1. Introduction

The interactions between electrons and light are indispensable to a range of technologies including radiation therapy [1], imaging the atomic structure of solids via nanoscale microscopy [2], and generating broadband, coherent light for free electron lasers [3]. Although the quantum interactions of free-space light with bound electrons have been studied extensively, the interactions of free electrons with confined light have only recently been made accessible by photon-induced near-field electron microscopy (PINEM) [4,5]. These interactions are realized by ensuring that the electrons and light are phase-matched in energy–momentum space, which allows for a wide range of phenomena including the Cherenkov and Smith–Purcell effects [6–8]. Using appropriately tailored light fields [9–11], the spatiotemporal and spectral properties of electron wavefunctions can be modulated via PINEM [12] in ways that are instrumental to a plethora of applications in medicine, electron acceleration, and radiation sources. The extent of modulation is determined by the strength of coupling between the electrons and light [4,5,10]. This coupling can be maximized using techniques such as shortening the pulse duration [13], amplifying the electric field overlapping with the electron [14], or extending the interaction length [13]. However, such designs are not yet commercially viable, which inhibits follow-on applications. This gap highlights the need for a scheme which can facilitate electron-photon coupling using an integrated platform. It would benefit investigations in quantum electrodynamics [15] as well as enable applications in electron energy loss spectroscopy [16] and PINEM [4].

A recent class of devices known as dielectric laser accelerators (DLAs) have shown tremendous potential in facilitating such interactions. DLAs can accelerate free electrons within a sub-millimeter interaction length to induce acceleration gradients in the range of GeV/m [17,18]. Compared to conventional radio-frequency particle accelerators, this represents an improvement of multiple orders of magnitude with respect to both performance and size. Therefore, DLAs herald a new class of devices which allow on-chip access to the interactions between electrons and light. Specifically, they offer the capability to optically supply, absorb, and modulate the kinetic energy of electrons. Using DLAs, these interactions could be pushed into the strong coupling regime [19,20], resulting in performance improvements that enable quantum entanglement phenomena between free electrons and confined photons [20–24]. In this context, the coupling strength per photon can be considered as a key performance indicator of a DLA because it determines the amount of energy transferred to/from the electron.

For the interaction to be maximized, the electron wavefunction and the light wave must be phase-matched along their overlap region [5]. This can be achieved by overlapping an electron beam (eBeam) and optical pulse at either perpendicular or parallel incidence to each other. In the scenario of perpendicular incidence, the eBeam traverses the alternating near-field of a grating in such a way that electrons are continuously accelerated by the positive phase-fronts of light along their trajectory [25]. Since this approach does not require light to be waveguided (unless imposed by the design [26,27]), a wide range of optical frequencies can be employed as long as the grating is thinner than its corresponding absorption depth [25]. Nevertheless, the nonzero incident angle of the light causes transverse near-fields which deflect the eBeam and therefore limit the scalability of this approach [28]. Alternatively, the scenario of parallel incidence employs an optical pulse co-propagating with the eBeam, which incurs negligible transverse deflection forces. This allows for longer coupling lengths [21] and cascaded, scalable designs [17,29]. The coupling strength can be maximized by amplifying the electric field per photon that is available to the electrons using techniques such as compressing the volume of the optical pulse into a waveguide mode [14], reducing the optical pulse duration [18], accessing whispering gallery modes [19], or resonating the field in a racetrack [11,21,30]. Such techniques have opened access to the quantized spectral broadening of the electron wavefunction [13] as well as pushed the interaction into the strong coupling regime [31] albeit with a high propagation loss. These advances and their limitations introduce the need for a scheme that optimizes the coupling efficiency in a commercially viable manner. It must be scalable [32] in order to act as the basis for practical devices that can maximize the kinetic energy gain and acceleration gradient. Such a scheme would support the vision of quantum electro-optic interactions and tabletop electron accelerators including follow-on applications.

In this work, we design a device to maximize the coupling between confined light and free electrons. It is designed as a Si photonic (SiP) integrated circuit operating at telecom wavelengths. Commercial SiP technology utilizes the complementary metal-oxide-semiconductor (CMOS) process flow to enable low-cost, large-scale deployment. Furthermore, by having Si as the core material, the design can benefit from the potential of dielectrics to host extended interactions resulting in a high excitation probability [21], and the high refractive index of Si which increases modal confinement as compared to Si₃N₄ [11,21,32]. Our proposed design allows for a modularized system employing well-developed circuit components that can be upgraded in a plug-and-play fashion. The key component is an exposed slot waveguide which hosts a supermode [33] to interact with co-propagating free electrons. Its input is a single-mode waveguide that connects to a sidewall of the slot via a mode evolution region (MER) [34,35]. Using a single input prevents the potential phase mismatch from two waveguides [14] as well as the imbalanced loss from their corresponding GCs. Additionally, building on the study in [14], we optimize the design parameters and characterize performance at two shorter wavelengths of 1310 nm and 1550 nm for which telecom-based components are readily available and the damage threshold is higher [36,37]. In doing so, we evaluate the coupling strength per photon over the interaction length and show that this efficiency metric directly maximizes the energy gain. Our scheme can also be used to maximize the acceleration gradient if required by the target application. It therefore offers a core building block on the roadmap towards optically manipulating the kinetic energy of free-electrons in a scalable, commercially-viable manner [38].

2. Device design

The specifications of the target design were constrained to ensure compatibility with an ultrafast transmission electron microscope (UTEM), which was previously used to test similar structures [13,31]. The steps that were taken to accommodate these compatibility constraints are described in the Supplement 1. Using the SiP platform, the device was designed with a Si core material grown on a SiO₂ buried oxide (BOX) layer without any cladding. A layout of the proposed device is shown in Fig. 1(a). Here, light is coupled into the chip from a surface, vertical grating coupler (GC) [39]. The coupled light is asymmetrically split [40,41] into two branches of which 10% is reflected by a Bragg grating (BG) to provide feedback during the alignment process and 90% is routed to the interaction region. The main component of the circuit is the slot waveguide shown in Fig. 1(b) and (d), which hosts an optical supermode to overlap with an eBeam at grazing incidence to the chip. It is connected to the circuit by a MER, shown in Fig. 1(c), which converts the fundamental transverse magnetic (TM₀) mode of from the GC into the required supermode of the slot (Fig. 1(c) and (e)). At the beginning and end of the interaction length, the cross-sectional geometry of the slot constrains the dimensions of the eBeam, as shown in Fig. 1(d).

 

Fig. 1. On-chip layout of the device which hosts an optical supermode to overlap with a free-electron beam. (a) Top view of the device layout consisting of an input grating coupler (GC) connected to a 90:10 splitter that sends 10% to a Bragg grating (BG) for alignment feedback and 90% to a slot waveguide via a mode evolution region (MER). An additional GC is placed at a fixed offset from the input GC for stronger feedback during coarse alignment. (b) Top-down view of the slot waveguide accommodating an eBeam that converges in the slot (the MER is not shown). (c) Top-down view of the MER with the simulated electric field distribution, E_0,Z, showing how the input TM₀ mode is converted to the supermode of the slot waveguide. (d) Front view of the slot waveguide at the point of entry or exit of the eBeam. (e) Front view of the slot waveguide with the simulated electric field distribution, E_0,Z, of the converted supermode.

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We design the slot waveguide to maximize the longitudinal electric field intensity of the optical supermode specifically in the region of overlap with incident free electrons. We target the device operation at wavelengths of 1310 nm and 1550 nm, corresponding to the lowest dispersion and insertion loss, respectively, of optical telecommunication systems. The required circuit components (i.e., GC, splitter, BG) are readily available at these center wavelengths. We neglect any limitations on the coupling bandwidth caused by dispersion effects [21] so the performance is only evaluated at these two wavelengths. The thickness of the Si layer, ${t_{\textrm{Si}}}$, was set as 220 nm corresponding to the standard thickness offered in the SOI platform. Within the overlap of the supermode with the eBeam, the coupling strength depends on the amplitude of its longitudinal electric field, ${E_\textrm{Z}}$, over the effective interaction length, ${L_{\textrm{eff}}}$. Using a dielectric like Si as the core ensures a relatively high damage threshold [21,36,42]. The high refractive index of Si also strongly confines the supermode and consequently ${E_\textrm{Z}}$, which is further amplified by compressing the mode volume in the slot [33]. The distribution of ${E_\textrm{Z}}$ for a given supermode is determined by the cross-sectional geometry of the slot represented by the widths of the gap, ${w_{\textrm{gap}}}$, between its sidewalls, ${w_{\textrm{Si}}}$. These dimensions determine the effective index of the supermode, ${n_{\textrm{eff}}}$, and its corresponding phase velocity, ${v_\textrm{p}}$, which set the electron velocity, ${v_\textrm{e}}$, such that ${v_\textrm{p}} = {v_\textrm{e}}$. They also determine the group velocity, ${v_\textrm{g}}$, and therefore the time period, $\Delta t$, over which the electron passes through the optical pulse of duration, ${T_{\textrm{pulse}}}$. We assume that the arrival times of the electron and optical pulse are synchronized such that they coincide in the middle of the interaction length. Therefore ${L_{\textrm{eff}}}$ is limited by $\Delta t$ as well as the decline in phase-matching as ${v_\textrm{e}}$ increases (and decreases) beyond ${v_\textrm{p}}$. Under these conditions, we characterize the three lowest order supermodes hosted by the slot waveguide (see Supplement 1 for details).

To find an optimum overlap between the electron and supermode, we emulate the process of aligning the eBeam to each supermode hosted by a specific waveguide geometry. This is done by varying its parameters to maximize the overlap and then calculating the corresponding coupling strength as shown in Fig. 2. The eBeam alignment is parametrized by its height from the BOX layer, ${h_\textrm{e}}$, and diameter, ${d_\textrm{e}}$, which set the divergence angle, ${\theta _\textrm{e}}$, based on the emittance of the eBeam for a given ${v_\textrm{e}}$ [12]. Together, these parameters geometrically limit the maximum interaction length, ${L_{\textrm{eff}}}$, as shown in Fig. 2(a). We approximate the spatial probability distribution of the electron wavefunction to be uniform within its spot size since energy can be exchanged at any $({x,y} )$ point in the overlap region. The figure of merit for the interaction is the coupling strength per photon [10,21], ${g_{\textrm{Qu}}}$, as shown in Fig. 2(b) and derived in the Supplement 1. The optimal distribution of ${E_\textrm{Z}}$ to overlap with the electron is shown in Fig. 2(c). During the interaction, the electron exchanges energy with the optical supermode in steps of ${\pm} \hbar \omega $, resulting in a quantized broadening of its kinetic energy spectrum [4,11,13], $\Delta {\varepsilon _\textrm{e}}$, and determines the acceleration gradient, G, as derived in the Supplement 1.

 

Fig. 2. Optimization of coupling strength per photon for the given slot waveguide dimensions. (a) Maximum achievable interaction length, ${L_{\textrm{eff}}}$, within the range of available eBeam diameters, ${d_e}$, and heights, ${h_e}$. (b) Coupling constant per photon, ${g_{\textrm{Qu}}}$, over the same parameters. A red dot indicates the maximum ${g_{\textrm{Qu}}}$. The blank regions indicate inadmissible parameters for which the eBeam was blocked by the sidewalls or BOX. The discontinuity in both plots is caused by switching the electron sources between thermionic and photoemission modes in the UTEM as explained in Section I of Supplement 1. (c) Distribution of the longitudinal electric field component, ${E_\textrm{Z}}$, of a supermode hosted by the slot waveguide. The solid and dotted red circles indicate the spot size at the focal point and at the ends of the slot, respectively. Dimensions of the slot waveguide are ${t_{\textrm{Si}}} = 220\; nm$, ${w_{\textrm{Si}}} = 300\; \textrm{nm}$, and ${w_{\textrm{gap}}} = 200\; \textrm{nm}$, which results in a ${g_{\textrm{Qu}}} = 0.4266$ for a supermode at $\lambda = 1550$ nm matching ${\varepsilon _\textrm{e}} = 196$ keV.

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The dependence of ${L_{\textrm{eff}}}$ on the eBeam parameters that are accommodated by the given slot dimensions is shown in the gradient of Fig. 2(a). Although we constrained ${L_{\textrm{eff}}} \le \; $500 µm to ensure that the device would fit into the sample holder of the UTEM, it was predominantly limited by the phase-mismatch caused by the accelerating and decelerating electrons. The discontinuity in Fig. 2(b) at a 20 nm diameter is caused by the different emittance of the eBeam when switching the electron source. The gradient of ${g_{\textrm{Qu}}}$ reveals its dependence on the alignment of the eBeam, or its misalignment tolerance. As ${h_\textrm{e}}$ increases, ${g_{\textrm{Qu}}}$ decays exponentially due to a proportional reduction in ${E_\textrm{Z}}$ [13]. It increases with a wider ${d_\textrm{e}}$ because the peaks of the supermode are split such that the ${E_\textrm{Z}}$ intensity is higher near the sidewalls than in the middle of the slot. This implies that having a tightly focused eBeam is not directly advantageous to the coupling strength (further validated by the characterization results in the Supplement 1). Counterintuitively, ${g_{\textrm{Qu}}}$ is not maximized at the longest ${L_{\textrm{eff}}}$ because of a decrease in the temporal overlap, $\mathrm{\Delta }t$, beyond the optimum ${L_{\textrm{eff}}}$. This is a physical implication of the finite durations of the electron and optical pulse. The interaction is also improved by the slight rotation of the nodes of the supermode, which increases ${E_\textrm{Z}}$ in the overlap.

Considering the damage threshold of Si and the losses of the GC, splitter, and MER, we calculate an optical pulse energy of 0.22 nJ and duration of 1 ps in the slot (see Supplement 1 for details). This enables a maximum ${g_{\textrm{Qu}}}$ of 0.4266 for the design case shown in Fig. 2, which produces an electron energy gain of 28.27 keV with an acceleration gradient of 1.05 GeV/m. A full characterization of the parameter space is presented in the Supplement 1.

3. Discussion

We evaluated the performance of the design based on its efficiency of inducing a quantum electro-optic interaction between the electron wavefunction and a classical electromagnetic field within the target experimental constraints. We therefore choose ${g_{\textrm{Qu}}}$ as the figure of merit. The practicality of our design enables it to be employed in a cascaded system by simply tuning the parameters of each device accordingly. In designing the device, we consider the following parameters (correlated quantities are indicated by a double arrow):

Our design process revealed a number of constraints, and identified ways of overcoming them, to further improve the coupling efficiency.

The damage threshold of Si limits G and correspondingly, the optical pulse energy and duration in the slot (see Supplement 1 for calculations). However, although G is limited by the damage threshold of Si, $\Delta {\varepsilon _\textrm{e}}$ is not. In this context, our scheme shows that the efficiency ${g_{\textrm{Qu}}}$ can be used to maximize $\Delta {\varepsilon _\textrm{e}}$ and therefore circumvent the limit on G. Hence, we maximize $\Delta {\varepsilon _\textrm{e}}$ by maximizing ${g_{\textrm{Qu}}}$ while still operating within the damage threshold of Si. Additionally, the damage threshold can be included in the design scheme by considering materials with higher thresholds (as indicated by their bandgap [36,37] [42]), account for the wavelength-dependent damage threshold, and cross-sectional areas (and corresponding MER loss) of every slot waveguide, and characterizing performance over different optical pulse energies and durations. Also, note that the energy in the optical pulse does not directly affect ${g_{\textrm{Qu}}}$, but it does increase the spectral broadening caused by the interaction (and therefore $G$). This affects the maximum interaction length at which phase-matching can be maintained, which in turn affects ${g_{\textrm{Qu}}}$.

Another limitation was the dephasing caused by ${v_\textrm{e}}$ changing beyond ${v_\textrm{p}}$. This is unavoidable if the target application requires maximum electron-photon coupling. However, if the goal is to maximize the electron acceleration (and conversely minimize deceleration), then the waveguide design could be modified so that ${n_{\textrm{eff}}}$ decreases along ${L_{\textrm{eff}}}$ such that ${v_\textrm{p}}$ increases along with ${v_\textrm{e}}$. This can be achieved by tapering the sidewalls of the slot [14], transitioning to a sub-wavelength grating [43,44], or cascading multiple such devices [17,30]. An edge coupler could also be incorporated at the end of the interaction to emit the co-propagating pulse with the electrons so that they continue to exchange energy beyond the chip. However, our simulations show that a lower ${n_{\textrm{eff}}}$ pushes the mode into the BOX layer thereby reducing its confinement and overlap with the eBeam, which implies that it cannot be reduced below the refractive index of the underlying BOX layer. This limitation can be overcome by suspending the waveguides or using a lift-off technique to separate the Si completely from the substrate. To accommodate a higher range of ${v_\textrm{p}}$, ${t_{\textrm{Si}}}$ could also be reduced to 145 nm. However, note that this weakens the confinement of the mode and consequently ${E_\textrm{Z}}$ in the overlap. A longer ${L_{\textrm{eff}}}$ would also require $\Delta t$ to be increased while maintaining that the electron coincides with the peak of the pulse envelope for as long as possible. It can be achieved by either increasing ${v_\textrm{g}} \to {v_\textrm{p}}$ to prevent dephasing or by increasing ${T_{\textrm{pulse}}}$ at the expense of lowering the optical power [13,18]. However, since ${v_\textrm{g}}$ depends on the material properties as well as dimensions, this parameter cannot be regulated as easily as ${v_\textrm{p}}$ and could be overcome by dispersion engineering the device [43] or incorporating other materials. Additionally, since ${g_{\textrm{Qu}}}$, and consequently $\Delta {\varepsilon _\textrm{e}}$ and G, are dependent on the energy, duration, and wavelength of the optical pulse, these laser settings may be used to further optimize the interaction as long as the fluence is less than the damage threshold of Si and SiO₂ [14,36] (as dscribed in the Supplement 1). Including ${T_{\textrm{pulse}}}$ as a parameter in the optimization process could also determine whether a longer pulse would improve ${g_{\textrm{Qu}}}$ or whether a shorter, higher power pulse is more efficient. The limits of ${T_{\textrm{pulse}}}$ depend on the laser source. Note also that a typical laser or eBeam source emits a finite spectrum rather than a single frequency or energy. So, the accuracy of our predictions can be improved by updating PINEM theory to consider the full interaction between an electron bunch and the range of frequencies in the optical pulse as well as phase-matching along the interaction length. The amount of energy reaching the slot waveguide can also be increased by minimizing the insertion loss of the circuit components, potentially at the expense of their bandwidth, however, this is not currently a limitation since the energy supplied by the laser is mainly limited by the damage threshold of Si. The GC could also be replaced by an edge coupler if permitted by the constraints of the application. Furthermore, the available ${E_\textrm{Z}}$ per photon can be amplified by looping around the ends of each sidewall to form a dual racetrack resonator configuration [30]. Such a configuration can recirculate the remaining optical energy of the pulse after the interaction and therefore harness constructive interference effects [21]. Additionally, the round-trip time of the racetracks can be matched to the repetition rate of a pulsed laser to synchronously pump the system resulting in a resonant amplification of the field strength [45]. If the efficiency is sufficiently high, then the device could even be operated with continuous-wave light [11]. Finally, our system can also potentially be modified by attaching a conductive coating to siphon any excess charge, thereby alleviating a critical bottleneck [11] in reducing the charging caused by the eBeam.

In comparison with previous designs and demonstrations, our proposed design achieves a high coupling efficiency by utilizing a co-propagating optical supermode in a Si slot waveguide at an optimal wavelength. This comparison is shown in Table 1.

Table 1. Comparative analysis of device designs enabling electron-photon coupling. Brackets indicate values which we calculated because they were not reported. Wg: waveguide, *: designed but not demonstrated, ⊥: perpendicular incidence, ||: parallel incidence

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The exchange of energy that occurs during the interaction can either accelerate the electron by absorbing energy from the optical mode as in the case of DLAs, or it can increase the energy in the optical mode by decelerating the electron as in the case of stimulated emission in a free electron laser. To produce an overall acceleration of the electron, its broadened energy spectrum must simply be skewed toward energy gain rather than loss. Our utilization of a commercially available SiP chip with a UTEM provides the basis for experiments that modulate the spatiotemporal properties of electrons for optical-electrical energy conversion or quantum information science. For example, the modulated electrons could directly be used in microscopy experiments that require attosecond electron pulses, using the electron coherence to probe the quantum state of matter [51], free-electron quantum-optical detection and free-electron homodyne detection [52], continuous-wave operation of laser-driven microscopes [11,53], enabling free-electron–bound-electron resonant interactions [54], and other such ideas. As a SiP-based design, it automatically benefits from the low-cost, high yield, and quick turnaround times of SiP technology. Additionally, SiP leverages commercial processes from the CMOS industry to produce highly accessible channels for the fabrication and testing of integrated devices, which has supported its widespread deployment in a variety of industries including optical communications, sensing, and computing. It has also facilitated an explosive growth in research and development resulting in the functionality of SiP devices being pushed toward their physical limitations. However, since electron-photon interactions are fundamental to the operation of opto-electronic devices, these limitations present an opportunity to investigate the physical mechanisms underlying the device operation. In this context, our work shows how SiP devices can be designed from the ground up to reimagine existing functionalities and explore new ones. For example, it improves upon existing DLA designs and can be used to investigate their quantum nature, which would provide insight into whether the electron energy spectrum exhibits quantized features [55] or whether this effect is eliminated by the limited coherence of the accelerated electrons [13]. It can also improve the prediction accuracy of electron energy gain and loss spectroscopy via electron-photon coincidence detection [31,56]. Furthermore, the ability to map entanglements onto photon-electron pairs via strong coupling could support a new class of devices which enable electron beams to be used in quantum information science [19,22,23,51,53]. Our scheme therefore offers a method to efficiently manipulate the coupling between free electrons and light on a SiP chip with the ability to optimize for either the energy gain or the acceleration gradient.

4. Conclusion

Our work reveals the potential of combining integrated photonics with electron microscopy. Using a known technology and experimental setup, we designed a device that optimizes electron-photon coupling along its interaction length in a technologically feasible approach. For a 0.22 nJ optical pulse of duration 1 ps in the slot, our design scheme achieves a coupling strength per photon, ${g_{\textrm{Qu}}}$, of 0.4266 resulting in a maximum energy gain of 28.27 keV at a wavelength of 1550 nm with a corresponding acceleration gradient of 1.05 GeV/m. Alternatively, the maximum acceleration gradient of 1.68 GeV/m is achieved at a wavelength of 1310 nm for which ${g_{\textrm{Qu}}}$ is 0.1928, resulting in an energy gain of 13.9 keV. Future work will involve testing on-chip devices and further developing the scheme to incorporate additional design parameters. By maximizing the electron energy gain via the coupling efficiency, our design scheme represents a core building block on the roadmap to realizing a portable, CMOS-based, integrated solution for quantum electro-optic interactions. It enables direct applications in electron acceleration, radiation sources, energy harvesting, and quantum information science.