🔬 mesoscale physics

Studying electron beam coherence using plasmon interference

This paper provides conclusive experimental evidence that cathodoluminescence signals from different lateral regions of an extended electron beam are mutually incoherent, whereas signals from distinct plasmonic scatterers within a single electron's evanescent field are coherent, offering a pathway to recover coherence through correlation measurements that erase electron trajectory information.

Original authors: Evelijn Akerboom, F. Javier García de Abajo, Albert Polman

Published 2026-08-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Evelijn Akerboom, F. Javier García de Abajo, Albert Polman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where tiny, invisible messengers zoom through space, carrying secrets about the materials they touch. These messengers are electrons, the same particles that power your electronics, but when they are fired at high speeds in a microscope, they behave like both solid bullets and rippling waves. This dual nature is the heart of quantum mechanics, a branch of physics that describes how the universe works at the smallest scales. Usually, we think of electrons as simple dots of negative charge, but when they move fast enough, they spread out like a wave, capable of being in many places at once. Scientists use these "electron waves" to take pictures of things too small to see with light, like viruses or the atoms inside a computer chip. But there's a catch: when these electron waves hit a material, they can make it glow with a special kind of light called cathodoluminescence (CL). The big question scientists have been asking is: does the "wave-ness" of the electron matter for this glow? If an electron wave is spread out over a wide area, does the light it creates from different spots interfere with itself like ripples in a pond, or does it just act like a bunch of separate, non-interacting bullets? Understanding this helps us build better microscopes and perhaps even new types of quantum computers that use light and electrons together.

This paper dives into that exact mystery, acting like a detective story to figure out when electron waves play nice together and when they don't. The researchers, working with a team of physicists, set up a series of experiments to see if they could make the light emitted by an electron beam interfere with itself. They started with a simple setup: a thin film of silicon nitride (a material often used in windows for microscopes) and a beam of electrons. First, they focused the electron beam into a tiny, sharp point, about 4 nanometers wide. Then, they deliberately "defocused" the beam, spreading it out into a fuzzy spot about 6 micrometers wide—roughly the width of a human hair. You might think that because the electron beam is a wave, spreading it out would create a beautiful, complex interference pattern in the light it emits, like the colorful swirls you see on a soap bubble. However, the results were surprisingly counterintuitive. The team found that even though the electron beam was wide and had the potential to be a coherent wave, the light it produced from the thin film did not show any interference. Instead, the light from the wide beam was just a simple, messy sum of all the little lights from every part of the beam, acting as if each tiny part of the electron was a separate, independent particle. It was as if the electron "knew" exactly where it hit the film, destroying any chance for the wave parts to dance together.

But the story doesn't end there. The researchers then switched to a more complex scene: two tiny, gold-coated pillars standing next to each other, separated by a small gap. They fired a single electron beam between these pillars. In this case, the electron's invisible "near field" (a sort of electromagnetic aura that surrounds the moving electron) could reach out and touch both pillars at the same time. Here, the magic happened. The light emitted from the two pillars did interfere, creating clear, rhythmic patterns in the data. This proved that a single electron can coherently excite two separate objects if they are close enough to be touched by its near field simultaneously. The key difference was that in the first experiment, the electron's position was too well-defined (it hit a specific spot on the film), so the "which-path" information was preserved, killing the interference. In the second experiment, the electron was in a superposition of interacting with both pillars at once, erasing the "which-path" information and allowing the wave nature to shine through.

The paper concludes by suggesting a way to force the first scenario to behave like the second. They propose a new experiment where, after the electron hits the material, you would measure the electron's path in a way that makes it impossible to tell where it started. If you could "erase" the information about which part of the wide electron beam hit the sample, you might be able to recover the interference patterns. This would be like taking a photo of a crowd of people, but then blurring the photo so much that you can't tell who stood where, suddenly making the whole crowd look like a single, unified wave. While this final idea is still a proposal and hasn't been built yet, the experiments they did perform provide solid proof: for a wide electron beam hitting a flat surface, the light adds up without interference, but for a single electron touching two nearby objects, the light dances in perfect sync. This helps scientists understand the limits of how we can use electron waves to control light, a crucial step for the future of nanotechnology.

Technical Summary: Studying Electron Beam Coherence Using Plasmon Interference

Problem Statement
While electron microscopes have long utilized the wave nature of electrons for structural imaging, the role of electron beam (e-beam) coherence in optical excitation processes, specifically cathodoluminescence (CL), remains an open question. Standard interpretations of electron energy-loss spectroscopy (EELS) and CL often treat the electron as a classical point charge. However, recent developments in photon-induced near-field electron microscopy (PINEM) and entanglement experiments suggest the electron acts as a quantum wave packet. A critical theoretical gap exists regarding the conditions under which CL excitation is sensitive to the lateral coherence of the e-beam. Specifically, it is unclear whether signals arising from different lateral regions of an extended electron wave interfere, or if they sum incoherently, and how "which-path" information constrains these observable interferences.

Methodology
The authors employ angularly resolved (AR) cathodoluminescence spectroscopy to investigate the coherence properties of electron-matter interactions under two distinct geometries:

Transition Radiation (TR) from an Extended Beam: The team measures AR-CL from a thin (15 nm) Si $_3$ N $_4$ film using a 30-keV electron beam. They compare two conditions: a tightly focused beam (~~4 nm spot) and a defocused beam (~~6 $\mu$ m spot). The defocused beam allows for the examination of whether lateral coherence across the beam width leads to interference in the emitted light.
Plasmonic Scatterers in the Evanescent Field: The team utilizes electron beam-induced deposition (EBID) to fabricate pairs of gold-coated nanopillars with varying heights and lateral separations. They remove the substrate beneath the pillars using focused ion beam (FIB) milling to eliminate transition radiation and substrate reflections. They perform AR spectral CL measurements while varying the beam position: grazing a single pillar, or passing between two pillars.
Data Analysis: The researchers analyze the angular and spectral distribution of the emitted light. To disentangle temporal components, they apply Fourier transforms to the spectral data, converting frequency-domain interference patterns into the time domain to identify time-of-flight (TOF) delays and phase relationships.

Key Contributions and Results

Incoherent Summation in Extended Beams: Contrary to the intuition that a laterally coherent defocused beam might produce coherent interference, the experiments on the Si $_3$ N $_4$ film demonstrate that the CL signal is an incoherent sum of transition radiation generated across the beam spot. Even though the electron beam possesses significant lateral coherence, the angular emission profile of the defocused beam is indistinguishable from that of the focused beam. Theoretical modeling confirms that without post-selection of the electron state, the excitation probability is an incoherent sum weighted by the transverse electron density.
Coherent Excitation via Evanescent Coupling: In the plasmonic pillar experiments, the authors observe clear interference patterns when a single electron passes between two pillars spaced within the range of its evanescent field. Fourier analysis reveals time-delay components corresponding to the electron traveling between the two structures. This confirms that a single electron can coherently excite spatially separated scatterers simultaneously, provided they are within the electron's near-field range. Crucially, this coherence arises from the simultaneous coupling of the electron's near field to both structures, not from the lateral coherence of the electron wave packet itself.
Loss of Interference with Separated Scatterers: When the pillars are separated beyond the extent of the electron's evanescent field and excited by a defocused beam (where the beam spot covers both pillars), the interference between the two pillars vanishes. The signal becomes an incoherent sum of the radiation from each pillar. This occurs because the "which-path" information (i.e., which pillar the electron interacted with) is preserved in the geometry, preventing the emergence of interference fringes.

Significance and Claims
The paper provides conclusive experimental evidence that the lateral coherence of an extended electron beam does not translate into coherent excitation of spatially separated emitters in a standard CL measurement. The authors assert that for interference to be observed in the far field, the "which-path" information of the electron trajectory must be erased.

The study establishes a clear distinction between two regimes:

Incoherent Regime: Signals from different lateral regions of an extended electron wave do not interfere; the total signal is an incoherent superposition.
Coherent Regime: Signals from structures coupled simultaneously within the evanescent field of a single electron remain mutually coherent and produce interference.

The authors propose a future measurement geometry to recover coherence across the e-beam spot: by correlating emitted light with post-selected electron states (e.g., using a diffracting structure like twisted bilayer graphene to erase spatial path information), it may be possible to retrieve interference from coherently excited CL signals across the coherent part of the beam. They note that while this is experimentally challenging, it is feasible and would further validate the quantum mechanical description of electron-light interactions.

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