Particles accelerators are important research tools in science. Diverse ways for charged particle acceleration have been used so far[1,2]. Radio-frequency (RF) accelerators, for instance, have been widely used for electron and ion acceleration. However, the acceleration gradient is limited due to electrical breakdown in acceleration structures. Hence, some methods for increasing the acceleration gradient have been extensively researched.

Dielectric laser acceleration (DLA) is one of the ambitious approaches to achieve higher acceleration gradients in accelerators[3-5]. In DLA, for example, a photonic crystal can be used to generate a very high accelerating field with a laser pulse if the laser pulse is injected into the photonic crystal. In this case, the phase matching between the electrons and the accelerating field is very important, and there are several ways to do it. One of the ways is using on-chip phase shifters[6]. However, their structure is very complicated and may lead to a severe beam loss in acceleration. Instead of this method, the phase matching can also be achieved by pulse front tilt of the injected laser in a simpler structure. Some studies have been conducted using this method with spatial and temporal laser chirp with external diffraction gratings[7,8]. However, we achieved the phase matching with pulse-front-tilt (PFT) using micro-scale prisms and periodic cavity without external diffraction gratings. Thus, we achieved an average acceleration field of 0.28 GV/m with the structure we have designed. The details are described in this paper.

The idea of DLA is associated with the Smith-Purcell effect[9], so the Smith-Purcell effect is reviewed here first. The Smith-Purcell effect is a phenomenon of electromagnetic wave emission from a diffraction grating when electrons pass over the diffraction grating surface very closely. As well known, both conductor and dielectrics can be used for the grating material. The emitted electromagnetic radiation is called the Smith-Purcell radiation and its dispersion relation depends on the angle between the grating surface and the radiation direction as follows:

where λ is the radiation wavelength, Λ is the grating period, n is the reflection order from the grating, β=v/c, v and c are the electron and light velocities, respectively. The Smith-Purcell effect is basically a kind of Cherenkov radiation, but no threshold velocity for electrons is required, unlike the Cherenkov radiation.

On the other hand, the inverse Smith-Purcell effect is the process in which an electromagnetic wave diffracted by the grating accelerates electrons. Early studies show that DLA with laminar-type holographic gratings can match the situation[10,11,12]. In this process, phase matching is one of the important concepts in DLA, which is crucial for gaining a high acceleration gradient. If a laser pulse is injected into the DLA structure with a tilted pulse front, it will generate a traveling electromagnetic wave in the longitudinal direction and can accelerate the injected electrons continuously, leading to a high energy gain. Here, the phase of the accelerating field by the input laser pulse is described by

where k_x is the wavevector in the longitudinal direction, x is the longitudinal coordinate, ω is the angular frequency of the accelerating field, t is the time, and ϕx0 is the initial phase.

It is also known that the phase matching condition for laser injection in the normal direction is given by[13]

where Λ′ is the DLA structure period and λ′ is the input laser wavelength. In the case of laser injection with a tilted pulse front, the phase matching condition in Eq. (3) should be adjusted depending on the tilt angle.

For acceleration, electrons should be injected by an injector, such as a modified scanning electron microscope (SEM) or tunneling electron microscope (TEM). Photocathode RF guns have also been used for electron injection[14].

The calculation of electric field components is required to know the acceleration gradient in the designed DLA structure. The finite-domain-time-differential (FDTD) simulations can show us how the input laser pulse behaves in the designed nanostructure. We can obtain time-dependent electric field components through the simulations.

For the simulations, we used the DLA structure, as shown in Fig. 1. A laser pulse is injected from the top direction in Fig. 1(a), and two laser pulses are injected from both directions in Fig. 1(b). Here, the laser field polarization is important, and the polarization direction should be in the x-direction for efficient electron acceleration. The injected laser is normally incident to the acceleration channel, and it is expected that the injected laser pulse would produce a longitudinal-directed accelerating electric field in the DLA. The laser beam parameters are based on the 1 TW/35 fs Ti:Sapphire laser system in our laboratory (λ′=800 nm). In these simulations, we used a nanostructure acceleration channel consisting of a Si with wg=40 nm, 360 nm in depth and, 1.4 μm in acceleration length. Along the acceleration channel, periodic cavities with height h are placed with a period of Λ= 180 nm.

Figure 1. (Color online) Schematic of the symmetric DLA structure. DLA acceleration cavities are placed with micro-prisms from both sides, where the input laser pulses are injected from both top and bottom directions. In the figures, h is the cavity height, θ is a prism angle of the micro prism, Λ is a period of the acceleration cavities, w_g is a gap in the acceleration channel, and the injected electrons are accelerated to the positive x-direction. The material of the substrate colored with red and the material of the cavity colored with cyan are Si. The micro-prism colored with white consists of SiO₂.

Here, we introduce a new component for DLA called a micro-prism. The micro-prism is a micro-scale right-angle prism fabricated with a lower refractive index than that of Si, for example, SiO₂. The injected laser's wavevector can be tilted by the micro-prism, and the laser beamlets arrive at the channel at different times. The longitudinal accelerating field E_x was calculated by simulations, and the results are shown in Fig. 2. As shown in the figures, the phase of the longitudinal electric field travels to the positive x-direction as time goes on, which is due to the micro-prism angle θ. The traveling electric field can accelerate electrons to high energies in the acceleration channel.

Figure 2. (Color online) Simulation results for the normalized longitudinal accelerating field E_x/E₀. The on-axis accelerating field is traveling along the acceleration channel as the time τ evolves. The simulations were conducted with 90 fs of the offset time. In these simulations, the input laser beam is polarized in the x-direction, its wavelength is 800 nm (Ti:sapphire laser), and the laser intensity is I₀ = 1.3 × 10¹¹ W/cm².

We also calculated the energy gain of an electron which is injected with an energy of 30 keV when the input laser intensity is I₀ = 1.3 × 10¹¹ W/cm². For the calculations, we made our own code using MATLAB to describe the equation of motion for a single electron. Figure 3(a) and (b) show the calculated electric fields in the acceleration channel, where E_x is the longitudinal field, E_y and E_z are transverse fields, and E₀ is the input laser field. As shown in Fig. 3(a), transverse electric fields are significant if a laser pulse is injected from only one side, while the transverse fields are greatly reduced if two laser pulses are injected from both sides, as shown in Fig. 3(b). In the case of Fig. 3(b), the longitudinal field strength is much higher than that of the transverse fields.

Figure 3. (Color online) Normalized electric field components in the DLA cavities for Fig. 1(a) and Fig. 1(b), respectively. E₀ is the maximum electric field of the input laser pulse.

We also investigated the effect of the cavity height h in the DLA structure. The peak longitudinal electric field E_x is shown in Fig. 4(a). The result shows that the longitudinal field is generally reduced as the cavity height h increases. At some heights, however, the longitudinal field has a peak. This can be explained by the fact that two counter-propagating laser pulses from both sides are overlapped inside the channel and cavities, leading to a stronger longitudinal acceleration field. A detailed process for this phenomenon is shown in the previous figure (Fig. 2). We also calculated the average acceleration gradient over the entire acceleration distance in the DLA structure, as shown in Fig. 4(b). The result also indicates that a special strong longitudinal gradient can be obtained for a specific cavity height.

Figure 4. (a) Normalized peak longitudinal electric field as a function of the cavity height h, and (b) average acceleration gradient as a function of the cavity height in the case of Fig. 2(b) with the input laser intensity of I₀ = 1.3 × 10¹¹ W/cm² and a micro-prism angle of θ = 14.3°.

The prism angle θ can affect the average acceleration gradient in the DLA structure, and its calculation result is shown in Fig. 5. Here, the laser intensity of I₀ = 1.3 × 10¹¹ W/cm² and the cavity height h=0.6 μm. As expected, the average acceleration gradient increases as the tilt angle increases, but there is an optimum angle. According to our calculations, the optimum angle is approximately 16.7°, as Fig. 5 shows.

Figure 5. Tilt angle θ and acceleration gradient for I₀ = 1.3 × 10¹¹ W/cm² and a cavity height of h = 0.6 μm. The result shows that the maximum acceleration gradient is approximately θ = 16.7°.

Based on the FDTD simulation results, we have designed an experimental setup for DLA. Fabrication of the DLA nanostructure is the most critical part in this experiment. It will be fabricated through a subtractive etching process on a silicon-on-insulator (SOI) wafer for the Si nanostructure and an additive process for the micro-prism. As the resolution of the structure is very high, the electron-beam (E-beam) lithography technique will be used for the subtractive process. The fabricated Si structure will serve as the periodic cavities and the acceleration channel. The additive process, on the other hand, will be done by atomic layer deposition (ALD)[15] or liquid phase deposition (LPD)[16]. The micro-prism will be made by SiO₂ through those processes.

For the experiment, we will use the 35 fs Ti:sapphire laser in our laboratory, which consists of a mode-locked Ti:sapphire oscillator, a Ti:sapphire regenerative amplifier, a pre-amplifier, and a pulse compressor in air. As Fig. 6 shows, the laser beam pulse will be divided into two pulses first, and then they will be injected into the DLA structure from both sides. Therefore, a strong longitudinal electric field is expected in the acceleration channel in the DLA. In this process, the focused laser spot size will be adjusted to have a maximum acceleration field, which will be higher than the injected laser field strength of E₀ = 109 V/m. The accelerated electrons will be injected to the magnetic spectrometer, yielding the electron beam energy and energy spread.

Figure 6. (Color online) Schematic of the DLA experimental setup. The laser pulse from the 1 TW/35 fs laser system is divided into two pulses, and they will be injected into the DLA structure from both sides.

In this study, we calculated electric fields inside the DLA structure when two laser pulses were injected with tilted pulse fronts by micro-prisms. We found that transverse electric field components are not so strong in DLA, but the strength of the longitudinal acceleration field is almost twice of the injected laser field strength. The strong longitudinal acceleration field can be used for electron acceleration with an average acceleration gradient of 0.28 GeV/m.

The research was initiated at Nano Photonics Laboratory, Konkuk University, and completed at Gwangju Institute of Science and Technology (grant number: NRF-2022R1A2C3013359).