Preamble

An Inverse-Cherenkov Dielectric Laser Accelerator Boosted by Surface Plasmon Polaritons

Dezheng Song,¹,† Zhuoer Hong,²,† Minghao Liu,¹ Hang Xu,² Senlin Huang,²,‡ and Liu Weihao¹,§

¹College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 211106, China
Key Laboratory of Radar Imaging and Microwave Photonics (Nanjing University of Aeronautics and Astronautics), Ministry of Education

²State Key Laboratory of Nuclear Physics and Technology & Institute of Heavy Ion Physics, School of Physics, Peking University, Beijing 100871, China

On-chip particle accelerators, especially dielectric laser accelerators utilizing the inverse Cherenkov effect (ICR-DLAs), show great potential for next-generation compact applications. However, single-sided laser-driven ICR-DLAs suffer from two key drawbacks: (1) the achievable acceleration gradient is notably weaker than the incident laser field due to reflection losses; and (2) asymmetric field distributions in the bunch channel cause particle deflection, reducing acceleration efficiency. To address these issues, we propose a novel surface plasmon polaritons (SPP)-based ICR-DLA design that integrates dual metal films into a silicon prism. This structure achieves two key improvements: (1) excitation of SPP significantly increases the acceleration gradient; and (2) the laser-induced surface wave is redistributed to form a symmetric field profile in the bunch channel, thereby minimizing deflection. The effectiveness of this design is validated through theoretical analysis and particle tracking simulations. This SPP-based approach marks a significant advance toward on-chip accelerator systems with high acceleration gradients and easily controllable laser-driven profiles.

Keywords: surface plasmon polaritons, inverse Cherenkov, particle accelerator

Introduction

Particle accelerators play a crucial role in scientific research and commercial applications. Conventional radio frequency accelerators are typically large in size, and their acceleration gradients are limited by the breakdown threshold of acceleration cells [1–4]. In contrast, dielectric laser accelerators (DLAs) can achieve a more compact footprint and higher acceleration gradients, demonstrating potential for on-chip integration. Among DLA schemes, the inverse Smith-Purcell DLA (ISP-DLA) and the inverse Cherenkov DLA (ICR-DLA) have attracted significant attention. The ISP-DLA utilizes a periodic grating to generate electromagnetic fields to accelerate an electron beam [2, 5–8], while the ICR-DLA employs the phase-matched evanescent wave produced by total internal reflection of a laser at a dielectric interface to provide energy gain to the electron beam [9–12]. In the ICR-DLA, prism-like structures are primarily used to induce total internal reflection. The achievable acceleration gradient mainly depends on the laser's field strength and the distance between the electron beam and the laser's total reflection interface, as determined by the exponential decay of the evanescent wave. In single-sided laser-excited ICR-DLAs, two critical challenges remain: the acceleration gradient remains several orders of magnitude weaker than the incident laser's electric field strength (typically GV/m), while field asymmetry within the channel readily causes particle beam deflection, leading to charge loss.

Recent studies on metal surface plasmons have demonstrated that plasmonic near-fields can be confined to nanoscale spatial dimensions and enhanced by several orders of magnitude relative to the optical field strength in the absence of plasmonic field localization [13]. Theoretical research has also reported that metallic metasurface plasmons designed for continuous electron acceleration can provide significantly higher acceleration gradients than the incident optical field. These studies show that metallic materials may also serve as excellent media for designing laser-driven accelerators [1, 14, 15]. In recent experiments, acceleration gradient enhancement in ISP-DLAs has been successfully achieved through surface plasmon polaritons (SPP) [16].

In this work, we present an innovative SPP-based accelerator structure that enhances electron acceleration fields through a dual-layer metallic mask integrated into a silicon prism configuration. The proposed design employs phase-matching conditions to excite surface plasmon polaritons in a seven-layer metal-dielectric stack, achieving exponential field enhancement within the metallic films. Our investigation begins with theoretical modeling and 2D electromagnetic simulations of this structure, showing excellent agreement between both approaches. Field calculations demonstrate that incorporating metallic films enhances the on-axis electric field in the electron channel by over 50%. Subsequent particle tracking simulations reveal that with an incident laser peak field of 1.1 GV/m, electrons gain approximately 29 keV over a 20 µm interaction length, yielding an average acceleration gradient of 1.45 GV/m. This represents a greater than 50% improvement in acceleration gradient compared to the non-metallic reference structure, while simultaneously demonstrating enhanced field stability and superior control over particle acceleration dynamics.

Model Description

Fig. 1 [FIGURE:1] schematically illustrates the proposed structure, whose layered architecture—from top to bottom—includes a silicon dielectric prism, a silver film, a silicon dielectric film, a high-vacuum particle channel, a second silicon dielectric film, another silver film, and a silicon dielectric substrate.

In this structural design, a p-polarized plane electromagnetic wave perpendicularly irradiates Surface I of a high breakdown threshold prism with a right-triangle cross-section. The base angle of the prism is θ. Due to the vertical incidence, the incident angle of the laser beam within the medium on Surface III is also θ. Free electrons move parallel to Surface V of the silicon dielectric film. The accelerating wave's phase velocity vp within the channel becomes synchronized with that of the evanescent wave produced by internal reflection at the prism hypotenuse. Then vp satisfies a specific relationship with the incident angle θ of the laser pulses [10]: vp = c/(n sin θ), in which n represents the refractive index of the prism material and c is the speed of light in vacuum. Given that the phase velocity of the surface wave vp < c, synchronization between the surface waves and the electrons can be achieved.

The accelerating wave within the electron channel is driven by laser-induced surface plasmon polaritons (SPP) excited at the channel boundaries via femtosecond laser pulse irradiation. This mechanism relies on precise wavevector matching between the longitudinal components of the propagating laser field across adjacent dielectric interfaces, which also enables the concurrent excitation of SPP at the upper and lower metal/dielectric interfaces (Surface IV and Surface VII) of the prism. The equation of wavevector matching is as follows [17]: k′′ = k′sin(θ) = εdk0sin(θ) = kspp, in which k′′ and k′ denote the wavevector components at different stages, θ is the incident angle, k0 is the wavevector in vacuum, ω is the angular frequency of the electromagnetic wave, and kspp is the wavevector of SPP.

Based on the analysis of electromagnetic characteristics, the constructed prism-metal film coupling structure, shown in Fig. 2 [FIGURE:2], can be approximated as a seven-layer dielectric-metal alternating model. Under the premise of satisfying specific boundary conditions and electromagnetic parameter constraints, this approximation effectively simplifies the complex three-dimensional structure while preserving the core physical mechanisms of SPP excitation and transmission.

Subsequent work will further analyze the electromagnetic field distribution in this model by deriving and calculating the transverse magnetic field component Hy for each layer via Maxwell's equations and material parameters. During the SPP excitation process, the Hy in each layer consists mainly of an incident wave and a reflected wave propagating along the transverse x-direction. Due to reflection at the surface, the fields in each layer should be expressed as the superposition of two opposite propagating waves in the x-direction. Adopting the general form of the time-harmonic field $\vec{H}(\vec{r}, t) = \vec{H}(\vec{r})e^{-i\omega t}$, the equation for Hy is given as follows: $Hy = H_ie^{ik_{spp}x - ik_0z - i\omega t} + H_re^{ik_{spp}x + ik_0z - i\omega t}$. Then the equations of Hy in each layer are as follows: $Hy_j = \tilde{H}^+j e^{ik^-}x - ik_jz - i\omega t} + \tilde{Hj e^{ik^-_j$ represent the amplitudes of the transmitted and reflected waves in the j-th layer, respectively. Considering boundary conditions, material parameters, and other factors, the calculation formula for the Hy matrix is derived and presented in Eq. A1 of the appendix, in which µ0 represents the vacuum permeability, ε0 is the vacuum permittivity, ε2 denotes the relative permittivity of silver (εAg), ε3 denotes the relative permittivity of silicon (εSi), and ε4 denotes the relative permittivity of vacuum (εVacuum = 1).}x + ik_jz - i\omega t}$, in which j = 1, 2, ..., 7, and Hyj denotes the Hy field in the j-th layer. The coefficients $\tilde{H}^+_j$ and $\tilde{H

Simulation

In this section, we first acquire electromagnetic field results within the bunch channel by finite difference time domain (FDTD) simulation in Lumerical. Owing to the structure's y-axis uniformity, two-dimensional simulations are performed in the x-z plane. The single-side excited laser possesses a central wavelength of 1030 nm and a peak electric field strength of 1.1 GV/m. Silicon is selected as the prism dielectric due to its mature fabrication processes, high breakdown threshold, and high refractive index $n = \sqrt{\varepsilon_d}$, which is critical for amplifying the incident wavevector $k'' = \varepsilon_d k_0 \sin\theta$ through internal reflection. This amplification ensures wavevector matching with the SPP wavevector kspp. Silver is preferred for exciting surface plasmons due to its high reflectivity, low absorption loss, superior plasmon resonance properties, chemical stability, ease of fabrication, and field enhancement effects [18]. Of particular note, at a laser wavelength of 1030 nm, the dielectric constant of silver used in this study is εAg = −48.54 + 3.13i, which is taken from the Palik Handbook given its greater suitability for nanostructure simulations [19]. In the simulations, perfectly matched layers are applied as boundary conditions on both the x- and z-axes to suppress numerical reflections. Subsequently, the simulated frequency-domain data are imported into the General Particle Tracer (GPT) to model particle dynamics in the time domain. The initial particle emission energy is set to 84 keV, with a Gaussian distribution in the x-z plane and a uniform distribution in the y-plane. In the particle-tracking simulations, three-dimensional space-charge effects are explicitly accounted for, and the number of macroparticles is set to 1000. Supplementary simulation parameters are detailed in Table 1 [TABLE:1].

Fig. 3 [FIGURE:3] compares the transverse magnetic field Hy curve from simulation with the calculation. The simulation results show excellent agreement with the theoretical predictions. The figure marks several surfaces (Surface V, VI, VII, III, IV), which represent internal boundaries of the model. For example, the bunch channel is situated between Surface V and VI, while the electric field of the metal film lies between Surface III and IV. As a result, the transverse Hy field within the bunch channel is approximately spatially symmetric about x = 0.

Fig. 4 [FIGURE:4] compares the spatial distribution of the transverse magnetic field Hy in metal films of varying thickness. Simulation results indicate that for a 50 nm thick metal film, the Hy field within the bunch channel not only achieves approximate symmetry about x = 0, but also shows higher field strength, thus striking a balance between both characteristics.

Fig. 5 [FIGURE:5] illustrates the electric field in the bunch channel of the structure with metal film. Simulation results show that the peak value of longitudinal electric field Ez within the bunch channel reaches 3.5 × 10⁹ V/m, representing a more than 50% increase compared to the state without metal film.

The frequency-domain electric field distribution at x = 74 nm, x = 0 nm and x = -74 nm with metal film are depicted in Fig. 6 [FIGURE:6]. The longitudinal electric field Ez shows symmetry about x = 0 and a relatively uniform distribution along the transverse direction with minimal fluctuations. Particles in the transverse direction experience non-uniform electric field strengths. The transverse electric field Ex is highly symmetric about the bunch channel center x = 0, where transverse electric field Ex is minimized due to the cancellation of upper and lower components, effectively suppressing transverse particle deflections during acceleration. However, due to the fact that the longitudinal electric field Ez and the transverse electric field Ex have the same phase in the transverse direction, the deflection of the particle beam is inevitable during its motion in the bunch channel.

Fig. 7 [FIGURE:7] shows the electric field in the bunch channel of the structure without metal film. The longitudinal electric field Ez decays rapidly in the negative x-direction, leading to non-uniform acceleration gradient distribution within the channel. The transverse electric field Ex not only decays in the transverse direction but also shows an asymmetric distribution along the channel center, a feature that increases the risk of particle deflection during acceleration.

Fig. 8 [FIGURE:8] compares the transverse electric field Ex and longitudinal electric field Ez for structures with and without the metal film against the position avgz. The metal film plays a crucial role in modulating the electric field distribution. Notably, the structure with the metal film demonstrates a stronger longitudinal electric field Ez and weaker average transverse electric field Ex compared to the film-free case. This shows that the metal film enhances particle beam acceleration efficiency while reducing beam deflection.

Fig. 9 [FIGURE:9] shows the trajectories of the entire particle beam as a function of particle position. To solve beam deflection, dynamic phase-driven synchronization is applied, which dynamically aligns beam phase with the accelerating wave, enhancing energy gain and pass ratio [20]. The bunch trajectory of the model with film, as shown in Fig. 9(a), demonstrates that through precise regulation of the dynamic synchronization between traveling-wave phase velocity and particle velocity, the particle beam experiences phase shifts in the acceleration channel while keeping particles within the acceleration region. In the initial stage, the particles are in the transverse defocusing phase (accompanied by longitudinal compression), causing the particle beam to defocus and deflect upward. With the phase slip during the acceleration process, the particles enter the transverse focusing phase (accompanied by longitudinal decompression), and the particle beam deflects downward and focuses. Ultimately, the particles enter the transverse defocusing phase (accompanied by longitudinal compression), and the particle beam deflects upward and most particles pass.

The bunch trajectory of the model without film, as shown in Fig. 9(b), shows greater deflection and a higher degree of divergence compared to the model with film, even though the dynamic phase-driven synchronization mechanism is equally applied.

Fig. 10 [FIGURE:10] provides the pass ratio for the entire electron bunch. The model with metal film achieves a bunch transmission efficiency of 76%. Compared with the model with the film, the transmission efficiency of the model without the film is only 10%. This is attributed to the uneven transverse field distribution within the channel, which causes inconsistent transverse forces on each particle and leads to inferior performance. The uneven transverse field distribution exacerbates particle trajectory divergence, thereby reducing the overall transmission efficiency in the structure without film.

In the model with the film, as shown in Fig. 11 FIGURE:11, most particles are accelerated to a final energy of 113 keV: particles that match the designed phase are successfully accelerated, while those with phase deviations are decelerated and eventually eliminated. This outcome is enabled by the proposed scheme's dynamic phase-driven synchronization mechanism, which regulates particles to ensure continuous energy gain. Specifically, the particles' kinetic energy increases steadily from 84 keV to 113 keV, resulting in a total gain of 29 keV and an average acceleration gradient of approximately 1.45 GV/m.

In the model without the film, as shown in Fig. 11(b), particles' kinetic energy rises steadily from 91 keV to 106 keV. The dynamic phase-driven synchronization mechanism regulates particles to ensure continuous energy gain, giving a total gain of 15 keV and an average acceleration gradient of 0.75 GV/m. Yet only 10% of particles reach 106 keV, which is attributed to the low particle pass ratio—most particles fail to sustain acceleration.

Fig. 12 [FIGURE:12] shows the energy distribution of particles after acceleration in the model with film, where the majority of the remaining particle bunch attains the maximum energy, indicating low energy dispersion.

Conclusion

In summary, we incorporate the surface plasmon polaritons mechanism into the ICR-DLA framework and design a novel SPP-based structure that incorporates dual metal films within a silicon prism configuration to enhance electron acceleration fields. In the proposed scheme, a symmetric field distribution and enhanced acceleration gradient can be achieved in the bunch channel around its central axis under single-side illumination conditions. Simulations demonstrate that particles can be accelerated from 84 keV to 113 keV with a transmission efficiency of over 75% within a 150-nm-wide and 20-µm-long bunch channel. This approach offers a compelling alternative for tabletop ultrafast electron accelerators, broadening the accessibility and applicability of particle acceleration technologies across diverse scientific and technological domains.

Acknowledgments

This work was supported by the Natural Science Foundation of China (No. 12475153), the National Key Laboratory Foundation of China (No. JCKYS2023LD5), the State Key Laboratory of Advanced Electromagnetic Technology (Grant No. AET 2025KF001), and the Postgraduate Research & Practice Innovation Program of NUAA (No. xcxjh20240405).

Appendix A: Multilayer Electromagnetic Propagation Matrix

The following matrix equation formulates the continuity conditions of transverse magnetic fields Hy at interfaces of the multilayer structure, derived by combining Maxwell's equations with boundary conditions. It relates incident, reflected, and transmitted fields to intra-layer field components, with phase delays and wave impedance terms characterizing wave propagation:

where $\mathbf{H} = [\tilde{H}_r, \tilde{H}^+_2, \ldots, \tilde{H}^+_6, \tilde{H}^-_6, \tilde{H}_t]^T$ is the field vector and $\mathbf{I}$ is the incident field vector.

$$
\begin{bmatrix}
-\frac{\varepsilon_0\varepsilon_2\omega}{k_2} & e^{-ik_2d_2} & -e^{-ik_2d_2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
\frac{\varepsilon_0\varepsilon_2\omega}{k_2} & e^{ik_2d_2} & e^{ik_2d_2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & -\frac{\varepsilon_0\varepsilon_3\omega}{k_3} & e^{-ik_3d_3} & -e^{-ik_3d_3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & \frac{\varepsilon_0\varepsilon_3\omega}{k_3} & e^{ik_3d_3} & e^{ik_3d_3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & -\frac{\varepsilon_0\varepsilon_4\omega}{k_4} & e^{-ik_4d_4} & -e^{-ik_4d_4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & \frac{\varepsilon_0\varepsilon_4\omega}{k_4} & e^{ik_4d_4} & e^{ik_4d_4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & -\frac{\varepsilon_0\varepsilon_3\omega}{k_3} & e^{-ik_3d_3} & -e^{-ik_3d_3} & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & \frac{\varepsilon_0\varepsilon_3\omega}{k_3} & e^{ik_3d_3} & e^{ik_3d_3} & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & -\frac{\varepsilon_0\varepsilon_2\omega}{k_2} & e^{-ik_2d_2} & -e^{-ik_2d_2} & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & \frac{\varepsilon_0\varepsilon_2\omega}{k_2} & e^{ik_2d_2} & e^{ik_2d_2} & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & \cos\theta & -\frac{k_1}{\varepsilon_0\varepsilon_1\omega} & \cos\theta & \frac{k_1}{\varepsilon_0\varepsilon_1\omega} \
0 & 0 & 0 & 0 & 0 & 0 & 0 & \cos\theta & \frac{k_1}{\varepsilon_0\varepsilon_1\omega} & -\cos\theta & \frac{k_1}{\varepsilon_0\varepsilon_1\omega}
\end{bmatrix}
$$

The incident field vector is given by: $[\tilde{H}_i \cos\theta, \frac{k_1}{\varepsilon_0\varepsilon_1\omega}\tilde{H}_i, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]^T$.

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