Design of a multichannel photonic crystal dielectric laser accelerator

Zhexin Zhao; Dylan S. Black; R. Joel England; Tyler W. Hughes; Yu Miao; Olav Solgaard; Robert L. Byer; Shanhui Fan

doi:10.1364/PRJ.394127

Abstract

To be useful for most scientific and medical applications, compact particle accelerators will require much higher average current than enabled by current architectures. For this purpose, we propose a photonic crystal architecture for a dielectric laser accelerator, referred to as a multi-input multi-output silicon accelerator (MIMOSA), that enables simultaneous acceleration of multiple electron beams, increasing the total electron throughput by at least 1 order of magnitude. To achieve this, we show that the photonic crystal must support a mode at the

Γ

point in reciprocal space, with a normalized frequency equal to the normalized speed of the phase-matched electron. We show that the figure of merit of the MIMOSA can be inferred from the eigenmodes of the corresponding infinitely periodic structure, which provides a powerful approach to design such devices. Additionally, we extend the MIMOSA architecture to electron deflectors and other electron manipulation functionalities. These additional functionalities, combined with the increased electron throughput of these devices, permit all-optical on-chip manipulation of electron beams in a fully integrated architecture compatible with current fabrication technologies, which opens the way to unconventional electron beam shaping, imaging, and radiation generation.

Miniaturization of accelerators is a rapidly growing field [1 - 4]. Among them, dielectric laser accelerators (DLAs) accelerate charged particles using the evanescent fields of dielectric structures driven by femtosecond laser pulses [4 - 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]. Recent progress in the integration of DLAs with photonic circuits has enabled the development of chip-scale accelerators, [13 - 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 .

Figure 1.Schematic of (a) a dual-pillar DLA and (b) a multichannel DLA. Two laser pulses (propagating in

\pm x

directions) incident on the DLA are indicated by the orange arrows. Electrons travel inside parallel channels along the

z

direction.

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 [23 - 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).

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y

Figure 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.

2 a

2 b

2 d

, 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 = 1 μm

a = 0.3 μm

b = 0.86 μm

, and

d = 0.2 μm

. 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.

x

g

L

β = 0.5

x

Figure 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

λ_{0} = 2 μm

, the magnitude of E field is shown in (b), while

E_{z}

E_{x}

, and

Z_{0} H_{y}

are shown in (c), where

Z_{0}

is free-space impedance.

m = 1

F_{z}

Figure 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.

g

A_{c}

Δ λ

Figure 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.

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.

Γ

x

Figure 6.(a) Band diagram of a photonic crystal deflecting structure with

L = 1 μm

a = 0.26 μm

b = 0.66 μm

, and

d = 0.2 μm

. 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).

x

Figure 7.Field distributions in the three-channel deflecting-mode MIMOSA under antisymmetric excitation. (a) Electric field amplitudes; (b) field components

E_{z}

E_{x}

, and

Z_{0} H_{y}

Figure 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.

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.

Figure 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.

Γ

Figure 10.(a) Band structure of the infinitely periodic photonic crystal underlying the electron centralizer. The periodicities in the

z

and

x

directions are

L = 1 μm

and

L_{x} = 1.52 μm

, and the dimensions of the rectangular pillar are

a = 0.26 μm

and

b = 0.56 μm

. 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

E_{z}

E_{x}

, and

Z_{0} H_{y}

are shown in (c).

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) .

Figure 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).

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.

Γ

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

In this appendix, we derive the expression of the electromagnetic fields inside multiple electron channels in the MIMOSA. The derivations partially follow that in Ref.?[42].

Assuming that the dielectric is nonmagnetic, linear, and isotropic, the electric field is a solution to Maxwell’s equations

? \times ? \times E = \frac{ω^{2}}{c^{2}} ? E ? j ω μ_{0} J,

(A1)

where

?

is the relative permittivity,

μ_{0}

is the vacuum permeability,

J

is the excitation current, and we consider the harmonic fields, with

\exp (j ω t)

time dependence. In the quasi-2D approximation, the structure is uniform along the

y

direction and incident plane waves are perpendicular to the

y

direction. Thus, the fields are invariant along the

y

direction and can be decomposed into TM and TE polarization. We further limit our study to the TM polarization (with nonzero

E_{x}

E_{z}

, and

H_{y}

components), which only generates forces in the

x - z

plane.

The fields inside the photonic crystal satisfy the Bloch theorem when

J = 0

. If the 2D photonic crystal is periodic in both the

x

and

z

directions, i.e.,

? (r) = ? (r + n_{x} L_{x} \hat{x} + n_{z} L \hat{z})

where

n_{x}

and

n_{z}

are arbitrary integers;

L_{x}

and

L

are the periodicity in the

x

and

z

directions, respectively; the fields satisfy

E (r) = U (r) \exp (? j k_{x} x ? j k_{z} z),

(A2)

where the periodic part

U (r) = U (r + n_{x} L_{x} \hat{x} + n_{z} L \hat{z})

, and

k_{x}

k_{z}

are Bloch wave vectors in the

x

and

z

directions, respectively.

The MIMOSA generally has finite number of channels and can be regarded as a 2D photonic crystal truncated in the

x

direction. Thus, in the MIMOSA, which is finite in the

x

direction and periodic along the

z

direction, i.e.,

? (r) = ? (r + L \hat{z})

, the fields have the following form:

E (r) = E_{p} (r) \exp (? j k_{z} z),

(A3)

where the periodic field

E_{p} (r) = E_{p} (r + L \hat{z})

. The periodic

E_{p}

can be further expanded into a Fourier series

E_{p} (r) = \sum_{m = ? \infty}^{m = \infty} e_{m} (x) \exp (? j k_{m} z),

(A4)

where

k_{m} = 2 π m / L

. Suppose only the

m

th-order diffraction is phase synchronized with the electron beam, i.e.,

\frac{ω}{β c} = k_{m} + k_{z} = \frac{2 π m}{L} + k_{z} .

(A5)

The forces generated by all other diffraction orders on the electron are averaged over one period to be zero. The synchronized electric fields have the form

E_{m} (r) = (\begin{matrix} e_{m, x} (x) \\ 0 \\ e_{m, z} (x) \end{matrix}) \exp (? j k_{m} z ? j k_{z} z),

(A6)

where

e_{m, x}

and

e_{m, z}

are the phasor notations. In the following derivation, we consider only this diffraction order and drop the diffraction order index

m

The synchronized field (phasor) inside the

n

th electron channel generally has the following form:

e_{z}^{n} (x) = d_{n} e^{? α (x ? x_{n})} + c_{n} e^{α (x ? x_{n})}, e_{x}^{n} (x) = \frac{k_{m} + k_{z}}{j α} [d_{n} e^{? α (x ? x_{n})} ? c_{n} e^{α (x ? x_{n})}],

(A7)

where

α = \sqrt{{(k_{m} + k_{z})}^{2} ? {(ω / c)}^{2}}

d_{n}

and

c_{n}

are amplitudes of the evanescent waves decaying away from the dielectric on two sides of the electron channel, and

x_{n}

is the center of the

n

th channel. Supposing the periodicity along the

x

direction is

L_{x}

and the center of MIMOSA is at

x = 0

, we find

x_{n} = [n ? (N + 1) / 2] L_{x}

. Due to the velocity of the electron being lower than the speed of light (

β < 1

), the wavevector

k_{m} + k_{z} = ω / β c > ω / c

, and

α

is always real. This indicates that the synchronized field inside the electron channel is always evanescent, and

α

characterizes the decay of the evanescent wave inside the electron channel. We can reformulate the phasor with hyperbolic cosine and sine functions:

e_{z}^{n} (x) = A_{c}^{n} ? \cosh [α (x ? x_{n})] + A_{s}^{n} ? \sinh [α (x ? x_{n})], e_{x}^{n} (x) = \frac{k_{m} + k_{z}}{j α} {A_{s}^{n} ? \cosh [α (x ? x_{n})] + A_{c}^{n} ? \sinh [α (x ? x_{n})]},

(A8)

where

A_{c}^{n} = d_{n} + c_{n}

and

A_{s}^{n} = d_{n} ? c_{n}

are the amplitudes of the cosh and sinh components.

Case 1: If the MIMOSA is excited by a plane wave polarized in the

z

direction and propagating in the

x

direction:

E_{inc} (r, t) = E_{0} ? \exp (j ω t ? j k_{0} x) \hat{z},

(A9)

where

k_{0} = ω / c

, the eigenmodes of the photonic crystal with

k_{z} = 0

and frequency

ω

can be excited. Suppose that only two eigenmodes with frequency

ω

and a Bloch wavevector (

\pm k_{x}

k_{z} = 0

) are excited. The field inside the truncated photonic crystal can be approximated by a sum of these two counterpropagating Bloch waves, which has the following form:

E (x, z) = a_{+} U (x, z; k_{x}, k_{z} = 0) \exp (? j k_{x} x) + a_{?} U (x, z; ? k_{x}, k_{z} = 0) \exp (j k_{x} x),

(A10)

where

a_{+}

and

a_{?}

are amplitudes of the two counterpropagating Bloch waves [Eq.?(A2)]. Thus, the synchronized field (phasor) should have the form

e_{1} (x) = a_{+} e_{p} (x; k_{x}) \exp (? j k_{x} x) + a_{?} e_{p} (x; ? k_{x}) \exp (j k_{x} x),

(A11)

where the subscript 1 highlights that this is the phasor for Case 1, and the periodic part

e_{p} (x; \pm k_{x}) = e_{p} (x + L_{x}; \pm k_{x})

Comparing Eqs.?(A8) and (A9), we find that

A_{c}^{n}

and

A_{s}^{n}

in Case 1 should have the form

A_{c, 1}^{n} = a_{+}^{c} \exp (? j k_{x} x_{n}) + a_{?}^{c} \exp (j k_{x} x_{n}),

(A12)

A_{s, 1}^{n} = a_{+}^{s} \exp (? j k_{x} x_{n}) + a_{?}^{s} \exp (j k_{x} x_{n}),

(A13)

where the coefficients

a_{+}^{c}

a_{?}^{c}

a_{+}^{s}

, and

a_{?}^{s}

are independent of

n

and have the following form:

a_{+}^{c} = a_{+} e_{p, z} (? \frac{\mod (N + 1, 2)}{2} L_{x}; k_{x}), a_{?}^{c} = a_{?} e_{p . z} (? \frac{\mod (N + 1, 2)}{2} L_{x}; ? k_{x}), a_{+}^{s} = \frac{j α}{k_{m}} a_{+} e_{p, x} (? \frac{\mod (N + 1, 2)}{2} L_{x}; k_{x}), a_{?}^{s} = \frac{j α}{k_{m}} a_{?} e_{p, x} (? \frac{\mod (N + 1, 2)}{2} L_{x}; ? k_{x}) .

(A14)

From Eqs.?(A12) and (A13), we find that generally the amplitudes of cosh and sinh components (

A_{c, 1}^{n}

and

A_{s, 1}^{n}

) are different from channel to channel in terms of both magnitudes and phases. However, when

k_{x} = 0

, the amplitudes of the cosh and sinh components in different electron channels become identical. This implies that even with single side drive, it is possible to achieve identical acceleration in multiple channels of the MIMOSA. Nevertheless, the condition

k_{x} = 0

is usually satisfied only at a single frequency. For broadband excitation (

～ 300 ?? fs

laser pulse [28]), the dual drive can selectively excite the desired mode and in general have larger bandwidth.

Case 2: If the MIMOSA is excited by a

z

-polarized plane wave propagating in the

? x

direction, i.e.,

E_{inc} (r, t) = E_{0} ? \exp (j ω t + j k_{0} x) \hat{z}

, the field is a mirror image of the field in Case 1 due to the mirror-

x

symmetry of MIMOSA:

e_{2} (x) = M_{x} [e_{1} (? x)] = a_{?} M_{x} [e_{p} (? x; ? k_{x})] \exp (? j k_{x} x) + a_{+} M_{x} [e_{p} (? x; k_{x})] \exp (j k_{x} x),

(A15)

where

M_{x}

represents a mirror-

x

operation on the vector such that

M_{x} [\begin{matrix} e_{x} \\ e_{y} \\ e_{z} \end{matrix}] = [\begin{matrix} ? e_{x} \\ e_{y} \\ e_{z} \end{matrix}] .

(A16)

Similar to Case 1,

A_{c}^{n}

and

A_{s}^{n}

in Case 2 are

A_{c, 2}^{n} = a_{?}^{c} \exp (? j k_{x} x_{n}) + a_{+}^{c} \exp (j k_{x} x_{n}),

(A17)

A_{s, 2}^{n} = ? a_{?}^{s} \exp (? j k_{x} x_{n}) ? a_{+}^{s} \exp (j k_{x} x_{n}) .

(A18)

Comparing Eqs.?(A12), (A13), (A17), and (A18), and recalling that

x_{n} = [n ? (N + 1) / 2] L_{x} = ? x_{N + 1 ? n}

, we find that

A_{c, 2}^{n} = A_{c, 1}^{N + 1 ? n}, A_{s, 2}^{n} = ? A_{s, 1}^{N + 1 ? n} .

(A19)

Case 3: If the MIMOSA is excited by two counterpropagating plane waves in the

x

direction with equal amplitudes and relative phase

θ

, i.e.,

E_{inc} (r, t) = E_{0} [\exp (? j k_{0} x) + \exp (j θ) \exp (j k_{0} x)] \exp (j ω t) \hat{z}

, the field in this case is a sum of the field in Case 1 and Case 2 with relative phase

θ

e_{3} (x) = e_{1} (x) + \exp (j θ) e_{2} (x) .

(A20)

Therefore, the amplitudes of the cosh and sinh components in the

n

th electron channel are

A_{c, 3}^{n} = A_{c, 1}^{n} + \exp (j θ) A_{c, 2}^{n} = a_{+}^{c} [\exp (? j k_{x} x_{n}) + \exp (j θ) \exp (j k_{x} x_{n})] + a_{?}^{c} [\exp (j k_{x} x_{n}) + \exp (j θ) \exp (? j k_{x} x_{n})],

(A21)

A_{s, 3}^{n} = A_{s, 1}^{n} + \exp (j θ) A_{s, 2}^{n} = a_{+}^{s} [\exp (? j k_{x} x_{n}) ? \exp (j θ) \exp (j k_{x} x_{n})] + a_{?}^{s} [\exp (j k_{x} x_{n}) ? \exp (j θ) \exp (? j k_{x} x_{n})] .

(A22)

With symmetric excitation, i.e.,

θ = 0

, the amplitudes of the cosh and sinh components are

A_{c, 3}^{n} = 2 (a_{+}^{c} + a_{?}^{c}) \cos (k_{x} x_{n}), A_{s, 3}^{n} = ? 2 j (a_{+}^{s} ? a_{?}^{s}) \sin (k_{x} x_{n}) .

(A23)

From Eq.?(A23), we observe that the channel-to-channel variance in the amplitudes of the cosh and sinh components is minimized when

k_{x}

is near zero. Therefore, to provide near-identical force for electrons traveling in different channels, the MIMOSA should operate near the

Γ

point (

k_{x} = 0

). When

k_{x}

is near zero, either real if the excitation frequency is within the band or purely imaginary if the excitation frequency is within the band gap, the phases of

A_{c, 3}^{n}

are identical for different channels. These are observed in the numerical study. We emphasize that these properties are crucial to provide near-identical acceleration for micro-bunched electrons with the same entering time.

Similarly, with antisymmetric excitation, i.e.,

θ = π

, the amplitudes of cosh and sinh components in different electron channels are

A_{c, 3}^{n} = ? 2 j (a_{+}^{c} ? a_{?}^{c}) \sin (k_{x} x_{n}), A_{s, 3}^{n} = 2 (a_{+}^{s} + a_{?}^{s}) \cos (k_{x} x_{n}) .

(A24)

The amplitudes of sinh components also have the same phase and small variation in magnitude for

k_{x}

near zero.

Since the studied MIMOSA has mirror-

x

symmetry, Eq.?(A19) implies that

A_{c, 3}^{n}

(

A_{s, 3}^{n}

) are symmetric (antisymmetric) with respect to

x = 0

under symmetric excitation, while

A_{c, 3}^{n}

(

A_{s, 3}^{n}

) are antisymmetric (symmetric) with respect to

x = 0

under antisymmetric excitation. These are consistent with Eqs.?(A23) and (A24).

We also find that a MIMOSA with high acceleration factor may not have high deflection factor when the dual drive excitation changes from symmetric to antisymmetric. From Eqs.?(A23) and (A24), it is obvious that the cosh component amplitudes under symmetric excitation are not necessarily the same as the sinh component amplitudes under antisymmetric excitation. This observation is aligned with the analysis in Ref.?[42].

In this appendix, we discuss the figure of merits of MIMOSA with large number of electron channels and the scaling rule of its bandwidth. In Section?3, we demonstrate a MIMOSA with three electron channels. Here we extend the number of electron channels to 10 with all other parameters fixed. With symmetric excitation and the proper electron phase to maximize the acceleration force, the longitudinal force distributions inside different channels are shown in Fig.?12(a). With another electron phase that maximizes the focusing force, the transverse force distributions inside different channels are shown in Fig.?12(b). The almost overlapping curves imply that the force distributions are almost identical from channel to channel. Indeed, the amplitudes of cosh components in different channels differ by less than 3%, and the amplitudes of cosh components dominate over that of sinh components [Fig.?12(c)].

Figure 12.MIMOSA with 10 electron channels. (a) and (b) show the longitudinal and transverse forces in different channels. The amplitudes of cosh and sinh components in different channels are shown in (c), and their wavelength dependence is shown in (d). Different colors indicate different channels. Due to the mirror-

x

symmetry, channels above

x = 0

are omitted in (d).

Figure 13.MIMOSA with high acceleration factor but small bandwidth. (a) The band structure of the photonic crystal (

L = 1 μm

a = 0.32 μm

b = 0.82 μm

d = 0.2 μm

). The red arrow points to the acceleration mode. The band containing the acceleration mode is highlighted in green. (b) The field distribution of the acceleration mode. (c) The pulse duration of the transform-limited Gaussian pulse, whose bandwidth matches the bandwidth of the MIMOSA, as a function of the number of channels.

The linear accelerators used for medical therapy usually operate on a dose rate around

1 ?? Gy \cdot \min^{? 1}

[46]. If we assume a typical phantom mass of 1?kg and a typical electron beam final energy of 6?MeV, the required average beam current is 2.8?nA. Supposing the femtosecond laser operates at 50?MHz repetition rate with 250?fs pulse duration [28,29], the corresponding number of electrons per laser pulse is

3.5 \times 10^{2}

. Recent progress in laser-driven nanotip electron sources has demonstrated this required number of electrons per laser pulse, with below

nm \cdot rad

emittance [47]. From the bandwidth analysis in Appendix?B (Fig.?5), we find that a seven-channel MIMOSA matches this pulse duration. If these electrons are distributed over the seven channels and 37 micro-bunches, which fall within the pulse duration of 250?fs, the average number of electrons per micro-bunch per channel is only 1.3.

To summarize, for medical applications, the required number of electrons per laser pulse is low due to the high laser repetition rate. With the multichannel DLA architecture and the micro-bunching technology, the required number of electrons per micro-bunch per channel is around 1.

In this appendix, we provide a brief discussion of the short-range wakefield effects and beam loading properties of the MIMOSA, although a comprehensive study of the wakefields in such multichannel DLAs is beyond the scope of this study due to the intrinsic complexity of Cheronkov radiation in photonic crystals [48].

To study the beam loading properties of the MIMOSA, we follow the discussion presented in Ref.?[5]. Suppose the unloaded gradient is

G_{0}

; the loaded gradient is

G_{L} = G_{0} ? N q k_{W} ? q k_{H}

, where

N

is the number of electron channels,

q

is the charge per channel per micro-bunch,

k_{W}

represents the loss factor due to the particle’s wakefield that overlaps with the excitation laser field, and

k_{H}

represents the nonoverlapping component of the wakefields such as the broadband Cherenkov radiation. We assume that the micro-bunches in different channels are in phase. Thus, their wakefield components that overlap with the laser field will add up coherently, which gives rise to the multiple of

N

in the expression of the loaded gradient. Although the Cherenkov radiation from the in-phase micro-bunches is also coherent at a particular frequency, the average effect over the broadband resembles an incoherent sum, due to their constructive interference over some frequencies and destructive interference over some other frequencies.

From Refs. [5,38], we find that

k_{W} = \frac{c Z_{C}}{4 λ^{2} β}

with

Z_{C}

being the characteristic impedance of the MIMOSA, and

k_{H} = \frac{c}{λ^{2}} Z_{H}

, where

Z_{H}

is the broadband wakefield impedance. The characteristic impedance is

Z_{C} = ξ^{2} \frac{β λ}{D} Z_{0}

, where

Z_{0} = \sqrt{μ_{0} / ?_{0}}

is the impedance of vacuum;

ξ

is the ratio of the unloaded gradient and the incident electric field, i.e.,

ξ = G_{0} / E_{0} ～ 0.5

; and D is the extension in the out-of-plane dimension. The broadband wakefield impedance depends on the Cherenkov radiation in the MIMOSA structure, which is intrinsically complicated and not discussed in detail in this study. The wakefield impedance can be approximated by a wide bunch, with extension

D

in the out-of-plane direction, traveling between two gratings with separation

2 d

. The impedance is approximately

Z_{H} = \frac{λ^{2}}{2 π R_{eff}^{2}} Z_{0}

and

R_{eff} = \sqrt{d D}

. Using the properties of the MIMOSA design presented in Section?3 and assuming

D = λ = 2 ?? μm

, the characteristic impedance is

Z_{C} = 47 ? Ω

and the broadband wakefield impedance is

Z_{H} = 600 ? Ω

For the seven-channel MIMOSA operating for medical applications discussed in Appendix?C, the reduction in acceleration gradient due to wakefields is 12?kV/m, much smaller than the unloaded gradient (

～ 0.25 ?? GV / m

). This observation confirms our anticipation that the wakefield effects are small in MIMOSAs for medical applications.

We also provide an estimation of the energy efficiency of MIMOSA and the optimal charge per bunch that achieves the maximal energy efficiency. The energy efficiency is

η = \frac{N q G_{L} L}{P_{L} τ_{pulse}}

, where

L

is the periodicity along the propagation direction,

P_{L} = L D E_{0}^{2} / Z_{0}

is the laser power incident on one period from two sides, and

τ_{pulse}

is the pulse duration.

Consistent with discussions in Appendix?C, we assume the laser pulse duration is

τ_{pulse} = 250 ?? fs

, and we take

N = 7

to match this pulse duration. For the MIMOSA operating for medical applications,

q = 1.3 e

and the energy efficiency is

1.1 \times 10^{? 6}

. Even when a train of 37 bunches is fed in the laser pulse, the energy efficiency is only

4.1 \times 10^{? 5}

. Such low energy efficiency suggests the demand for recycling the laser pulses.

Nevertheless, the energy efficiency of MIMOSA can be improved significantly with a proper choice of charge per micro-bunch. To maximize the efficiency, the optimal charge is

q_{opt} = \frac{G_{0}}{2 (N k_{W} + k_{H})}

[5,39] and the optimal efficiency is

η_{opt} = \frac{1}{1 + 4 β Z_{H} / (N Z_{C})} \frac{τ_{c}}{τ_{pulse}}

, where

τ_{c}

is the duration of an optical cycle. With the aforementioned impedance of the MIMOSA design, we find that

q_{opt} = 2.18

fC, which corresponds to

1.36 \times 10^{4}

electrons per micro-bunch, and

η_{opt} = 0.58 %

. Furthermore, if a train of micro-bunches is fed into the MIMOSA structure, the efficiency can be increased further. For a train of 37 bunches, which falls within the pulse duration of 250?fs, the efficiency becomes 21%, under the approximation that the long-range interaction of the wakefields is negligible due to the low-resonance nature of the MIMOSA.

In this appendix, we discuss the space-charge effects of an attosecond electron bunch injected into the MIMOSA and the required external focusing field to avoid emittance growing.

An ideal scenario for beam coupling to the MIMOSA architecture is a line of point emitters spaced at the channel separation triggered in unison by a single laser pulse. Such a linear array of electron sources can be produced using nanotip field emitters etched in silicon as discussed in Ref.?[49]. Consistent with recent DLA experiments [34,42], the three-channel MIMOSA has 15 periods in the electron propagation direction, whose geometric parameters are given in Section?3. We use the particle tracking code General Particle Tracer (GPT) to simulate the normalized transverse emittance of a short electron bunch under the influence of space-charge effects in this scenario. For the tracking simulation, we assume that each channel has an incident micro-bunch of 500 attoseconds FWHM duration, consistent with recently demonstrated optically bunched beams [29,50,51]. A transverse normalized emittance of 0.1?nm is assumed with a bunch charge of 2 fC, consistent with the estimated optimal bunch charge for efficient beam loading calculated in Appendix?D.

Due to a combination of space-charge repulsion, emittance pressure, and transverse defocusing in the narrow accelerating channel, strong transverse focusing forces are required for beam confinement. The normalized focusing force K may be interpreted as the linear focusing term appearing in a

z

-dependent paraxial ray equation of the form

x^{''} (z) = ? K^{2} x (z)

. We note that the minimal focusing strength for optimal emittance preservation and near 100% particle transport in Fig.?14 (

K^{2} = 4 \times 10^{12} ?? m^{? 2}

) is of the order of magnitude estimated in Ref.?[4] as being required for focusing of subrelativistic electron beams in DLA structures. For the present tracking simulation we use a solenoidal magnetic field

B

oriented in the

z

direction, which provides a transverse focusing force

K = e B / 2 m c β γ

. The minimum emittance point in Fig.?14(a) corresponds to a field of

B = 3.8 ?? kT

, which is impractically large. However, compatible laser-driven focusing techniques capable of forces of this magnitude, which have been proposed in Refs.?[19,20] and recently demonstrated in Ref.?[29], can be readily adapted to the MIMOSA architecture. However, a full implementation of such a ponderomotive focusing scheme lies beyond the scope of the present paper. Although these results were calculated for all three channels, in Fig.?14 we show only the result for the center channel of the MIMOSA device. Since the electromagnetic design successfully equalizes the accelerating fields in the channels, the results for the other two channels are nearly identical, and displaying all three on one plot would provide no additional information. Also of note is that the uncompensated emittance growth (at

K = 0

) in Fig.?14(a) is significantly larger in the

x

coordinate due to the fact that the dominant transverse defocusing is in this dimension as predicted by Eq.?(4) and Fig.?4.

$Particle tracking simulations of 500 attoseconds FWHM bunch with initial emittance of 0.1 nm and charge of 2 fC showing (a) final emittance after propagating through the 15 μm MIMOSA structure and (b) corresponding fraction of transmitted particles as functions of externally applied focusing field K. Only the center channel of the MIMOSA is shown since the results for the three channels are nearly identical.$

Figure 14.Particle tracking simulations of 500 attoseconds FWHM bunch with initial emittance of 0.1 nm and charge of 2 fC showing (a) final emittance after propagating through the 15 μm MIMOSA structure and (b) corresponding fraction of transmitted particles as functions of externally applied focusing field K. Only the center channel of the MIMOSA is shown since the results for the three channels are nearly identical.

$Particle tracking simulation of (a) centroid angular deflection ⟨x′⟩ and ⟨y′⟩ and (b) fraction of transmitted particles for a 20 μm long three-channel MIMOSA operating in deflection mode. The initial beam parameters are the same as for the acceleration case considered in Fig. 14, with space charge and external focusing turned off. Deviation of the deflection curve from a linear relationship is due to truncation of particles by the aperture of the channel at higher incident field.$

Figure 15.Particle tracking simulation of (a) centroid angular deflection

⟨ x^{'} ⟩

and

⟨ y^{'} ⟩

and (b) fraction of transmitted particles for a 20 μm long three-channel MIMOSA operating in deflection mode. The initial beam parameters are the same as for the acceleration case considered in Fig. 14, with space charge and external focusing turned off. Deviation of the deflection curve from a linear relationship is due to truncation of particles by the aperture of the channel at higher incident field.

In this appendix, we study the electron beam dynamics in the MIMOSA functioning as a deflector. The three-channel deflector has 20 periods in the electron propagation direction and geometric parameters mentioned in Section?4. We investigate the deflection angle and particle transmission for each channel using GPT, where the simulation parameters are the same as those for Fig.?14. The results in Fig.?15 confirm that the beam dynamics in three channels are almost identical. The deflection angle scales linearly with the incident field until a portion of the particles are truncated by the channel aperture. Moreover, the deflection angle and transmission, under a specific incident field, can be optimized by tuning the length of the deflector.