Enhanced photon emission from free electron excitation of a nanowell

Rapid strides in nanofabrication have led to a wealth of promising techniques for generating and shaping light on a compact scale.^1–5 Methods of nanoscale photon emission include nanolasers,^6–11 quantum dots,^12–14 whispering gallery microcavities,^15,16 and free electron light sources.^17–25 In free electron light sources, emission can occur through a broad range of mechanisms including Smith–Purcell radiation,¹⁷ Compton scattering,²¹ Čerenkov radiation,²⁶ and transition radiation.²⁷ Free electron sources have attracted attention for their potential tunability, directionality, and compactness.^18,20 On the contrary, the efficiencies that can be achieved by free electron sources have not been systematically explored. It has recently been shown that fundamental upper limits on the emission rate can be expressed semi-analytically,²² but whether and how these limits can be reached by realistic devices are significant open questions.

One of the ways to extract light from free electrons is light emission by electrons impinging on a nanostructure,²⁸ which can include a number of underlying mechanisms, including the excitation and outcoupling of localized surface plasmons (LSPs) and surface plasmon polaritons (SPPs). In recent years, there have been numerous studies involving the imaging and detection of light emission from various nanostructures, including periodic gratings,^29,30 spheres,^31–33 triangles,³⁴ annular gratings,^35,36 wires,^37,38 and holes.^39,40 Other similar works use the term electron-induced radiation emission (EIRE) to describe similar emission mechanisms.^28,41–43 Among the many possible device configurations for extracting light emission from free electrons, directing electrons through a nanohole is one of the simplest due to simplicity of design for practical applications and minimization of disruption to electron beam, avoidance of possible material damage, and allowing for some degree of light confinement to enhance light–matter interaction. This makes the nanohole a key building block for photonic and plasmonic devices.^44,45

In this work, we show that the light emission from a free electron scattering off a nanohole can be optimized to within 90% of the theoretical upper limit²² in the visible to near-infrared regime. This is accomplished by encircling the nanohole with a nanoridge to form a structure we call the nanowell. Using three-dimensional (3D) finite-difference time-domain (FDTD) simulations, we demonstrate over two orders of magnitude enhancement from the nanowell over optimized alternative designs, including the bullseye and hole–groove structures. Our analysis shows that the strong light emission from the nanowell can be ascribed to a combination of scattering of the incident electron’s electromagnetic field (transition radiation) and the excitation and re-radiation of LSP modes along the ridge of the nanowell.

Such structures may find applications as on-chip light sources in nanophotonic circuits and other settings. Although there have been previous studies on gold nanorings and nanotubes^46–48 and plasmonic ring cavities⁴⁹ showing field enhancements when the structures were illuminated by light, these studies did not consider the case of incident free electrons. The current rapid advancement in the development of compact electron sources, such as dielectric laser accelerators,⁵⁰ plasma laser accelerators,⁵¹ and field emitter arrays,^52–54 may in the future allow for the integration of compact electron sources together with nanoscale radiation emitters.

We simulate light emission from the interaction of individual electrons with freestanding structured silver (Ag) films using three-dimensional (3D) finite-difference time-domain (FDTD) simulations. The FDTD method solves Maxwell’s equations with no approximations apart from spatial and temporal discretization and is implemented in the Lumerical FDTD Solutions software package.⁵⁵ The electron source is represented by a time-delayed dipole chain^23,56,57 with electric dipoles strung together along the linear trajectory of the electron and their dipole moments modulated smoothly in time to approximate the dipole moment of an electron moving at a constant velocity (a raised-cosine filter is used to turn the dipole amplitudes on and off gradually). By integrating the Poynting vector over a wide surface parallel to the surface of the Ag film, we derive the total emission probability Γ as well as its spectrum dΓ/dλ. The refractive index for Ag is derived from experimental studies,⁵⁸ fitted using an analytical multicoefficient model.⁵⁵ Further details about the simulation parameters are given in the supplementary material (Sec. SI1).

To verify the correctness of the simulation setup, including the dipole chain approximation, we performed simulations on a freestanding unstructured Ag film of 100 nm thickness. Over the range of electron energies used in this work (on the order of 10 keV), the simulated emission probabilities agree well with previously derived analytical formulas for transition radiation,^30,59 as shown in the supplementary material (Sec. SI2).

We then compared the light emission from several different types of patterned films. As shown in Fig. 1(a), the structures are etched on the freestanding Ag film of thickness 100 nm with a hole of diameter 100 nm through which the electron passes. The first design, which we call a nanowell, is the primary focus of this paper. It consists of a thin circular ridge of width 10 nm surrounding the hole, followed by a wider groove of width 200 nm (the effects of varying the various structure parameters will be discussed later). The second design consists of a bare hole, representing the case of minimal structuring of the Ag film. The third design, a hole–groove structure, is like a nanowell with a much wider ridge, consisting of a 80 nm groove etched 140 nm away from the hole. The fourth and final design is a bullseye structure consisting of concentric grooves surrounding the central hole. The bullseye shown in Fig. 1 has seven grooves of depth 80 nm and periodicity 100 nm. The structural parameters for all of these designs have been optimized for total emission probability, as described below.

The electron is sent through the hole at normal incidence. Previous authors have noted that light emission from nanoholes can depend strongly on the axial distance of the electron beam from the center of the hole, with substantially stronger light emission for off-axis beams that pass closer to the walls of the hole.^18,39,60 In Fig. 1(b), we plot the total emission probability Γ vs axial distance for an electron energy of 10 keV, where Γ calculated by integrating dΓ/dλ over a wavelength range of 400–900 nm. The nanowell is found to significantly outperform the other structures (bare hole, hole–groove, and bullseye), particularly for on-axis incidence. The emission probability for the nanowell is also much less sensitive to the axial distance, increasing by a factor of ∼8 from the center to the edge of the hole, compared to a variation of up to two orders of magnitude for the other structures. All the structures analyzed here, including the nanowell, have been optimized to achieve maximal total emission probability subject to their individual design constraints (see below on the hole–groove and bullseye structure descriptions). Figure 1(c) plots a snapshot of the instantaneous field intensity during on-axis passage of the electron through the nanowell, showing the excitation of and subsequent radiation from surface plasmons.

Figure 1(d) shows the emission probability spectrum for the four structures, with the electron beam placed 25 nm off-axis. The off-axis case is considered here so that the radiation from the other structures is non-negligible. For on-axis incidence, the results are skewed even more in nanowell’s favor [see the supplementary material (Sec. SI3)].

Over the wavelength range of 500–950 nm, the nanowell clearly outperforms the bare hole, hole–groove, and bullseye structures. However, for wavelengths below 500 nm, the nanowell’s performance drops below the other designs. One reason for this is that all the structures, including the nanowell, have been explicitly optimized for the total emission probability over the 400–900 nm range (i.e., visible and near-infrared wavelengths), a regime in which LSP coupling is dominant. By contrast, the less-structured hole–groove and bare hole designs are less able to exploit the LSP modes, and their emission peaks are close to the bulk plasmon frequency for Ag near 330 nm. The peaks in the bullseye emission spectrum can be attributed to the LSP modes localized to the edge of the hole and bullseye ridges, but the total emission probability is far less than the optimized nanowell [see Fig. 1(b)]. Field profiles for the bare hole, hole–groove, and bullseye structures are shown in the supplementary material (Sec. SI4).

It is useful to compare these emission spectra to a theoretical upper bound derived by convex optimization of the formula for the electromagnetic work done.²² The bound has the form

where $χ̄̄$ is the material susceptibility, $Finc=(Einc,Z0Hinc)T$ denotes the electromagnetic fields of the incident electron, Z₀ is the impedance of free space, and ξ_rad = η(1 − η) is a radiative efficiency factor derived from η = dΓ/dΓ_loss, the ratio of emission probability dΓ to electron energy loss dΓ_loss. The gray dashes in Fig. 1(d) indicate the values of dΓ_max/dλ for a bare hole, calculated using the frequency-dependent refractive index of silver and a loss ratio η = 4.5 × 10⁻⁴ obtained from electron energy loss simulations [see the supplementary material (Sec. SI4)]. In the near-infrared regime (∼800 nm), the emission spectrum of the nanowell can approach 90% of the bound (see below). By comparison, the emission spectra of the other three designs do not approach the bound anywhere in the 500–950 nm range.

The nanowell, hole–groove, and bullseye structures featured in Fig. 1 have been optimized to maximize their total emission probability [see the supplementary material (Sec. SI5)]. The film thickness and the diameter of the central hole are kept constant. For the hole–groove structure, we swept over the groove depth h (10–100 nm) and the hole-to-groove distance d (100–600 nm). The groove width w was kept constant at 80 nm as we found that the dependence of emission on w is weak. Similarly, the optimization of the bullseye was performed by sweeping over the radial periodicity P (100–500 nm) and the groove depth h (60–95 nm), subject to the constraint that the diameter of the outermost ring does not exceed 2.5 µm (for computational tractability) and with the grooves and ridge widths taken to be equal. In particular, this parameter range covers the bullseye structures previously studied in the literature.^{35,36,60–62} Due to the constraint on the radius of the outermost ring in the optimization sweep, larger values of P specify a smaller number of rings, but we separately verified that the optimized structure indeed has higher emission than the large-P bullseyes with at least four rings [see the supplementary material (Sec. SI6)]. Most previously studied bullseye structures have typically consisted of 4–8 rings.

It is noteworthy that the bullseye, even after optimization, provides significantly lower light emission than the nanowell, considering that the bullseye structures have been extensively investigated for the imaging of plasmonic modes.^35,36 This can be attributed to the fact that even though the bullseye structures are used for imaging and single quantum dot emission,^63–65 their efficiency as a free-electron excited radiative source does not appear to have been fully investigated. (Although the bullseye considered here has a hole in the center, unlike the ones previously studied, the hole does not substantially affect the character of the light emission.)

Figure 2 shows how the light emission of the nanowell varies with the various structure parameters, particularly the hole radius R, ridge width d, groove width w, groove depth h, and incident electron energy. In Fig. 2(a), we see that the optimized emission holds over a relatively wide range of groove depths and widths. Figure 2(b) shows that changing the hole radius R does not significantly shift the optimal choice of ridge width d from around 8 nm, whereas the emission probability increases with decreasing hole radius R. (We chose d = 10 nm and R = 50 nm, since excessively small feature sizes would be impractical to fabricate.) Moreover, Fig. 2(c) shows that the groove depth can be used to tune the emission spectrum by changing the wavelength of the emission peak. As the groove depth increases, the peak intensity strengthens, bringing the emission peak to within 90% of the theoretical bound.

In Fig. 2(d), we see that changing the electron energy has little effect on the peak wavelength. This is in stark contrast to other methods of generating free electron radiation, including Compton scattering²¹ and Smith–Purcell radiation,¹⁷ where the frequency peaks are usually strongly dependent on the electron energy. This suggests that the photon emission in our scenario is largely governed by resonances of the material structure, independent of the excitation source similar to many other free electron-based light sources; however, Fig. 2(d) does show stronger overall emission at larger electron energies.

The enhanced light emission from the nanowell appears to be tied to the formation of LSPs on the surface of the ridge. In Fig. 3(a), we compare the responses of the structure to a free electron and an electric dipole placed at 25 nm off-axis and on the same level as the surface of the metal. The similarities between the two curves corroborate our previous discussion about the emission being enhanced by structural resonances independent of the driving source. Figure 3(b) shows the field intensity distributions at the resonance peaks (with the dipole source), which indicate the formation of surface plasmons on both sides of the ridge. This is reminiscent of studies on thin hollow cylindrical rods, which have found that the dispersion relations in those structures predict resonance modes at longer wavelengths.^46–48,66 Similar resonances can be excited by alternative illumination conditions, such as plane waves.⁴⁶ For the 800 nm peak, the intensity distribution appears to be uniform over the entire edge of the ridge, indicating that this is a zero-order LSP mode. Both resonances are strongly confined to the ridge of the nanowell, as shown in the right panels of Fig. 3(b).

Despite the apparently strong confinement of the fields to the ridge, the presence of the ridge alone is insufficient to explain the strong light emission. In Fig. 3(c), we plot the emission probability for different radii R′ for the overall structure, ranging from R′ = 1.25 to R′ = 0.165 µm [the latter corresponds to the case where only the ridge exists without any groove, as shown in the lower panel of Fig. 3(d)]. As R′ decreases, the light emission diminishes and, eventually, blueshifts in the extreme case where only the ridge is left. This implies that the performance of the nanowell depends on the combination of the groove and ridge. This finding is confirmed by frequency domain simulations (see the supplementary material, SI7).

We have shown that light emission from free electrons passing through a thin film nanohole can be enhanced by as much as 100 times in the presence of a nanowell consisting of a simple ridge and groove. The nanowell significantly outperforms the other structures, including bullseye structures, and achieves emission probabilities within 90% of a previously derived theoretical limit.²² The emission peak can be tuned over visible and near-infrared wavelengths by changing the depth of the groove.

The predicted enhancement of the light emission by a nanowell is suitable for experimental validation. Although the small features of the circular nanoridge (10 nm thickness or less) may pose difficulties for fabrication, the robustness of the resonant peaks may persist even if there are some fabrication-induced irregularities in the structure. We have performed additional simulations with surface roughness on the Ag film and/or roughness in the ridge and groove; for experimentally realistic roughness amplitudes, we find that the emission signal does not exhibit significant degradation [see the supplementary material (Sec. SI9)]. Moreover, the ridge width can be increased to around 30 nm, which would still leave an emission enhancement factor of 30–50 relative to the other structures.

An observation of strongly enhanced emission from a nanowell could lead to a wide range of possible applications, such as the biological and chemical sensing of molecules,⁶⁷ for which the nanowell design is especially advantageous due to its high surface-area-to-volume ratio.⁶⁸ Another intriguing application is the use of free electron-driven nanowell-based light sources as part of nanophotonic circuits. Thanks to the advancements in compact electron sources,^50–54 such devices could be achieved in the near future. Finally, deeper nanowell structures could be used in electron-plasmon scattering experiments²¹ for efficient plasmon excitation on the surface of the ridge walls.

See the supplementary material for more in-depth discussions of the parameters of simulations, approximation of the transition radiation, optimization of hole–groove and bullseye structures, and frequency domain simulations that support our findings.

This research was supported by the Agency for Science, Technology and Research (A*STAR) Science and Engineering Research Council (Grant No. A1984c0043) and Singapore MOE Academic Research Fund Tier 3 Grant No. MOE2016-T3-1-006, Tier 1 Grant Nos. RG187/18 and RG148/20, and Tier 2 Grant No. MOE2019-T2-2-085. L.J.W. acknowledges the support of the Nanyang Assistant Professorship Startup Grant. A.N. acknowledges the A*STAR Singapore International Graduate Award (SINGA) scholarship. The authors acknowledge fruitful discussions with Y. Yang and Y. J. Tan.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.