Study of a Graphene Surface Plasmon Polariton-Based Dielectric Laser Accelerator

Abstract

Due to their high breakdown threshold and acceleration gradient, dielectric laser accelerators (DLAs) have become an important technical direction of accelerator miniaturization. In this study, an electron accelerator scheme based on graphene surface plasmon polaritons (SPPs) is proposed. The grating was designed to be etched on the silica surface in the simulation, and a layer of graphene was modeled to cover the surface of the medium. The incident laser light in the simulation was configured to be coupled by the grating to generate surface plasmon polaritons (SPPs) on the graphene surface. According to the simulation results, a relatively large acceleration channel aperture and long acceleration length could be formed on the graphene surface using a mid-infrared laser; this provides a technical solution for increasing the beam current of a DLA. A 53.375 THz laser was incident on the surface of the accelerating structure to carry out tracking calculations on a 10 MeV electron beam. For the 100 $\mu$m accelerating structure, an energy gain of 0.105 MeV was achieved, and the acceleration gradient reached 1.05 GeV/m.

1. Introduction

Particle accelerators are critical devices for exploring the microscopic characteristics of matter, with significant applications in high-energy physics, chemical materials, biomedicine, and other fields. With the continuous advancement of science and technology, researchers in various fields have raised higher demands for the performance of particle accelerators; consequently, many large-scale accelerators, such as the Large Hadron Collider and the Relativistic Heavy Ion Collider, have been constructed. Large accelerators require extensive construction areas; while they enhance beam energy, they also significantly increase costs. Therefore, the miniaturization of particle accelerators has long been a key development direction in accelerator technology. A high acceleration gradient is pivotal to the miniaturization of particle accelerators. To this end, researchers have explored various high-gradient accelerator technologies, including dielectric laser accelerators (DLAs) [1], plasma wake field accelerators [2], laser wake field accelerators [3], and two-beam accelerators [4]. Different high-gradient acceleration mechanisms exhibit unique advantages, such as high current, high beam quality, and low manufacturing costs, which determine their distinct application domains. Due to recent advancements in laser and material technologies, significant breakthroughs have been achieved in DLA technology [5]. Notably, the adoption of Bragg reflection structures has doubled the energy conversion efficiency [6], positioning DLAs as a leading candidate for future high-gradient accelerators. As a novel high-gradient compact accelerator, DLAs are poised for widespread adoption in medical and industrial applications [7] and hold potential for future collider technologies [7]. To date, demonstration devices for DLAs mainly use dielectric grating accelerator structures, which can be divided into two types: single-sided [8] and double-sided gratings [9]. Due to the open structure of single-sided gratings, the utilization efficiency of laser pulse energy is low, and the double-sided grating structure has a small electron beam channel size, which limits its applicability in large bunches and heavy-ion accelerators. Since the discovery of abnormal fringes on grating surfaces in 1902 by Wood, the principle of plasmons has been established after a century of research and exploration. Surface plasmon technology has been rapidly developed, and key breakthroughs have been made in the fields of photodetection and optoelectronic integration [10]. Plasmons are the collective oscillations of electrons resulting from the interaction between light and matter. The excitation of plasmons can effectively confine a laser pulse on a metal surface while forming a strong electric field. To this end, an accelerator structure based on surface plasmon polaritons (SPPs) is proposed. SPPs are used to bind the energy of a laser pulse on the surface of a structure, reducing the reflectance and improving the energy use efficiency of the laser pulse. This accelerator provides a new scheme for the acceleration of large bunches and heavy ions.

2. Principle

Since plasmons cannot be excited in free space, three conventional excitation methods can be used for surface plasmons: grating coupling, prism coupling, and near-field excitation [11]. In this study, we used the grating-coupled excitation method to excite SPPs. Plasmons are generated at the medium-metal interface. Their vertical penetration depth (Y direction) is shallow in the metal region but extends deeper into the dielectric layer. Evidently, electron beams cannot be accelerated within the metal or dielectric material. Therefore, we propose structuring the metal layer as a thin film to enable its electric field to penetrate into a vacuum. In this configuration, electrons traversing the thin metal film surface are accelerated by the surface electric field. The plasmonic electric field decays exponentially along the Y direction and is wavelength-dependent. Therefore, the laser wavelength should not be excessively short; otherwise, the penetration depth of the surface electric field into vacuum becomes insufficient, hindering the maintenance of a sizable beam channel. Graphene, a novel two-dimensional material, exhibits exceptional optical and electrical properties [12]. Recent studies have highlighted its potential for surface plasmon polariton (SPP) applications [13,14,15,16]. Compared to noble metal-based plasmon polaritons, graphene plasmon polaritons demonstrate superior mode confinement, longer propagation distances, and unique electrical/chemical tunability [17], alongside higher breakdown thresholds. Therefore, we employed grating coupling to excite SPPs at the graphene–medium interface, as illustrated in Figure 1. In this study, we started with the Cooper equation to calculate the transformation relationship between the conductivity and frequency of graphene under different energy potentials. In addition, we selected 5.617 $\mu$m as the laser wavelength. Based on the calculated conductivity of graphene, the interaction lengths of the surface electric field with the beam in the Y- and Z-directions were calculated. The obtained graphene conductivity was imported into the medium model in electromagnetic field simulation software to obtain the distribution of the surface electric field. Finally, charged particle beam tracking calculation software was used to obtain the charged particle beam parameters.

2.1. Graphene Conductivity and Grating Structure

According to the Cooper formula [18,19], the graphene permittivity can be expressed as follows: $\sigma (\omega ,{\mu }_c,\Gamma ,\mathrm{T})={\sigma }_{\mathrm{intra}}+{\sigma }_{\mathrm{inter}}$

$${\sigma }_{\mathrm{intra}}={\displaystyle \frac{-\mathrm{i}{e}^2}{\pi {\hslash }^2(\omega +2\mathrm{i}\Gamma )}}{\int }_0^{\infty }\xi [{\displaystyle \frac{\partial f_d\left(\xi \right)}{\partial \xi }}-{\displaystyle \frac{\partial f_d(-\xi )}{\partial \xi }}]d\xi$$

(1)

$${\sigma }_{\mathrm{inter}}={\displaystyle \frac{\mathrm{i}{e}^2(\omega +2\mathrm{i}\Gamma )}{\pi {\hslash }^2}}{\int }_0^{\infty }{\displaystyle \frac{f_d(-\xi )-f_d\left(\xi \right)}{{(\omega +2\mathrm{i}\Gamma )}^2-4{(\xi /\hslash )}^2}}d\xi$$

(2)

where e, ℏ, and $k_B$ are universal constants related to the electron charge, Planck’s constant, and Boltzmann’s constant, respectively; ${\mu }_c$ is the chemical potential; and $f\left(\xi \right)={\left[exp\left({\displaystyle \frac{\xi -{\mu }_c}{k_{B}T}}\right)+1\right]}^{-1}$ is the Fermi Dirac distribution. We assume intrinsic losses of $\Gamma$ = 0.1 MeV in this paper. This value is based on the theoretical estimation of the maximum mobility in graphene [20]. In this regard, the graphene conductivity curve in the 10 to 70 terahertz (THz) frequency band is created, as shown in Figure 2a,b. The relationship between the conductivity and permittivity of graphene is as follows: ${\epsilon }_{\mathrm{g}}=1+{\displaystyle \frac{i{\sigma }_g}{{\epsilon }_0\omega t_g}}$[21], where $t_g$ is the thickness of graphene. In this study, 20 graphene layers were used to obtain the graphene permittivity curve in the 40–70 THz frequency band. The propagation constant $\beta$ on the graphene surface can be expressed as follows [20]: $\beta =k_0\sqrt{1-{\left({\displaystyle \frac{{\eta }_0{\sigma }_g}{2}}\right)}^2}$,where ${\eta }_0$ is the intrinsic impedance of free space, and ${\eta }_0=\sqrt{{\mu }_0/{\epsilon }_0}\approx 377 \Omega$, where $k_0$ is the wave vector in vacuum. From these equations, the relationship between the wave vector of the surface plasmon and the wave vector of the incident light is shown in Figure 2d. In this study, we used the grating structure. The excitation of SPPs needs to satisfy specific conditions. When the laser light is incident on the surface of a periodically changing medium, corresponding diffracted waves are generated. When the diffraction wave vector is superimposed on the incident light wave vector and matches the plasmon wave vector, the surface electrons resonate, thereby exciting the plasmons. When the grating SPPs are excited, the incident laser light and the periodic structure have the following relationship, where $k_{spp}$ and $k_0$ are the plasmon wave vector and the incident laser wave vector, respectively, and N is the diffraction order of the grating.

$$k_{spp}=k_{0}sin\theta \pm N{\displaystyle \frac{2\pi }{\Lambda }}$$

(3)

When the resonant excitation condition is satisfied, the wavelength of the surface plasmon is related to the grating constant. A smaller grating constant causes the wavelength of the plasmon to be much shorter than that of incident light, thereby breaking the diffraction limit. However, in this study, the wavelength of the surface plasmon cannot be too short because the attenuation length of the surface plasmon in the X- and Y-directions is positively correlated with the wavelength; specifically, a shorter attenuation length results in a smaller action area between the surface electric field and the beam current, which in turn limits the bunch charge amount and reduces the acceleration gradient. In Figure 2d, at a frequency of 50 THz, the ratio between the wave vector of the plasmon and the wave vector of the incident light is close to 1. If the grating constant is 8.1 µm, the diffraction order is N = 1. As a result, we predict that the incident angle of the laser is approximately 16 degrees.

2.2. Interaction Length

Due to the electrochemically adjustable characteristics of the graphene acceleration structure, the electrical characteristics of graphene can be changed by adjusting the surface applied voltage, and a specific accelerating electric field can be generated on the graphene surface to adjust the action length. The localization of the optical field distribution of SPPs is usually described by its longitudinal attenuation length, which is related to the vertical wave vector $\beta$. A larger attenuation length decreases the electric field of the surface plasmon. Therefore, in optoelectronic devices based on surface plasmon control technology, the goal is often to shorten the vertical attenuation length and strengthen the locality of the light field distribution to improve the effects of information transfer and control [22]. In the graphene dielectric laser accelerator structure, we aim to obtain SPPs with high electric field intensities. However, the bunch has a certain dimension; if the vertical attenuation length is too small, the transverse acceptance is too small to accelerate large bunches, and the electron beam cannot be effectively accelerated. A balance between the surface electric field and the attenuation length is needed. The propagation distance of the surface plasmon in the z direction is $L_z=\mathrm{Im}\left({\displaystyle \frac{1}{2\ast k_{spp}}}\right)$, and the penetration depth in the Y-direction is $L_{\mathrm{y}}=\mathrm{Im}\left({\displaystyle \frac{1}{\sqrt{k_{spp}^2-k_0^2}}}\right)$. According to these formulas, we can obtain the propagation lengths of the model in the Y- and Z-directions. Since the imaginary part of the propagation constant of the plasmon is close to zero at 47 THz, Lz has a maximum value at this frequency. As shown in Figure 3, the model has a longer propagation length in the Z-direction and Y-direction in the low-frequency range. In the context of high incident laser frequencies, both of the Lz and Ly exhibit a decremental trend in correlation with the frequency increase. This phenomenon poses a significant challenge for the formation of a sizable acceleration aperture (Ly is less than 1 $\mu$m) for graphene-based laser acceleration structures, particularly when the incident laser frequency exceeds 60 terahertz. Conversely, within the frequency range inferior to 30 THz, the interaction length is observed to augment as the frequency diminishes, culminating in a relatively pronounced interaction length within the lower frequency spectrum. However, the low-frequency range (below 10 THz) is in the THz laser region. To date, high-power THz lasers are complicated and expensive. The complexity and price contribute to our reasoning for the selection of a 50 THz (close to 47 THz) infrared laser as a light source. At 50 THz, the beam channel size $L_y$ is 8 $\mu$m, and the acceleration length $L_z$ is 250 $\mu$m. In the calculations in Figure 3, a 20-layer graphene structure is used. Since the conductivity of graphene has a linear relationship with the number of layers, if the number of graphene layers is further reduced, the intensity of the graphene surface electric field is reduced; however, the beam channel and acceleration length are further increased. Thus, we can increase the horizontal acceptance and acceleration length by reducing the number of graphene layers. According to the results in Figure 3, the beam channel can reach 100 microns, and the acceleration length can reach the order of 100 mm (horizontal acceptance and acceleration increase by 20 times) by reducing the number of graphene layers from 20 to 1, which is much greater than those of existing dielectric laser accelerators; thus, the scheme has good application potential in improving the current intensity of dielectric laser accelerators.

3. FDTD Simulation

To further investigate plasmons on the graphene surface, we incorporate the graphene conductivity into the finite-difference time-domain (FDTD) material model and simulate the structure shown in Figure 1. When the chemical potential of graphene is below 300 meV, its low conductivity ensures that variations in electrochemical potential (within 10–70 THz) have negligible simulation impact, while the number of graphene layers plays a dominant role. For chemical potentials exceeding 300 meV, the imaginary part of conductivity rises sharply, leading to a rapid reduction in the interaction distance. This reduction is detrimental to long-distance electron beam acceleration. Hence, we selected graphene with an electrochemical potential of 100 meV for simulations. Based on previous estimations, a laser frequency near 50 THz incident at nearly 16$°$ should yield a pronounced absorption peak. We performed a parametric scan around 15$°$ to measure graphene surface absorption rates at varying incident angles. At a laser incident angle of 21$°$, a stronger absorption peak emerges at 53 THz, aligning with the predictions in Section 2. To identify the optimal plasmon excitation structure, we performed parametric scans of the grating constant, substrate thickness, and laser incident angle. For laser excitation at 53.375 THz, with a substrate thickness of 2.5 $\mu$m, the grating tooth height was set to 3.205 $\mu$m. A grating constant of 6.41 $\mu$m achieves absorptance values approaching 1 over a broad range, enabling efficient SPP generation. The size structures and laser parameters are listed in

Table 1. Accelerator structures and incident laser parameters.

Parameter	Value	Note
W	3.205 $\mu$m	1/2 grating constant
$\Lambda$	6.41 $\mu$m	Grating constant
H	5.705 $\mu$m	Base thickness + h
h	3.205 $\mu$m	Grating tooth height
$\theta$	21 deg	Angle of incidence
n	2.5 $\mu$m	Reflective index
${\mu }_c$	100 meV	Chemical potential
f	53.375 THz	Laser frequency

.

Using the above graphene grating structure, we conducted a simulation analysis on the graphene surface electric field. The graphene surface electric field shows an exponential attenuation distribution in the Y-direction, and the propagation length in the Y-direction is on the order of microns; this result is approximately the same as the propagation length estimated in Section 2. In the Z-direction, due to the longer transmission length and shorter simulation length, the exponential attenuation distribution is not evident. The theoretically predicted damage threshold is close to 250 mJ/cm² [23], and the single-pulse laser damage threshold of 50 femtoseconds is 3 TW/cm² [24]. Thus, irradiation is carried out using a laser with a 50-femtosecond pulse duration and an electric field amplitude of 1 GV/m. Based on the relationship between the pulse fluence (laser energy per unit area) and electric field amplitude $W_s\approx {\left|E\right|}^2\ast {\sigma }_{\mathrm{w}}/2\eta$ [25], The incident laser energy to which graphene was subjected was well below its damage threshold. When the electric field intensity of the incident laser was 1 GV/m, the simulation results of the graphene surface electric field were normalized, as shown in Figure 4d.

4. Theoretical Calculations of the Surface Electric Field and Particle Tracking

Since the incident laser is linearly polarized in the YZ plane, the electromagnetic field on the graphene surface can be expressed as follows:

$$\begin{array}{c}E=(0,E_y,E_z)e^{ikz}{e}^{-{\displaystyle \frac{z}{L_z}}}{e}^{-{\displaystyle \frac{y}{L_y}}}\hfill \\ B=(B_y,0,0)e^{ikz}{e}^{-{\displaystyle \frac{z}{L_z}}}{e}^{-{\displaystyle \frac{y}{L_y}}}\hfill \end{array}$$

(4)

where $e^{-{\displaystyle \frac{z}{L_z}}}$ and $e^{-{\displaystyle \frac{y}{L_y}}}$ indicate that the surface electric field decays exponentially along the Z- and Y-directions, respectively. As the acceleration length Z is significantly shorter than the plasmon propagation length $L_z$, the term $e^{-{\displaystyle \frac{z}{L_z}}}$ approximates 1. Maxwell’s equations are as follows:

$$\left\{\begin{array}{c}\nabla \times E=-{\displaystyle \frac{\partial B}{\partial t}}\hfill \\ \nabla \times H=J_f+{\displaystyle \frac{\partial D}{\partial t}}\hfill \end{array}\right.$$

(5)

The differential expression of the electromagnetic field can be written as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial E_z}{\partial y}}-{\displaystyle \frac{\partial E_y}{\partial z}}=i\omega B_x\hfill \\ {\displaystyle \frac{\partial B_x}{\partial z}}-{\displaystyle \frac{\partial B_z}{\partial x}}=-i{c}^{-2}\omega E_y\hfill \\ {\displaystyle \frac{\partial B_y}{\partial x}}-{\displaystyle \frac{\partial B_x}{\partial y}}=-i{c}^{-2}\omega E_z\hfill \end{array}\right.$$

(6)

The analytical solutions for the surface electromagnetic field are derived as

$$\left\{\begin{array}{c}{E}_y=k{L}_y{E}_0{e}^{-{\displaystyle \frac{y}{L_y}}}{e}^{-{\displaystyle \frac{z}{L_z}}}sin(kz-\omega t+\phi )\hfill \\ E_z=E_0{e}^{-{\displaystyle \frac{y}{L_y}}}{e}^{-{\displaystyle \frac{z}{L_z}}}cos(kz-\omega t+\phi )\hfill \\ B_x=L_y{\displaystyle \frac{\omega }{c^2}}{E}_0{e}^{-{\displaystyle \frac{y}{L_y}}}{e}^{-{\displaystyle \frac{z}{L_z}}}sin(kz-\omega t+\phi )\hfill \end{array}\right.$$

(7)

On the basis of the above analytical expressions for the electromagnetic field, we used a General Particle Tracer to perform simulation calculations. The electron bunch comprised 1000 particles with an emittance of $\pi$ $\mu$m × $\mu$ rad, positioned 1 mm above the graphene surface. Since an acceleration plan for relativistic particles was used, the initial energy was 10 MeV, and the Lorentz factor was 20.56. After passing through the 100 $\mu$m acceleration region, the energy of the electron beam reached 10.105 MeV. According to the simulation results, the traverse emittance of the beam significantly increased, as shown in Figure 5a,b. The normalized velocity of the beam in the Y-direction increased by three orders of magnitude, mainly because the transmission length of the electric field $E_y$ on the graphene surface was relatively short. In addition, the electric field components $E_y$ at different heights were quite different. The increase in the emittance was relatively large; however, due to the high acceleration gradient, the needed acceleration length was very short, and the traverse dimension of the bunch did not significantly change, as shown in Figure 5c,d. After being accelerated for 329.3 femtoseconds, the beam reached a point near 100 $\mu$m in the acceleration structure, where Figure 6 displays the energy spread distribution of the beam at this moment. In Figure 6, the darker the color, the more particles are present at that energy distribution. The vast majority of particles receive an energy gain of 0.1 MeV. Due to the acceleration phase, electrons at the head and tail of the beam receive less energy gain; therefore, the energy spread distribution of the beam in the figure exhibits a mountain-like shape. The Lorentz factor of the beam increases from 20.55 to 20.77, and the energy gain reaches 0.105 MeV. The vertical height represents the statistical result of the particle number. Here, most particles in the bunch obtained good acceleration results, the energy gain of the bunch was 0.105 MeV, and the accelerating gradient was 1.05 GeV/m. In Figure 7, no significant increase is observed in the transverse size as the cluster passes through the 100 $\mu$m acceleration zone. Ez is much larger than Ey, and most of the laser energy is converted into an accelerating electric field. Eighty percent of the laser energy is transformed into a useful acceleration electric field.

5. Discussion

The energy gain and acceleration gradient serve as fundamental benchmarks for evaluating the performance of laser-driven particle accelerators, dictating both their practical utility and technological feasibility. Conventional RF-based systems, limited to gradients of 10–50 MV/m, necessitate kilometer-scale infrastructures for high-energy applications. Recent advancements in silicon photonic crystals and plasmonic metallic gratings have pushed gradients to GV/m. In contrast, dielectric laser accelerators exploit electric fields to surpass 1 GV/m, enabling tabletop-scale implementations.

In 2014, Jacob Scheuer proposed utilizing metal surface plasmon mechanisms for electron beam acceleration [26]. In 2020, researchers at Stanford University and SLAC National Accelerator Laboratory achieved a milestone: a maximum energy gain of 0.915 keV over 30 $\mu$m, corresponding to an acceleration gradient of 30.5 MeV/m [27]. In 2023, Tomáš Chlouba and his team made a leap forward in the field by demonstrating a nanophotonic electron accelerator. This advanced device achieved a remarkable maximum coherent energy gain of 12.3 keV, representing a substantial 43% increase in energy from the initial 28.4 keV to 40.7 keV [28]. Also in 2023, Li Jianqi’s team at the Beijing National Laboratory for Condensed Matter Physics presented a design for metallic material-based on-chip laser-driven accelerators that showed a remarkable electron acceleration capability, as demonstrated in ultra-fast electron microscopy investigations. Due to the presence of surface plasmon enhancement and nonlinear optical effects, the maximum acceleration gradient can reach as high as 0.335 GeV/m [29]. In this study, the experiments revealed a fourfold amplification of the electric field near the surface compared to the incident field, confirming the efficacy of surface plasmon enhancement in electron acceleration. The results indicate that the acceleration efficiency drops sharply when the electron beam-grating distance exceeds 0.3 $\mu$m, precluding the acceleration of clusters larger than 0.3 $\mu$m in diameter. Recently, Leon Brückner from Friedrich-Alexander-Universität Erlangen-Nürnberg demonstrated the dielectric laser acceleration of electrons driven with a 10 µm laser in a silicon dual-pillar structure. They observed the acceleration of 27 keV electrons by 1.4 keV, corresponding to a 93 MeV/m acceleration gradient [30]. Limited by the size of the grating spacing (1 $\mu$m), large beam clusters cannot pass through the acceleration channel.

Drawing from the calculations presented in Section 4, this scheme exhibits a superior acceleration gradient of 1.05 GeV/m compared to current dielectric laser accelerators, suggesting that it possesses significant advancement and potential. In this paper, based on the material properties of graphene, we calculated the transverse propagation length of surface plasmons (in the Z-direction). By selecting the incident laser wavelength, incident angle, and dielectric constant of graphene, we expanded the range of the accelerating electric field (in the Y-direction), enabling the device to accelerate larger-sized beam clusters. According to our calculations, further reducing the number of graphene layers can further increase the range of the accelerating electric field in this direction, even reaching the hundred-micrometer level, providing possibilities for the realization of micro–nano-accelerators for high-flux electrons, protons, and heavy ions. Furthermore, in contrast to the previously mentioned studies, this study adopted a traveling wave acceleration method, which has different phase-matching conditions with the standing wave acceleration method, similar to the dielectric prism accelerator [31,32]. To maintain the continuous acceleration of the electron beam, it is necessary to make Vp equal to Ve. For relativistic electron beams, Ve is approximately equal to the speed of light, c. As analyzed in the first chapter, the phase velocity of surface plasmon propagation in this structure is close to the speed of light, thus matching the phase velocity with the electron velocity, i.e., using the grating required for the generation of surface plasmons to achieve a slow-wave structure.

6. Conclusions

Owing to graphene’s electrical properties, the surface electric field distribution can be tuned via external voltage modulation. In future studies, we will further study the effect of a controllable voltage on the surface electric field distribution and the effect of acceleration on electron beams. Compared with existing DLAs, the graphene DLA has a larger beam channel, can accelerate beams with a greater amount of charge, and has a longer acceleration length. With the development of high-power THz laser techniques, graphene DLAs have the potential to overcome the technical difficulties pertaining to micro–nano-accelerators with low current and be developed into high-current micro–nano-accelerators.