Design of a multi-channel photonic crystal dielectric laser accelerator

In this Appendix, we derive the expression of the electromagnetic fields inside multiple electron channels in the MIMOSA. The derivations partially follow that in [36].

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

$$\nabla\times\nabla\times\mathbf{E}=\frac{\omega^{2}}{c^{2}}\epsilon\mathbf{E}-j\omega\mu_{0}\mathbf{J},$$

(5)

where $\epsilon$ is the relative permittivity, $\mu_{0}$ is the vacuum permeability, $\mathbf{J}$ is the excitation current, and we consider the harmonic fields, with $\exp(j\omega t)$ time dependence. In the quasi-2D approximation, the structure is uniform along y-direction and incident plane waves are perpendicular to y-direction. Thus, the fields are invariant along 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 xz-plane.

The fields inside the photonic crystal satisfy the Bloch theorem when $\mathbf{J}=0$. If the 2D photonic crystal is periodic in both x and z direction, i.e. $\epsilon(\mathbf{r})=\epsilon(\mathbf{r}+n_{x}L_{x}\mathbf{\hat{x}}+n_{z}L\mathbf{\hat{z}})$ where $n_{x}$ and $n_{z}$ are arbitrary integers, $L_{x}$ and $L$ are periodicity in x and z directions respectively, the fields satisfy

$$\mathbf{E}(\mathbf{r})=\mathbf{U}(\mathbf{r})\exp(-jk_{x}x-jk_{z}z),$$

(6)

where the periodic part $U(\mathbf{r})=U(\mathbf{r}+n_{x}L_{x}\mathbf{\hat{x}}+n_{z}L\mathbf{\hat{z}})$, and $k_{x}$, $k_{z}$ are Bloch wave vectors in 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 x-direction and periodic along z-direction, i.e. $\epsilon(\mathbf{r})=\epsilon(\mathbf{r}+L\hat{z})$, the fields have the following form:

$$\mathbf{E}(\mathbf{r})=\mathbf{E}_{p}(\mathbf{r})\exp(-jk_{z}z),$$

(7)

where the periodic field $\mathbf{E}_{p}(\mathbf{r})=\mathbf{E}_{p}(\mathbf{r}+L\hat{z})$. The periodic $\mathbf{E}_{p}$ can be further expanded into a Fourier series,

$$\mathbf{E}_{p}(\mathbf{r})=\sum_{m=-\infty}^{m=\infty}\mathbf{e}_{m}(x)\exp(-jk_{m}z),$$

(8)

where $k_{m}=2\pi m/L$. Suppose only $m^{\textrm{th}}$ order diffraction is phase synchronized with the electron beam, i.e.

$$\frac{\omega}{\beta c}=k_{m}+k_{z}=\frac{2\pi m}{L}+k_{z}.$$

(9)

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:

$$\mathbf{E}_{m}(\mathbf{r})=\begin{pmatrix}e_{m,x}(x)\\ 0\\ e_{m,z}(x)\end{pmatrix}\exp(-jk_{m}z-jk_{z}z),$$

(10)

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 $n^{\textrm{th}}$ electron channel has the following form generally,

$\displaystyle\begin{split}e_{z}^{n}(x)&=d_{n}e^{-\alpha(x-x_{n})}+c_{n}e^{\alpha(x-x_{n})},\\ e_{x}^{n}(x)&=\frac{k_{m}+k_{z}}{j\alpha}[d_{n}e^{-\alpha(x-x_{n})}-c_{n}e^{\alpha(x-x_{n})}]\end{split}$

(11)

where $\alpha=\sqrt{(k_{m}+k_{z})^{2}-(\omega/c)^{2}}$, $d_{n}$ 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 $n^{\textrm{th}}$ channel. Suppose the periodicity along 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 electron being lower than speed of light ($\beta<1$), the wave vector $k_{m}+k_{z}=\omega/\beta c>\omega/c$, and $\alpha$ is always real. This indicates that the synchronized field inside the electron channel is always evanescent and $\alpha$ characterizes the decay of the evanescent wave inside the electron channel. We can reformulate the phasor with hyperbolic cosine and sine functions:

$\displaystyle\begin{split}e_{z}^{n}(x)=\,&A_{c}^{n}\cosh[\alpha(x-x_{n})]+A_{s}^{n}\sinh[\alpha(x-x_{n})],\\ e_{x}^{n}(x)=\,&\frac{k_{m}+k_{z}}{j\alpha}\{A_{s}^{n}\cosh[\alpha(x-x_{n})]\\ &+A_{c}^{n}\sinh[\alpha(x-x_{n})]\}\end{split}$

(12)

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 z-direction and propagating in x-direction:

$$\mathbf{E}_{inc}(\mathbf{r},t)=E_{0}\exp(j\omega t-jk_{0}x)\mathbf{\hat{z}},$$

(13)

where $k_{0}=\omega/c$, the eigenmodes of the photonic crystal with $k_{z}=0$ and frequency $\omega$ can be excited. Suppose that only two eigenmodes with frequency $\omega$ and Bloch wave vector ($\pm k_{x}$, $k_{z}=0$) are excited. The field inside the truncated photonic crystal can be approximated by a sum of these two counter-propagating Bloch waves, which has the following form:

$\displaystyle\begin{split}\mathbf{E}(x,z)=&a_{+}\mathbf{U}(x,z;k_{x},k_{z}=0)\exp(-jk_{x}x)\\ &+a_{-}\mathbf{U}(x,z;-k_{x},k_{z}=0)\exp(jk_{x}x),\end{split}$

(14)

where $a_{+}$ and $a_{-}$ are amplitudes of the two counter-propagating Bloch waves (Eq. 6). Thus, the synchronized field (phasor) should have the form:

$\displaystyle\begin{split}\mathbf{e}_{1}(x)=\,&a_{+}\mathbf{e}_{p}(x;k_{x})\exp(-jk_{x}x)\\ &+a_{-}\mathbf{e}_{p}(x;-k_{x})\exp(jk_{x}x),\end{split}$

(15)

where the subscript 1 highlights that this is the phasor for Case 1, and the periodic part $\mathbf{e}_{p}(x;\pm k_{x})=\mathbf{e}_{p}(x+L_{x};\pm k_{x})$.

Comparing Eq. (12) and (15), we find that $A_{c}^{n}$ and $A_{s}^{n}$ in Case 1 should have the form:

	$\displaystyle A_{c,1}^{n}$	$\displaystyle=a_{+}^{c}\exp(-jk_{x}x_{n})+a_{-}^{c}\exp(jk_{x}x_{n}),$		(16)
	$\displaystyle A_{s,1}^{n}$	$\displaystyle=a_{+}^{s}\exp(-jk_{x}x_{n})+a_{-}^{s}\exp(jk_{x}x_{n}),$		(17)

where the coefficients $a_{+}^{c}$, $a_{-}^{c}$, $a_{+}^{s}$, and $a_{-}^{s}$ are independent of $n$ and have the following form:

$\displaystyle\begin{split}a_{+}^{c}&=a_{+}e_{p,z}(-\frac{mod(N+1,2)}{2}L_{x};k_{x}),\\ a_{-}^{c}&=a_{-}e_{p.z}(-\frac{mod(N+1,2)}{2}L_{x};-k_{x}),\\ a_{+}^{s}&=\frac{j\alpha}{k_{m}}a_{+}e_{p,x}(-\frac{mod(N+1,2)}{2}L_{x};k_{x}),\\ a_{-}^{s}&=\frac{j\alpha}{k_{m}}a_{-}e_{p,x}(-\frac{mod(N+1,2)}{2}L_{x};-k_{x}).\end{split}$

(18)

From Eqs. 16 and 17, 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 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 ($\sim$ 300 fs laser pulse [23]), 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 $-x$-direction, i.e. $\mathbf{E}_{inc}(\mathbf{r},t)=E_{0}\exp(j\omega t+jk_{0}x)\mathbf{\hat{z}}$, the field is a mirror image of the field in Case 1 due to the mirror-$x$ symmetry of MIMOSA.

$\displaystyle\begin{split}\mathbf{e}_{2}(x)=\,&M_{x}[\mathbf{e}_{1}(-x)]\\ =\,&a_{-}M_{x}[\mathbf{e}_{p}(-x;-k_{x})]\exp(-jk_{x}x)\\ &+a_{+}M_{x}[\mathbf{e}_{p}(-x;k_{x})]\exp(jk_{x}x),\end{split}$

(19)

where $M_{x}$ represents a mirror-$x$ operation on the vector such that

$$M_{x}\begin{bmatrix}e_{x}\\ e_{y}\\ e_{z}\end{bmatrix}=\begin{bmatrix}-e_{x}\\ e_{y}\\ e_{z}\end{bmatrix}.$$

(20)

Similar to Case 1, $A_{c}^{n}$ and $A_{s}^{n}$ in Case 2 are

	$\displaystyle A_{c,2}^{n}$	$\displaystyle=a_{-}^{c}\exp(-jk_{x}x_{n})+a_{+}^{c}\exp(jk_{x}x_{n}),$		(21)
	$\displaystyle A_{s,2}^{n}$	$\displaystyle=-a_{-}^{s}\exp(-jk_{x}x_{n})-a_{+}^{s}\exp(jk_{x}x_{n}).$		(22)

Comparing Eqs. 16, 17, 21, and 22, and recall that $x_{n}=[n-(N+1)/2]L_{x}=-x_{N+1-n}$, we find that

$\displaystyle\begin{split}A_{c,2}^{n}&=A_{c,1}^{N+1-n},\\ A_{s,2}^{n}&=-A_{s,1}^{N+1-n}.\end{split}$

(23)

Case 3: If the MIMOSA is excited by two counter propagating plane waves in x-direction with equal amplitudes and relative phase $\theta$, i.e. $\mathbf{E}_{inc}(\mathbf{r},t)=E_{0}[\exp(-jk_{0}x)+\exp(j\theta)\exp(jk_{0}x)]\exp(j\omega t)\mathbf{\hat{z}}$, the field in this case is a sum of the field in Case 1 and Case 2 with relative phase $\theta$.

$\displaystyle\mathbf{e}_{3}(x)=\mathbf{e}_{1}(x)+\exp(j\theta)\mathbf{e}_{2}(x)$

(24)

Therefore, the amplitudes of cosh and sinh components in the $n^{\textrm{th}}$ electron channel are:

	$\displaystyle\begin{split}A_{c,3}^{n}=\,&A_{c,1}^{n}+\exp(j\theta)A_{c,2}^{n}\\ =\,&a_{+}^{c}[\exp(-jk_{x}x_{n})+\exp(j\theta)\exp(jk_{x}x_{n})]\\ &+a_{-}^{c}[\exp(jk_{x}x_{n})+\exp(j\theta)\exp(-jk_{x}x_{n})],\end{split}$			(25)
	$\displaystyle\begin{split}A_{s,3}^{n}=\,&A_{s,1}^{n}+\exp(j\theta)A_{s,2}^{n}\\ =\,&a_{+}^{s}[\exp(-jk_{x}x_{n})-\exp(j\theta)\exp(jk_{x}x_{n})]\\ &+a_{-}^{s}[\exp(jk_{x}x_{n})-\exp(j\theta)\exp(-jk_{x}x_{n})].\end{split}$			(26)

With symmetric excitation, i.e. $\theta=0$, the amplitudes of cosh and sinh components are

$\displaystyle\begin{split}A_{c,3}^{n}&=2(a_{+}^{c}+a_{-}^{c})\cos(k_{x}x_{n}),\\ A_{s,3}^{n}&=-2j(a_{+}^{s}-a_{-}^{s})\sin(k_{x}x_{n}).\end{split}$

(27)

From Eq. 27, we observe that the channel-to-channel variance in the amplitudes of 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 $\Gamma$ 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 to micro-bunched electrons with the same entering time.

Similarly, with anti-symmetric excitation, i.e. $\theta=\pi$, the amplitudes of cosh and sinh components in different electron channels are

$\displaystyle\begin{split}A_{c,3}^{n}&=-2j(a_{+}^{c}-a_{-}^{c})\sin(k_{x}x_{n}),\\ A_{s,3}^{n}&=2(a_{+}^{s}+a_{-}^{s})\cos(k_{x}x_{n}).\end{split}$

(28)

The amplitudes of sinh components also have the same phase and small variation in magnitude for $k_{x}$ near zero.

Since the studied MIMOSA have mirror-$x$ symmetry, Eq. 23 implies that $A_{c,3}^{n}$ ($A_{s,3}^{n}$) are symmetric (anti-symmetric) with respect to $x=0$ under symmetric excitation, while $A_{c,3}^{n}$ ($A_{s,3}^{n}$) are anti-symmetric (symmetric) with respect to $x=0$ under anti-symmetric excitation. These are consistent with Eqs. 27 and 28.

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 anti-symmetric. From Eqs. 27 and 28, it is obvious that the cosh component amplitudes under symmetric excitation are not necessarily the same as the sinh component amplitudes under anti-symmetric excitation. This observation is aligned with the analysis in [36].

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 the main text, we demonstrate a MIMOSA with 3 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. 11(a). With another electron phase that maximizes the focusing force, the transverse force distributions inside different channels are shown in Fig. 11(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 amplitude of cosh components dominate over that of sinh components (Fig. 11(c)).

Nevertheless, with increasing number of channels, the bandwidth of the MIMOSA decreases. From Fig. 11(d), we find that the bandwidth decreases to 16.6 nm in the 10-channel MIMOSA. The bandwidth approximately scales with the inverse of 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 ($\Delta\lambda$), in nm, as a function of number of channels ($N$) is estimated as $\Delta\lambda(\text{nm})=144/(N-1.33)$ when $N\geq 4$, as shown in Fig. 12(a). If we choose a transform limited Gaussian pulse with central wavelength at 2 $\mu$m and bandwidth matching the bandwidth of the MIMOSA, the relation between the pulse duration ($\tau$) and the number of channels is linear (Fig. 12(b)). The estimated scaling rule is $\tau(\textrm{fs})=40.8\times N-54.4$ for $N\geq 4$, where $\tau$ is the full width at half maximum pulse duration in femto-second. 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.

We study further the scaling coefficient in the relation between the matched pulse duration ($\tau$) and the number of channels ($N$), which is generally linear for large $N$,

$$\tau=\alpha_{1}N+\alpha_{2},$$

(29)

where the scaling coefficient $\alpha_{1}$ determines how fast the bandwidth narrows as the number of channels increases. To achieve broadband operation and large number of channels simultaneously, a small scaling coefficient is favorable. We find that this scaling coefficient depends on the photonic crystal and is related to the band structure of the underlying photonic crystal along $\Gamma X$ direction, specifically, related to the band containing the acceleration mode and the band gap adjacent to the acceleration mode. When the band is close to a flat band, i.e. the band has small curvature near $\Gamma$ point, a small change in the excitation frequency leads to large change in $k_{x}$ of the mode, which results in narrow bandwidth. On the other hand, when the excitation frequency is within the band gap near the acceleration mode, field decays exponentially inside the MIMOSA from the out-most pillars, which increases the field variation from channel to channel. Since the penetration depth is positively correlated to the band curvature near $\Gamma$ point [39], a large band curvature, or even a linear dispersion relation, near the $\Gamma$ point is favorable. Therefore, to achieve small scaling coefficient $\alpha_{1}$, the underlying photonic crystal should have large band curvature, ideally a linear dispersion relation, near the $\Gamma$ point. We take this design rule into account when we optimize the geometric parameters of the MIMOSA. As shown in Fig. 2(c) and 5(a) of the main text, the bands near the acceleration mode (red arrow in Fig. 2(c)) and the deflection mode (blue arrow in Fig. 5(a)) have nearly linear dispersion relation.

To emphasize the importance of small bandwidth scaling factor in a MIMOSA, We show a counter-example of a MIMOSA with high acceleration factor but narrow bandwidth due to the small band curvature. As shown in Fig. 13(a), the photonic crystal with rectangle pillars ($L=1$ $\mu$m, $a=0.32$ $\mu$m, $b=0.82$ $\mu$m, $d=0.2$ $\mu$m) support an acceleration mode with normalized frequency around 0.5. From the field distribution of the acceleration mode (Fig. 13(b)), the acceleration factor is as high as 0.53. However, the band containing the acceleration mode has small curvature, as highlighted in green in Fig. 13(a). As a result, the bandwidth decreases fast as the number of channels increases. Figure 13(c) shows the pulse duration of the transform limited Gaussian pulse with the same bandwidth as the MIMOSA. We estimate the scaling is $\tau(\textrm{fs})=279\times N-819$, where the scaling coefficient $\alpha_{1}=279$ is much larger than that of the MIMOSA shown in Fig. 2 of the main text ($\alpha_{1}=40.8$).

The linear accelerators used for medical therapy usually operate on a dose rate around 1 $\textrm{Gy}\cdot\textrm{min}^{-1}$ [40]. If we assume a typical phantom mass of 1 kilogram and a typical electron beam final energy of 6 MeV, the required average beam current is 2.8 nA. Supposing the femto-second laser operates at 50 MHz repetition rate with 250 fs pulse duration [24, 23], 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 $\textrm{nm}\cdot\textrm{rad}$ emittance [41]. From the bandwidth analysis in Appendix B (Fig. 12), we find that a 7-channel MIMOSA matches this pulse duration. If these electrons are distributed over the 7 channels and 37 micro-bunches, which falls 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 multi-channel 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 MIMOSA, although a comprehensive study of the wakefields in such multi-channel DLA is beyond the scope of this study due to the intrinsic complexity of Cheronkov radiation in photonic crystals [42].

To study the beam loading properties of the MIMOSA, we follow the discussion presented in [1]. Suppose the unloaded gradient is $G_{0}$, the loaded gradient is $G_{L}=G_{0}-Nqk_{W}-qk_{H}$, 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 [1, 32], we find that $k_{W}=\frac{cZ_{C}}{4\lambda^{2}\beta}$, where $Z_{C}$ is the characteristic impedance of the MIMOSA, and $k_{H}=\frac{c}{\lambda^{2}}Z_{H}$, where $Z_{H}$ is the broadband wakefield impedance. The characteristic impedance is $Z_{C}=\xi^{2}\frac{\beta\lambda}{D}Z_{0}$, where $Z_{0}=\sqrt{\mu_{0}/\epsilon_{0}}$ is the impedance of vacuum, $\xi$ is the ratio of the unloaded gradient and the incident electric field, i.e. $\xi=G_{0}/E_{0}\sim 0.5$, 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 $2d$. The impedance is approximately $Z_{H}=\frac{\lambda^{2}}{2πR_{\textrm{eff}}^{2}}Z_{0}$ and $R_{\textrm{eff}}=\sqrt{dD}$. Using the properties of the MIMOSA design presented in Section III and assuming $D=\lambda=2$ $μ$m, the characteristic impedance is $Z_{C}=47$ $\Omega$ and the broadband wakefield impedance is $Z_{H}=600$ $\Omega$.

For the 7-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 ($\sim 0.25\textrm{GV/m}$). 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 achieve the maximal energy efficiency. The energy efficiency is $\eta=\frac{NqG_{L}L}{P_{L}\tau_{\textrm{pulse}}}$, where $L$ is the periodicity along the propagation direction, $P_{L}=LDE_{0}^{2}/Z_{0}$ is the laser power incident on one period from two sides, and $\tau_{\textrm{pulse}}$ is the pulse duration.

Consistent with discussions in Appendix C, we assume the laser pulse duration is $\tau_{\textrm{pulse}}=250$ fs, and take $N=7$ to match this pulse duration. For the MIMOSA operating for medical applications, $q=1.3e$ and the energy efficiency is $1.1\times 10^{-6}$. Even a train of 37 bunches are fed in the laser pulse, the energy efficiency is only $4.1\times 10^{-5}$. Such low energy efficiency suggest the demand of 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 $q_{\textrm{opt}}=\frac{G_{0}}{2(Nk_{W}+k_{H})}$ [1, 33] and the optimal efficiency is $\eta_{\textrm{opt}}=\frac{1}{1+4\beta Z_{H}/(NZ_{C})}\frac{\tau_{c}}{\tau_{\textrm{pulse}}}$, where $\tau_{c}$ is the duration of an optical cycle. With the aforementioned impedance of the MIMOSA design, we find that $q_{\textrm{opt}}=2.18$ fC, which corresponds to $1.36\times 10^{4}$ electrons per micro-bunch, and $\eta_{\textrm{opt}}=0.58\%$. 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 wake fields is negligible due to the low resonance nature of 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 [43]. Consistent with recent DLA experiments [36, 29], the 3-channel MIMOSA has 15 periods in the electron propagation direction, whose geometric parameters are given in Sec. III. 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 incident on it a microbunch of duration 500 attoseconds full-width at half maximum (FWHM), consistent with recently demonstrated optically bunched beams [44, 24, 45]. 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 $x^{\prime\prime}(z)=-K^{2}x(z)$. 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. [2] as being required for focusing of sub-relativistic 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 $K=eB/2mc\beta\gamma$. 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 [17, 16] and recently demonstrated in [24], 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 2 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.