Efficient diffractive phase optics for electrons

Electron diffraction gratings can be used to imprint well-defined phase structure onto an electron beam. For example, diffraction gratings have been used to prepare electron beams with unique phase dislocations, such as electron vortex beams, which hold promise for the development of new imaging and spectroscopy techniques for the study of materials. However, beam intensity loss associated with absorption, scattering, and diffraction by a binary transmission grating drastically reduces the current in the beam, and thus the possible detected signal strength it may generate. Here we describe electron-transparent phase gratings that efficiently diffract transmitted electrons. These phase gratings produce electron beams with the high current necessary to generate detectable signal upon interaction with a material. The phase grating design detailed here allows for fabrication of much more complex grating structures with extremely fine features. The diffracted beams produced by these gratings are widely separated and carry the designed phase structure with high fidelity. In this work, we outline a fabrication method for high-efficiency electron diffraction gratings and present measurements of the performance of a set of simple prototypical gratings in a transmission electron microscope. We present a model for electron diffraction gratings that can be used to optimize the performance of diffractive electron optics. We also present several new holograms that utilize manipulation of phase to produce new types of highly efficient electron beams.


Introduction
Scanning transmission electron microscopy (STEM) has recently offered a large number of critical insights into the structure and behavior of materials at the atomic scale [1][2][3]. As a result of several decades of advancements in electron optics, modern STEM instruments use precisely controlled electric and magnetic fields to prepare angstrom-sized 60-300 keV electron probe beams with currents on the order of nano-amperes [4,5]. The focused probes have an approximately Gaussian intensity distribution and a regular phase profile determined by aberrations.
Following the development of atomic-scale STEM, there has been a surge of interest in using shaped probe beams with carefully designed phase structure and non-Gaussian intensity distributions. The interaction of such a shaped probe beam with a material can offer more information than is available through the use of a traditional probe beam [6,7]. For example, electron vortex beams [8][9][10] may soon be employed to accomplish atomic resolution spin imaging [9,11] with STEM. Holographic diffraction gratings [9,10,12,13] and phase platesboth material [8] and magnetic [14,15]-have been successfully used to imprint the helical phase that defines a vortex beam.
Electron probe beams with well-defined phase structure can be produced via diffraction from a nanofabricated holographic diffraction grating placed in the probe-forming aperture of a STEM instrument [6] (see figure 1). A second aperture lower in the column can be used to isolate one of the diffracted probes and use it to form an image. One can, in general, define the two-dimensional pattern s x y ( , ) of a holographic diffraction grating by interference of a wavefunction of interest Ψ x y ( , ) with a reference wave [16] Ψ Ψ = + s x y x y x y ( , ) ( , ) ( , ) .
(1) ref 2 For example, to produce an electron vortex beam with  m orbital angular momentum in the first diffraction order of a linear grating periodic in the x-direction, one uses the two-dimensional pattern 1 2 e e 1 cos( ). ( 2 ) im ikx 2 A physical diffraction grating can imprint this pattern onto a transmitted complex wave, either through a modulation of amplitude or phase. In the case of a phase grating, the optical path length is varied according to the two-dimensional pattern s x y ( , ), usually by varying the thickness of transparent grating material. We will return to the theory that guides the design of holographic diffraction gratings, and in particular optimal three-dimensional structure, in section 4.
Other methods exist for imprinting a spiral phase dislocation onto electron beams; both material phase plates [8] and magnetic nanowire phase plates [14,15] add a spiral phase to an electron passed through them. However, production of pure spiral phase states with integer topological charge m is a significant challenge with these techniques. The topological charge imprinted depends on electron beam energy and is highly sensitive to fabrication errors and fringing fields, respectively. Diffractive electron optics offer the advantage that arbitrary phase structure can be imprinted on the electron beam with high fidelity and independent of beam energy. Furthermore, these devices can simultaneously produce multiple probe beams with complementary phase dislocations for dichroism techniques [6,17].
There are two common challenges that must be addressed before diffractive electron optics may be widely adopted for use in electron microscopy. First, diffraction gratings must produce sufficiently intense diffracted beams so that information carried by a diffracted probe is measurable. Second, the multiple diffracted probe beams must be sufficiently separated in the specimen plane for use in the study of materials with varied shapes and sizes. For example, When placed at the condenser aperture plane, the patterned circular area of the transparent grating has the same beam-defining properties as a regular circular condenser lens aperture. The diffracted beams can either be isolated using a second aperture, or the entire set of beams can be scanned across a local sample feature, providing multiple images of the object each containing unique information. (Bottom image) TEM image of multiple diffracted STEM probes from a μ 50 m-wide forkdislocation grating at the specimen plane. several recent diffractive structures used to create electron vortex beams [9,12,13] are composed of patterns of slits milled all the way through a relatively thick foil that is opaque to electrons. These structures operate by selectively subtracting beam current through high-angle scattering in the material and transmitting the rest through the slits. Thus, these structures behave as binary amplitude gratings, which can only place a maximum of 10.1% of the incident electrons into the first diffraction order. This is particularly troublesome for STEM applications, in which beam current must be maximized in order to generate a detectable signal of interest over background noise. These electron amplitude gratings furthermore have small separation between diffracted beams; it is thus difficult to isolate signal generated by a particular beam. An amplitude grating must be sufficiently thick so as to be opaque to electrons. This restricts the smallest lateral feature size that can be patterned, which consequently limits the diffraction angle. One recent demonstration of electron vortex production employed an amplitude diffraction grating which produced a beam separation of μ 3 rad [9], or, equivalently, a realspace separation on the order of 5 nm in the specimen plane of a modern high-resolution STEM instrument.
To address these challenges, in our work developing electron vortex beams [10,18] we have devoted considerable effort towards developing electron-transparent phase grating structures [19] that modulate the phase of the electron wave rather than the amplitude. Here we present a detailed study of these diffractive phase optics for electrons. We discuss the materials and nanofabrication method, performance measurements, and a model for these devices that incorporates the effects of both phase and amplitude modulation. Here we primarily discuss simple straight gratings periodic in one dimension, as shown in figure 2, in order to elucidate the role of several basic grating structure parameters on diffraction efficiency. Optimization of the nanofabrication process for these simple gratings can then be applied to fabricate diffraction holograms which produce electron beams with non-trivial transverse wavefunctions Ψ x y ( , ). In figure 3, we demonstrate four gratings which produce diffracted beams with various different phase dislocations.

Electron diffractive phase optics design goals
To be broadly useful for novel electron microscopy techniques, diffractive electron optics should efficiently diffract beam current into a desired diffraction order, and produce diffraction orders that are well-isolated from one another.

Diffraction efficiency
In emerging STEM techniques using beams with engineered phase dislocations, it is important to maximize current in the probe beam. The efficiency of the diffraction hologram producing the beam thus becomes a primary concern. The diffraction hologram can be designed to generate just one electron probe beam, or simultaneously produce two or more probe beams with complementary phase dislocation properties. For example, a forked grating hologram with a symmetric thickness profile simultaneously produces pairs of electron probe beams with phase dislocations that are equal in magnitude but opposite in sign. It is expected that these two beams with opposite wavefront topologies can be used to provide dichroic image contrast [6,17]. While in some applications it is desirable to produce only these two beams, in other instances it is desirable to also have a conventional electron probe beam with no phase dislocation to use for a reference signal. In all of these circumstances, the thickness profile of the hologram can be tailored to maximize the current in the electron beam and diffract it into one or more desired beams. It is also possible to form a blazed diffraction hologram that produces only one probe beam with a particular magnitude and/or sign of phase dislocation.
There are several important independent figures of merit for the diffraction efficiency of a holographic grating. Absolute diffraction efficiency η n i ( ) , transmitted diffraction efficiency η n t ( ) , and relative diffraction efficiency η ′ n n ( ) are each useful for characterizing total detectable current, total detectable current as a fraction of the theoretical maximum for a given grating design, and detectable current relative to the noise produced by inelastic scattering in the grating, respectively.
The absolute diffraction efficiency is defined as where I n is the current in the nth diffraction order, and I inc is the total current incident on the diffraction grating. Potential electron microscopy applications require a maximal beam current in the +1st diffraction order, in which case η +  For material electron diffractive phase optics, some loss of electrons due to inelastic scattering is unavoidable. The total coherently transmitted beam current I trans , equal to the sum of currents in all diffraction orders, is always less than the incident beam current, = ∑ < I I I n n trans inc , and this is largely independent of the parameters of the particular groove shape. So to compare just the effects of groove shape and depth on diffraction efficiency independently from the effects of different thicknesses of material, it is then useful to define a transmitted diffraction efficiency describing the intensity in a particular diffraction order relative to the integrated intensity of all coherently transmitted electrons For STEM EELS with diffracted probes, quantitative analysis of an energy loss spectrum depends not just on the overall current in the beam, but also on the isolation of signal from a single diffracted probe. Adjacent diffraction orders may contribute to background signal. Thus, for such an application, the relative diffraction efficiency, defined as the ratio of intensity of the nth order to the ′ n th order, η ′ In most of the applications proposed to date for shape probe beams, the probe of interest is in the = ± n 1 order, and the next most intense probe is the ′ = n 0 order. We will explicitly discuss the relative efficiency η 1 (0) of gratings produced in this work. Each of these figures of merit for efficiency represents a unique property of the STEM probes produced by a grating; absolute efficiency η n i ( ) measures total detectable current in the nth diffracted order, transmitted efficiency η n t ( ) is a good measure of total detectable current as a function of theoretical maximum, and relative efficiency η ′ n n ( ) affects the signal-to-noise ratio for a measurement involving the nth probe where noise from the ′ n th is a concern. During review of this manuscript, Grillo et al reported 25% efficiency for the first-order diffracted probe of their phase diffraction gratings [20]; we note that this was a transmitted efficiency. Using our model developed in section 4, we estimate that the 120 nm thick excess silicon nitride material supporting the grating structure incoherently scattered roughly 80% of the incident intensity; after accounting for this intensity loss, we calculate that the absolute efficiency of this grating was approximately η = 5% As we shall see in section 4, the diffraction efficiency of a grating is a function of the shape and depth of grooves, the projected mean inner potential of the grating material, and the electron beam energy. The theoretical maximum efficiency of a sinusoidal pure phase grating is 33.9% [21]. For electron sinusoidal phase gratings made of silicon nitride (Si 3 N 4 ), our model predicts that a physical groove depth of about 33 nm achieves this maximum. Utilizing high resolution FIB milling we have consistently fabricated gratings with η > 20% i 1 ( ) .

Diffraction order separation
A diffractive optical element in a STEM application produces in general multiple probe beams, and there must be sufficient angular separation between them such that the signal they each generate can be isolated. To meet this design goal and provide large free space diffraction angles, electron diffractive optical elements should be fabricated with as small a feature size as possible without sacrifing pattern fidelity.
For electrons of de Broglie wavelength λ transmitted through a grating-like diffraction hologram with pitch d, where typically λ ≪ d, the angular separation Δθ between diffracted beams is In the specimen plane of a STEM, the real-space physical separation Δx between diffracted probe spot centers at the specimen plane is where M is the magnification of the lower probe-forming STEM optics (not the image magnification) and z is the physical distance between the diffraction hologram and the specimen plane. Alternatively, in terms of the effective camera length L of the lower probe-forming optics, the spot separation is In the TEAM 1 instrument at the National Center for Electron Microscopy, a grating with pitch = d 83 nm installed in the second condenser lens aperture produces diffracted 300 keV (λ = 1.97 pm) probe beams separated by Δx = 43 nm in the specimen plane (see figure 1).
We note that it is easier to achieve larger angular separation between diffraction orders using diffractive phase optics. A distinct advantage of phase gratings over amplitude gratings is that they can be fabricated with much finer feature sizes, and so can produce much wider separation between diffracted beams. Free-standing amplitude-type gratings must be sufficiently robust to support the mass of the relatively thick, electron-opaque material; this condition limits the minimum lateral feature size of such a structure. Amplitude-blocking diffractive optics demonstrated to date [9,12,13] have a minimum periodic feature size on the order of μ 1 m. On the other hand, phase gratings can be much thinner than amplitude gratings, and can be fabricated on an electron-transparent supportive membrane [19]. In [10] we demonstrated phase gratings with 75 nm period, and have since fabricated gratings with periodic feature sizes down to 20 nm [22].

Nanofabricated diffractive electron optics
To achieve our stated design goals for efficient electron diffractive optics, we explored a diverse array of nanofabrication techniques for imprinting the phase grating onto electron-transparent materials. High-quality gratings may be produced with high-resolution focused ion beam (FIB) milling, electron beam-induced deposition (EBID), and electron beam lithography (EBL); the choice of technique places some limitations on the structure of the gratings produced but is primarily a question of practical considerations. In this work, we consider prototypical electron diffractive optics produced by FIB milling, as FIB instruments are present in many TEM labs and can be used safely to produce an electron diffraction grating with minimal training. Some specific considerations for nanofabrication of gratings with FIB are detailed in the appendix.

Phase grating model for optimizing diffraction efficiency
To compare the measured diffraction efficiencies of our holographic phase gratings to theoretical limits, we developed a model for mixed phase/amplitude gratings for electrons. The structure of the grating is a thin membrane with a modulated thickness h x y ( , ). Figure 4 shows the thickness patterns of several fabricated grating measured with atomic force microscopy (AFM). The transmission function describing the effects of this structure on transmitted electron wavefunctions is is the effective complex wavenumber of the electron within the material. The amplitude decay coefficient α describes the effects of inelastic and high angle scattering, and V 0 is the mean inner potential of the grating material. C depends only on the energy of the beam [23]. If such a grating is illuminated by an incident electron wave ψ i , then the transmitted electron wavefunction immediately behind the grating is The far field diffracted wave can be computed with the Fraunhofer approximation as shown in the appendix (C.13).
In the case of simple linear phase gratings with uniform periodicity in only one direction, x, the modulated thickness of the material can be described as an array of grooves n where h is the average thickness of the material grating, d is the pitch, or center-to-center groove spacing, and g x y ( , ) describes the profile of the grooves. The offset b cancels any residual constant from the infinite sum. These parameters, as well as the groove depth Δh and width w, are shown in figure 4. If the incoming wave ψ i is a normally-incident plane wave (C.8), the diffracted wave amplitude from this linear grating may be written as here λ is the de Broglie wavelength and V is a plane wave normalization factor. The relative amplitude of the nth diffraction order, c n , depends upon the specific groove profile g x y ( , ). A general form for computing c n from any arbitrary periodic structure is described in detail in (C.23)-(C.25). The transmitted diffraction efficiency η n t ( ) defined in (4) is calculated as n t n ( ) 2 Note that the absolute diffraction efficiency η n i ( ) , defined in (3) as a fraction of the incident beam, is less than (13) due to high-angle scattering where h is the average thickness of the grating over the back surface, as illustrated in figure 4. Cross sections (figure 2) and AFM profiles (figure 4) of the surface of our gratings indicate that individual grooves have a somewhat Gaussian shape. Earlier findings suggest that singlepixel-wide FIB-milled trenches have a remarkably Gaussian profile [24], so we model our gratings as a periodic sum of Gaussian grooves. The profile of an individual groove is x 2 2 2 with amplitude A, characteristic width σ; the profile of the entire grating is described by the periodic sum in (11) with this grating shape g(x). The groove depth Δh and groove width w are related to the parameters A, σ, and b in (C.36)-(C.37) and (C.40). Groove width is defined as the full width of the groove at half the maximum depth as a fraction of the pitch d.
With structure parameters describing the groove shape (depth Δh and width w) and material properties (absorption coefficient α and mean inner potential V 0 ), we can calculate the expected relative diffraction amplitudes c n for a grating according to (C.23). We can thus reliably model the diffraction efficiency with a small number of measured parameters. While the diffracted wave could also be computed without any model numerically, the analytical calculation of c n as a function of model parameters allows us to build an intuitive map of diffraction efficiency as a function of groove shape.
We paramaterized the average groove shape of every grating in all of our fabrication arrays using the Gaussian groove model to determine the width w and depth Δh. In figure 6, we used these average groove dimensions to categorize each grating (location in figure) and express the measured electron diffraction efficiency as a color value. These measured values compare well to the theoretical diffraction efficiency calculated using our model (background color).
The variations in efficiency between gratings with nearly identical trench depth and width, seen most clearly in figure 7, are primarily due to gallium implantation from the FIB. We have observed up to 10 atomic % gallium concentrations in milled areas of our diffraction gratings with energy-dispersive x-ray spectroscopy composition analysis. We do not account for any modification of the mean inner potential V 0 of the material in the mill process. We expect that the lower-than-expected first-order transmitted efficiency η ± t 1 ( ) and higher-than-expected zerothorder transmitted efficiency η t 0 ( ) observed for gratings with a trench depth Δ = ± h 20 5 nm are also due to gallium implantation. However, as gallium concentration is not easy to measure, and as this variation in efficiency is specific to only small range of gratings fabricated with FIB and does not affect our prediction for peak efficiency, we have chosen not to include extra parameters in our model to characterize this additional variation. We can use this simple model, which we have shown to predict efficiency with reasonable accuracy, to make prescriptions for the fabrication of highly efficient gratings.
The analysis summarized by figure 6 provides a map that we used to explore a space of many interdependent parameters used for the nanofabrication process. We see in figure 6 that any grating fabricated with a depth Δh between 27 nm and 40 nm and a full width at half max > w 0.40 will produce first-order diffracted beams with transmitted efficiency η > 30% . The most efficient grating we fabricated indeed lies within this range. Table 1 offers a comparision of predicted and measured efficiencies for this grating. The predictions of the Gaussian groove model compare well with the diffraction efficiencies we measured from actual gratings.
This model provides a useful tool in the future design and fabrication of nanoscale electron phase gratings. For example, figure 3 shows how the diffraction efficiency of the holograms can be manipulated to produce several different types of electron beams with engineered phase and intensity.

Conclusion
Here we described our studying and optimizing the efficiency of electron diffractive phase optics for use in high resolution electron microscopy. We described a process for fabricating spatially coherent nanoscale material phase gratings, and characterized the performance of simple prototypical phase gratings in a TEM. We developed a theoretical model for optimizing the design of the gratings for particular uses. We demonstrated diffractive phase gratings with 80 nm periodic feature sizes covering areas μ 50 m in diameter, and demonstrated grating periods down to 20 nm. We demonstrated phase gratings that produce equally intense −1st, 0th, and +1st diffraction order beams with 25% of the transmitted beam current each, gratings with suppressed 0th order beams (intensities in the 1st order up to 5 times greater) and blazed gratings that place 27% of the incident beam current into the +1st diffraction order. When installed in a TEM operated at 300 keV, these gratings are capable of providing multiple angstrom-scale electron probes with precisely manufactured phase dislocations. We demonstrate example nano fabricated holograms in which this diffraction efficiency is controlled in order to efficiently convert an incident electron beam into various engineered electron beams. This technology now makes possible shaped STEM probe techniques designed to produce more information about magnetically ordered, superconductor, semiconductor and biomolecular materials.
To determine the intensity of one diffraction order, we first measured noise in the image; we calculated the average background intensity I b in a part of the micrograph where no signal intensity was present. We then subtracted this background intensity from the entire image, then set to zero all pixels which had an intensity less than this. Finally, to measure the intensity of the nth diffraction order, we centered a circular measurement region with a diameter equal to the spacing between orders over the nth diffraction spot. We then measured the total integrated intensity inside the circle. We performed this measurement on an FEI 80-300 Titan TEM at 300 keV in low-angle diffraction at a camera length of 104 m. We used a gun lens strength of 6 and a spot size of 8, a μ 150 m C2 aperture to limit the extent of the incoming wave, and an illuminated area of μ 104 m. We then limited the outgoing wave with a μ 10 m objective aperture. The incident, transmitted and relative diffraction efficiencies of the beams formed by the grating are independent of aperture size and shape when absolute efficiency is measured as a function of incident intensity after the aperture [25]. We used a defocus of − × − 21.37 10 6 so that the spots did not saturate the detector but were still well spaced.

Appendix B. Nanofabrication considerations
To meet our stated design goals for efficient electron diffractive optics, we explored a large number of nanofabrication techniques for imprinting the phase grating onto electron-transparent materials. We have found a combination of grating material, charge alleviation layer, and patterning technique that provides good results, and discuss each of these separately in the following sections.

B.1. Grating material
While there are many electron-transparent materials that can be used for this purpose, we choose to use silicon nitride membranes because of its mechanical robustness, thermal stability under electron illumination, and availability. We used silicon nitride membranes from several manufacturers, ranging in thicknesses from 15 nm to 100 nm thick, suspended over windows from μ 80 m across to 2 mm. Low-stress 30-100 nm thick silicon nitride membranes purchased from SPI Supplies, Inc. performed best under long-term mill processing. We note that in general, a membrane thickness on the order of twice the final mill depth minimizes beam amplitude loss due to high-angle inelastic scattering while maintaining mechanical stability. In particular, we found that while 30 nm and 50 nm thick membranes block roughly 20% and 50% of the incident beam intensity, respectively, these standard thickness membranes routinely provide the greatest absolute diffraction efficiency for a 300 keV beam.

B.2. Nanofabrication optimization
We have applied several nanofabrication techniques for patterning electron diffractive optics, including high-resolution FIB milling, EBID, and EBL. We will concentrate our description here on the FIB-milling technique, since FIBs are present in many TEM labs and this maskless fabrication technique provides a quick method for making electron gratings. In particular, we used an FEI Helios NanoLab FIB to mill all gratings described in this work.
Modern FIB instruments offer many different parameters that can be used for milling the electron diffraction hologram pattern. The ion beam dwell time, number of passes, milling scan direction, beam current, ion beam convergence angle, addressable pixel spacing, total ion dose, and pattern complexity all play a large and interconnected role in the quality of the final grating. An in-depth discussion and review of these effects is provided in [24]. We performed a systematic exploration of this complex parameter space in order to find the best combination for manufacturing efficient gratings for electrons.
We created a series of arrays of linear diffraction gratings in which we systematically varied total ion dose, dwell time, number of passes, patterning order, and beginning membrane thickness. These dose-arrays were placed in a field emission TEM operated at 300 keV. Low angle electron diffraction patterns were recorded under identical illumination conditions for each grating. Examples of these diffraction patterns are shown in figure 5. The diffraction spots were defocused in order to examine the uniformity of the intensity of each beam-darker areas indicate an unwanted variation in groove width and depth across the grating area. We then measured the surface topology of each grating using an AFM with a small diameter tip (figure 4).
Many FIBs provide the ability to mill a pattern using a bitmap image or by direct programmable control of the beam path. We find that when using the bitmap patterning method, the pixel spacing of the magnified bitmap image should be an integer multiple of the minimum pixels spacing [24] of the FIB in order to avoid artifacts in the final structure due to nonuniform ion dosing. The scan direction of the beam should be chosen such that the slow scan axis is perpendicular to the grooves. While we have made decent gratings using both raster and serpentine scans during milling, we find best results when using a 'vector scan' technique [24]; taking full programmable control of the beam path such that one complete groove can be milled before moving on to the next element. For most purposes, we found that ten passes, reversing the milling order each time, resulted in the highest quality gratings. However, when milling patterns covering large areas, settling and movement of the silicon nitride membrane reduces the quality of the pattern-in these cases it is best to reduce the number of passes. We find that ion beam currents below 10 pA produce the finest features, but larger currents on the order of 20-50 pA are necessary to complete a pattern larger than μ 20 m diameter without major thermal drift over the course of the mill. For the pattern milled in figure 2, the ion beam current was set to 10 pA and the number of passes was 80 with a pixel size of 3.3 nm.

B.3. Charge alleviation layer
The nitride gratings must be coated with a thin conductive layer to alleviate charge. We have experimented with using sputtered Ni, Ti, Cr, and Au, thermally evaporated C, and Pt from ion beam induced deposition (IBID). We find that 15 nm of amorphous carbon sufficiently minimizes charging and causes little absorption in the transmitted beam. However, when the carbon-coated grating is placed in an upper condenser lens of a TEM and exposed to a beam over the course of several weeks, the carbon can migrate into the grating trenches and decrease the absolute diffraction efficiency. A 5-10 nm layer of Au deposited on a 1 nm Cr adhesion layer leads to a slightly lower absolute diffraction efficiency η i 1 ( ) , due to absorption and scattering, but the diffraction efficiency of such gratings remain stable for weeks under exposed conditions in the beam path of the TEM. We find that Pt deposited by IBID contains significant amounts of carbon which can migrate under electron beam exposure, but the grating diffraction . efficiency can be renewed by regular plasma cleaning. In all cases where a metal film is used, enhanced electron scattering decreases the absolute diffraction efficiency η i 1 ( ) and contributes to an unwanted background signal. However, a diffractive structure composed of silicon nitride grooves capped with a metal layer, produced either with FIB or IBID, modulates both the amplitude and phase of transmitted electrons. Such a mixed amplitude-phase grating can almost entirely suppress the 0th (undiffracted) order (see figures 5(d) and (e)). Parameter space map showing the predicted (background color) and measured (squares) first-order (top) and zeroth-order (bottom) transmitted diffraction efficiencies for electron phase gratings as a function of the depth Δh and width w of grooves. Each square data point represents a phase grating fabricated in a silicon nitride membrane using a unique combination of FIB milling parameters. Each squareʼs position describes the associated gratingʼs groove depth (horizontal axis) and width (vertical axis) measured by AFM scans, and the square color is the diffraction efficiency of the respective order measured by TEM diffraction. The labeled data points correspond to the gratings that produced the diffraction patterns in figure 5. The background color map represents a model that assumes symmetric grooves with uniform Gaussian-shaped depth cross sections and plane wave illumination. Both measured and predicted efficiencies in each plot share the same color scale denoted by the right-hand bar.

B.4. Hologram uniformity, spatial coherence and quality of imprinted phase
Spatial variations in the width and depth of grooves across the grating affect the amplitude uniformity inside each diffracted beam. These spatially dependent errors of FIB-milled gratings are primarily due to secondary dynamic processes such as redeposition, charging, heating, and membrane relaxation associated with the incidence of ions on the substrate surface. Large-area patterns with very fine pitch typically take several hours to mill, and changes in substrate tension or temperature can cause the substrate to drift at speeds on the order of nanometers per minute, which noticeably impacts the spatial coherence of the resulting structure. However, with an appropriate choice of a moderate ion beam current, a fewer number of patterning passes, and the application of a conductive layer to the membrane to prevent local charging, we have successfully produced spatially coherent gratings with 10 3 grooves over areas several tens of microns in diameter. We are in the process of developing a quantitative measure of spatial coherence.
Appendix C. Theoretical diffraction efficiency from a general mixed amplitude/phase grating The diffractive electron optical elements described in this work are thin membranes with a modulated thickness h x y ( , ). As the both the de Broglie wavelength of electrons and the maximum thickness of an electron-transparent membrane in a transmission electron microscope are necessarily small relative to the grating period, the thin grating condition under which wave interference due to propagation inside the grating material is negligible [21] λ ≪ h d (C.1) 2 is necessarily satisfied. Therefore, if an electron diffractive grating is illuminated by an incident electron wave ψ i , then the transmitted electron wavefunction immediately behind the grating is Let us examine how this wave propagates. Far from the grating, the outgoing electron wavefunction behind a grating can be described by the Fraunhofer formula [26]  (C.14) x y Equation (C.13) can be easily modified to take into account the shape of a finite-size aperture which limits the incident intensity; the diffraction efficiencies we will calculate, however, are unaffected by aperture shape and size [25]. Letʼs then rewrite t x y ( , ) so that we can compute the Fourier transform easily. In terms of the We now have t x y ( , ) written in terms of plane waves, but not as a linear superposition of plane waves. We can write the transmission function more simply if we perform the product. As the product of two plane waves with wave vectors k 1 and k 2 is another plane wave whose wave vector is the sum of the first two, we can rewrite a product of a sum of plane waves as a sum of plane waves whose coefficients are products over all terms whose total wave vector is constant. Each set s n describes one path which terminates at q f ; c n is calculated as the sum of products of the coefficient γ j m ,