1. Introduction

Despite the essential physical distinctions between electrons and light waves, their time-independent wavefunctions can be expressed as solutions of the scalar Helmholtz equation, if we ignore their spin properties. This indicates that electron beams can feature transverse modes with structured phase fronts in the paraxial approximation, analogous to optical beams [1]. The structured electron beams have recently been generated [2–8], and this idea is attracting considerable attention as it can provide innovative applications. For example, electron vortex beam, which is a paraxial electron wave with a quantized phase term exp (iℓφ), where ℓ is an integer and φ is the azimuthal angle, is expected to be a novel tool for magnetically or chirally dependent microscopic analysis [9]. However, only a few practical demonstrations of vortex beams have been presented [3, 10, 11], partly owing to the low generation efficiencies of the elements developed so far.

Typically, structured electron beams can be generated and manipulated using a holographic diffraction grating (hologram). All the electron holograms reported so far were fabricated using focused ion beam (FIB) milling because of its high-resolution milling capability. However, these holograms were mostly inefficient binary-amplitude holograms because they were thick and metal-coated resulting in significant reduction of electron transmission. The first-order diffraction efficiency of a binary-amplitude hologram is known to reach only a maximum of 10.13%, compared with the expected 40.53% efficiency of an ideal (lossless) binary-phase hologram, assuming the duty cycle of the grating slits is 0.5. Moreover, FIB milling is generally time consuming and costly. Although several groups have reported thin electron phase holograms with higher diffraction efficiencies [12–15], their performance is proportional to the technical difficulty of the FIB process. Further, the mass-production for future applications would be very difficult, because the membrane elements’ performance all too easily degrades owing to electron irradiation, even under normal conditions [16, 17].

Herein, we demonstrate a new method of fabricating electron phase holograms by combining femtosecond laser ablation [18] and laser interference processing [19]. This manufacturing technique is faster than FIB because all the grating slits are fabricated via single-shot irradiation of a femtosecond laser pulse. Furthermore, the flexibility in designing holographic pattern using FIB milling is substantially maintained by the phase modulation of a laser beam using a computer-controlled spatial light modulator (SLM).

2. Phase hologram material

To create electron phase holograms, we used thin single-crystal silicon membranes (US100-C35Q33, SiMPore Inc.) with sufficiently high electrical conductivity to prevent charge-up of the hologram. The phase shift of electron waves in a non-magnetic material is given by [20]

Additionally, this membrane contributes to high electron transmittance because the inelastic mean free path λ_mfp for 200-keV electrons in silicon is 150 nm [22, 23], which is significantly larger than the membrane thickness. The transmission coefficient is expected to be

For an ideal (a = 1) binary-phase hologram containing a modulation reiteration of 0.5 duty cycles, the first-order diffraction efficiency is η_±1 = 40.53%, which is the ratio of the first-order diffraction beam power to the incident beam power. The electron transmittance of this membrane is 0.89² ≈ 0.79 if there are no other losses; thus, diffraction efficiency of around 36% is expected.

3. Electron phase hologram fabrication

To demonstrate our technique, we fabricated a fork-shaped electron phase hologram, which is commonly used to generate electron vortex beams. Figure 1 illustrates our interference processing method. Herein, an object beam in the fundamental Gaussian transverse mode is converted to an optical vortex using a SLM. Figure 2(a) shows the applied phase modulation pattern, which has a topological charge of |m | = 1. The object beam is focused onto the silicon membrane with the clear aperture size of 30 µm in diameter. Because the focusing lenses are configured as a 4f system, laser energy was delivered even at the beam center on the membrane, which corresponds to the singular point of the vortex beam. The membrane is processed by a laser pulse with a fork-shaped interference pattern generated using the object and reference beams. Herein, the experimentally observed threshold fluence was about 0.1 J/cm².

 

Fig. 1 A schematic of the experimental two-beam interference processing setup. The laser source (Spirit One, Spectra-Physics) delivers 40 µJ and 311 fs pulses with the center wavelength of 1,041 nm. The linearly polarized output beam is collimated to the 1/e ² width of 6 mm by a beam expander (not shown) and separated into object and reference beams using a polarizing beam splitter (PBS). The object beam is modulated using a spatial light modulator (LCOS-SLM, Hamamatsu Photonics K. K.) and focused onto a membrane mounted on a 3-axis linear stage by a convex lens of 750-mm focal length (L1) and an objective lens of 10-mm focal length (L2). These lenses are configured as a 4f system, which comprises a cascade of two Fourier transforms. The reference beam is focused using a convex lens of 300-mm focal length (L3) at an incidence angle of about 45 degrees. A half-wave plate placed before the PBS (HWP1) is used to control the branching ratio of the laser power. The polarization direction of the reference beam is rotated to match that of the object beam using a half-wave plate placed after the PBS (HWP2). An optical delay is inserted into the reference beam path to synchronize the two laser pulses on the membrane. Fabrication is conducted in atmosphere at room temperature and atmospheric pressure.

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Figure 2(b) shows a scanning electron microscope (SEM) image of a fabricated hologram. Herein, the grating slit interval was about 1.5 µm, resulting from a laser wavelength of 1,041 nm and a crossing angle of 45 degrees between the object and reference beams. Although the top and bottom slits were slightly distorted, this had a negligible effect on electron vortex generation because the incident electron beam was limited to irradiating the central area. In addition, Fig. 2(c) shows a scanning probe microscope (SPM) image of a fabricated hologram. Because the surface of each grating bar was flat within the 4.5-nm height resolution of the SPM, the hologram was strongly suggested to be binary-shaped.

Figure 2(d) shows a bright-field image of the hologram, obtained using a transmission electron microscope (TEM) equipped with a field-emission gun (JEM-2100F, JEOL Ltd.) operating at 200 keV. Here, the diffraction contrast (indicated by arrows) suggests that the electron transmittance was lower in the unprocessed area than in other regions, probably due to high-angle scattering processes that are sensitive to the orientations of atomic planes or strings (e.g., Bragg scattering or dynamical scattering by the channeling effect) because of the high crystallinity of the single-crystal silicon membrane. However, as shown later, the function as an electron phase hologram was sufficiently realized.

 

Fig. 2 Images of fabricated electron holograms. (a) Phase pattern applied to the SLM to generate an optical vortex beam. (b) SEM image of a fabricated hologram, created by forming a fork-shaped holographic pattern on the membrane. The scale bar represents 10 µm. (c) SPM image of a fabricated hologram. The height resolution of the SPM was 4.5 nm. The hologram measured here was different from that used for the SEM observation. (d) Bright-field TEM image of the hologram shown in (b), where the arrows indicate areas of diffraction contrast.

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4. Electron diffraction observation and characterization

The electron diffraction patterns produced using the fabricated hologram were observed in a high-dispersion diffraction mode of the TEM, in which a long camera length of 80 m was applied because the diffraction angle θ of the hologram was very small: θ ≈ 2.51 pm/1.5 µm = 1.7 µrad. The diffraction patterns were obtained from a circular region of diameter 8 µm at the center of the hologram. Figure 3(a) shows the observed diffraction pattern. Herein, the zeroth-order diffraction in the center exhibits a unimodal intensity profile with an Airy disk, on either side of which we can clearly see two doughnut-shaped intensity profiles, strongly suggesting the generation of electron vortices. Figure 3(d) shows the relative efficiency δ_q, which is defined as the q-th-order diffraction beam power divided by the total beam power for all diffraction orders. The experimental relative efficiencies ${\delta }_q^{\exp }$ (black bars) indicate that the zeroth- and first-order diffraction powers were equal.

To estimate the topological charge, similar to optical vortices, we converted the electron vortices into the Hermite–Gaussian mode by applying a cylindrical-lens-like phase modulation using a stigmator in the TEM [24, 25]. The bimodal intensity profiles shown in Fig. 3(b) suggest that the electron vortices at the first-order had topological charges of |m| = 1. In addition, the orthogonal directions of the two profiles (dashed lines) suggest that the charges had opposite signs.

Next, we compared the experimental results with numerical simulations. For simplicity, we assumed that the electron transmittance through the membrane region of the hologram was spatially uniform. We defined the transmission function of the hologram as follows:

The electron diffraction pattern shown in Fig. 3(c) was obtained by calculating the Fraunhofer diffraction ${\left|ℱ\left[\psi ({\mathbf{r}}_{\perp })\right]\right|}^2$, where ψ(r_⊥) = T(r_⊥)ψ₀(r_⊥) is the electron wavefunction just after passing through the hologram, ψ₀(r_⊥) is the wavefunction for electrons incident on the hologram, and ℱ[·] denotes the Fourier transform. Here, the calculated results are in good agreement with the experimental results shown in Fig. 3(a). The simulated relative efficiency ${\delta }_q^{\text{sim}}$ is also in good agreement with the experimental efficiency ${\delta }_q^{\exp }$, except for the slight asymmetry shown in Fig. 3(d). In contrast, the simulation for a binary-amplitude hologram (a = 0) shows that the relative efficiency ${\delta }_q^{\text{amp}}$ of the first-order diffraction beam is much lower.

 

Fig. 3 Electron diffraction results for the fabricated hologram. (a) Experimental electron diffraction pattern, observed via TEM in a high-dispersion diffraction mode. The diffraction orders are shown above each pattern. (b) Mode-converted diffraction pattern, obtained by applying cylindrical-lens-like phase modulation. The two orthogonal dashed lines indicate the orientations of the two Hermite–Gaussian profiles. (c) Simulated electron diffraction pattern, based on the holographic mask pattern shown in the SEM image in Fig. 2(b). (d) Relative efficiencies δ_q of each diffracted electron beam to the transmitted electron beam power. Here, ${\delta }_q^{\exp }$ (black bars), ${\delta }_q^{\text{sim}}$ (gray bars), and ${\delta }_q^{\text{amp}}$ (white bars) are the experimental results, simulation results, and simulation results for a binary-amplitude hologram, respectively.

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Figure 4 shows the calculated electron probability distributions for ${\left|{\psi }_0({\mathbf{r}}_{\perp })\right|}^2$ and ${\left|\psi ({\mathbf{r}}_{\perp })\right|}^2$ under the same simulation conditions, namely a = 0.42. Here, we estimated the power attenuation ratio γ_att as

Consequently, we estimated the first-order diffraction efficiency ${\eta }_{\pm 1}={\gamma }_{\text{att}}\times {\delta }_{\pm 1}^{\exp }$ to be greater than 19%. Notably, this is the minimum efficiency possible in our simulation. Indeed, when we instead used a = 0.89 and Δϕ = 0.75π, we obtained an efficiency η_±1 of 27%.

 

Fig. 4 Calculated probability distributions for incident electrons, ${\left|{\psi }_0({\mathbf{r}}_{\perp })\right|}^2$ (left), electrons just after passing through the hologram, ${\left|\psi ({\mathbf{r}}_{\perp })\right|}^2$ (right). The grating bars show an electron transmittance of about 0.18.

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5. Discussion and conclusion

In this study, the estimated first-order diffraction efficiency η_±1 ≧ 19% is almost twice the maximum value that can be obtained by binary-amplitude holograms (10.13%), suggesting that the phase modulation action by the hologram provided a significant contribution. We also estimated the diffraction efficiency from the experimental results and obtained a value greater than that from the amplitude hologram, although finding a way to accurately determine the diffraction efficiency via TEM remains a task for future work. If we are to reduce scattering losses and increase diffraction efficiency, it will also be important to utilize a more appropriate phase retarder material, such as amorphous carbon, which is widely used in phase-contrast electron microscopy [26].

Although the mask quality in the present demonstration was relatively low compared with that of FIB-milled holograms, we were still able to successfully generate electron vortices and directly measure the topological charge. Although our ability to fabricate vortex holograms with higher charges is currently limited by the spatial resolution of the laser process, using a shorter-wavelength laser source should help to resolve this issue in the future. Furthermore, our laser interference processing technique could be used to fabricate not only holographic diffraction gratings but also holographic zone plates, simply by changing one optical beam to a coaxial multi-ring pattern [27]. As demonstrated in a previous study [28], such holographic zone plates can be beneficial if used at the aperture position of a condenser lens in a TEM, where the electron beam’s wavefront has finite curvature. It could also be interesting to utilize multiple-beam interference to create more complex two-dimensional lattice structures [29].

In conclusion, we have fabricated electron phase holograms via single-shot interference laser processing. To the best of our knowledge, this is the first demonstration of an electron phase hologram fabrication process suitable for mass-production. We have also successfully generated electron vortex beams with first-order diffraction efficiencies of more than 19% using such phase holograms. This single-shot two-dimensional processing technique should enable us to provide electron phase diffractive elements for a wide range of research and industrial fields, and we hope it will make a significant contribution to the electron optics field.