). Each element consists of a layered structure similar to the one
described by Matsumoto and Tonomura for a single phase shifting element [39].
An aspect ratio of approximately 2 was chosen to avoid lensing, and a total
diameter of the active area of 50 $\mu$m assures that a modern electron
microscope can coherently illuminate the whole device.

Figure 1: Sketch of the working principle of the phase plate (a). Only 2
pixels are drawn. 3D render of the setup (b) and the main components,
including the phase plate (c), the voltage sources (d), and the phase
controller computer (e). A reference bar of 30 $\mu$m is presented in (c).

One considerable advantage of this phase plate design lies in the relatively
low voltage (in the mV range) required to induce a phase shift of 2$\pi$. This
avoids high electric field breakdown issues in the nanoscale features of the
chip and has the benefit that readily available voltage sources, which are
simultaneously precise, stable, low power, fast and reliable, can be used.

###  2.2 Characterization

To experimentally examine the projected potential profile, phase
reconstruction based on the Gerchberg-Saxton (GS) algorithm [40] was performed
on a set of images of the phase plate, where each pixel is excited with
increasing electrostatic potential (48 pixels, 11 voltage levels, 528 images
in total). For the characterization, the phase plate is inserted in the sample
plane of an FEI Tecnai Osiris S/TEM operating at 200 kV and illuminated with a
parallel electron beam. The images are taken from the back focal plane of the
objective lens (diffraction mode), while the objective lens is largely
defocused so that the detector can capture the near-field diffraction pattern
of the phase plate. This experiment is aimed to characterize the projected
potential on the phase plate when varying the phase inside each pixel in a
range between 0 and $2\pi$. A rough estimation of the voltage corresponding to
a $2\pi$ phase shift was first found by assigning a gradually increasing
voltage to half of the pixels randomly and repeatedly. Theoretically, a $2\pi$
phase shift should not result in any difference in the diffraction pattern
formed by the phase plate. Thus a visual inspection of the voltage at which
the pattern shows the least variation over time is a reasonable estimation of
the value at which the pixels yield a $2\pi$ phase shift. Once this voltage
$V_{2\pi}$ was found, a series of images with 11 different potentials equally
spread between 0 and $V_{2\pi}$ was taken for each pixel.

The defocused condition was specifically chosen so that outgoing waves from
the electrodes interfered strongly with each other, and the phase difference
between separate neighboring wavelets is significantly encoded in the recorded
intensity images (see supplementary). This choice of detection plane was
preferred over recording at an in-focus condition that interferes all of the
wavelets together for several reasons. First of all, at the right focus, the
transmitted electrons are concentrated in a very small region (less than
$1~{}\%$ of the size of the recorded defocused images), and creating a high
enough camera length to sufficiently sample such patterns on a pixelated
camera for phase retrieval is not trivial. On top of that, the inversion
invariant nature of the wave intensity in the reciprocal space would also
challenge obtaining a unique reconstruction and greatly hinder the retrieval
algorithm’s convergence.

The result of the reconstruction is summarized in Figure 2. The phase response
of all pixels, as they were individually excited, is fitted using a linear
function, representing the phase sensitivity of that pixel to the applied
voltage. A phase sensitivity matrix can be constructed showing the phase
sensitivity of pixel $i$ upon exciting pixel $j$. The phase sensitivity matrix
in Figure 2 shows a strong response on the diagonal, meaning that the excited
pixel is the only one showing a significant linear phase shift against the
applied voltage. An average phase sensitivity of 0.075 rad/mV is found, which
translates to a theoretical phase resolution of approximately
3$\cdot$10${}^{-3}~{}\pi$ according to the smallest step size provided by an
ideal 16-bit DAC (maximum $2.5$ V, smallest step $2.5\times 2^{-16}$ V). The
error matrix, also shown in Figure 2, indicates response deviation from the
expected linear behavior, mainly resulting from imperfections in the phase
retrieval process, such as the finite pixel size and non-ideal detector
response. These can cause a difference between the recorded intensity and the
actual waveform. The error is calculated by the root mean square error of the
fitted result, which is found, at maximum, to be $3\%$ of $2\pi$ (0.19 rad),
while on average less than $0.5\%$ of $2\pi$ (0.027 rad).

Figure 2: Phase sensitivity matrix and the corresponding root mean squared
error of the linear fitting.

Besides the expected response of the phase plate, it is equally important to
characterize any non-ideal behavior. The inhomogeneity describes the phase
deviation within the pixel area from the ideal constant, homogeneous
expectation. We evaluate the standard deviation of the reconstructed phase
within each activated pixel and find it to be $<1.7\%$ of $2\pi$. The cross-
talk refers to the phase response within a pixel region caused by the voltage
applied to another pixel. We estimate this as the maximum linear response of a
non-excited pixel as a function of any other excited pixel. The off-diagonal
lines found exactly 12 pixels away from the main diagonal in both matrices in
Figure 2 indicate that the strongest cross-talk is, unsurprisingly, found
between neighboring pixels due to how the pixels are ordered in the matrix
(see supplementary). The cross-talk is measured to be $<0.012$ rad/mV, which
amounts to $15\%$ of the response of the excited pixel. In summary, the
inhomogeneity only creates phase error much less than $\frac{2\pi}{10}$, which
is generally accepted as very good in light optics [41, 42], while the cross-
talk is clearly the biggest contributor to a non-ideal response. This behavior
could be significantly improved in the next design iteration, where an
additional top-ground layer could shield the effects from neighboring pixels
and the conductive tracks leading to those pixels.

Characterizing the temporal response of the phase plate is also important for
applications that rely on rapid switching between different electron probe
shapes or phase configurations. Since the phase shift results from the
projected potential in the electrodes, the response of the phase plate can be
characterized by the time required to build up the potential. With the
criterion of phase error $<\frac{2\pi}{10}$, the response time is measured to
be less than $1.3~{}\mu s$ for reaching from $10~{}\%$ to $90~{}\%$ of
$V_{2\pi}$ and is entirely dominated by electronics.

##  3 Application Examples

###  3.1 Designer Electron Waveforms

To demonstrate the capability and visualize the effects of a freely
programmable phase plate, we recorded the far-field diffraction patterns of
various phase-modulated electron waves in a TEM (Figure 3). These patterns
form rather complex configurations compared to ones formed by commonly-used
round apertures, even when all phase plate elements are at ground potential.
This is due to the amplitude modulation created by the set of holes, which
produces highly delocalized tails. Previous theoretical research points out
that the proportion of the electrons in these tails is directly related to the
fill factor (% of the electron wave not blocked by the material of the phase
plate) of the probe forming aperture [43]. Although improvement has been made
on the fill factor (current design approximates 30%, while the proof of
concept 2x2 version from 2018 [31] had only 17%), a large proportion of the
electrons can still be expected in the tails.

Figure 3: Realization of various electron quantum states. The three rows of
figures, from top to bottom, are the phase configurations set on the phase
plate, the simulated probe shapes, and the resulting experimental probe
images, respectively. Note the excellent agreement between expected and
obtained results showing successful arbitrary wavefront shaping.

A close comparison between experimental intensity profiles and simulations is
found. From Figure 3, columns (b-e) show a phase shift of $\pi$ applied to
half of the total pixels with different patterns; therefore, the original
single intense spot in the diffraction pattern is split into multiple parts
due to destructive interference. Double-spots (b), quadruple-spots (c), and
even a duodecuple-spot (d) consisting of six $0-\pi$ pairs are shown. By
taking into account the radial distribution of the rings, a checkerboard-like
pattern (e) can be created. These patterns cover a few of the 48-dimensional
Hadamard basis [44], which defines an orthogonal basis consisting entirely of
pixels with either 0 or $\pi$ phase. Lastly, (f) shows the result of a vortex
setup with an orbital angular moment equal to 1 [45]. This is done by creating
a phase ramp from 0 to $2\pi$ in the azimuthal direction. The vortex can be
verified by the signature singularity point at the center of the resulting
probe approximating one member of the Laguerre-Gaussian orthogonal basis set
[46].

The phase plate can also create a phase profile imitating geometric optical
elements and aberrations. Typically they can be modeled by a phase shift that
follows a Zernike polynomial in the angle with respect to the optical axis
[47]. How faithfully the phase plate can recreate such polynomials at
different angles has been discussed theoretically in detail by Vega Ibáñez et
al. [43] and relates to parameters such as the order of aberration, the fill
factor, number of pixels, and pixel shape. Here, a defocus effect (second-
order in angle) is introduced by either the conventional electromagnetic
objective lens of the microscope or by the phase plate to demonstrate this
concept. The resulting probe shapes are shown in Figure 4, respectively. The
two rows show good resemblance with each other up to 200 nm defocus. Further
defocusing causes a steep phase ramp within the area of the individual pixels,
which can not be faithfully reproduced anymore by the phase plate. For this
reason, the phase plate can obviously not replace an actual (round) lens of
any significant strength.

Figure 4:  Defocused probes formed by defocusing the microscope lenses (top
row) and the phase plate (bottom row) at 300 kV acceleration voltage and an
opening angle of 1 mrad. Note the close similarity of both, showing that the
phase plate can mimic the action of a round lens to up to 200 nm defocus.

###  3.2 Object Sampling with Different Wavefunctions

Electron microscopy is a process of sampling an unknown material with an
electron wave. Once the incident wave interacts with the examined object, the
information from the object is imprinted on the wave by creating changes in
amplitude, phase, and the creation of inelastic scattering signals. When the
measurement result of the interaction between the object and a beam with a
given electron waveform provides insufficient information about the sample,
different waves can be used to interrogate the object. For example, in-line
holography [48, 49, 50] is done by recording the intensity of a beam while
varying the phase by changing the defocus of the objective lens. STEM
essentially describes a process to accumulate information about the material
by a dense sampling while spatially scanning a localized electron beam. In
both cases, multiple measurements while changing the incoming electron wave
enriches the acquired information and eliminates confusion that can sometimes
not be resolved with a measurement process that only uses a beam with a single
static waveform.

Such multi-waveform sampling schemes rely entirely on the ability to alter the
wavefunction of the beam electron states. Even though some form of modulation
of the wavefunction is present in any electron microscope (e.g., defocus, beam
tilt, beam shift, or aberration correctors), they often rely on
electromagnetic elements, which can suffer from slow settling times and
hysteresis effects. For example, in the acquisition of through focal series
images, an update rate in the order of seconds to minutes is typically applied
to induce small focal changes in the objective lens[51, 52].

The phase plate presented here can update to an entirely new pattern in a few
$\mu$s without hysteresis so that complex sampling schemes can be realized
efficiently. For instance, the phase plate can cycle over a few different
wavefront settings for each probe position in a STEM recording. Compared to
through focal TEM acquisition, where the focus is changed between recording
image frames, we could now update multiple focus levels for each probe
position in a STEM scan, providing, e.g., increased depth of field. This
dramatically reduces the difficulty of realigning each image, especially in
cases of severe sample drift, and also avoids inconsistencies caused by
contamination building up on the sample over time.

Changing defocus is just one of the possible wavefunctions to sample an object
of interest. Being a non-orthogonal change to the beam wavefunction, it could
be argued that this is not even an optimal choice of basis. The adaptability
and rapid response of the phase plate can be extended to a wide variety of
orthogonal basis sets that can be specifically chosen to efficiently encode
selected knowledge about the sampled object into the probing electron waves.

This concept is widely used in light microscopy and serves as an important
cornerstone for techniques such as stimulated emission depletion (STED)
microscopy [53, 54, 55] and switching laser mode microscopy (SLAM) [56, 57].
Two or more waveforms sequentially illuminate the sample, and the sharp
feature created by the difference between the illuminating waves can be
exploited to increase the resolution of the final image.

The same concept can now be applied to electron microscopy with a two-fold
beneficial effect. Indeed, changing between a probe state with and without
orbital angular momentum will slightly improve image resolution due to
differential imaging with both probes (super-resolution). But more
importantly, this method also cancels the long probe tails arising from the
amplitude modulation of the pixel shapes, as these tails are nearly identical
for both probe wavefunctions. This is a far more critical effect as it
dramatically increases the practical resolution that can be obtained even when
the fill factor of the phase plate is not ideal and shows a way to
significantly outperform the results presented earlier for the single waveform
aberration correction prospects of programmable phase plates for electrons
[43]. The result of this differential scheme is demonstrated with high-angle
annular dark field (HAADF)-STEM simulation (Figure 5). Electron probes, as
results of the far-field diffraction of three illuminating wave functions,
created by a phase plate with zero phase, vortex phase, and a conventional
round aperture, are used to scan a single-layer hexagonal boron nitride, with
200 keV electron beam energy, a spherical aberration $C_{3}$ of 1.2 mm and
operating at Scherzer defocus, in agreement with a typical uncorrected TEM
instrument. The convergence angles of the electron probes are set to 9.5 and
11 mrad for the round aperture and the phase plate, respectively. We select a
larger opening semi-angle for the phase plate since its capability to correct
aberrations yields an optimal imaging condition at 11 mrad. The subtraction of
the vortex image from the plain phase plate is then presented as the
difference image.

Figure 5: Simulated ADF images of various probe shapes (see the insets) and
their Fourier transforms. The line profiles (orange lines) are taken at the
position of the white dashed line in each image. Note that the intensity
profile in the round aperture image is halved for better presentation and that
the black line in the difference image indicates zero, while in other images,
zero is set at the bottom of the figures.

The simulated images are then juxtaposed to illustrate the effect of the
tails, and an intensity profile (orange line) is drawn across each image (at
the position of the white dashed lines). Both images from the phase plate have
non-zero intensity in the vacuum area (the left half of the simulation box)
due to the tails’ interaction with the crystal. The profile from the image
formed with the round aperture shows much faster decay as the intensity
distribution of an aberrated Airy probe is more concentrated. The difference
image demonstrates good cancellation of this false background, and the
intensity profile quickly converges to zero, with small fluctuations due to
slight differences between the tail configuration of the two probes. This
result shows that the phase plate can indeed provide an excellent tail effect
cancellation when alternating between a flat phase and vortex phase probe. The
resolution is significantly improved over the non-corrected round aperture at
the expense of some signal loss related to the fill factor and the loss of
low-frequency sample information. This demonstrates the potential for
aberration correction with a device that is significantly smaller ($<5$ mm),
lighter, faster ($\mu$s), more energy efficient ($<5$ W) and requires far less
stringent control over the precision of the voltage/current sources as
compared to current multipole correctors.

###  3.3 Adaptive Optics

Using the fast and hysteresis-free phase programming offered by the
electrostatic phase plate opens the attractive possibility of adaptive optics.
As a proof of concept, such a setup is realized (fig. 6). An algorithm
repeatedly reshapes the electron probe with the phase plate in order to reach
a higher variance in the high-angle annular dark field (HAADF) image, which is
taken as a figure of merit that links with ’image sharpness’ [58].

Figure 6: Schematic of the adaptive probe correction with phase plate. The
flowchart displayed in (a) demonstrates the fundamental steps of the algorithm
being employed. The HAADF images before (b) and after (c) the correction are
shown below.

The algorithm sequentially adds phase modifications from a list of discretized
low-order Zernike polynomials to the latest best-performing phase
configuration. Zernike polynomials are chosen since they exhibit close
similarity to common aberrations in the electron microscope and form a
complete, orthogonal basis. A HAADF image is consequently recorded with every
new probe. If the variance is higher in the new image, the current best is
replaced with this new variation. Once all the configurations are tested,
their magnitudes in terms of phase value are reduced by half for a further
refinement step. The process is demonstrated by inserting the phase plate in
the C2 aperture of a probe and image-corrected FEI-Titan operating at 300 kV
in microprobe mode with a convergence angle of 1 mrad (to minimize the effect
of aberrations and partial coherence effects). The HAADF image is taken from a
gold cross-grating test sample with a deliberately introduced initial defocus
of approximately 1 $\mu$m. The result of the correction is shown in figure
6(c). The process converges after 32 iterations with a sharper resulting
image, even though 1 $\mu$m defocus cannot be entirely compensated by the
phase plate due to the steep phase profile. The result shows the feasibility
of counteracting the lens defocus automatically. The process takes
approximately 1 minute, but this time is currently dominated by sub-optimal
software handshaking between scan engine control, image readout, and phase
plate control, and can be dramatically improved in the future. As an estimate,
with the assumption that an update can be made by evaluating a minimum area of
100x100 pixels at 1 $\mu$s dwell time (a reasonable dwell time to produce
HAADF images with an acceptable noise level), the update rate for the
correction scheme would be 1 kHz. This frequency is easily within reach of the
phase plate, which currently offers a maximum update rate of 100 kHz, limited
by the electronics. This would result in an adaptively optimized image within
10 ms, which would be a small fraction of the time to take, e.g., a full
1024x1024 frame. Of course, this time depends on the beam current, as enough
image quality is required to make good decisions on the next step. Further
work is needed to evaluate the best goal function and most optimum control
loop, but the proof of concept demonstrates the scheme’s feasibility.

This process could bring significant benefits for the automation of microscopy
experiments. Automatic data acquisition and feature identification are widely
used for life science research and quality control in the semiconductor
industry. With them, the analysis of large amounts of samples can be done
without operator intervention, and the demonstrated probe correction scheme
can be utilized for maintaining the quality of the optical system over a much
longer operation time.

This iterative optimization process can also be extended to any technique in
electron microscopy where a specific quantifiable property is related to the
shape or phase of the electron probe. For example, in electron energy loss
spectroscopy, the intensity of a specific plasmon peak can be tracked while
reshaping the electron probe until the optimal probe shape selectively
highlights the corresponding plasmon mode [59].

###  3.4 Phase programmed Ptychography

Besides shaping a focused electron beam, the phase modulation capability of
the phase plate operating under parallel-beam conditions can bring new
opportunities for other microscopy applications. For example, coherent
diffraction imaging and ptychography can benefit from using the phase plate as
a ”modulator” or a ”diffuser” to break symmetry in the illuminating beam and
thus increase the robustness and convergence rate of the reconstruction. The
benefit of a modulator has been widely reported and studied in the field of
light microscopy[60] and electron microscopy [61, 62, 63]. Among the reported
realizations of ptychography in electron microscopy, with or without a
modulator, the reconstruction of the complex object relies on repeated
sampling at different locations of the object, with the criterion that the
illuminating beam partially overlaps with the sampling at a nearby position.
This overlap creates the so-called ”information redundancy” [64, 65], which
eliminates the twin-image artifact [66] that originates from the central
symmetry of the illuminating beam. On the other hand, such symmetry can be
easily broken by a random phase configuration introduced by the phase plate
instead of the displacement of the beam or the sample.

We hereby demonstrate this concept by performing phase reconstruction on
simulated diffraction patterns from a target pure phase object (Figure 7a).
The diffraction patterns are generated by different illuminating waves, formed
with a round aperture, the phase plate set to zero, and one randomly generated
phase configuration. The phase reconstruction is again based on the GS
algorithm, and the resolved objects are obtained after 50 iterations. The
results are shown in Figure 7(b-d). Neither a round aperture nor a zero phase
plate could generate a convincing reconstruction result, as the geometry of
both apertures is centrosymmetric. However, introducing a random phase
configuration increases the reconstruction quality significantly despite the
sample being only illuminated at one beam position.

Figure 7: Simulated ptychographic phase reconstruction from recorded
diffraction patterns with various illuminating beams. (a) The ground truth
phase image of the object. (b-d) Reconstruction results from illuminating
beams formed by a conventional round aperture, flat phase plate, and phase
plate with random phase configuration, respectively. The dark region indicates
the opaque part of the aperture. Note the significant improvement in phase
reconstruction quality when the incoming beam is phase randomized. As the
object is only illuminated once, reconstruction is only possible in those
areas where the amplitude is not zero.

The amplitude modulation of the phase plate inevitably results in missing
information about the reconstructed object, which could be filled by moving
the beam or the sample to illuminate the whole region of interest at least
once. It should be noted here that the phase plate is placed in front of the
sample, and all electrons interacting with it are recorded. This means that
the limited fill factor does not reduce the electron dose efficiency nor
increase beam damage on the sample.

##  4 Conclusion

We report the successful realization of arbitrary wavefront shaping of
electrons with a novel 48-pixel programmable electrostatic phase plate. The
phase plate is capable of introducing a phase shift of more than $60\pi$, as
well as fine-tuning the phase value with step size as small as
3$\cdot$10${}^{-3}~{}\pi$ for 300 keV coherent electron beams. Cross-talk
between pixels was shown to be $<15\%$ and can be improved further with better
shielding electrode geometries. This brings modern adaptive light optics
concepts into the domain of electron beam instruments. The rapid response of
the device allows up to 100 kHz update rates, making it possible to do on-the-
fly auto-tuning of differential contrast schemes without a noticeable
recording time penalty for the user. The examples demonstrate the potential
for a rich field of emerging applications offered by the phase degree of
freedom. Immediate use cases focus on electron microscopy, but other electron
beam instruments, such as, e.g., e-beam lithography or semiconductor
inspection tools, could also profit significantly from this realization. With
an even broader perspective, we demonstrate here the arbitrary preparation of
coherent quantum states that might be exploited in novel quantum
information/computing schemes over a much wider range of electron energies
than the ones demonstrated here.

## Acknowledgements

All authors want to thank Gert de Bont for providing the graphics in figure 1
and Stijn Van den Broek for never-ending support with the focused ion beam
instrument.

#### Funding information

This project is the result of a long-term effort involving many different
sources of funding: JV acknowledges funding from an ERC proof of concept
project DLV-789598 ADAPTEM, as well as a University IOF proof of concept
project towards launching the AdaptEM spin-off and the eBEAM project,
supported by the European Union’s Horizon 2020 research and innovation program
FETPROACT-EIC-07-2020: emerging paradigms and communities. This project has
received funding from the European Union’s Horizon 2020 research and
innovation program under grant agreement No 823717 – ESTEEM3 and via The
IMPRESS project from the HORIZON EUROPE framework program for research and
innovation under grant agreement n. 101094299. FV, JV, and AB acknowledge
funding from G042820N ’Exploring adaptive optics in transmission electron
microscopy.’ CPY acknowledges funding from a TOP-BOF project from the
University of Antwerp.

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##  Appendix A Defocused Images of the Phase Plate

The defocused images of the phase plate with a voltage applied to one pixel in
each ring and the difference map between images with and without excited
pixel. The changes in the interference patterns imply a local phase shift
which is revealed by the phase reconstruction algorithm.

##  Appendix B Pixel Index

Positions of the pixels and their corresponding indices.