Introduction

Particle accelerators have attracted a lot of interest over the past years ranging from medical imaging, therapy, and fundamental sciences1,2. Radio frequency (RF)-powered devices are the conventional choice for the accelerating elements3. However, its large size, high input power, and costly infrastructures limit its utility and accessibility to broader scientific communities. The growing desires for on-chip accelerators, portable medical devices, and radiotherapy machines motivate us to explore alternative technologies that are more compact and cost-effective4,5,6. Recently, multiple small-scale accelerator concepts have been shown, such as laser-plasma accelerators, terahertz-driven accelerators, and dielectric-laser accelerators. Dielectric-laser accelerators (DLAs)5,7 powered by femtosecond lasers are a promising option, owing to the high damage threshold in the dielectric materials8, modern ultrashort pulse lasers, and nanofabrication technologies9. It supports a few GV m−1 10 field gradient (~10 GV m−1 for SiO211 and ~3 GV m−1 12 for Si) inside a microstructure.

Over the past 30 years, various setups have been proposed to optimize the acceleration process13,14,15,16,17,18. Many fundamental functions required in an on-chip particle accelerator, such as acceleration, bunching, deflection, and focusing have already been demonstrated experimentally using DLAs19,20,21,22. However, it is still a remaining challenge to accelerate the electrons in the non-relativistic regime with ultrashort pulses. A chirped grating acceleration structure was proposed to enhance the interaction length23, where a phase shift is introduced onto the electric field, allowing the electron to avoid the deceleration cycle13. However, the acceleration process is still limited by the pulse duration τ. For uncorrelated, laterally impinging pulses, a 100 fs pulse, for example, would result in v0τ = 6 μm with initial velocity v0 = 0.2c. A pulse with a longer duration may be used to enlarge the interaction length, but this requires larger input energy at given electric field strength. The damage threshold fluence of the acceleration structure material prohibits such an approach or requires resort to a lower field amplitude.

A tree-network waveguide approach24 redistributes the entire acceleration into multiple series of interactions with short laser pulses; however, it is overly complex for practical implementations. Currently, among DLAs, the common approach to implement short laser pulses for a long acceleration length is by utilizing a pulse-front-tilted (PFT) laser pulse14,18,25,26. The PFT scheme brings in a delay of the pulse along the particle acceleration direction x (see Fig. 1 for coordinates definition), making the short laser pulses at a given location x arrive simultaneously with the electron. However, the PFT scheme can only match the driving laser with a fixed electron velocity, which fulfills \(\tan \alpha =c/v\), where α represents the PFT angle. As a result, a walk-off occurs between the laser pulse and sub-relativistic electrons when the velocity increases due to acceleration.

Fig. 1: Illustration of the spatio-temporal coupling (STC) scheme.
figure 1

Panel a shows the illustration of the proposed acceleration configuration. In this symmetric configuration, two identical input laser pulses illuminate the grating simultaneously at position 1 (P.1). The electron interacts with counter-propagating fields at position 2 (P.2) in between the acceleration structures. The acceleration structure has period w and thickness h. The focal length of the lens is denoted by f. Panel b represents three different cases of input optical pulses. The group delay dispersion and the third-order dispersion are denoted by GDD and TOD, respectively. The temporal distribution at P.1 converts to the spatial distribution along x at P.2.

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The PFT corresponds to a linear laser pulse arrival time along x. However, the electron velocity changes drastically during the acceleration. To overcome the sub-relativistic acceleration difficulty, the laser pulse needs to catch up with the rapidly changing electron, i.e., a continuously changing PFT angle, resulting in a curved laser intensity front.

In this letter, we propose an all-optical spatio-temporal coupling (STC) controlled driving laser pulse, which changes its tilt angle according to the increasing velocity of the electron (see Fig. 1). It is combined with a chirped dielectric structure23. A mid-infrared laser (10 μm), which can be generally achieved via optical parametric chirped-pulse amplification27,28, is used in this simulation. A longer wavelength will lead to higher breakdown threshold due to the multiphoton ionization and permit larger apertures for higher charge. The proposed scheme converts the temporal manipulation of the laser pulse into a spatially varying delay, which can be achieved by manipulating the group delay dispersion (GDD, Φ2) and third-order dispersion (TOD, Φ3). The STC scheme extends the interaction length and enhances the kinetic energy gain. Moreover, it retains high flexibility in the optical operations for creating the driving laser pulse.

Results and discussion

Description of the setup

The configuration we propose is shown in Fig. 1a. Due to a symmetric setup with counter-propagating (x-polarized) pulses, the magnetic fields cancel out at the channel center (P.2), and the electric fields add up. In Fig. 1b, the sketches of three different cases of spectral phase-induced PFTs are presented.

In order to analyse the STC in detail, we look into two aspects. One is the perfect phase-matched situation shown in Fig. 2 where the electron with 0.1 eV initial kinetic energy, typical excess energy for photoelectron29, is used. In the perfect phase-matched situation, the driving electric field Re[E(x, t)] in Eq. (4) (see the “Methods” section) is replaced by ∣E(x, t)∣, where ∣∣ represents the absolute value. In other words, the electron is assumed to remain on the peak of the field. This corresponds to an ideal acceleration structure where no dephasing between the driving field and the electron occurs and thus no longitudinal focusing is required. This gives us insights on the maximum kinetic energy gain with the given parameters. The other is the realistic situation where acceleration results of a 20 keV electron with a specific acceleration structure are shown in Figs. 3 and 4. This gives realistic guidance to future experimental work. The results with different initial electron energy can be found in Supplementary Note 6.

Fig. 2: Simulation for the perfect phase-matched pulse with 0.1 eV electron.
figure 2

Panel a shows the kinetic energy gain as a function of the group delay dispersion (GDD, Φ2) and third-order dispersion (TOD, Φ3), where the maximum kinetic energy gain ~0.6 MeV is represented by the white dot. With the parameters represented by the white dot, the peak field strength E0 = 2 × 2.4 GV/m and a factor of 0.7 is considered for the evanescent field effect. The envelope of the electric fields ∣E(x, t)∣ before interacting with the acceleration structure are presented in (b, c), where Φ2 = 3.9 × 10−2 ps2, Φ3 = 0, and 1.1 × 10−3 ps3, respectively. One can see that the GDD leads to a constant pulse-front-tilt along x and TOD modifies the pulse-front-tilt along x. The white dashed line indicates the electron injection position \(x=-0.45{\sigma }_{{{{{{{{\rm{FWHM}}}}}}}}}^{\prime}\), and \({\sigma }_{{{{{{{{\rm{FWHM}}}}}}}}}^{\prime}\) is the full-width-half-maximum of the beam size at location P.2 in Fig. 1a.

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Fig. 3: Acceleration of a 20 keV electron with a specific acceleration structure.
figure 3

Panel a shows the delay of the laser pulse induced by the first period of the acceleration structure. The length of the first period is denoted as w1. Panel b shows the phase (\({\tan }^{-1}\){Im[E(x, t)]/Re[E(x, t)]} − ω0t, where ω0 is the center frequency of the laser pulse) along the electron trajectories in c, d (black curves) where the electric field distribution for spatio-temporal coupling (STC) and pulse-front-tilt (PFT) is shown as function of x and t. Note that the electric field presented is for location P.2 with a fixed z. It can be seen that for the spatio-temporal coupling scheme, the electric field carries a curved phase-front, whereas the pulse-front-tilt scheme has a flat phase-front.

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Fig. 4: Comparisons among the spatio-temporal coupling (STC), pulse-front-tilt (PFT), and no-tilt schemes for the electron with 20 keV initial kinetic energy.
figure 4

Panel a shows the kinetic energy gain ΔE with peak field strength E0 = 2 × 2.25 GV/m. The perfect phase-matched STC scheme with the same parameters is shown by the blue dashed curve as a reference. Panel b shows instantaneous electric field experienced by the electron. Panel c shows a comparison of the tilt angle along the acceleration direction x, where the PFT scheme has a constant tilt angle, while the STC has an adapting one. The instantaneous PFT angle is derived from the electron velocity (solid blue line in (a)). The parameters used are Φ2 = 4.5 × 10−2 ps2 and Φ3 = 9.4 × 10−4 ps3. At P.2 the pulse duration \({\tau }_{{{{{{{{\rm{FWHM}}}}}}}}}^{\prime}=500\) fs, and the beam size \({\sigma }_{{{{{{{{\rm{FWHM}}}}}}}}}^{\prime}=0.16\) mm.

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Results

In Fig. 2a, the kinetic energy gain of a slow electron (0.1 eV) with a perfect phase-matched electric field is presented as a function of the GDD and TOD. A factor of 0.7 is included to take into consideration of the evanescent field effect, which is an approximate average of the structure constants of common DLA cells in the range 0.1 to 0.6 MeV. The maximum kinetic energy gain is ~0.6 MeV. It can be seen that the STC scheme is particularly advantageous for extremely low initial electron energies, i.e., where the velocity changes over a large range. The electric field envelopes ∣E(x, t)∣ are presented in Fig. 2b, c, where the white dashed line represents the electron injection position \(x=-0.45{\sigma }_{{{{{{{{\rm{FWHM}}}}}}}}}^{\prime}\) and \({\sigma }_{{{{{{{{\rm{FWHM}}}}}}}}}^{\prime}\) is the full-width-half-maximum of the beam size at P.2. In addition, the electric fields at larger t values (t > 0) arrives later than that with smaller t values (t < 0). Note that for the convenience of the representation, in Fig. 2b, c x = 0, t = 0 is chosen to be where the peak intensity of the laser beam locates. In Figs. 3 and 4, the initial electron acceleration location with the white dashed line is denoted as x = 0. It can be seen that the GDD and TOD drastically influence the PFT shape. The TOD modifies the PFT along x, leading to a continuously matching PFT for the entire electron acceleration.

For the STC scheme, the beam size and pulse duration at P.2 largely depend on the parameters of the entire system. We focus on a 2D+1 (x, z, t) model where the electric fields in the frequency domain and the time domain are connected by Fourier transform \(E(x,z,t)={{{{{{{\mathcal{F}}}}}}}}[E(x,z,\omega )]\). Note that we use the complex notation for the electric fields and only the positive half of the spectrum is used, i.e., ω > 0. The electric field used to calculate the electron acceleration is \({{{{{{{\rm{Re}}}}}}}}[E(x,z,t)]\), where “Re” represents taking the real part. The incident electric field in the frequency domain before the grating at P.1 follows the expression30,31

$${E}_{1}(x,0,\omega )R=\; {A}_{1}\exp (-ik{x}^{2}/{q}_{1})\exp (-\Delta {\omega }^{2}{\tau }^{2}/4)\\ \times \exp [i({\Phi }_{2}\Delta {\omega }^{2}/2+{\Phi }_{3}\Delta {\omega }^{3}/6)],$$
(1)

where A1 is a constant representing the amplitude, \({q}_{1}=i\pi {\sigma }_{1}^{2}/{\lambda }_{0}\) is the q-parameter for a Gaussian pulse, σ1 is the beam size, k = 2π/λ0 is the wave vector, λ0 = 10 μm is the center wavelength, τFWHM = 100 fs is the transform-limited pulse duration (full-width-half-maximum), \(\tau ={\tau }_{{{{{{{{\rm{FWHM}}}}}}}}}/\sqrt{2\ln 2}\), Δω = ω − ω0, ω0 = 2πc/λ0, Φ2 is the GDD, and Φ3 is the TOD. The electric field for electron acceleration at P.2 is constructed by two steps.

First, the electric field reflects on the grating, propagates through the lens, and arrives at the acceleration structure. These are calculated analytically via the ABCDEF matrix method32 (see Supplementary Note 1). The analytical expression of the electric field right before the acceleration structure is shown as the following:

$${E}_{2}(x,2f,\omega )= \; {A}_{2}\exp [-ik{(x-\beta \Delta \omega f)}^{2}/{q}_{2}]\exp (-{\tau }^{2}\Delta {\omega }^{2}/4)\\ \times \exp [i({\Phi }_{2}\Delta {\omega }^{2}/2+{\Phi }_{3}\Delta {\omega }^{3}/6)],$$
(2)

where β is the angular dispersion induced by grating at P.1, and f is the focal length of the lens. The choice of β can be found in Supplementary Note 2. The parameters A2 and q2 are amplitude and the q-parameter at P.2, which depend on β and f (see explicit expression in Supplementary Note 1). Note that Eq. (2), is the expression at the focal point, i.e., the propagation distance after the lens is f. In principle, the propagation distance between the lens and the acceleration structure is f − h. We found that the extra propagation distance h has a minor influence on acceleration results with our parameter choices since f ~ cm and h ~ μm. The analytical expression as a function of propagation distance can be found in Supplementary Note 1. Moreover, we assume perfect lenses where the focal length for all frequencies are the same. Thus, we present the electric field at P.2 at the focal length as shown in Eq. (2). Second, each acceleration structure period wn is iteratively calculated with the electron acceleration process23,33. In other words, upon entering the acceleration structure, the electron velocity v0 is used to calculate the first acceleration structure period, w1 = λ0v0/c. The accelerator structure introduces a x-dependent delay onto the driving field as shown in Fig. 3a. Without the loss of generality, a smooth flat-top function is used as an approximation of the shape of the delay. With the acceleration structure material as silicon (nSi = 3.5)34 and initial kinetic energy 20 keV, the first period of the acceleration structure w1 = 2.7 μm. For each period of the acceleration, both vacuum and the pillar section take 50% length of the entire period as shown in Fig. 3a. In our work, the maximum phase difference of the pulse at the vacuum and tooth/pillar regime within one period of the acceleration structure is taken as π. With the parameters of Figs. 3 and 4, acceleration results of the optimal design, i.e., the perfect phase-matched pulse are presented in Fig. 6 in Supplementary Note 4. The specific design is beyond the focus of this work and can be found in the work of Niedermayer et al.17,26. The phase of the electric field that the electron experiences, i.e., phase deviation the structure needs to be designed to correct, is shown in Fig. 3b.

After traveling through distance w1, the new electron velocity is used to calculate the next acceleration structure period w2, and this process repeats till the end of the acceleration. The evanescent field effect of each period is calculated by \(\exp [-0.5l\sqrt{{(2\pi /{w}_{n})}^{2}-{k}^{2}}]\), where l = 1 μm is the gap distance between the two facing acceleration structures. The evanescent field decay factor varies from ~0.3 to ~0.7 in Fig. 3c. The PFT used for comparison in Fig. 3b, d is defined in the work of Hebling et al. and Wei et al.14,25. In Fig. 3c, d, the electric field along the acceleration direction x versus the time t at P.2 (a fixed z) is plotted. It can be seen that the STC has a curved phase-front whereas the PFT scheme has a flat phase-front. More comparisons between the PFT and STC schemes are presented in Supplementary Note 3. Due to the continuously changing intensity front, the electron stays within the pulse in the STC scheme for the entire acceleration process. In contrast, for the PFT scheme, the electron walks off immediately with the pulse.

Discussion

Figure 4 presents the comparisons among the STC scheme, PFT scheme, and the no-tilt (direct transverse injection without any pulse-front-tilt) scheme, where the acceleration results of the STC and PFT schemes are outcomes of the electric fields presented in Fig. 3c, d, respectively. The optical lasers at P.2 of the three schemes are chosen to have the same beam size and pulse duration. The kinetic energy gains are presented in Fig. 4a. The perfect phase-matched STC scheme with the same parameters is shown by the blue dashed curve. The perfect phase-matched case for all three cases can be found in Fig. 6 in Supplementary Note 4. The peak electric field strength illuminating on the acceleration structure before considering the evanescent field effects is 2.25 GV/m from each side. Figure 4a indicates that a matching PFT enhances the acceleration energy drastically. The STC scheme should show greater advantages with higher acceleration field strength. The instantaneous electric fields the electron experiences along the acceleration position x are shown in Fig. 4b. It can be seen that for the PFT and no-tilt schemes, the electron walks off with the pulse imminently whereas, for the STC scheme, the electron sees the acceleration field for a longer interaction length. In Fig. 4c, the PFT angles are presented. The black curve is presented as a reference, where the instantaneous PFT angle is calculated from the electron velocity, i.e., \(\tan ({{{{{{{\rm{angle}}}}}}}})=c/v(x)\). The PFT angle of the PFT scheme is a constant c/v0. Note that the acceleration structure extends over the entire x, the relatively moderate electron energy increment after x > 0.15 mm is due to the decreasing electric field strength.

In all the calculations presented in this letter, we assume a constant distribution along y dimension with the beam size σy = 0.2 mm and calculate the total input energy as 0.48 mJ = 0.5cε0σy∬∣E(x, 0, ω)∣2dxdω = 0.5cε0σy∬∣E(x, 2f, ω)∣2dxdω, where the ε0 is the vacuum permittivity.

Conclusion

We present an all-optical-controlled scheme for non-relativistic electron acceleration in the DLA via the spatio-temporal coupling controlled driving pulse. It is promising especially for the acceleration of non-relativistic electrons with high electric field strength, where the electron velocity varies drastically during the acceleration process. The STC shows the possibility of high precision PFT angle control by converting the temporal variation into a spatial manipulation, which highly relaxes the nano-scale fabrication precision and increases the feasibility of implementing such a scheme with dielectric structures.

Owing to the continuously matching PFT, STC provides long interaction length and high kinetic energy gain. The optical configuration enables unique continuous tunability of the optical intensity front shape by changing the GDD and TOD. The scheme is a general method that could potentially be applied to driving fields of other wavelengths. Our results bring possibilities to portable electronic devices and table-top acceleration experiments.

The STC scheme enables high flexibility of the optical system elements. There is no constrain of the focal length, as long as the electron interaction point and the grating are positioned at each side of the lens’ focal points. In addition, this scheme enables high tunability since the PFT is defined by Φ2 and Φ3, which can be controlled by a commercially available acoustic-optical modulator. It offers independent programmable adjustment of GDD, TOD, and higher-order dispersion on-the-fly. Meanwhile, the adjustment of GDD and TOD does not influence the pre-aligned optical system. It provides the possibility of fine adjustment of even higher-order dispersion through electron feedback. Machine leaning35,36 can also be implemented into the system to optimize the beam properties. Most importantly, the TOD modifies PFT along the x dimension, resulting in a curved PFT that enables a continuously matching driving field to the electron beam for the entire acceleration process, which is crucial for high energy acceleration with short acceleration lengths. This largely enhances the flexibility of the experimental implementations.

Methods

This proof-of-concept demonstration is based on a single electron model. Due to relatively low space charge at the DLA injection for existing experiments, we ignore space charge effects in our model completely. Detailed information on intensity effects in DLAs, mostly dominated by wake fields analysis can be found in the work of Egenolf et al.37. The Leap-Frog method is used to numerically calculate the relativistic electron position and velocity as the following

$$\frac{\partial x}{\partial t}=v$$
(3)
$$\frac{\partial p}{\partial t}=-e{{{{{\rm{Re}}}}}}[E(x,t)],$$
(4)

where e > 0 is the elementary charge, \(p=mv/\sqrt{(1-{v}^{2}/{c}^{2})}\) is the electron momentum. The electric field E(x, t) is the resulting field from E2(x, t) at P.2 after adding the periodic delay caused by the acceleration structure/spatial grating (see Fig. 3c, d). In the expression of the laser electric field, the Gouy phase is not included for two reasons. One is that the Rayleigh range is larger than the acceleration structure/interaction length. The other is that the electron and the laser propagate perpendicularly to each other. Thus, the electron interacts with the laser pulse at different x with the same z. Thus, the Gouy phase is negligible, even when the focus is not exactly at the electron trajectory. Note that the position of the incident electron is at the exact gap center of the acceleration structure, where no net magnetic field or deflecting field exists. The electron injection location is at \(x=-0.45{\sigma }_{{{{{{{{\rm{FWHM}}}}}}}}}^{\prime}\) in Fig. 2b, c marked by the white dashed line. The injection time is synchronized at the peak of the temporal envelope at \(x=-0.45{\sigma }_{{{{{{{{\rm{FWHM}}}}}}}}}^{\prime}\). The initial injection electric field phase is chosen such that the final kinetic energy gain is the maximum. We chose that for simplicity, in accordance with the consideration to accelerate only a single electron at perfectly determined initial position. Longitudinal stability at finite pulse length and tolerance toward laser amplitude errors is achieved by choosing the synchronous phase off-crest, while alternating between the positive and negative flank provides both longitudinal and transverse stability17,26. This comes with a slight lowering of the gradient by a factor \(\cos ({\phi }_{s})\approx 0.5...0.8\), which we do however not consider in this study, since it affects all laser pulse schemes in the same way. Our numerical integration method takes <60 s for a single CPU on a standard PC (Intel(R) Core (TM) i5-10400 CPU @ 2.9 GHz). The convergence and stability tests can be found in Supplementary Notes 5, 6, and 7, respectively.