Dynamic control of laser driven electron acceleration in a photonic structure using programmable optical pulses

Abstract

Progress in optical techniques has made precision control of the phase profile in optical pulses common and accessible in scientific laboratories. Carefully shaping the field profile of a laser pulse can be used to master the dynamics of electrons traveling in photonic accelerating structures, such as the ones obtained by precisely aligning two dielectric gratings. Here, we show that by applying a liquid-crystal mask to program the phase and amplitude of an infrared laser pulse in combination with a pulse front tilt scheme, it is possible to implement dynamic control on a laser accelerator. This results in a nearly limitless live tuning capability of the accelerator beam dynamics, allowing the demonstration of software-based correction of structure and optical front imperfections, implementation of transverse focusing schemes, control of the energy and charge of the output beam, and ultimately optimization of the interaction length, leading to measured energy gains of up to 0.55 MeV.

Control of the phase field is essential to enhance the efficiency of dielectric laser accelerator schemes. Here, the authors introduce a software-tunable approach to the optimization and control of the electron beams by combining spatial light modulators with pulsefront tilted beams.

Introduction

Particle accelerators have become a mainstay of modern science and technology. High-energy colliders enable the discovery of new particles, while synchrotrons and free-electron lasers are crucial tools for the next generation of chemistry, biology and material sciences^1,2. In addition, a large number of medical and industrial applications of relativistic beams continue to drive the development of compact and cost-effective solutions^3,4. Conventional accelerators use radiofrequency fields on the order of 10-100 MV/m to accelerate particles, but the cavities that support these fields are large and expensive, as are the power sources. With the advent of dielectric materials that can withstand GV/m fields, higher gradient photonic structures that can be mass-produced using modern nanofabrication techniques have become of great interest⁵. Due to the extremely small size of the accelerating channels, GV/m field intensities can be achieved using modest (μJ to mJ) pulse energies from widely available compact laser sources. Not only do these dielectric laser accelerators (DLAs) promise to shrink accelerator dimensions, fabrication and operation costs by several orders of magnitude, they also enable the generation of electron bunches with sub-nm normalized emittances and attosecond characteristic time scales^6,7.

A key challenge in using lasers to directly drive dielectric accelerator structures lies in the ability to manipulate the electromagnetic field to control the transverse and longitudinal dynamics of charged particles⁸. Precise tuning of the phase of the field experienced by the electrons is essential for maximizing particle trapping, ensuring prolonged resonant interaction, and controlling the transverse dynamics. In experimental efforts, the initial demonstration of energy modulation mediated by simple gratings quickly gave way to the demonstration of higher gradients, longer interactions, and the introduction of on-chip power delivery and focusing elements^9–13. Different materials and drive laser wavelengths have also been explored^14–17. In order to mitigate the large defocusing fields which accompany high acceleration gradients, alternating phase focusing (APF) and ponderomotive focusing schemes have been proposed^18,19, and demonstrated^20,21. In all previous implementations, phase control has been experimentally achieved either by manipulating multiple independent laser pulses or breaking the translational symmetry of pillar structures by varying the period or adding physical phase-resetting gaps^22–24.

Meanwhile, spatial light modulators (SLM) have become critical tools in quantum optics, microscopy and holography,^25–28 due to their off-the-shelf capability to provide programmable, nearly arbitrary shaping of the phase and/or amplitude of laser pulses. In this paper, we leverage the versatility offered by liquid-crystal-based SLM to demonstrate tunable control of the acceleration dynamics in a DLA based on a simple fused silica dual-grating structure illuminated with a pulse-front-tilted (PFT) laser beam. (Fig. 1a)²⁹. Injecting the drive laser at ≈90 degrees with respect to the propagation axis of the electrons allows us to lengthen the temporal overlap region of laser and electrons and at the same time to project the tunable spatial modulation onto the electrons as they travel through the structure, as illustrated in Fig. 1a. This approach enabled extending the interaction to multi-mm lengths, testing of transverse and longitudinal phase space dynamics control schemes, and led to the demonstration of up to 0.55 MeV energy gains.

Fig. 1. Conceptual design of a DLA accelerator with programmable phase control.

a Cartoon drawing of the pulse-front-tilted laser incident on an asymmetric fused silica dual-grating structure. b Optical system used to generate arbitrary phase profiles on a laser with PFT. Imaging planes are located at the PFT grating, SLM, and DLA, such that PFT is transferred to the DLA and phase profiles in the electron trajectory direction (z) are transferred temporally to the electron beam. A cylindrical lens between the SLM and DLA allows for decoupling between x (transverse to the electrons) and z so that the SLM can be used simultaneously for phase and amplitude control. c Example phase masks that can be programmed onto the SLM to manipulate the electron beam. The final applied mask (right image) is the sum of (i) a z-dependent fresnel lens, (ii) a phase matching mask, and (iii) a nonlinear effect compensation mask.

Limiting initially the discussion to longitudinal dynamics, the energy modulation in a DLA can be written as a simple integral of the field experienced by the electrons along their trajectory, $\mathrm{\Delta }E=\mathfrak{Re}\left[q\kappa {\int }_{-L/2}^{L/2}{E}_0\left(z,t\right)e^{i\varphi \left(z\right)}dz\right]$. The structure factor, κ, measures how efficiently the structure converts the incident field into an accelerating gradient; L is the structure length, and E₀(z,t) represents the laser field amplitude. Maximum energy modulation corresponds to the case when the phase ϕ(z) is constant along the interaction such that the resonant phase matching requirement is met, i.e. the electrons do not slip in the accelerating wave when traveling in the structure. In order to appreciate the level of control on the accelerator enabled by the SLM, we can identify the different contributions to the total laser phase ϕ(z) = ϕ_TW(z) + ϕ_SLM(z) + ϕ_NL(I₀, z) as the traveling wave, SLM-controllable term and nonlinear contribution, respectively. The traveling wave phase is defined as ${\varphi }_{TW}=k_{g}z-\frac{k_{0}z}{\beta \left(z\right)}$ where β(z) is the normalized electron velocity along the structure and k_g and k₀ are the wavenumbers of the grating and the laser, respectively. In all simulations and experiments described in this paper, these wavenumbers are equal and constant—the gratings employed to form the structure are simple untapered transmission gratings with a periodicity equal to the drive laser wavelength (i.e. k₀ = k_g). For our relativistic beams the initial normalized velocity value β₀ is near 1, and phase matching can be obtained varying the incident angle, θ_I, of the laser on the structure or equivalently adding a linear phase ramp ${\varphi }_{SLM}=k_0\sin \left({\theta }_I\right)z$ with the SLM; this additional phase term tunes the phase velocity of the accelerating wave and can be used to match it to the electron velocity in the structure³⁰. As a result of the the DLA-induced energy gain, the velocity of the particles β(z) can change along the interaction, which can be compensated by a z-dependent slope θ_I(z) = θ_I0 + Δθ(z), where in the small angle approximation ${\theta }_{I0}\simeq \frac{1}{{\beta }_0}-1$. As an example, for 8 MeV ballistic (i.e. constant velocity) electrons, the initial matching angle θ_I0 = 1.73 mrad, while Δθ(z) can be used as a tuning parameter to compensate for DLA acceleration-induced dephasing with the SLM phase. Additionally, the use of slab transmission gratings in conjunction with high-intensity laser pulses results in Kerr effect dephasing ϕ_NL = n₂I₀(z)k_gΔy. Here, Δy refers to the thickness of the bulk fused silica (1 mm), n₂ is the nonlinear index of refraction, and I₀(z) is the intensity profile of the laser. Both dephasing contributions due to the acceleration and due to Kerr effects have been observed to strongly limit the interaction length in previous DLA experiments^11,31. While other approaches hard-code into the structure the phase variation needed to keep the acceleration in resonance, this is accompanied by significant flexibility limitations^20,23,24,32.

In our scheme, we utilize a single laser drive illuminating a DLA structure orthogonally with respect to the electron beam propagation direction (Fig. 1a). A pulse-front-tilt is applied to the laser pulse to extend the interaction³³, which would otherwise be limited to the 100-fs laser pulse length. Programmable control of the optical field seen by the electrons is accomplished by using a reflective 2D liquid-crystal array between the grating and the DLA (see methods). In the z-dimension (parallel to the electron propagation) the phase applied by the SLM is relay-imaged to the DLA. The SLM tunability can be used to directly apply on the incoming laser arbitrary phase profiles ϕ_SLM in order to flatten the phase experienced by the electrons and satisfy the resonant condition. Sample phase masks corresponding to the different contributions to compensate the acceleration via θ_I(z) and compensate nonlinear dephasing ϕ_NL are shown in Fig. 1c,ii/iii. Note that due to the 2π limit on phase retardation imposed by the SLM, the applied masks are not simple smooth profiles. For instance, a linear tilt across the SLM is generated according to $\text{mod}\left(k_{0}z\sin \left(\theta \right),2\pi \right)$, so that larger angles result in an increasing number of phase jumps across the SLM screen.

In addition to phase control, the scheme allows to use the same single programmable SLM element to manipulate the amplitude of the field seen by the electrons. This is accomplished inserting a cylindrical lens (labeled fx in Fig. 1b) to break the imaging condition in x such that applying a variable Fresnel lens profile ${\varphi }_F\left(x,z\right)=-\frac{k_0}{2}\frac{x^2}{f_x\left(z\right)}$ at the SLM plane results in direct control over the laser transverses size and hence the field intensity at the DLA channel. An example of such a mask profile is shown in Fig. 1c, i. Crucially, the accelerator tuning enabled by the SLM extends beyond simple resonant phase matching. As detailed later, the full control of the 2D liquid-crystal array allows for advanced phase and amplitude shaping, which can be used to manipulate transverse beam dynamics and engineer regions of strong energy modulation followed by low-field drift sections along the structure. This can be exploited both to locally optimize the acceleration in a particular region of the structure as well as enhance the beam microbunching process to improve the capture and output beam quality.

Results

We experimentally demonstrate programmable control of DLA acceleration using the 8.7 MeV electron beam from the UCLA Pegasus photoinjector (see Section “Spatial and temporal alignment”)³⁴. The structure is an asymmetric dual-grating structure formed by precisely aligning a pair of custom-fabricated transmission gratings to a 1.2 μm gap (see Section “Structure”), where the structure factor is κ = 0.05³⁵. The primary observable is the energy spectrum of the electron beam after it has passed through the structure and interacted with the laser field. Laser on and laser off electron spectra are then recorded on either the wide acceptance or high resolution spectrometer screens (see Section “Spatial and temporal alignment”) and compared to extract the maximum energy gain or loss experienced by the particles.

In order to best phase match to the accelerating electrons, the SLM phase profile is first optimized in segments, varying the linear tilt (or phase ramp) in 1.7 mm sections of the DLA and recording the measured change in the electron beam energy ΔE as a function of the applied Δθ_I as shown in Fig. 2a). In this way, we can exceed the interaction length limitations of the simpler flat-phase profile as used by Crisp et al.³¹. In areas designated for minimal interaction, we deliberately introduce a Δθ = 3 mrad offset, ensuring that the field in that portion of the DLA remains non-resonant with the electrons. In addition, the intensity is reduced by more than a factor of 4 in these same regions by applying a variable focal length Fresnel lens in the x-direction. Corresponding laser profiles on a phosphor screen adjacent to the DLA are viewed with an overhead camera and reported in Fig. 2b. In the right panel, we show the extracted amplitude and applied phase masks on such a laser, indicating that any observed energy gain can be attributed only to the high-intensity region. Each star in Fig. 2a corresponds to the optimal Δθ_I at that structure location. The energy gain is non-uniform at each z position owing to the underlying non-uniform distribution of the non-modulated spot, shown in Fig. 2b. We then construct a complete phase map θ_I(z) by stitching together the locally optimal Δθ_I values along the structure.

Fig. 2. Phase Tuning for Maximal Energy Gain Throughout Dielectric Structure.

a Each lineout corresponds to the illumination of 1.7 mm of structure, centered about the designated z location. The rest of the laser is defocused and >3 mrad detuned from resonance. The energy gain is non-uniform along z due to the variation of the optical intensity along the long laser axis. b Overhead images of laser profiles as seen on a phosphor screen. Each laser profile corresponds to a lineout of (a). c Phase mask and amplitude profile for the illumination of the highlighted 1.7 mm long region of the structure. d Maximum energy gain as a function of illumination length, comparing the curved, tuned phase profile indicated in solid lineouts to a linear tilt, indicated by the dashed lineouts. e Electron spectrum for the largest obtained energy modulation, obtained with the use of a piezo mirror in place of the SLM.

The benefits of this approach are clearly shown by comparing the energy gain as a function of the DLA accelerator length in Fig. 2d (note that the optimal Δθ_I vs. z curve was observed to be sensitive to day-to-day small changes in e-beam alignment, energy and laser setup, so the data in this plot corresponds to a slightly different tune than a). The interaction length is adjusted as before by varying the Fresnel lens, but in this case, we illuminate increasing lengths of the structure with high intensity with each step. The SLM is set to apply the optimized matched phase profile. The result is a marked increase in the energy gain compared with the flat-phase case, corresponding to a longer (up to 4 mm) phase-matched interaction over the grating structure.

Noticing an approximately linear variation of the optimal incident angle over the structure length, we found that by properly adjusting the lens positions in the optical system, we could obtain an incoming quadratically curved phase front that would remove the z-dependence in the SLM-applied θ_I(z) map. In this configuration, it became possible to replace the SLM with a piezo-controlled mirror to maintain the capability to fine-tune to the resonant incident angle θ_I0. Overcoming the severe reflection losses and fluence limitations associated with the use of the SLM enabled us to maximize the intensity incident on the DLA accelerator. With the piezo mirror in place and the lenses tuned to flatten the θ_I(z) map, at an incident field of 6 GV/m, a DLA-record peak energy gain of 0.55 ± 0.05 MeV (see Fig. 2e) was observed.

With the implementation of the SLM scheme, accelerator tuning is not limited simply to resonant condition matching, and we can explore the application of SLM-based phase and amplitude control to the study of different transverse focusing schemes. Due to the micron-scale aperture, the transverse acceptance of a DLA is inherently limited, which is often stated as one of the main drawbacks of this advanced accelerator technique. To make matters worse, strong defocusing forces (∝κk₀E₀ or MT/m) are present when the particles are accelerated at phases where the longitudinal motion is stable⁵. Nevertheless, it has been pointed out recently that proper shaping of the DLA fields themselves can lead to particle confinement^18,19. Multiple approaches have been proposed, generally taking advantage of the fact that at phases different from the accelerating phase the DLA fields are strongly focusing. So far, the realization of these schemes required building custom nanofabricated structures with carefully designed phase jumps. This restricts the range of usable beam energies and input laser intensities, and is not fully scalable to larger structures as manufacturing and laser profile errors stack up²⁰. Leveraging the live control offered by the SLM, we employ our setup to test and compare two such approaches, namely ponderomotive focusing and alternate phase focusing (APF). In the experiment, due to pointing jitter and the mismatched phase space of the incoming e-beam, particle throughput was highly variable; consequently, we were not able to distinguish the expected increase in integrated transmitted current from the larger shot-to-shot charge fluctuations of the portion of the electron beam not temporally overlapping with the laser. Nevertheless, we observed clear signatures of both of these focusing schemes in the energy spectra of the beams, providing insights into the underlying beam dynamics.

Ponderomotive focusing relies on exciting different spatial harmonics in the structure so that while the resonant harmonic is used for acceleration, the non-resonant spatial harmonic provides an average transversely focusing force¹⁹. In practice, multiple spatial harmonics can be realized by adding to the phase mask a sinusoidal modulation with amplitude A_p and period T_p. By the Jacobi-Anger expansion $e^{i{A}_p\sin \left(\frac{2\pi z}{T_p}\right)}=\sum J_n\left(A_p\right)e^{i\frac{2\pi n}{T_p}z}$, this phase modulation results in a spectrum of spatial harmonics with different wavenumbers $k_n=k_g+\frac{2*\pi n}{T_p}$ and hence different phase velocities. The phase lineout along the DLA (lower panel) and the corresponding laser intensity distribution (upper panel) when this oscillatory phase contribution is superimposed to the matched quadratic phase mask are shown in Fig. 3a. The measured intensity profile of the laser highlights the need to minimize the SLM phase discontinuities since they translate into undesirable gaps in intensity (see Methods “Spatial light modulator”)³⁶.

Fig. 3. Ponderomotive Excitation of a DLA accelerator.

a Laser profile (upper) and corresponding phase profile (lower) across the structure for the Δθ = − 2 mrad, A_p = 1.0 rad, T_p = 390 μm case. (Note: Simulations indicate that the transverse asymmetry in the laser intensity profile is an artifact resulting from a minor misalignment of the SLM-applied Fresnel lens combined with a limited phase modulation range.) b The laser is tuned to the matched laser phase, and then the linear phase tilt Δθ is scanned using the SLM over a ±5 mrad range. Several simulation traces are plotted corresponding to the prediction of a simple ballistic model for the DLA acceleration over different interaction lengths L. The laser intensity is normalized for each interaction length to produce the same maximal energy gain at Δθ = 0. The dashed lines indicate interaction lengths that do not match the observed energy modulation peak width. c,d Energy modulation as a function of Δθ, for a variety of ponderomotive phase contributions. In (c), T_p = 390μm. In (d), A_p = 1 rad. e Maximum energy modulation for the three largest spatial harmonics (at Δθ = 0 and ±2 mrad) respectively.

By adding a constant Δθ offset to the applied phase mask, it is possible to measure the energy modulation as a function of the phase velocity detuning and thus visualize the spatial harmonics in the accelerating wave as they become resonant with the beam. When no phase modulation is added, the scan reveals a single peak at Δθ = 0 (Fig. 3b). The dependence of the energy gain on the angular detuning Δθ follows a sinc-like function, the width of which is inversely proportional to the interaction length. The same is true for the peak widths in the ponderomotive excited case. By minimizing the root mean square error between a ballistic interaction model, normalized for equal energy modulation, and the data, we estimate an effective interaction length of 2.5 ± 0.5 mm.

In Fig. 3c we report the result of the same measurement when ponderomotive sinusoidal phase modulations with different T_p and A_p are applied. Up to the n =  ± 2 modes are clearly visible, especially for larger modulation amplitudes. For ∣n∣ > 3, the resonant energy gains approach the noise floor; this is also partly due to the diffraction efficiency of the SLM greatly decreasing at larger phase slopes. We see that the positions of the excited modes at Δθ = nk_p/k_g agree perfectly with the expectation (3d). In Fig. 3e we also show that the relative amplitude of the peaks at the n = 0, ± 1 harmonics matches the predicted behavior J_n(A_p) from the Jacobi-Anger expansion. A larger phase modulation amplitude results in more power and therefore larger energy exchanges for the n ≠ 0 modes. In practice, the choice of A_p must be made on the basis of necessary focusing power for stable acceleration¹⁹.

Simulations indicate that ponderomotive focusing can be very effective at confining the particles to the accelerating gap, and our data demonstrates that this approach is well suited to be implemented with the SLM, but the transverse confinement comes at the expense of energy gain; for example, when using the n = − 1 mode for acceleration, the vast majority of the input energy will go to the n = 0 mode for focusing and the electron beam is therefore accelerated only at a relatively small fraction of the maximum gradient. As proposed in previous designs^19,29 and also in the Supplementary Fig. 1 of this paper, stable acceleration conditions require ponderomotive phase oscillation amplitudes A_p between 0.2 and 0.6, thus achieving utilizing less than 10–30% of the field amplitude for acceleration. If one wants a higher effective acceleration gradient while still confining the beam, an alternate phase focusing (APF) scheme can be used.

In an APF scheme, particles are continuously accelerated at a constant, often large, fraction of the maximum possible gradient proportional to $\cos \left({\varphi }_{APF}\right)$ where ϕ_APF is the synchronous phase. The phase actually jumps repeatedly between  ± ϕ_APF so that the field experienced by the particles alternates between longitudinally focusing, transversely defocusing and longitudinally defocusing, transversely focusing regions. Overall, this forms a so-called FODO lattice which can stably confine a particle beam. In our setup, we test the APF concept by adding to the matched SLM phase profile a series of discrete phase jumps of amplitude  ± ϕ_APF where the sign switches twice each cell length λ_APF as pictured in the inset of Fig. 4a. We plot the maximum observed acceleration vs. the step height 2ϕ_APF. In this dataset, λ_APF = 0.85 mm, so every 0.425 mm there is a phase jump. The observed energy modulation aligns well with our model, showing again that we are able to coherently accelerate a particle bunch at a programmed resonant phase. By increasing Δϕ_APF up to π, we almost completely eliminate energy gain in this multi-mm interaction.

Fig. 4. Longitudinal Manipulation.

a Energy gain as a function of Δϕ_apf, shown in the inset. b Simulation results showing the fraction of electrons that reach 90% of the resonant energy gain as a function of the resonant energy gain, colored according to Δϕ, the actual tunable parameter in this scheme. The inset shows an example phase and amplitude input with a 1 mm buncher followed by APF confinement and acceleration. c Top panel shows the energy gain vs. Δϕ_b for a 1.28 mm interaction, where the first 0.64 mm is accelerated at the matched phase, and a phase jump Δϕ_b is added for the second 0.64 mm. In the bottom panel, an initial 1.7 mm section is used to accelerate the beam, followed by a 1.28 mm drift section and the rest of the 5 mm structure is used to shift the energy spectrum for different Δϕ_b. At the top (bottom), the beam has a starting energy of γ = 17(12). Error bars show the standard deviation of 5 shots at each ϕ_b. Solid lines indicate simulation results. Here, gain refers to the maximal observed energy gain relative to the nominal energy, and loss to the maximal observed loss. d Phase and amplitude for scans shown in c.

In order to increase not just the number of particles transmitted, but also the quality of the accelerated beam, it is important to prepare the longitudinal phase space of the beam to avoid injection over all phases of the accelerating optical wave^37,38. This can be done by controlling the amplitude of the fields in the DLA channel and introducing a drift region where an initial energy modulation can quickly convert into density modulation, bunching the beam. To demonstrate the principle, we first illuminate a 1.28 mm region of the structure, with the first 0.64 mm illuminated at constant phase (Fig. 4d). Here, we define maximal gain as the maximal observed energy gain relative to the nominal energy, and loss as the maximal observed loss. When we vary the phase of the latter half, Δϕ_b, we see that at a π phase shift, much of the energy gain and loss on the 8.7 MeV electron beam is canceled out, as we would expect in a ballistic case where phase evolution is minimal (Fig. 4c, d, top). The imperfect cancellation can be attributed to varying phase slippage for different particles. In this case, there is no clear shift in the energy spectrum. By lowering the injection beam energy to 6.2 MeV and introducing a drift section, the results are very different. Due to increased dynamic dephasing, at no point in a scan over Δϕ_b does the energy gain come close to fully canceling. Instead, the gain and loss regions of the spectrum are shifted depending on Δϕ_b (Fig. 4c, d, bottom). Comparison with simulations indicate that this shift is due to an induced sub-wavelength density modulation which can be exploited for significantly higher quality output electron beams with decreased energy spread. We attribute the reduced modulation depth in the experimental data compared to simulations to the finite resolution of the SLM, which introduces phase smoothing and intensity gaps at the sharp transition points required by this scheme.

Discussion

In summary, we have shown that by utilizing a combination of a PFT illumination scheme with an SLM, it is possible to fine-tune the phase resonant condition in the DLA accelerator and significantly increase the interaction length and energy gain with respect to previous DLA implementations. We demonstrated acceleration for distances larger than 4 mm, with as much as 0.55 MeV energy gain. The flexible setup also allowed us to test out different focusing schemes; further investigation with better-matched injection beams should be performed to test how much particle confinement is increased utilizing these schemes. All of this is possible due to the immense tunability of the accelerating laser field, enabled not by complicated custom nanofabricated structures but instead by commercial SLM technology.

The potential for future experiments is showcased in simulations that combine the phase and amplitude control schemes demonstrated in our experiment in an ideal setup where we assume a matched input beam, a 10 GV/m incident field and a significantly larger structure factor up to 0.12 (see Supplementary Table 1). Figure 4b, Supplementary Fig. 2, and Supplementary Fig. 3 show the results of these simulations where an input beam of total energy 6.1 MeV is focused and bunched using an APF scheme, indicating that over 20% of the particles entering the structure could be accelerated by up to 2.4 MeV in only 5 mm, 1 mm of which is used entirely for bunching. We can see the large effect that APF focusing is having on particle throughput in these simulations, as lowering Δϕ_APF greatly increases the energy gain at the expense of particle throughput. In the Supplementary Fig. 1, we also present an ideal simulation case for the ponderomotive focusing scheme. Comparing the output of these simulations, it is clear that the potential for APF to provide both higher acceleration gradients and more particle throughput for similar incident laser intensities. Both approaches were experimentally investigated in our work, as we note that the ponderomotive focusing scheme is implemented via a smoother phase profile with fewer phase jumps, making it more compatible with SLM-based optical control methods.

The setup presented here enabled rapid testing of the most essential elements of creating a scalable accelerator. The soft, live control enabled by the SLM allows for the tuning of beam energy, transmitted beam current, and energy gain as required by the application^39–41. The scheme is also an almost perfect canvas for utilizing machine learning optimization techniques⁴². In fact, while static optics may offer higher power handling, the SLM provides a crucial layer of active feedback necessary to correct for inevitable experimental imperfections in the laser wavefront, structure fabrication, and alignment. This methodology allows for the manipulation of the electron phase space without the need for precise a priori knowledge of these errors, effectively enabling ’live’ tuning of the matched phase irrespective of the specific cause of dephasing. When scaling to longer structures and larger energy gains, fabrication errors in structure or phase inhomogeneity in the laser will inevitably stack up, and this independence of the tuning from the cause of error will only become more crucial. Programmable optical components allow for arbitrarily complicated phase and amplitude structures to be tested, which could also help to bridge the gap between the sub-relativistic and relativistic regimes, which is critical for the development and future operation of on-chip accelerators.

Methods

Electron Beam Measurement

For this experiment, the electron beam has a charge ranging from 0.2 to 5 pC and is first accelerated by a radiofrequency photoelectron gun before being boosted by the LINAC to a total energy of 8.7 MeV (6.2 MeV for Fig. 4b) (Fig. 5). It is then focused by a series of 6 quadrupoles to a spot size of σ_x = 23 μm × σ_y = 54 μm at the DLA plane. The transmitted beam is then transported by a solenoid and additional quadrupoles to the end of the beamline, where the energy spectrum is viewed by an ICCD camera after a dipole spectrometer on one of two screens. At the closer screen, the horizontal dispersion (i.e. the linear transport matrix element which relates the horizontal offset at the screen with the relative momentum deviation) is 10 cm, while at the far screen we have a horizontal dispersion of 32.8 cm; the far screen thus has higher energy resolution but a smaller spectral field of view. The close screen has a spatial resolution of 44 μm/pixel, and we therefore find a calibration of 3.85 keV/pixel for the close screen while operating at a total energy of 8.7 MeV, as in the case of the maximum gain reported in Fig. 2.

Fig. 5. Pegasus Beamline and Dielectric Laser Accelerator Schematics.

a Pegasus beamline at UCLA. The electrons are photoemitted in the gun, then accelerated in the linac and then focused by quadrupole magnets into the DLA structure. After the laser interaction they are transported to the dipole spectrometer where the energy spectrum can be observed on one of two YAG crystals using an ICCD camera. b Exploded view of the structure assembly. c View and dimensions of one of the two gratings in the structure.

Similarly to ref. ³¹, the maximum energy gain and loss are extracted from the spectrometer profiles by defining a figure of merit defined as FOM = $\frac{⟨S_1⟩-⟨S_0⟩}{{\sigma }_0}$, where 〈S₁₍₀₎〉 are the observed spectra averaged over at least 5 laser on (off) shots and σ₀ is the standard deviation of the laser off shots. To determine the maximum gain/loss at the tails of the spectrum where the modulated electron density nearly blends into the noise floor, we require at least 3 consecutive pixels to have a signal-to-noise ratio of at least FOM > 1.5.

For the ponderomotive focusing datasets, the high dispersion spectrometer screen was used to increase energy resolution. As a consequence of the relatively low energy acceptance, there is a cutoff on the maximum energy modulation observable for the central n = 0 mode. However, by increasing the structure gap and decreasing the incident field strength, we are still able to verify that the relative excitation amplitude of the ±1 modes agrees well with our model as shown in Fig. 3b.

An Integrating Current Transformer (ICT) was used to measure the input charge as having ≈12% fluctuations; comparison of the measured input charge and transmitted charge indicates that pointing and energy jitter decouple these variables (linear fit R² = 0.41). As a result, we normalize all spectra to the transmitted, not input, charge, allowing us to make direct comparisons of laser off-laser on shots to extract energy gain/loss without the influence of charge variations.

Structure

The structure is composed of two fused silica gratings of constant 780 nm periodicity manufactured by Ibsen Photonics. The gratings were designed and simulated using the finite-difference time-domain software Lumerical to provide the optimal field structure for long interactions within the limits of single side illumination³⁵. While a dual drive has shown improved control on the symmetry of the fields in the gap, for experimental simplicity, we elect to operate with a single laser drive and tuned the teeth height of each grating separately to compensate for the asymmetric power delivery and minimize the transverse kick of the DLA structure.

The upper grating has a tooth height of 450 nm, reducing the diffraction efficiency compared to the lower, 650 nm grating. This symmetrizes the fields within the structure for certain gap-offset combinations, resulting in an accelerating cosh mode. The gratings are 1 mm thick, 5 mm long and 9 mm wide. The thickness of the grating serves as a filter; any electrons transmitted to the spectrometer screens pass through the DLA gap. The gratings are independently mounted with a gap between them adjustable over a range of 10 μm using a 3 piezo motor stage (Fig. 5).

The structure is assembled using diode laser interferometry to ensure sub-micron flatness as described in ref. ⁴³. In the same way, the relative rotation is tuned with sub-microradian accuracy. The full assembly is mounted in vacuum on a stage with motors controlling the x, y, $x^{\prime }$, and $y^{\prime }$ axes. In-situ transmission scans which measure particle count as a function of piezo value confirm that unless otherwise noted, the structure gap was equal to 1.2 μm. For our input electron beams, this results in only 1.6% transmission, so that the charge collected at the spectrometer screen is typically less than 1 fC. When the gap size is set to 1.2 μm, using the energy modulation induced by a short, non-PFT laser pulse, we measure the structure factor to be κ = 0.05 in agreement with the simulation predictions.

The implementation of dual illumination or a Distributed Bragg Reflector would greatly improve the acceleration gradient. As discussed in Crisp et al.³⁵, asymmetric illumination of a symmetric structure introduces a deflection force, which for high gradient interactions, results in particle loss. As a result, in this experiment, we deliberately lowered the design structure factor to enable long interactions and utilized a larger than wavelength gap for the same purpose. Dual illumination would allow for the use of higher efficiency gratings and a smaller gap size, both of which increase structure factor; for example, the structure used by Peralta et al.⁹ had a structure factor of up to 0.24, but there was no stable accelerating channel.

Drive Laser

A 20 mJ, 780 nm, 100 fs FWHM Ti:sapphire laser is split 9:1, with the low energy arm redirected to a frequency tripling path used to generate the UV which drives the photocathode in the RF gun. The majority is used to drive the DLA, with a 2:1 telescope immediately before the 600 ln/mm PFT generating grating⁴⁴. The laser is incident on the grating at a 27.9° angle so that the diffracted beam is along the grating normal in order to minimize temporal distortions over the beam profile⁴⁵. A 2-lens telescope images the grating onto the SLM with a magnification of M = 0.55; another 2-lens system with M = 0.85 images the SLM onto the DLA. These specific magnifications were chosen to best decouple the PFT angle θ_PFT and pulse length τ. The total magnification is defined by the electron velocity, v = βc, such that $\beta =\cos \left({\theta }_{PFT}\right)/\sin \left({\theta }_{PFT}+{\theta }_I\right)$ is satisfied. Since λ₀ = λ_g, θ_I < 5 mrad, and θ_PFT = 45° is required for group velocity matching.

The optical system is designed with 3 imaging planes: at the location of the 600 ln/mm diffraction grating used to generate PFT, at the SLM plane, and at the final DLA plane. The PFT angle is verified using an interferometric measurement between the PFT laser and a copropagating reference pulse⁴⁶. The use of 2 lenses between the PFT grating and SLM allows the PFT angle to be (mostly) decoupled from the imaging condition, allowing for the pulse to remain short whilst the PFT angle is tuned. A cylindrical lens in the second stage means that we do not image in x, allowing the Fresnel lens phase mask to modulate the intensity of the laser while technically operating the SLM in a phase-only configuration. The chromatic aberrations of the Fresnel lens are mitigated by the relatively long (5 cm) Rayleigh range of the laser spot size at the waist.

Spatial Light Modulator

We use a commercial liquid crystal on silicon reflective SLM with a 1920 × 1080 pixel array with >93% fill factor. The optical axis is at 45°. We use a half-wave plate before and after the SLM, such that the laser polarization is on the same axis, so we are in a phase-only modulation configuration. The laser is incident at a 10° angle from the normal to the liquid-crystal panel, introducing minimal phase errors. Also, to mitigate interference from the zero-order reflection, we always superimposed a small linear phase ramp on the SLM modulation to spatially separate the desired beam from interaction region artifacts.

The SLM has a 12.5 mm by 7.1 mm panel active area, which the laser overfills transversely, corresponding to the z electron propagation axis, but not vertically. The SLM has a measured 55% reflectance and a theoretical diffraction efficiency of above 50% for the phase masks used. The total energy throughput of the optical transport is limited by the 138 mJ/cm² damage threshold of the SLM.

The SLM does not have enough phase range to encompass the entire desired mask, causing jumps along the laser profile at all points where the phase is reset. Unfortunately, imaging the first-order diffraction of the SLM onto the DLA plane implies that at each phase jump, there is a corresponding gap in intensity (Fig. 3a), leading to a degradation of the DLA performance. Many of the phase jumps are due to the optimization of the resonant phase profile, but some more are introduced by the implementation of the ponderomotive and APF focusing schemes. Interestingly, due to the gentler phase variation, the ponderomotive focusing scheme is only responsible for a minimal number of these phase jumps, while for APF, this is a more serious issue, which is exacerbated by the requirement to propagate the laser through 1 mm of fused silica. Nonlinear effects at the points where there are large variations in intensity result in large nonlinear phase shifts, posing significant challenges to retaining phase coherence for the full structure length. Thinner substrates could greatly mitigate this problem.

Spatial and temporal alignment

Spatial alignment is obtained by viewing a 45° phosphor screen with an overhead CCD camera. The electron beam is focused to a small spot and then transmitted through the structure. The tunability of the structure allows us to maximize transmission by tuning the pitch with a 10 μm gap and then closing the gap to the optimal run condition of 1.2 μm. When lower structure factors are desired the structure gap can be increased.

Preliminary timing overlap is obtained by placing a photodiode at the interaction plane. A delay stage prior to the main optical table to scan the relative time-of-arrival in 16 fs steps, much lower than the decay time of the photodiode. Femtosecond-scale synchronization is obtained using a reference laser pulse without any pulse front tilt such that the bulk of the electron beam is energy modulated by the DLA, even if only for the 100-fs spatial extent of the laser pulse. After this, an unmatched grating (900 ln/mm) is used to generate θ_PFT = 56° to establish electron energy gain and verify temporal overlap, again with a limited interaction length, before swapping to the final 600 ln/mm grating, which provides θ_PFT = 45°.

Simulation

Simulations are done using a modified version of the Matlab-based numerical code SHarD by Ody et al.⁴⁷. The code accepts as input arbitrary phase and amplitude profiles for the laser and decomposes the field into a large number (typically > 100 different harmonics, then it uses a Runge-Kutta method to update the particle position and momentum. The particles can be traced throughout their transport, and are eliminated when they hit the structure walls. The structure can be defined simply by a gap, length, and structure factor, or by implementing a full complete model of the diffraction from the top and bottom gratings using complex coefficients⁴⁸.

To model our experimental setup, we use a 5000 particle electron beam with a normalized emittance of 200 nm, σ_y = 30 μm, and total energy 6.1(8.7) MeV and 0.05% energy spread. To save on computing time, we track only the 1.6% of particles initially within the 1.2 μ aperture of the DLA. For the simulations in Fig. 1c, d and Fig. 4 we use a more idealized electron beam with 1 nm normalized emittance, σ_y = 1μm, and total energy of 6.1 MeV. For all simulations, particles are assumed to be injected at uniformly random phase over longitudinal phase space. The bunching section in the idealized simulation is designed to maximize the number of particles trapped at the accelerating phase at z = 1 mm. The ponderomotive design follows the stability criterion laid out by Naranjo et al.¹⁹, and the APF lattice design follows the method from Niedermeyer et al.¹⁸. In both cases, optimal focusing requires the periodicity of the added phase mask to increase as the energy of the beam increases.

Author contributions

Conceptualisation: P.M. and R.J.E. Data curation: S.C. Formal analysis: S.C. Investigation: S.C., R.J.E., A.O., and P.M. Visualisation: S.C. Validation: S.C., R.J.E., A.O., and P.M. Project administration: P.M. Writing-original draft: S.C. and P.M. Writing-review: S.C., R.J.E., A.O., and P.M. Writing-editing: S.C. and P.M. Supervision: P.M.

Peer review

Peer review information

Nature Communications thanks Roy Shiloh and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The datasets collected during this study in the Zenodo repository at 10.5281/zenodo.19687832. Further information is available from the corresponding authors upon request.

Code availability

The code that supports the findings of this study is available from the corresponding authors upon reasonable request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-75233-z.