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Abstract
In dielectric laser acceleration, nanostructures etched into silicon are used to convert free-space ultrashort laser pulses, incident from the side and parallel to the wafer substrate, to accelerate particles. This current approach is experimentally challenging and, as it turns out, not quite necessary for most experiments and practical applications. Here, we experimentally demonstrate and numerically verify the efficacy of top-illuminated structures, and measure a maximum acceleration gradient of 49.2 ± 3.1 MeV/m. We discuss how, in practice, this approach proves superior to the current standard in the field, and expect it to become the definitive choice for nanophotonic particle laser acceleration.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Soon after the invention of the laser, in 1962, Shimoda [1] suggested using a gas-filled cavity to accelerate electrons using optical fields. In the same year, Lohmann submitted an IBM tech-note [2] and filed a patent for a “particle accelerator utilizing coherent light” [3], which included a description of two dielectric gratings facing each other, and complementary coherent, in-phase, counter-propagating lasers. The device is placed in a vacuum chamber and an electron emitter directs a stream of electrons into a gap between the two gratings, such that an electron propagates normal to the grating teeth and in parallel to the substrate. When the laser pulses are properly aligned and synchronized, energy is transferred to such an electron through interaction with the generated evanescent near-fields, and the electron accelerates. The prominent advantage of producing particle accelerators out of dielectric structures and using ultrashort lasers to drive them, rather than traditional radio-frequency metallic cavities, is the damage threshold of the construction material. While metallic cavities are usually operated in the 25-50 MV/m range, dielectric laser-acceleration structures have been shown to yield an acceleration gradient of 1.8 GV/m and withstand field strengths of up to 9 GV/m [4], which directly implies the length of such a linear accelerator could be shrunken down by almost two orders of magnitude.
The dielectric grating scheme was successfully realized in two experiments in 2013, utilizing single- and double-gratings [5,6], nano-fabricated into fused silica. The high damage threshold of the dielectrics was not fully employed in these proof-of-concept experiments, however. One difficulty in achieving this is the nonlinear optical effects that distort the high-intensity laser pulse as it is focused down into and through the bulk material. Interest in the field of dielectric laser acceleration (DLA) has been growing ever since, with different structures, techniques, and applications proposed and investigated [7,8].
Almost twenty years following Lohmann, Palmer [9] reviewed, proposed, and discussed different novel accelerating structures for a laser-driven grating linear accelerator [10]. The research culminated in a proposal for the study of laser acceleration of electrons using micrograting structures at the accelerator test facility in Brookhaven national labs [11]. In that proposal, submitted in 1989, it is interesting to find actual measurements of accelerated electrons near gratings and the so-called “colonnade” structures. The latter are the origin of the popular “dual pillar” structures, see Fig. 1, which first resurfaced in 2015 [12] and are mainly used in DLA research today. It is favored over the double-grating scheme because of two important benefits: first, it completely resolves the requirement of nanometer-accurate wafer alignment and bonding, since the pillars are inherently precisely positioned in one nano-fabrication step; second, the distance in the structure material through which the laser must traverse is reduced to the pillar diameter, which is typically a few hundred nanometers, annulling any nonlinear optical effects–assuming the laser impinges from the side of the colonnade.
Fig. 1. Sketch of the two approaches (a) top-illumination and (b) single-side illumination, based on scanning electron microscope images of the structures used in the experiment. The diagonal lines in the bottom right side of each panel indicate the edge of the mesa (see fabrication process). The exact geometry and distance of this edge from the channel plays a major role in the side-illuminated approach, see text. Red color indicates laser illumination. Cyan arrow indicates electron propagation direction.
Indeed, while the geometry of the colonnade and dual-pillar structures is equivalent–a set of cylindrical pillars arranged in two columns with a vacuum gap, or channel, in between, the principal difference between the two is the laser illumination. Since 2015, any experiment employing dual-pillars used the side-illumination approach. Two variations are evident: using one, or two incident laser beams. In the dual-sided illumination, two perfectly synchronized, counter-propagating laser beams are incident from both sides of the dual-pillar structure, which implies that the resulting near-fields in the channel are tunable. This enables, for example, adjusting the phase of the super-imposed fields to achieve symmetric or anti-symmetric conditions, thereby allowing one to control and systematically probe the phase-space behavior of the electron pulse [13]. The difficulty in such an experimental setup is inherent in its complexity: alignment of the two laser beams in terms of incidence angle, spot size and beam divergence, intensity, jitter, and temporal overlap, is not straightforward. One could trade-off the on-demand tunability by using one incident laser and adding a distributed Bragg mirror to the structure [14], see Fig. 1(b); the mirror is arranged by alternating vacuum and silicon slabs, so only 4 slabs are required to achieve virtually 100% reflection. However, the mirror only mimics dual-sided illumination: the reflected beam has already traversed through the structure once, losing some power. Additionally, if very short laser pulses are desired (below few tens of femtoseconds) then the reflected pulse would spatially interfere with the incident one only partially, though this, along with diffraction of the reflected beam are negligible in current experiments.
In this work, we return to Palmer’s proposal of illumination of the dual-pillar structures directly from above, as shown in Fig. 1(a). It follows naturally from the structure’s symmetry that the transverse fields inside the channel would also exhibit transverse symmetry, eliminating the short-comings of the single-side-illuminated mirror approach, and also the inherent complexities of the dual-sided illumination setup. In particular, we will now compare the performance of the top illumination and single-sided illumination schemes, both numerically and experimentally.
2. Theory
While in the side-illumination approach, one can approximate an analytical solution of the fields by simply calculating the diffracted components of a wave through a periodic thin grating [15,16], in the top-illumination scheme the laser field impinges from vacuum onto two columns of pillars, which may also be assumed periodic in the electron propagation direction (z axis in Fig. 1), but certainly finite in the orthogonal transverse direction. Furthermore, the evanescent near-fields’ intensity and formation, particularly in the vertical direction (y) along the pillars’ height, now depends on the height of the pillars and the reflectivity of the substrate. A theoretical treatment of this problem in the sense of approximate analytical expressions is difficult, and is not in the scope of this paper. Instead, we design the nanophotonic structures’ parameters according to basic principles and optimize them numerically.
The Widerøe or synchronicity condition in DLA implies that during interaction, the electron’s normalized velocity $\mathrm{\beta }$ and the phase velocity of the chosen spatial harmonic $\textrm{m}$ of the diffracted incident electromagnetic wave match, which, when the laser is incident normal to the structure’s periodicity, may be expressed as
where $\mathrm{\Lambda }$ is the period of the structure and $\mathrm{\lambda }$ the illumination central wavelength. The energy gain of a synchronous particle per period can then be calculated [17] fromFor the purpose of this work, we choose ${\mathrm{\varphi }_\textrm{s}}\textrm{ = }\mathrm{\pi }$ such that the synchronous electron accelerates on-crest (samples the peak of the electric field), which implies that its energy gain $\mathrm{\Delta} \textrm{W}$ is maximal; however, at the same time we would like to keep the periodicity of the structure fixed. Over an infinite, periodic structure, the synchronous electron would accelerate, effectively changing the synchronous phase or slipping from the initially designed one, and eventually no net acceleration can occur [10]. However, if the structure is short enough such that phase slippage is minimal, $\textrm{cos}({{\mathrm{\varphi }_\textrm{s}}} )\approx \textrm{1}$, we can approximate the total energy change of the synchronous electron over a structure of length $\textrm{L}$ as
where we define ${\textrm{e}_\textrm{1}}$, the first harmonic of the field, as the synchronous field, which is the peak field accelerating the particle. If we do not use the periodic simplification and instead taper the structure, Eq. (3) is no longer valid and the analysis becomes more involved.In our experiment, an electron pulse is much longer than one optical cycle, and as such spans all phases rather than just the synchronous one. As a result, following the interaction and under ideal experimental conditions, we expect to measure a symmetric energy modulation. However, as will be evident from the experimental results, this is true only for low field strengths.
3. Simulations and experimental procedure
The experimental setup has been previously described in great detail in [20]. Briefly, ultraviolet 257 nm, 150 fs full-width half-max (FWHM) laser pulses are used to trigger a Schottky emitter in a scanning electron microscope (SEM) generating short electron pulses. These are accelerated in the SEM column to an energy of 28.4 keV while elongating in time to about 600 fs. Concurrently, infra-red $\textrm{1}\textrm{.93}\; \mathrm{\mu}\textrm{m}$, 680 fs FWHM pulses are synchronized to impinge on the structure. The laser beam cross-section is elliptic, approximately Gaussian, with its waist set to $\textrm{76}\; \mathrm{\mu}\textrm{m}$ by $\textrm{8}\; \mathrm{\mu}\textrm{m}$ (1/e2), its long axis aligned to the length of the structure. The electron energy is measured using a magnetic spectrometer with a resolution of ∼100 eV, connected to a micro-channel plate (MCP) and CMOS camera reading the phosphor screen behind the MCP. Temporal overlap is achieved using a high-resolution delay stage, thus enabling us to measure a delay-scan: energy modulation against temporal delay between the laser and electron pulses.
To compare between the two approaches, namely top- and single-sided-illumination, we designed two structures as shown in Fig. 2. The top-illuminated structure (Fig. 2(a)) includes a 40-period symmetric dual-pillar structure roughly $\textrm{25}\; \mathrm{\mu}\textrm{m}$ long, and Bragg mirrors arranged on both sides. By numerical simulations, we have found that the existence of the mirrors increases the amplitude of the near-fields in the channel. They are comprised of 10 slabs each, the positions of each from the other calculated according to the Bragg condition; the distance of the first slab to the channel is optimized numerically. The structure geometry affects the power conversion factor of the incident propagating field to the different harmonics: in the present case, the pillars were designed to be round with a diameter of 400 nm; however, an optimization of the structure, for example by replacing the circular pillars with elliptical ones, or resorting to nanophotonic inverse-design [21], may produce higher gradients.
Fig. 2. Synchronous field simulations of top- and side-illuminated $\textrm{3}\; \mathrm{\mu}\textrm{m}$-high structures. SEM images of the top (a) and side (f) structure, with 619 nm pillar period, insets showing additional dimensions. (b), (d), (g), (i) $\textrm{|}{\textrm{e}_\textrm{1}}\textrm{|}$ xz-plane cross-sections $\textrm{2}\textrm{.5}\; \mathrm{\mu}\textrm{m}$ above the substrate. (b), (c), (g), (h) show the longitudinal component $\textrm{|}{\textrm{e}_{\textrm{1z}}}\textrm{|}$ along z. (d), (e), (i), (j) show the transverse component $\textrm{|}{\textrm{e}_{\textrm{1x}}}\textrm{|}$ along x. Cross-sections of the structures are marked in translucent white with black border. Black-framed clear rectangles in (h), (j) are projections of the position of the pillars not cut by the cross-section plane. Dashed lines mark the position of the cross-sections relative to each other and the structure. In-ward pointing crosshairs in (c), (e), (h), (j) depict optimal injection height in the xy-plane for high gradient and symmetric transverse forces. Color-scale is in arbitrary units, spanning a $\mathrm{\pi }$ phase shift, the contrast of which has been rescaled for visibility; however, the scale of each column of panels matches.
The design and optimization process are repeated for the side-illuminated structure (Fig. 2(f)), in which case we have used a previously-optimized $\textrm{15}\; \mathrm{\mu}\textrm{m}$-long structure design: the pillar diameters are 300 nm in the transverse direction and 400 nm in the longitudinal (electron propagation) direction. Here, the two pillar columns are shifted by half a period with respect to each other and the Bragg mirror is now closer to the center of the channel [22]. Additional dimensions for both structures are depicted in Fig. 2.
The fields were calculated in a unit-cell using the Ansys Lumerical 3D electromagnetic simulator software [23], with periodic boundary conditions in the z direction (see Fig. 2). In simulation, the laser wavelength is set to $\textrm{1}\textrm{.93}\; \mathrm{\mu}\textrm{m}$. The fields were then loaded into the particle tracking software General Particle Tracer [24], which we used to simulate the resulting energy modulation. The electron beam parameters were 100 pm rad transverse normalized emittance and 0.5 eV FWHM Gaussian energy spread, as expected in the experiment. The exact position and minimum size of the electron beam have practically no effect on our current goal, of reaching the maximum energy gain. In order to sample all laser phases and avoid complex trajectory effects, it is sufficient to simulate a uniformly-distributed, one optical cycle (6.45 fs) electron pulse spatially covering the entire channel height. We note here that since each sample was etched in a separate process to achieve the different heights, the transverse dimensions of each structure can slightly differ from the design, which was taken into account in the simulations.
Example cross-sections of the simulated fields are also shown in Fig. 2. Here, we compare a $\textrm{3}\; \mathrm{\mu}\textrm{m}$-high top (Fig. 2(a)–2(e)) and side illumination (Fig. 2(f)–2(j)) structures. The nearfield profiles presented were filtered along the longitudinal (periodic) z-direction, to include only the first harmonic or synchronous field, ${\textrm{e}_\textrm{1}}$, which is the dominant phase-synchronous contribution to the interaction with the electrons. It is clear that the fields are not uniformly distributed on the channel aperture, which suggests that the injection position (along the pillar height) is of utmost importance to achieve the highest gradient. Figure 2(b)–2(c) and Fig. 2(g)–2(h) show the longitudinal field component ${\textrm{e}_{\textrm{1z}}}$ in the x-z Fig. 2(b) and 2(g) and x-y Fig. 2(c) and 2(h) planes, the cross-sections of which are marked by the gray dashed line. The corresponding transverse field component ${\textrm{e}_{\textrm{1x}}}$ is shown in Fig. 2 in the x-z Fig. 2(d) and 2(i) and x-y Fig. 2(e) and 2(j) planes. The color scale indicates a $\mathrm{\pi }$ phase-shift (between red and blue) of the laser; both phases can be used for acceleration, provided that the electrons are injected at the correct time.
It is evident that the top-illumination field distributions are transversely symmetric, unlike the side-illumination case. Moreover, the top-illuminated pillars exhibit a periodic set of amplitude peaks along the pillar height, with the longitudinal fields appearing strongest around $\textrm{2}\textrm{.5}\; \mathrm{\mu}\textrm{m}$ (top hotspot), and between $\textrm{0}$ and $\textrm{0}\textrm{.5}\; \mathrm{\mu}\textrm{m}$ (bottom hotspot), the latter having a slightly higher amplitude. The peaks are a result of the pillar acting vertically as a nano-resonator, yielding modes dependent on the pillar height. Furthermore, the bottom hotspot is enhanced by the constructive interference of the reflected field from the substrate (Fresnel reflection of the incident field–a factor of 1.55 for our parameters). In practice, the optimal electron beam injection position in terms of acceleration gradient would be chosen at the top hotspot, because nearer the substrate effects such as charging and alignment may prove challenging to overcome. Complementary to the longitudinal fields, the transverse fields also exhibit a higher amplitude peak towards the substrate.
In the case of side-illumination, reflection from the Bragg mirror similarly acts to enhance the field in the channel but with a 100% reflection (a factor of 2, field-wise), which would ultimately explain the high acceleration gradient in comparison to top-illumination. In Fig. 2(h) and 2(j), it would appear (as naively expected), that for side-illumination the fields are distributed nearly-uniformly along the pillar height. In fact, in simulation, we have had to take special care to arrive at this condition: since the Gaussian laser beam impinges on the substrate or mesa edge (bottom right corner in Fig. 1(b); the mesa itself allows us to focus the laser parallel to the substrate - along x), it diffracts and breaks up before it reaches the pillars. Thus, this distribution is quite heavily dependent on the laser incidence angle, wavelength, and the distance and shape of the mesa edge from the pillars. In an additional simulation (not shown), different only by changing the distance of the edge to the pillars by $\textrm{1}\; \mathrm{\mu}\textrm{m}$, the distribution Fig. 2(h) and 2(j) changed to two peaks unevenly distributed along the channel height, indicating a tight tolerance and therefore inherent instability in experiment. Moreover, the field distribution is always asymmetrically distributed along the x-direction, which is a result of the horizontal asymmetry of the structure. The asymmetry, as previously discussed, is inherent in the geometry, and implies that over many periods the electron beam will experience accumulated deflection. Consequently, the beam would skew and may eventually steer most of the electrons away from the center axis and into the pillars, resulting in loss of the particles. Indeed, we have not been able to exactly reproduce the simulation curve in experiment, as will be discussed in the next section.
Lastly, it is important to note that the fields shown here are the real part of the Fourier-transform of the impulse response of the structure, at the design wavelength. In a transient analysis, the laser pulse impinges on the structure and reflects from the substrate (or Bragg mirror). This picture implies that there is a temporal dependence of the electron acceleration on the interference of the incoming and reflected beams. However, in general, we use laser pulses of many cycles, and inject the electron pulse such that it samples the peak, time-averaged interference for maximum acceleration.
4. Fabrication process
The structures were fabricated from a 1-5 ohm-cm phosphorus-doped silicon in the <100> orientation. The patterns are written by a 100-kV electron-beam lithography machine (Raith EBPG 5200) on a 400 nm thick negative resist (ma-N2405). The developed wafer was then etched by a cryogenic reactive ion etching technique to a depth of $\textrm{3}\; \mathrm{\mu}\textrm{m}\; \pm \; 0\textrm{.1}\; \mathrm{\mu}\textrm{m}$ (Oxford Instruments Plasmalab100). In a second step, a mesa mask was formed using laser lithography (Heidelberg Instruments DWL66+), by exposing a $\textrm{6}\; \mathrm{\mu}\textrm{m}$ thick positive resist (AZ 4562). Following development, a Bosch process yielded the final mesa height of $\textrm{30}\; \mathrm{\mu}\textrm{m}\; \pm \; 1\; \mathrm{\mu} \textrm{m}$ (see bottom right corners in panels of Fig. 1). The remaining resist on the structures was then removed by immersing the wafer into a Piranha solution for 5 min.
5. Results and discussion
We fabricated and measured three different heights for the top-illuminated structures: 620 nm, $\textrm{1}\textrm{.2}\; \mathrm{\mu}\textrm{m}$, and $\textrm{3}\; \mathrm{\mu}\textrm{m}$, which we compare to a $\textrm{3}\; \mathrm{\mu}\textrm{m}$-high side-illuminated structure. Example of a delay scan is shown in Fig. 3(e) for the top-illuminated $\textrm{1}\; \mathrm{\mu}\textrm{m}$ structure, measured at an incident laser peak field of 442 MV/m, corresponding to a peak intensity of $\textrm{2}\textrm{.6} \times 1{\textrm{0}^{\textrm{10}}}\; \textrm{W/c}{\textrm{m}^\textrm{2}}$. Assuming the laser pulse and electron pulse are Gaussian, we fit the electron energy modulation signal, which reflects their overlap, with a Gaussian as well (white line) with 90% prediction bounds (white dashed line). The measured maximum energy gain is taken as the location of 95% of the highest measured energy, while the error bars mark the resolution of our spectrometer. In effect, this metric excludes the remaining 5% signal as noise. The fitted maximum of the overlap (green circle) is plotted for each incident laser peak field in Fig. 3(a)–3(d), along with the calculated bounds, theoretical curve, and full 3D particle-tracking simulation results.
Fig. 3. Experimental and simulation results. Green dots mark experimental maximum energy gain peaks, analyzed as explained in the text. “Peak field” refers to the peak amplitude of the incident laser field. Dashed linear curve is the linear fit according to Eq. (4). Gradient shaded patches depict 100 eV energy steps from the maximum energy gain (for clarity, a total of 300 eV are shown here). Top-illumination with structure heights of: (a) 620 nm, (b) 1.2 µm, (c) 3 µm. (d) Side-illumination of a 3 µm high structure. The simulation curve is higher than in experiment, presumably due to the complex shape of the substrate edge, which could not be modeled accurately. (e) Example $\textrm{1}\; \mathrm{\mu}\textrm{m}$ top-illumination structure delay-scan at a laser incident peak field of 442 MV/m, with Gaussian fit (white line), 90% prediction bounds (dashed white line), and Gaussian maximum (green dot).
In simulation (shaded areas), the top-most boundary of the shaded area represents the highest energy gain, while each of the shaded patches indicate a 100 eV difference from the maximum energy gain. Hence, our measurements lie within 300 eV of the simulated results, and fall in line with the theoretical prediction of Eq. (3) (black dashed line).
In order to match Eq. (3) to experiment, we fit a linear curve to find the proportionality factor ${\mathrm{\alpha }_\textrm{1}}$, or structure factor, relating the full incident peak electric field $|{{\textrm{E}_{\textrm{peak}}}} |$ induced by the laser from free-space to the synchronous field $|{{\textrm{e}_\textrm{1}}} |$:
Table 1. Structure performance comparison
Table 1 compares the performance of the different structures. The most important figure of merit is ${\mathrm{\alpha }_\textrm{1}}$, which gives us the normalized efficiency of the structure in accelerating particles, and can be related to the first-order coefficient of the Fourier series expansion of the diffracted field, due to the grating. As we previously stated, the side-illumination structure is optimized for high acceleration gradient, and indeed boasts ${\mathrm{\alpha }_\textrm{1}}\textrm{ = 0}\textrm{.139}$, higher than the values for the top-illuminated structures. Two important factors can explain this: First, the field reflection from the substrate is estimated to be 55% while the reflection from the Bragg mirror is 100%. This could be circumvented by a different fabrication process incorporating a more reflective substrate or coating it with different layers forming a Bragg mirror. Second, the height of the pillars should be fine-tuned such that the constructive interference from the field reflected from the substrate interferes optimally at the top hotspot. Variation is already apparent between the three top-illuminated samples.
The damage threshold of all structures, under our laser parameters, is similar – an incident laser peak field of roughly 500 MV/m. While our structures are based on silicon, which is easily processed and has relatively no charging issues, one could switch to a different material and achieve about an order of magnitude larger damage threshold [25], or enhance the damage threshold by applying an additional hydrogen annealing step to the fabrication process [26].
In the last row of the table we calculated the corresponding gradient, which can be used as an estimate of the usable acceleration gradient in future dielectric structure design. With respect to the acceleration gradient, the 620 nm and $\textrm{3}\; \mathrm{\mu}\textrm{m}$ structures underperform by roughly 50% to the side-illuminated structure. The $\textrm{1}\textrm{.2}\; \mathrm{\mu}\textrm{m}$ top-illuminated structure is outperformed by only 35% in acceleration gradient, which is, however, far outweighed by the inherent advantages of working with top illuminated structures both in experiment and practical devices. We expect additional optimization of the structures to diminish this difference even further, for example, as previously mentioned, using photonic inverse design [21].
6. Conclusion
Though the work of Palmer was the basis of the current dual-pillar nano-structures used in DLA, up until now these have only been side-illuminated. In this work, we demonstrated in both experiment and simulation how top-illumination, as first suggested by Palmer [9] in 1962, has a significant practical advantage arising from the symmetry of the structure and the opportunity to use only one laser beam. The field formation along the pillar height introduces an interesting new avenue to explore: the manipulation of the vertical direction which is considered “invariant” under the side-illumination approach. One immediate application of top illumination would be the confinement of the electron beam in 3D, and specifically in the vertical direction – a requirement that must be satisfied in order to extend DLA stages beyond several hundred micrometers. This type of confinement in the vertical direction has already been treated in theory and simulation [27], however, only for the side-illumination approach, which then requires additional complications such as using silicon-on-insulator to generate the required field distributions, barring the usage of dielectrics with higher damage thresholds such as fused-silica. With the top-illumination approach, these extra fabrication steps, along with the required experimental precision and similar practical difficulties could possibly be avoided: simply by illuminating from the top.
Funding
Bundesministerium für Bildung und Forschung (05K19WEB); European Research Council (616823, 884217); Gordon and Betty Moore Foundation (GBMF4744).
Disclosures
The authors declare no conflicts of interest.
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