1. INTRODUCTION

Miniaturization of accelerators is a rapidly growing field [1–4" target="_self" style="display: inline;">–4]. Among them, dielectric laser accelerators (DLAs) accelerate charged particles using the evanescent fields of dielectric structures driven by femtosecond laser pulses [4–8" target="_self" style="display: inline;">–8]. Due to the high damage threshold of dielectric materials in the near-infrared, the acceleration gradient of DLAs can be more than 10 times higher than conventional radio-frequency accelerators [9–12" target="_self" style="display: inline;">–12]. Recent progress in the integration of DLAs with photonic circuits has enabled the development of chip-scale accelerators, [13–18" target="_self" style="display: inline;">–18], which promise to further increase the compactness and practicality of this scheme. For subrelativistic and moderately relativistic electrons, focusing provided by laser-driven techniques, such as alternating phase focusing [19,20], enables stable electron beam transport and acceleration for long distances.

To promote the application of DLAs in both fundamental science and medical therapy [4,21,22], it is important to deliver high electron currents. However, since the geometric dimensions of currently existing DLA designs are on the wavelength scale, where the wavelength corresponds to that of their near-infrared drive lasers, it is intrinsically challenging to deliver an electron beam with high current through a single narrow electron channel as currently used in DLAs [19]. Motivated by this challenge, we explore a photonic crystal DLA architecture that has multiple electron channels (Fig. 1). We show that straightforward design principles lead to the electromagnetic fields inside different channels being almost identical, which enables simultaneous acceleration or manipulation of N phase-locked electron beams, increasing the total current by a factor of N.

Fig. 1. Schematic of (a) a dual-pillar DLA and (b) a multichannel DLA. Two laser pulses (propagating in ±x directions) incident on the DLA are indicated by the orange arrows. Electrons travel inside parallel channels along the z direction.

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2. DESIGN PRINCIPLES

Typical DLAs consist of a pair of dielectric gratings such as the dual silicon pillar DLAs illustrated in Fig. 1(a) [12]. When the laser beams illuminate the gratings, the near fields can be used to accelerate the electron beam that travels in the gap between two gratings. Although the height of the electron channel can be several micrometers, its width is limited to the submicrometer scale, due to the evanescent nature of the near fields. The small channel width limits the total electron current. To bypass this constraint, we propose a DLA architecture that consists of multiple parallel electron channels as shown in Fig. 1(b). The idea of simultaneous acceleration of multiple parallel beams has also been studied in millimeter wave accelerators [2325" target="_self" style="display: inline;">–25]. However, the underlying physics and design principles of multichannel DLAs are fundamentally different from those of the millimeter-wave accelerators. Since each row of dielectric grating in the multichannel DLA is identical, this DLA structure is a finite photonic crystal. In contrast to previous single channel copropagating photonic crystal waveguide accelerators [26,27], our approach incorporates a multichannel design and operates based on side illumination to increase the total current. Consistent with Ref. [28], we focus on silicon pillars and refer this design as a multi-input multi-output silicon accelerator (MIMOSA).

We find that the essential characteristics of the MIMOSA are captured in the band structure and eigenmode properties of the underlying infinite photonic crystal. For simplicity, we study a two-dimensional photonic crystal with a rectangular lattice (Fig. 2), where the dielectric pillars extend uniformly in the y direction. Previous studies confirm the validity of using a two-dimensional calculation to describe a photonic crystal with pillar height larger than the wavelength [28,29]. In a two-dimensional photonic crystal, the fields can be decomposed into TE (with Ey, Hx, and Hz nonzero) and TM (with Ex, Ez, and Hy nonzero) polarization [30]. Since only the TM polarization provides acceleration, we study only the TM polarization in this work. The unit cell and the band diagram of the photonic crystal underlying the MIMOSA are illustrated in Figs. 2(a) and 2(c), respectively. We assume that the electrons travel along the z direction and are centered around x=0 and that the incident light propagates along the x direction. Therefore, the incident laser can excite eigenmodes lying on the ΓX direction in the reciprocal space [Fig. 2(b)]. Furthermore, the mode at the Γ point has the same phase in each unit cell. Thus, to ensure that the electron beams in different channels get the same acceleration, the mode at the Γ point should be excited dominantly. Therefore, the frequency ω of the eigenmode at the Γ point should match the frequency ω0 of the incident light. In addition, in order to satisfy the phase synchronization condition for the DLA [29,31], we need to have where c is the speed of light, β=v/c where v is the speed of electron, L is the periodicity along the z direction, and m is the diffraction order. In typical DLAs, the amplitude of the first-order diffraction is stronger than that of higher-order diffraction, so we take m=1 in the synchronization condition.

Fig. 2. (a) Illustration of the photonic crystal where the unit cell is highlighted in the dashed red box. The electron propagation direction is indicated by the gray arrow. 2a, 2b, 2d, and L represent pillar length, pillar width, gap width, and the periodicity in the electron propagation direction, respectively. (b) The reciprocal Brillouin zone. (c) Band diagram of the TM mode of the photonic crystal with L=1μm, a=0.3μm, b=0.86μm, and d=0.2μm. Big (small) black dots represent eigenmodes with odd (even) mirror-z symmetry, and red (blue) dots represent eigenmodes with even (odd) mirror-x symmetry. (d), (e) Field profiles of the eigenmodes at the Γ point with normalized frequency 0.500 and 0.411, respectively.

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Additionally, the unit cell of typical photonic crystals may have certain symmetries. In the demonstration shown in Fig. 2, the mirror-x and mirror-z symmetries of the unit cell require that the acceleration mode should also have certain symmetries. To accelerate the electron traveling along x=0, the mode should be symmetric with respect to the mirror plane x=0 [red dots in Fig. 2(c)]. And to couple to plane waves propagating in the x direction, the mode should have odd mirror-z symmetry. In other words, the photonic crystal underlying the MIMOSA should support an eigenmode at normalized frequency (frequency normalized by c/L) β with odd mirror-z and even mirror-x symmetry; below we refer to such a mode as an acceleration mode. This condition on the symmetry of the acceleration mode, together with Eq. (1), represents one of the main contributions of this study and is referred to as the band structure condition below. It can be satisfied through tuning the geometrical parameters of the dielectric pillar.

Moreover, the figure of merit of the MIMOSA can also be derived from eigenmode analysis of the underlying photonic crystal. This “acceleration factor" g [3234" target="_self" style="display: inline;">–34] is the maximal acceleration gradient at the center of the electron channel divided by the maximal electric field inside the dielectric material. From the field profile of the acceleration mode [Fig. 2(d)], which is computed for a structure that is infinitely periodic along both the x and z directions, we can calculate the acceleration factor, which turns out to be close to the acceleration factor of the MIMOSA, where the structure is finite along the x direction.

Based on the discussions above, the MIMOSA design procedure is summarized as follows. (1) Given the electron speed and the frequency of the excitation laser, the periodicity (L) of the photonic crystal along the electron propagation direction is determined by Eq. (1). (2) The periodicity along the transverse direction and the shape of the dielectric pillar in the unit cell can be tuned such that the band structure condition is satisfied and the width of the electron channel exceeds a value set by experimental considerations. (3) Using the acceleration factor of the acceleration mode as the objective function, the unit cell of the photonic crystal is optimized with the constraints in (1) and (2). (4) The procedure above involves only the study of an infinitely periodic photonic crystal. After such a photonic crystal is designed, we can confirm the design by simulating a photonic crystal with a finite number of periods as we will discuss in Section 3.

3. DEMONSTRATION AND ANALYSIS

Using the design procedure as discussed above, we demonstrate the design of a MIMOSA for electron speed 0.5c (β=0.5) with rectangular silicon pillars and a central laser wavelength at 2 μm (λ0=2μm). The same design principles apply to other electron speeds, pillar shapes, dielectric material systems, and laser wavelengths. The periodicity of the photonic crystal is determined by the phase synchronization condition L=βλ0=1μm [29]. We denote the half length and width of the rectangular pillar as a and b, respectively, and half width of the electron channel as d. Larger channel width generally results in higher beam current but lower acceleration gradient due to the exponential decay of the near fields. To compromise between the requirements of high acceleration gradient and wide channel width, we choose d=0.2μm, consistent with previous studies [12,28]. By tuning a, b, we find that with parameters a=0.3μm and b=0.86μm, the photonic crystal supports an acceleration mode at normalized frequency 0.5 [Fig. 2(c)] with high acceleration factor. The mode profile is shown in Fig. 2(d), from which the inferred acceleration factor is g=0.51.

To verify our design principles, we truncate the photonic crystal in the x direction to specify a finite number of channels and perform a full wave simulation [35]. For demonstration purposes, we limit the number of electron channels to N=3. The field distributions are shown in Fig. 3, where the two driving plane waves are symmetric with respect to x=0. The fields inside different electron channels are almost identical and strongly resemble the eigenmode shown in Fig. 2(d), calculated for an infinite photonic crystal. From Fig. 3(b), we find that the largest E field is inside the vacuum rather than inside the dielectric, which contributes to the high acceleration factor [34]. This holds when the rectangle pillars have rounded corners with small radius [33].

Fig. 3. (a) Schematic of the dual drive simulation. The dashed box highlights the unit cell, the red arrows represent the illuminating lasers, and the gray arrows indicate the electron propagation direction. Under in-phase and equal amplitude illumination at wavelength λ0=2μm, the magnitude of E field is shown in (b), while Ez, Ex, and Z0Hy are shown in (c), where Z0 is free-space impedance.

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When the synchronization condition Eq. (1) is satisfied for m=1, and in the limit of a perfectly rigid electron beam, only the first diffraction order interacts with the electron beam [31]. The corresponding longitudinal field distribution of the first diffraction order has the following form inside the electron channel [12,28]: where Acn and Asn are the amplitudes of the “cosh” and “sinh” components, respectively, in the nth electron channel, α=(2π/L)2−(ω/c)2 characterizes the decay of the evanescent wave inside the electron channel, and xn is the center of the nth channel. Acn and Asn depend on the incident waves and are discussed in detail in Appendix A. From Maxwell’s equations, we can get Ex and Hy. The synchronized Lorentz force F=q(E+v×B) on the electron is and Fy(x)=0, where q=−e is the electron charge, the electron phase ϕ0=ωt0 is the phase of the oscillating field experienced by an electron entering the structure at time t0 with respect to the reference particle entering at t=0, and γ=1/1−β2 is the familiar relativistic factor [29,31]. Equations (3) and (4) suggest that the cosh component can provide acceleration (deceleration) or focusing (defocusing), depending on ϕ0, while the sinh component can provide transverse deflection.

Figure 4(a) shows the acceleration force Fz where we choose ϕ0 such that Fz is maximized at x=0. Again choosing ϕ0 such that the force is maximized, the transverse force Fx is shown in Fig. 4(b), which focuses the electron to the channel center [29]. The longitudinal forces in different channels differ by less than 7%, which confirms that the acceleration in different channels is almost identical. This holds even better as the number of channels increases, for instance to 10, as shown in Appendix B.

Fig. 4. (a) Longitudinal and (b) transverse force distribution inside each electron channel. (c) Amplitudes of the cosh and sinh components in each channel at central frequency and (d) their frequency dependence.

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The acceleration factor g, derived from the cosh component of the MIMOSA, is |Acn|/Emax in the nth electron channel, where Emax is the maximal electric field inside the dielectric. We find the cosh component to be dominant over the sinh component in each channel. Moreover, the cosh components have similar amplitudes in different channels. Acn/Emax=0.512, 0.494, and 0.512 in channels 1, 2, and 3, respectively [Fig. 4(c)]. These acceleration factors agree with the prediction made by eigenmode analysis, which predicts g=0.51. Due to the mirror-x symmetry of the excitation, Acn (Asn) in channels 1 and 3 are equal in amplitude with same (opposite) sign. Moreover, Acn are in phase with the symmetric excitation (Appendix A). Therefore, the electrons with the same entering time experience almost identical forces in different channels.

The bandwidth of the MIMOSA is 62 nm, within which the difference between Ac in each channel and Ac in the central channel at central frequency is less than 10% [Fig. 4(d)]. Such bandwidth corresponds to 95 fs pulse duration of a transform-limited Gaussian pulse at 2 μm. With such a pulse and a fluence half of the damage fluence of silicon (0.17J/cm2 [36]), the predicted acceleration gradient can reach 0.56 GeV/m.

However, as the number of electron channels increases, the bandwidth of the MIMOSA decreases. The bandwidth approximately scales inversely with the number of pillars, since the MIMOSA can be regarded as an optical resonator where the stored energy scales linearly with the number of pillars while the energy leakage rate remains roughly constant. For the studied MIMOSA, the bandwidth (Δλ), in nanometers (nm), as a function of the number of channels (N) is estimated as Δλ(nm)=144/(N−1.33) when N≥4 as shown in Fig. 5(a). If we choose a transform-limited Gaussian pulse with central wavelength at 2 μm and bandwidth matching the bandwidth of the MIMOSA, the relation between the pulse duration (τ) and the number of channels is linear [Fig. 5(b)]. The estimated scaling rule is τ(fs)=40.8×N−54.4 for N≥4, where τ is the full width at half-maximum (FWHM) pulse duration in femtoseconds (Appendix B). Therefore, due to the bandwidth scaling rule, the number of electron channels cannot be arbitrarily large and should be chosen to match the bandwidth of the excitation pulse. For the subpicosecond optical pulses, the laser-induced damage fluence remains approximately constant [14,37], which is about 0.17J/cm2 for bulk silicon [36]. Therefore, the maximal electric field scales as inverse square root of the laser pulse duration, i.e., Emax∼1/τ. Thus, as N increases, the maximal electric field decreases, which functions as a limiting factor for the number of channels. To achieve 0.3 GeV/m acceleration gradient at half of the damage fluence, the maximal number of channels is 10, with a matched pulse duration of 350 fs. Thus, increase of total current by 1 order of magnitude can be achieved without sacrificing the acceleration gradient.

Fig. 5. (a) Bandwidth of the MIMOSA versus number of electron channels. The geometric parameters are the same as those studied in Section 3. (b) The corresponding pulse duration of a transform-limited Gaussian pulse with central wavelength 2 μm and a bandwidth matching the bandwidth of the MIMOSA.

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Due to the broadband nature of the MIMOSA, we anticipate the long-range wakefield effects to be insignificant. The short-range wakefield effects and beam loading properties are similar to those in dual-grating DLAs [5,38,39], and a brief discussion is presented in Appendix D. The space-charge effect in MIMOSA is similar to that in previous single-channel DLAs [4,17], which is briefly discussed in Appendix E. Moreover, for applications like medical therapy, the required average number of electrons per micro-bunch per channel is only slightly above 1 (Appendix C). Therefore, both the space-charge and wakefield effects are negligible in the MIMOSA operating for medical applications.

To match the phase synchronization condition as the electrons get accelerated, we can gradually change the geometric parameters of the MIMOSA unit cell such that the band structure condition [Eq. (1)] is always satisfied. The tapering of grating periods demonstrated previously [40,41] can be applied to MIMOSA designs straightforwardly to achieve continuous phase velocity matching. The shape of dielectric pillar can also be tapered such that the frequency of the acceleration mode at the Γ point continuously matches the central frequency of the driving laser. Furthermore, since the electromagnetic field distributions are almost identical from channel to channel, the alternating phase focusing [19] can also be applied to MIMOSA for long-distance acceleration with stable beam transport.

4. DEFLECTOR

Instead of providing multichannel acceleration, the MIMOSA can be designed to manipulate electron beams in many other ways. In this section, we consider a MIMOSA designed for simultaneous deflection of N electron beams [28,42]. The design procedure is almost the same as that of an accelerating-mode MIMOSA. In contrast to the acceleration mode, the deflection mode in the photonic crystal with rectangular pillars has odd mirror-x symmetry. The figure of merit of the deflector is the ratio between the deflection gradient at channel center and the maximal electric field inside the dielectric, i.e., As/Emax. We start with the photonic crystal shown in Section 3 to demonstrate the design procedures (2) and (3). Although the photonic crystal we show in Section 3 supports a deflection eigenmode at frequency around ω=0.5×2πc/L, eigenmode analysis shows that As/Emax is low. Nevertheless, the deflection mode at frequency ω=0.411×2πc/L has high As/Emax. To satisfy the band structure condition, we tune the geometric parameters of the pillar to a=0.26μm and b=0.66μm such that the normalized frequency of this deflection mode equals β [Fig. 6(a)]. We find that the field distributions of the eigenmode remain qualitatively unchanged as indicated by comparing the deflection mode before [Fig. 2(e)] and after [Fig. 6(b)] the parameter tuning.

Fig. 6. (a) Band diagram of a photonic crystal deflecting structure with L=1μm, a=0.26μm, b=0.66μm, and d=0.2μm. The symbols have the same meaning as in Fig. 2(c). The field profiles of the deflection mode at the Γ point with normalized frequency 0.5 (indicated by the blue arrow) are shown in (b).

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To validate the design of the photonic crystal electron deflector, we truncate the photonic crystal in the x direction to have three electron channels and perform a full wave simulation of such finite-width structure. The two driving plane waves propagating in the x direction are set to have odd mirror symmetry with respect to x=0. The field distributions are shown in Fig. 7, which are almost identical from channel to channel. The longitudinal and transverse Lorentz forces calculated from the field distributions are shown in Figs. 8(a) and 8(b), respectively. Since the relative phase between the peaks of longitudinal and transverse forces is π/2, the longitudinal forces almost vanish when the transverse forces peak. The deflection force has a cosh-like shape inside the channel, and the channel-to-channel variance is within 7% [Fig. 8(b)]. The amplitudes of the cosh and sinh components are shown in Fig. 8(c). The fields inside the electron channels are dominantly sinh components, where As/Emax=0.412, 0.398, 0.412 in channels 1, 2, and 3, respectively. Such a multichannel electron deflector also has large bandwidth as indicated by Fig. 8(d). Its bandwidth shares a similar scaling rule with the number of electron channels as accelerating-mode MIMOSAs. This photonic crystal electron deflector is a natural extension of the laser-driven electron deflectors, which were recently experimentally investigated [28,29]. A particle tracking study is provided in Appendix F.

Fig. 7. Field distributions in the three-channel deflecting-mode MIMOSA under antisymmetric excitation. (a) Electric field amplitudes; (b) field components Ez, Ex, and Z0Hy.

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Fig. 8. Longitudinal and transverse force distributions in a deflecting-mode MIMOSA are shown in (a) and (b), respectively, with the proper electron phase that maximizes each force. (c) Amplitudes of the cosh and sinh components in different channels at central frequency; (d) their frequency dependence.

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5. CENTRALIZER

The MIMOSA can be designed to achieve more complicated functions. In this section, we further explore its functionalities by studying the modes in MIMOSA with different field distributions from channel to channel. As an example, we demonstrate a three-channel centralizer, which deflects the electron beam on the two outside channels to the central channel (Fig. 9). This functionality provides the possibility of combining multiple electron beams (super-beam) to form a high-brightness, high-current beam [43]. The centralizer preserves the emittance of the super-beam while allowing the beamlets traveling in the individual channel to collide at a fixed point. The total emittance is at best a linear addition of the emittance of the individual beamlets.

Fig. 9. Schematic of a multichannel centralizer. With symmetric excitation, the transverse forces inside the electron channels are indicated by the small red arrows. The gray arrows indicate the trajectories of electron beams.

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The underlying photonic crystal is chosen such that the excitation frequency is within the band gap of the photonic crystal along the ΓX direction. Figure 10(a) shows the band structure of a photonic crystal with L=1μm, a=0.26μm, b=0.56μm, and d=0.2μm. The excitation frequency (ω0=0.5×2πc/L) is within the band gap of the photonic crystal.

Fig. 10. (a) Band structure of the infinitely periodic photonic crystal underlying the electron centralizer. The periodicities in the z and x directions are L=1μm and Lx=1.52μm, and the dimensions of the rectangular pillar are a=0.26μm and b=0.56μm. The green arrow indicates that the incident frequency is within the band gap. With symmetric excitation, the electric field amplitudes are shown in (b), while the nonzero field components Ez, Ex, and Z0Hy are shown in (c).

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With symmetric dual drive excitation [Figs. 10(b) and 10(c)], the fields generally have larger magnitudes in the outside channels and smaller magnitudes in the central channel. The force distributions inside different channels are shown in Figs. 11(a) and 11(b). The electron phase should be chosen such that the transverse forces are maximized. We find that, at this electron phase, the electron beam in the bottom channel is deflected up while the electron beam in the top channel is deflected down. The electron beam traveling along the central channel experiences a small focusing force. Therefore, this device can centralize the electron beams towards the central channel. The amplitudes of the cosh and sinh components are shown in Fig. 11(c). We can find that the fields inside the central channel have only the cosh component while the fields in the two outer channels have dominantly sinh components with equal magnitudes and opposite signs. This centralizer is also with broadband as suggested by the relatively flat frequency dependence shown in Fig. 11(d).

Fig. 11. MIMOSA functioning as a centralizer. (a) and (b) show the longitudinal and transverse forces, respectively, with the electron phases that maximize the longitudinal or transverse forces. The amplitudes of the cosh and sinh components are shown in (c), and their frequency dependence is shown in (d).

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6. DISCUSSIONS AND CONCLUSIONS

The diverse functionalities of MIMOSA, as discussed above, enable sophisticated control of multiple electron beams. For example, the MIMOSA can provide a platform to study the interference of phase-locked electron beams with the added capability of acceleration, attosecond-scale bunching, and coherent deflection. Furthermore, the interaction between multiple phase-locked electron beams with a photonic-crystal-based radiation generator can potentially further boost the radiation generation [44]. The experimental demonstration of MIMOSA is also under investigation.

In conclusion, we here propose a DLA architecture based on photonic crystals that enables simultaneous acceleration of multiple electron beams and has the potential to increase the total beam current by at least 1 order of magnitude. We find that the characteristics of the MIMOSA can be inferred from the band structure and eigenmodes of the underlying photonic crystal, which provides a simple approach to designing such MIMOSA structures. The underlying photonic crystal should support an eigenmode at the Γ point with normalized frequency β. Numerical studies confirm that the field distributions in different channels are indeed almost identical and the acceleration factor is qualitatively consistent with the eigenmode prediction. We further extend the principle to design electron deflectors and other electron manipulation devices based on photonic crystals. Our study opens new opportunities in dielectric laser accelerators and, in general, nanoscale electron manipulation with lasers.

7 Acknowledgment

Acknowledgment. The authors acknowledge all contributors in the ACHIP collaboration for their guidance and comments.