# Spatiotemporal Electron-Beam Focusing through Parallel Interactions
with Shaped Optical Fields

F. Javier García de Abajo<EMAIL_ADDRESS>ICFO-Institut
de Ciencies Fotoniques, The Barcelona Institute of Science and Technology,
08860 Castelldefels (Barcelona), Spain  ICREA-Institució Catalana de Recerca i
Estudis Avançats, Passeig Lluís Companys 23, 08010 Barcelona, Spain  Claus
Ropers  Max Planck Institute for Multidisciplinary Sciences, 37077 Göttingen,
Germany  4th Physical Institute – Solids and Nanostructures, University of
Göttingen, 37077 Göttingen, Germany

###### Abstract

The ability to modulate free electrons with light has emerged as a powerful
tool to produce attosecond electron wavepackets. However, research has so far
aimed at the manipulation of the longitudinal wave function component, while
the transverse degrees of freedom have primarily been utilized for spatial
rather than temporal shaping. Here, we show that the coherent superposition of
parallel light-electron interactions in separate spatial zones allows for a
simultaneous spatial and temporal compression of a convergent electron wave
function, enabling the formation of sub-Ångström focal spots of attosecond
duration. The proposed approach will facilitate the exploration of previously
inaccessible ultrafast atomic-scale phenomena in particular enabling
attosecond scanning transmission electron microscopy.

##  I Introduction

Fourier analysis shows that strongly peaked waveforms can be obtained by
superimposing a large number of phase-locked frequency components. This
ubiquitous principle underpins pulsed mode-locked lasers and is leveraged to
synthesize attosecond light pulses by combining high harmonics generated in
atomic gases Paul  _et al._ (2001); Corkum and Krausz (2007); Krausz and
Ivanov (2009) or solid-state targets Li  _et al._ (2020). Likewise, attosecond
electron bunches were formed through interaction with the near fields induced
by laser scattering at a periodic structure followed by electron propagation
in free space Sears  _et al._ (2008). In addition, optical near-field
interaction and dispersive propagation were predicted to produce attosecond
wavepackets in the wave function of individual electrons Feist  _et al._
(2015), as later demonstrated in experiments using laser scattering by
nanostructures Priebe  _et al._ (2017); Morimoto and Baum (2018) and also
through stimulated Compton scattering in free space Kozák  _et al._ (2018).

Temporal compression of free-electron beams (e-beams) has a long tradition in
the context of accelerator physics and electromagnetic wave generation Sears
_et al._ (2005, 2008); Gilmour (2011); Andrews and Brau (2004); Emma  _et al._
(2010); Hemsing  _et al._ (2014); Pellegrini  _et al._ (2016); Gover  _et al._
(2019); Ryabov  _et al._ (2020). In an intuitive picture, exposure of the
e-beam to electromagnetic fields induces a momentum modulation that causes a
periodic compression into subcycle bunches upon dispersive propagation of the
electron ensemble. By subsequently interacting with gratings Smith and Purcell
(1953) or undulators Pellegrini  _et al._ (2016), bunches containing a large
number of electrons $N$ can produce radiation by acting in unison, giving rise
to directed emission with an intensity $\propto N^{2}$ in what is known as
superradiance Urata  _et al._ (1998). This mechanism is widely used in
radiation sources operating over spectral ranges extending from microwaves in
klystrons Gilmour (2011) to x-rays in free-electron lasers Andrews and Brau
(2004); Emma  _et al._ (2010); Pellegrini  _et al._ (2016); Gover  _et al._
(2019).

Figure 1: Temporal compression of a free electron through parallel
interactions with light. (a) We consider an incident electron separated into
two paths, each of them interacting with light as described by the respective
coupling coefficients $\beta_{1}$ and $\beta_{2}$. The two paths are then
recombined at a focal point. (b) Spectrum of the incident electron state
before (top) and right after (middle) interacting with light, along with the
spectrum produced at the focal point by path mixing (bottom). (c) Temporal
profile at the focal point, showing a comb of electron probability density
with the same period $\tau=2\pi/\omega$ as the employed light. The interaction
coefficients in (b) and (c) are optimized to achieve maximum temporal
compression. We also show results for an optimally compressed single-
path/interaction configuration. Figure 2: Optimum temporal compression. (a) An
increasing number $N$ of parallel interactions with light leads to a narrowing
of the probability density peak. The maximum magnitude of the required
coupling coefficient is indicated for each $N$ by color-matching labels.
Optimum results are obtained for a propagation distance $z=z_{T}/4$ in all
cases. (b) Degree of coherence ${\rm DOC}_{m}$ for optimally compressed
electrons after interaction with $N$ parallel zones. We consider harmonic
orders $m=1-22$. Figure 3: Focal spot profile under temporal compression. (a)
We consider a set of $N$ equal-area concentric circular zones, each of them
delivering a uniform light-electron coupling coefficient. (b-d) Upon
optimization of the latter for electron temporal compression at the focal spot
${\bf r}=0$ using $N=30$ zones, we still find a Gaussian-like spatial profile
(b) of short duration, as revealed in cuts along both longitudinal (c) and
transverse (d) directions. (e) A selection of focal region frames near the
time of maximum electron probability reveals a short duration compared with
the optical period $\tau$. The pulse duration is shown to decrease with
increasing $N$. All distances are scaled to $L=\lambda_{e}/{\rm NA}$, where
$\lambda_{e}$ is the de Broglie electron wavelength and NA is the numerical
aperture (outer-zone diameter divided by zone-focus distance).

Electron compression can also be accomplished through the coherent evolution
of each individual free-electron wave function after interaction with intense
optical fields, provided that the level of spatial and temporal coherence of
both electrons and light is sufficiently high. For sufficient laser
intensities, multiple photon exchanges take place, in analogy to low-energy
electron scattering by illuminated atomic targets Weingartshofer  _et al._
(1977, 1983). In the context of ultrafast transmission electron microscopy,
this type of process has attracted considerable interest in the form of
photon-induced near-field electron microscopy (PINEM) of optical near-field
distributions Barwick  _et al._ (2009); García de Abajo  _et al._ (2010);
Feist  _et al._ (2015); Piazza  _et al._ (2015); Kfir  _et al._ (2020); Wang
_et al._ (2020); Talebi (2020). In PINEM, electrons emerge in states
comprising a superposition of energy sidebands equally spaced by the photon
energy. Besides imaging, the quantum-coherent phase modulation underlying the
inelastic light-electron scattering process was predicted Feist  _et al._
(2015) and experimentally shown Priebe  _et al._ (2017); Morimoto and Baum
(2018) to produce longitudinal shaping and attosecond bunching. In the
momentum representation, the velocity dispersion across the sidebands
translates into relative phase differences that accumulate as the probes
propagate, developing a periodic train of temporally compressed electron
pulses analogous to the Talbot effect Priebe  _et al._ (2017); Di Giulio and
García de Abajo (2020); Tsarev  _et al._ (2021).

While a single PINEM interaction is fundamentally limited to produce just a
moderate level of temporal compression Zhao  _et al._ (2021); Kfir  _et al._
(2021); Di Giulio  _et al._ (2021), close to perfectly confined pulses are
predicted to be formed from a sequence of PINEM interactions separated by
free-space propagation Yalunin  _et al._ (2021). In a separate development,
following the demonstration of ponderomotive phase plates for electron
microscopy Schwartz  _et al._ (2019), the possibility of realizing a
customizable modulation of the transverse electron wave function profile was
proposed using PINEM Konečná and García de Abajo (2020) and ponderomotive
García de Abajo and Konečná (2021) light-electron interactions, and recently
realized in separate experiments Madan  _et al._ (2022); Mihaila  _et al._
(2022). Conceivably, the coherent superposition of electron waves that have
undergone distinct PINEM interactions in the transverse plane should grant us
access into a much wider range of electron wave functions such as, for
example, states that comprise tailored spatiotemporal compression.

In this work, we theoretically demonstrate that inelastic electron–light
interaction can simultaneously produce spatial and temporal compression.
Specifically, we consider a convergent electron wave produced by the objective
lens of a scanning transmission electron microscope and study the effect of
the interaction with light at a plane preceding the focal plane. For
relatively simple transverse field profiles reminiscent of zone plates, we
predict the formation of sub-Ångström focal spots of attosecond duration. In
particular, a high level of compression is achieved with the superposition of
only two wave function components (Fig. 1), while more complex profiles (Figs.
2 and 3) enable a stronger temporal compression without substantially
compromising the spatial focusing performance of the microscope. Our work
holds potential for the study of ultrafast phenomena at the atomic scale,
including highly nonlinear and subcycle charge and lattice dynamics.

##  II Results and discussion

Right after interaction with monochromatic light of frequency $\omega$
characterized by a space- and time-dependent electric field amplitude ${\bf
E}({\bf r}){\rm e}^{-{\rm i}\omega t}+{\rm c.c.}$, the wave function of an
electron moving along the $z$ direction with average velocity $v$ becomes
García de Abajo  _et al._ (2010); Park  _et al._ (2010); García de Abajo and
Di Giulio (2021)

$\displaystyle\psi({\bf r},t)=\psi^{\rm inc}({\bf
r},t)\sum_{\ell=-\infty}^{\infty}\alpha_{\ell}({\bf r}){\rm e}^{{\rm
i}\ell\omega(z-vt)/v},$ (1)

where $\psi^{\rm inc}({\bf r},t)$ is the incident wave function. The sum in
Eq. (1) extends over the net number of exchanged photons $\ell$, corresponding
to an electron energy change $\ell\hbar\omega$ and having an associated
transition amplitude

$\displaystyle\alpha_{\ell}({\bf r})=J_{\ell}\big{[}|\beta({\bf
R})|\big{]}\,{\rm e}^{{\rm i}\ell\,{\rm arg}\\{-\beta({\bf R})\\}}\,{\rm
e}^{-2\pi{\rm i}\ell^{2}z/z_{T}},$ (2)

which is expressed in terms of a single electron-light coupling parameter

$\displaystyle\beta({\bf
R})=\frac{e}{\hbar\omega}\int_{-\infty}^{\infty}dz\,E_{z}({\bf r})\,{\rm
e}^{-{\rm i}\omega z/v}.$ (3)

We note that Eq. (3) depends on the transverse coordinates ${\bf R}=(x,y)$.
This result assumes an initial energy spread much smaller than $\hbar\omega$,
which is in turn negligible compared with the average kinetic energy
(nonrecoil approximation). In addition, a phase $\propto\ell^{2}$ is
incorporated to include the effect of velocity dispersion with a
characteristic Talbot distance Di Giulio and García de Abajo (2020)
$z_{T}=4\pi m_{\rm e}v^{3}\gamma^{3}/\hbar\omega^{2}$, where
$\gamma=1/\sqrt{1-v^{2}/c^{2}}$.

In the spatial representation, the wave function is modulated in time with the
optical period $\tau=2\pi\omega/v$ imposed by the light frequency $\omega$.
This allows for a quantification of the achieved level of temporal compression
through the so-called degree of coherence Kfir  _et al._ (2021); Di Giulio
_et al._ (2021)

$\displaystyle{\rm DOC}_{m}({\bf
r})=\bigg{|}\\!\\!\int_{0}^{\tau}\\!\\!\\!\\!dt\,|\psi({\bf r},t)|^{2}\,{\rm
e}^{{\rm i}m\omega
t}\bigg{|}^{2}\\!\\!\bigg{/}\bigg{|}\\!\\!\int_{0}^{\tau}\\!\\!\\!\\!dt\,|\psi({\bf
r},t)|^{2}\bigg{|}^{2}\\!\\!,$ (4)

which measures the ability of the electron to excite an optical mode localized
at a position ${\bf r}$ and characterized by a harmonic frequency $m\omega$.
The ideal compression associated with the point-particle limit is thus
corresponding to ${\rm DOC}_{m}({\bf r})=1$ for all $m$’s.

For a single light-electron interaction, one finds the analytical expression
Zhao  _et al._ (2021); Di Giulio  _et al._ (2021); Yalunin  _et al._ (2021);
Tsarev  _et al._ (2021) ${\rm DOC}_{m}=J_{m}^{2}[4\beta\sin(2\pi mz/z_{T})]$.
For $m=1$, a moderate maximum value of ${\rm DOC}_{1}\approx 0.34$ [see Fig.
1(b)] is obtained by adjusting the optical field strength to satisfy the
condition $|\beta|=e_{1}/4\sin(2\pi z/z_{T})$, where $e_{1}\approx 1.841$ is
the absolute maximum of the $J_{1}$ Bessel function. As stated above, even
tighter compression can be reached by sequential interactions Yalunin  _et
al._ (2021). However, harnessing the transverse degrees of freedom to produce
tailored temporal as well as spatial structuring has yet to be explored.

Here, we leverage the coherent superposition of electron wave function
components undergoing separate parallel interactions with light in distinct
zones for far-reaching spatiotemporal control. Replacing the coefficients in
Eq. (2) by a weighted sum with various values of $\beta$ yields a powerful set
of additional control parameters. In a simple picture, temporal compression
can be achieved if $\alpha_{\ell}$ becomes independent of $\ell$ over a
certain range such as $-L\leq\ell\leq L$, for which the wave function becomes
$\psi\propto\sum_{\ell=-L}^{L}{\rm e}^{{\rm
i}\ell\zeta}=\sin[(L+1/2)\zeta]/\sin(\zeta/2)$, such that ${\rm
DOC}_{m}\approx(1-|m|/L)^{2}$ approaches the perfect compression limit for
$L\gg 1$. The question we ask is whether near-unity, $\ell$-independent
coefficients can be obtained by superimposing parallel electron-light
interactions, such that they become

$\displaystyle\alpha_{\ell}({\bf r})={\rm e}^{-2\pi{\rm
i}\,\ell^{2}z/z_{T}}\sum_{i}a_{i}\,J_{\ell}(2|\beta_{i}|)\,{\rm e}^{{\rm
i}\ell{\rm arg}\\{-\beta_{i}\\}}$ (5)

for a given set of weighting coefficients $a_{i}$, with a common dispersive
phase $-2\pi z/z_{T}$ proportional to the distance $z$ from a given
interaction plane to a common focal spot toward which the electron is
converged.

The superposition of wave function components from two parallel zones [Fig.
1(a)] is already enough to produce a substantial improvement in temporal
compression corresponding to ${\rm DOC}_{1}\approx 0.84$ not  and illustrated
by the sharp wave function profile plotted in Fig. 1(c), where it is compared
with the single-interaction result. The spectral distribution of the wave
function resulting from this superposition cannot be achieved with a single
interaction [Fig. 1(b)]; it rather approaches a more continuous spectral
distribution, as also observed for sequential interactions Yalunin  _et al._
(2021). This exemplifies a method to produce any designated combination of
sideband amplitudes by superimposing a sufficient number of parallel
interactions. Incidentally, we impose real coefficients $a_{i}$ in Fig. 1 (see
below), for which the optimum solution involves moderate values of the
coupling coefficients $\beta_{i}$ [see Fig. 1(b)].

As a practical zone-plate-type configuration, we consider an ${\bf
R}$-dependent light-electron interaction taking place at a plane situated
within the pole piece gap in an electron microscope and before the focal
plane, although similar designs could operate by placing the plate at other
places along the electron column. Under the paraxial approximation, and
assuming axially symmetric coupling coefficients $\beta({\bf R})$ with respect
to the e-beam axis at ${\bf R}=0$, the wave function near the focal region is
found to take the form of Eq. (1) with coefficients (see details in the
Appendix)

$\displaystyle\alpha_{\ell}({\bf r})\approx{\rm e}^{-2\pi{\rm
i}\,\ell^{2}z/z_{T}}$ $\displaystyle\int_{0}^{\rm NA}\\!\\!\\!\theta
d\theta\,J_{0}(2\pi\theta R/\lambda_{e})\,{\rm e}^{-{\rm
i}\pi\theta^{2}z/\lambda_{e}}$ $\displaystyle\times
J_{\ell}\big{[}2|\beta(\theta)|\big{]}\,{\rm e}^{{\rm i}\ell{\rm
arg}\\{-\beta(\theta)\\}},$ (6)

where NA is the numerical aperture (set to 0.02 in this work), and
$\lambda_{e}$ is the electron wavelength. This expression is accurate within
the paraxial approximation for the wide range of geometrical parameters
encountered in currently available electron microscopes (see Appendix).
Incidentally, both spherical and chromatic aberration can easily be included
in our formalism, but are not considered here for simplicity. For
concreteness, we consider a stepwise distribution of $\beta(\theta)$
parameters (with the ${\bf R}$ dependence now absorbed in the paraxial angle
$\theta$, see Appendix), which could be achieved by projecting a
correspondingly shaped laser beam on an electron-transparent film at an
oblique angle Vanacore  _et al._ (2018) or, alternatively, by a weakly focused
laser beam illuminating a film featuring a stepwise thickness profile
consisting of concentric circular zones [see Fig. 3(a)]. This configuration
reduces Eq. (6) to Eq. (5), with coefficients $a_{i}$ determined by
restricting the $\theta$ integral to each of the $i$ zones.

We numerically optimize temporal compression at the focal spot ${\bf r}=0$
(where the coefficients $a_{i}$ become real) by finding the maximum of ${\rm
DOC}_{1}({\bf r}=0)$ not  through the steepest-gradient method. Separating the
circle defined by the NA in the interaction plane into a total of $N$ equal-
area zones [Fig. 2(a)], the coefficients $a_{i}$ are made independent of $i$.
Under these conditions, the optimum focal electron wave function, represented
over an optical period at ${\bf r}=0$ in Fig. 2(b), becomes increasingly
sharper as $N$ is increased (see Table A2 in the Appendix for a subset of the
obtained optimum values of $\beta_{i}$). Again, attainable values of the
coupling coefficients $\beta_{i}$ are obtained. Interestingly, optimum results
are obtained for a propagation distance $z=z_{T}/4$ (e.g., $z\approx 2.5\,$mm
for 60 keV electrons and 4 eV photons), which renders odd $\ell$ terms in
quadrature relative to even terms. The corresponding degree of coherence is
calculated analytically for each set of $\beta_{i}$ values (see Appendix) and
plotted as a function of the number of zones $N$ and the harmonic order $m$ in
Fig. 2(c). A monotonic increase is observed for each order $m$ as $N$
increases, and in particular, we find ${\rm DOC}_{m}>0.8$ for $|m|\leq 5$ with
$N\geq 15$.

The optimum solution at ${\bf r}=0$ is compatible with spatial focusing, as
revealed by the analysis presented in Fig. 3. Remarkably, a good level of
temporal compression is simultaneously obtained within a finite spatial region
covering the focal spot under the configuration presented in Fig. 3(a). At the
time of maximum electron density [Fig. 3(b)], the focal spot is laterally
confined within a region $R\lesssim\lambda_{e}/2\,{\rm NA}$ (e.g., $\approx
1\,$Å for 100 keV electrons and NA=0.02) for $N=30$, whereas it extends over a
longitudinal range $|z|\lesssim\lambda_{e}/({\rm NA})^{2}$ ($\approx 10\,$nm).
When examining the temporal profile of cross sections passing by the focal
spot and oriented along the transverse [Fig. 3(c)] and longitudinal [Fig.
3(d)] directions, we observe an overall level of compression similar to the
optimized behavior at the spot center. A similar preservation of spatial
focusing is observed with other values of $N$, while this parameter is
primarily affecting temporal compression. For illustration, we present focal
spot movies [Fig. 3(e)] revealing the emergence of a sharp electron density
distribution within an interval spanning 20% of the optical period for $N=2$,
while shorter focal splot durations are obtained with larger values of $N$.

##  III Conclusion

In summary, we predict the formation of subnanometer-attosecond spatiotemporal
electron probes by molding the transverse electron wave functions through
PINEM-like interactions with spatially separated optical fields structured in
relatively simple zone profiles. This approach is generally compatible with
spatial electron focusing in scanning transmission electron microscopy, where
the required optical fields could be directly projected on an electron-
transparent plate. Alternatively, a simpler configuration could rely on
illumination by a broad light beam, supplemented by lateral structuring of the
plate (e.g., a dielectric film coated with metal and forming a layer of
laterally varying thickness; see Appendix). While we have considered
monochromatic light, such that a long electron pulse is transformed into a
train of attosecond pulses spaced by the optical period, more general
illumination conditions relying on broadband fields could be employed to
obtain individual electron pulses with much wider temporal separation. As an
extrapolation of these ideas, we envision the formation of arbitrary
spatiotemporal electron profiles consisting, for instance, of several
individual probes at tunable positions and instants to realize subnanometer-
attosecond electron-electron pump-probe spectroscopy. A currently attainable
light-electron pump-probe scheme could consist in triggering strongly
nonlinear processes in a sample through ultrafast laser pulse irradiation,
whose fast evolution within a sub-optical-cycle timescale could be probed by
compressed electrons such as those here investigated.

## ACKNOWLEDGMENTS

We thank V. Di Giulio, A. Feist, J. H. Gaida, and S. V. Yalunin for insightful
discussions. This work has been supported in part by the European Research
Council (Advanced Grant 789104-eNANO), the European Commission (Horizon 2020
Grants FET-Proactive 101017720-EBEAM and FET-Open 964591-SMART-electron), the
Spanish MICINN (PID2020-112625GB-I00 and Severo Ochoa CEX2019-000910-S), the
Catalan CERCA Program, and Fundaciós Cellex and Mir-Puig, and the Humboldt
Foundation.

## APPENDIX

###  III.1 Paraxial free-electron focusing

We consider a monochromated free-electron beam (e-beam) of kinetic energy
$E_{0}$ moving along the positive $z$ direction. Starting with the electron
wave function $\psi({\bf R}^{\prime},z_{0})$ at a plane $z=z_{0}$ [Fig.
A4(a)], where ${\bf R}^{\prime}=(x^{\prime},y^{\prime})$ denotes the
transverse position coordinates, we can construct the wave function $\psi({\bf
R},z_{1})$ at a different plane $z=z_{1}$ after free propagation by first
projecting on components of transverse wave vector ${\bf Q}=(q_{x},q_{y})$ and
then incorporating the $z$ dependence through a longitudinal wave vector
$q_{z}=\sqrt{q_{0}^{2}-Q^{2}}$, where $q_{0}=m_{\rm e}v\gamma/\hbar$ is the
total electron wave vector corresponding to the specified kinetic energy
$E_{0}=m_{\rm e}c^{2}(\gamma-1)$, written in terms of the velocity
$v=c\sqrt{1-1/(E_{0}/m_{\rm e}c^{2}+1)^{2}}$ and the Lorentz factor
$\gamma=1/\sqrt{1-v^{2}/c^{2}}$. More precisely,

$\displaystyle\psi({\bf R},z_{1})$ $\displaystyle=\int\frac{d^{2}{\bf
Q}}{(2\pi)^{2}}\,{\rm e}^{{\rm i}{\bf Q}\cdot{\bf R}+{\rm
i}q_{z}(z_{1}-z_{0})}\int d^{2}{\bf R}^{\prime}{\rm e}^{-{\rm i}{\bf
Q}\cdot{\bf R}^{\prime}}\psi({\bf R}^{\prime},z_{0})$
$\displaystyle\approx\frac{-{\rm i}q_{0}}{2\pi\,(z_{1}-z_{0})}{\rm e}^{{\rm
i}q_{0}(z_{1}-z_{0})}\int d^{2}{\bf R}^{\prime}{\rm e}^{{\rm i}q_{0}|{\bf
R}-{\bf R}^{\prime}|^{2}/2(z_{1}-z_{0})}\psi({\bf R}^{\prime},z_{0}),$ (A7)

where the second line is obtained by analytically integrating over ${\bf Q}$
in the paraxial approximation $q_{z}\approx q_{0}-Q^{2}/2q_{0}$.

Figure A4: Free-space electron propagation. (a) We consider free-space
propagation of a monochromatic electron of kinetic energy $E_{0}$ from a plane
$z=z_{0}$ to a plane $z=z_{1}$, where the wave function is
$\psi(x^{\prime},y^{\prime},z_{0})$ and $\psi(x,y,z_{1})$, respectively. The
e-beam is assumed to be paraxial and moving toward the positive $z$ direction.
(b) We are interested in the propagation of a beam focused at the $z=z_{1}$
plane through a lens placed before $z=z_{0}$. (c) PINEM interaction is
incorporated in the $z=z_{0}$ plane through a light-electron coupling
coefficient $\beta(x^{\prime},y^{\prime})$ that depends on the transverse
coordinates and produces a coherent electron comb of energies
$E_{0}\pm\ell\hbar\omega$, separated from the incident one by multiples of the
photon energy $\hbar\omega$.

We are interested in an e-beam prepared to be focused at ${\bf r}=0$ by an
axially symmetric, aberration-free lens placed before the $z=z_{0}$ plane
[Fig. A4(b)], so we write the incident wave function as $\psi({\bf
R}^{\prime},z_{0})=\psi_{0}{\rm e}^{-{\rm i}q_{0}R^{\prime 2}/2f}$, where
$\psi_{0}$ is an overall amplitude coefficient and $f$ is the focal distance
relative to such plane. Introducing these elements in Eq. (A7), the electron
wave function near the focal region reduces to

$\displaystyle\psi({\bf r})=\frac{-{\rm i}\psi_{0}q_{0}}{f+z}\,{\rm e}^{{\rm
i}(q_{0}R^{2}/2)/(f+z)}\,{\rm e}^{{\rm i}q_{0}(f+z)}\int_{0}^{R_{\rm
max}}\\!\\!\\!R^{\prime}dR^{\prime}\,J_{0}\big{[}q_{0}RR^{\prime}/(f+z)\big{]}\,{\rm
e}^{-{\rm i}(q_{0}zR^{\prime 2}/2f)/(f+z)},$ (A8)

where the longitudinal coordinate $z=z_{1}-z_{0}-f$ is referred to the focal
plane and the upper limit of the radial integral ${R_{\rm max}}\approx
NA\times f$ is limited by the numerical aperture (NA$\lesssim 0.02$ in typical
transmission electron microscopes).

###  III.2 Time-dependent focused beam after coherent PINEM interaction

We now introduce an ${\bf R}^{\prime}$-dependent PINEM interaction with light
of frequency $\omega$ at the $z=z_{0}$ plane [Fig. A4(c)], expressed through a
coupling coefficient $\beta({\bf R}^{\prime})$ García de Abajo and Di Giulio
(2021). It is then convenient to explicitly write the time-dependent wave
function, which, for the incident monochromatic electron, reads $\psi^{\rm
inc}({\bf r}){\rm e}^{-{\rm i}E_{0}t/\hbar}$. Right after the noted
interaction, the wave function at $z=z_{0}$ becomes

$\displaystyle\psi({\bf R}^{\prime},z_{0},t)=\psi^{\rm inc}({\bf
R}^{\prime},z_{0})\sum_{\ell=-\infty}^{\infty}J_{\ell}\big{[}2|\beta({\bf
R}^{\prime})|\big{]}\,{\rm e}^{{\rm i}\ell{\rm arg}\\{-\beta({\bf
R}^{\prime})\\}}\,{\rm e}^{-{\rm i}(E_{0}+\ell\hbar\omega)t/\hbar}.$ (A9)

Each $\ell$ component in Eq. (A9) needs to be propagated to the focal region
according to Eq. (A7), but with the electron wave vector $q_{0}$ replaced by
$q_{\ell}=q_{0}+\ell\omega/v-2\pi\ell^{2}/z_{T}+\cdots$, corresponding to the
modified electron energy $E_{0}+\ell\hbar\omega$, where $z_{T}=4\pi m_{\rm
e}v^{3}\gamma^{3}/\hbar\omega^{2}$ is the so-called Talbot distance Di Giulio
and García de Abajo (2020). Assuming axial symmetry in $\beta({\bf
R}^{\prime})\equiv\beta(R^{\prime})$ and applying Eq. (A8) to propagate the
wave function in Eq. (A9) [i.e., considering again an incident focused beam
characterized by a wave function $\psi^{\rm inc}({\bf
R}^{\prime},z_{0})=\psi_{0}{\rm e}^{-{\rm i}q_{0}R^{\prime 2}/2f}$], the time-
dependent electron wave function in the focal region reads

$\displaystyle\psi({\bf r},t)$ $\displaystyle=\frac{-{\rm
i}\psi_{0}}{f+z}\sum_{\ell}q_{\ell}\,{\rm e}^{{\rm
i}(q_{\ell}R^{2}/2)/(f+z)}\,{\rm e}^{{\rm i}q_{\ell}(f+z)-{\rm
i}(E_{0}/\hbar+\ell\omega)t}$ (A10)
$\displaystyle\quad\quad\quad\times\int_{0}^{R_{\rm
max}}\\!\\!\\!R^{\prime}dR^{\prime}\,J_{0}\big{[}q_{\ell}RR^{\prime}/(f+z)\big{]}\,{\rm
e}^{-{\rm i}(q_{\ell}zR^{\prime
2}/2f)/(f+z)}\,J_{\ell}\big{[}2|\beta(R^{\prime})|\big{]}\,{\rm e}^{{\rm
i}\ell{\rm arg}\\{-\beta(R^{\prime})\\}}.$

This expression can be simplified by assuming parameter ranges that encompass
a broad set of experimental conditions, such as $E_{0}\geq 1\,$keV,
NA$\lesssim 0.02$, $f\gtrsim 1\,$mm, and a photon wavelength $\lambda_{0}=2\pi
c/\omega\sim 1\,\mu$m, for which the electron wavelength is
$\lambda_{e}<39\,$pm$\,\ll\lambda_{0}$, the Talbot distance is $z_{T}\gtrsim
0.2\,$mm, and the radial and longitudinal extensions of the focal spot are
delimited by $R\lesssim\lambda_{e}/{\rm NA}$ and $|z|\lesssim\lambda_{e}/({\rm
NA})^{2}$. Neglecting phase contributions of the order of
$\lambda_{e}/\lambda_{0}$, $z/f$, and $R/R_{\rm max}$ (see detailed analysis
in Table A1), we find

$\displaystyle\psi({\bf r},t)\approx C\sum_{\ell}\alpha_{\ell}({\bf r})\,{\rm
e}^{{\rm i}\ell\omega(f+z-vt)/v},$ (A11a) $\displaystyle\alpha_{\ell}({\bf
r})={\rm e}^{-{\rm i}\,\ell^{2}\phi}\int_{0}^{\rm NA}\\!\\!\\!\theta
d\theta\,J_{0}(q_{0}R\,\theta)\,{\rm e}^{-{\rm
i}q_{0}z\,\theta^{2}/2}\,J_{\ell}\big{[}2|\beta(\theta)|\big{]}\,{\rm e}^{{\rm
i}\ell{\rm arg}\\{-\beta(\theta)\\}},$ (A11b)

where $\phi=2\pi\,f/z_{T}$ is a propagation phase accounting for velocity
dispersion in different $\ell$ contributions, we have changed the variable of
integration to $\theta=R^{\prime}/f$ (limited by ${\rm NA}\approx{R_{\rm
max}}/f$), the $R^{\prime}$ dependence of the PINEM coefficient is indicated
through $\beta(\theta)$, and the coefficient $C=-{\rm i}\,\psi_{0}q_{0}f\,{\rm
e}^{{\rm i}q_{0}(f+z+R^{2}/2f)}{\rm e}^{-{\rm i}(E_{0}/\hbar)t}$ has a
constant modulus (independent of ${\bf r}$ and $t$).

Electron |  |  $q_{\ell}=q_{0}[1+\ell\eta-\ell^{2}\eta^{\prime}+\cdots]$ — $\eta=\omega/vq_{0}=(c/v)\lambda_{e}/\lambda_{0}\ll 1$ — $\eta^{\prime}=2\pi/q_{0}z_{T}$
---|---|---
momentum |  |  — $\eta^{\prime}/\eta=2\pi v/\omega z_{T}=(v/c)\lambda_{0}/z_{T}=(c/2v\gamma^{2})(\lambda_{e}/\lambda_{0})\ll 1$
expansion |  |  — $z_{T}=4\pi m_{\rm e}v^{3}\gamma^{3}/\hbar\omega^{2}=2(v\gamma/c)^{2}(\lambda_{0}^{2}/\lambda_{e})$ — $q_{0}=m_{\rm e}v\gamma/\hbar=2\pi/\lambda_{e}$
Focal region |  |  $R\lesssim g\lambda_{e}/({\rm NA})$ — $|z|\lesssim g^{\prime}\lambda_{e}/({\rm NA})^{2}$ — ${\rm NA}\approx{R_{\rm max}}/f$ — $g,g^{\prime}\sim 1$
Approximations |  | ${\color[rgb]{1,0,0}(q_{\ell}R^{2}/2)/(f+z)}=(q_{0}R^{2}/2f)[1+\ell\eta-\ell^{2}\eta^{\prime}+\cdots][1-z/f+z^{2}/f^{2}+\cdots]\approx q_{0}R^{2}/2f$
|  |  $+O\big{[}\eta q_{0}R^{2}/2f=\pi(c/v)R^{2}/f\lambda_{0}\lesssim\pi g(c/v)(R/{R_{\rm max}})(\lambda_{e}/\lambda_{0})\ll 1\big{]}$ — $R/f\ll R^{\prime}/f\leq{\rm NA}\ll 1$
|  |  $+O\big{[}q_{0}|z|R^{2}/2f^{2}=\pi|z|R^{2}/f^{2}\lambda_{e}\lesssim\pi g(|z|/f)(R/{R_{\rm max}})\ll 1\big{]}$ — $|z|/f\ll 1$ — $R/\lambda_{0}\ll 1$
|  | ${\color[rgb]{1,0,0}q_{\ell}RR^{\prime}/(f+z)}=(q_{0}RR^{\prime}/f)[1+\ell\eta-\ell^{2}\eta^{\prime}+\cdots][1-z/f+z^{2}/f^{2}+\cdots]\approx q_{0}RR^{\prime}/f$
|  | $+O\big{[}\eta q_{0}RR^{\prime}/f=2\pi(c/v)RR^{\prime}/f\lambda_{0}<2\pi g(c/v)\,\lambda_{e}/\lambda_{0}\ll 1\big{]}$
|  | $+O\big{[}q_{0}|z|RR^{\prime}/f^{2}=2\pi|z|RR^{\prime}/f^{2}\lambda_{e}\lesssim 2\pi g(|z|/f)(R^{\prime}/{R_{\rm max}})\leq 2\pi g(|z|/f)\ll 1\big{]}$
|  | ${\color[rgb]{1,0,0}(q_{\ell}zR^{\prime 2}/2f)/(f+z)}=(q_{0}zR^{\prime 2}/2f^{2})[1+\ell\eta-\ell^{2}\eta^{\prime}+\cdots][1-z/f+z^{2}/f^{2}+\cdots]\approx q_{0}zR^{\prime 2}/2f^{2}$
|  | $+O\big{[}\eta q_{0}|z|R^{\prime 2}/2f^{2}=\pi(c/v)|z|R^{\prime 2}/f^{2}\lambda_{0}\leq\pi(c/v)|z|({\rm NA})^{2}/\lambda_{0}\lesssim\pi g^{\prime}(c/v)\lambda_{e}/\lambda_{0}\ll 1\big{]}$
|  | $+O\big{[}q_{0}z^{2}R^{\prime 2}/2f^{3}\leq q_{0}z^{2}({\rm NA})^{2}/2f\lesssim g^{\prime}q_{0}\lambda_{e}|z|/2f=\pi g^{\prime}|z|/f\ll 1\big{]}$
|  | ${\color[rgb]{1,0,0}q_{\ell}(f+z)}=q_{0}(f+z)(1+\ell\eta-\ell^{2}\eta^{\prime}+\cdots)=q_{0}(f+z)+q_{0}(f+z)\ell\eta-q_{0}(f+z)\ell^{2}\eta^{\prime}+\cdots$
|  |  $\approx q_{0}(f+z)+(f+z)\ell\omega/v-2\pi f\ell^{2}/z_{T}+\cdots$ — $q_{0}|z|\eta^{\prime}=2\pi|z|/z_{T}<\pi g^{\prime}(c/v\gamma)^{2}\,\lambda_{e}^{2}/(\lambda_{0}\times{\rm NA})^{2}\ll 1$
Table A1: Details of the approximations made to transform Eq. (A10) into Eqs.
(A11).

The wave function in Eqs. (A11) is periodic in time with a period
$\tau=2\pi/\omega$ determined by the light frequency. The time-averaged
probability density then reduces to

$\displaystyle\frac{1}{\tau}\int_{0}^{\tau}dt\,|\psi({\bf
r},t)|^{2}=|C|^{2}\sum_{\ell}|\alpha_{\ell}({\bf r})|^{2}.$

In addition, upon integration over transverse coordinates, this quantity
yields an electron current proportional to

$\displaystyle\frac{1}{\tau}\int d^{2}{\bf R}\int_{0}^{\tau}dt\,|\psi({\bf
r},t)|^{2}=\pi|\psi_{0}|^{2}R_{\rm max}^{2},$ (A12)

where we have used the equation
$\int_{0}^{\infty}xdx\,J_{\ell}(ax)J_{\ell}(a^{\prime}x)=\delta(a-a^{\prime})/a$
to reduce the double sum over sidebands to a single one, and then applied the
identity $\sum_{\ell}J_{\ell}^{2}(a)=1$. Reassuringly, the result in Eq. (A12)
is independent of the longitudinal position $z$, as expected from the
conservation of electron probability.

It is convenient to discretize the $\theta$ dependence of $\beta(\theta)$ by
considering $N$ concentric circular zones, such that $\beta(\theta)=\beta_{i}$
is uniform within each zone $\theta_{i-1}<\theta<\theta_{i}$, with
$i=1,\cdots,N$ and $\theta_{N}={\rm NA}$. We also define $\theta_{0}=0$ to
refer to the intersection with the axis of rotational symmetry. This allows us
to rewrite Eq. (A11b) as

$\displaystyle\alpha_{\ell}({\bf r})={\rm e}^{-{\rm
i}\,\ell^{2}\phi}\sum_{i}a_{i}\,J_{\ell}(2|\beta_{i}|)\,{\rm e}^{{\rm
i}\ell{\rm arg}\\{-\beta_{i}\\}},$ (A13a) $\displaystyle
a_{i}=\int_{\theta_{i-1}}^{\theta_{i}}\\!\\!\\!\theta
d\theta\,J_{0}(q_{0}R\,\theta)\,{\rm e}^{-{\rm i}q_{0}z\,\theta^{2}/2}.$
(A13b)

In particular, at the focal point ${\bf r}=0$ the expansion coefficients are
$a_{i}=(\theta_{i}^{2}-\theta_{i-1}^{2})/2$, real and proportional to the
areas of the concentric zones. Equations (A13) provide a simple prescription
to parametrize any arbitrary $\beta(\theta)$ profile by taking a sufficiently
large number of zones $N$. In the present work, we consider moderate values of
$N$ under the assumption that the coupling coefficient is made uniform in each
zone.

###  III.3 Evaluation of the degree of coherence

We consider an electron modulated as shown in Eqs. (A11). When one is
interested in the subsequent electron interaction with a specimen, the
coherence factor that is defined as Kfir  _et al._ (2021); García de Abajo and
Di Giulio (2021)

$\displaystyle M_{m}({\bf r})=\int_{0}^{\tau}dt|\psi({\bf r},t)|^{2}\,{\rm
e}^{{\rm i}m\omega t}$

provides a measure of its ability to excite an optical mode of frequency
$m\omega$ (a harmonic $m$ of the light frequency) localized at a position
${\bf r}$. In the process carried out in the main text to optimize the
temporal compression of the electron, we maximize the degree of coherence Kfir
_et al._ (2021)

$\displaystyle{\rm DOC}_{m}({\bf r})=|M_{m}({\bf r})/M_{0}({\bf r})|^{2},$
(A14)

which determines the enhancement in the excitation probability relative to an
unmodulated electron. In the limit of a point particle, we have ${\rm
DOC}_{m}({\bf r})=1$ for all $m$’s. In practice, when maximizing ${\rm
DOC}_{1}(0)$ (i.e., for $m=1$ at the focal spot), we obtain compressed
electron pulses in which ${\rm DOC}_{m}({\bf r})$ is also enhanced for other
values of $m$ within an extended focal region, as illustrated by Figs. 2 and 3
in the main text.

Considering light-coupling coefficients structured in a set of concentric
circular zones, we start from Eq. (A11a) to write the coherence factor as

$\displaystyle M_{m}({\bf r})=|C|^{2}\sum_{\ell}\alpha_{\ell}({\bf
r})\alpha_{\ell+m}^{*}({\bf r}).$

A useful expression can be found by expanding $\alpha_{\ell}({\bf r})$ as
shown in Eq. (A13a) and then making use of Graf’s theorem (see Eq. (9.1.79) of
Ref. Abramowitz and Stegun (1972)) to evaluate the $\ell$ sum. This leads to

$\displaystyle M_{m}({\bf
r})=|C|^{2}\sum_{ii^{\prime}}a_{i}a_{i^{\prime}}^{*}\,{\rm e}^{{\rm
i}m(m\phi-\varphi_{i^{\prime}}+\chi_{ii^{\prime}})}J_{m}(2b_{ii^{\prime}}),$
(A15)

where $\varphi_{i}={\rm arg}\\{-\beta_{i}\\}$,
$b_{ii^{\prime}}=\sqrt{|\beta_{i}|^{2}+|\beta_{i^{\prime}}|^{2}-2|\beta_{i}\beta_{i^{\prime}}|\cos\xi_{ii^{\prime}}}$
with $\xi_{ii^{\prime}}=\varphi_{i}-\varphi_{i^{\prime}}+2m\phi$, and we
define the phase $\chi_{ii^{\prime}}$ to satisfy the equations
$b_{ii^{\prime}}\sin\chi_{ii^{\prime}}=|\beta_{i}|\sin\xi_{ii^{\prime}}$ and
$b_{ii^{\prime}}\cos\chi_{ii^{\prime}}=|\beta_{i^{\prime}}|-|\beta_{i}|\cos\xi_{ii^{\prime}}$.
Under uniform illumination (i.e., $\beta_{i}$ independent of $i$), direct
inspection of Eq. (A15) leads to the single-PINEM result García de Abajo and
Di Giulio (2021) $M_{m}=(\sum_{i}|a_{i}|^{2})|C|^{2}{\rm i}^{m}{\rm e}^{-{\rm
i}m\,{\rm arg}\\{-\beta_{1}\\}}J_{m}[4|\beta_{1}|\sin(m\phi)]$, for which the
maximum degree of coherence is ${\rm DOC}_{m}=J_{m}^{2}(\zeta_{m})$, obtained
with $|\beta_{1}|=\zeta_{m}/|4\sin(m\phi)|$, where $\zeta_{m}$ is the maximum
of $J_{m}(x)$ (e.g., $\zeta_{1}\approx 1.8412$ for $m=1$, which leads to ${\rm
DOC}_{1}\approx 0.3386$).

As an interesting configuration, we consider two concentric zones ($N=2$) with
the central one having $\beta_{1}=0$ (i.e., no interaction with light), for
which Eqs. (A14) and (A15) produce

$\displaystyle{\rm
DOC}_{1}=\Delta^{2}\frac{2J_{1}(2|\beta_{2}|)\sin\phi+\Delta
J_{1}(4|\beta_{2}|\sin\phi)}{1+\Delta^{2}+2\Delta J_{0}(2|\beta_{2}|)},$

where $\Delta=a_{2}/a_{1}$ is taken to be real. This expression has an
absolute maximum of ${\rm DOC}_{1}\approx 0.513$ for $\Delta\approx 2.303$,
$|\beta_{2}|\approx 1.694$, and $\phi\approx 0.284$, already exceeding the
single-PINEM result.

$N$ | $|\beta_{1}|$ | ${\rm arg}\\{\beta_{1}\\}$ | $|\beta_{2}|$ | ${\rm arg}\\{\beta_{2}\\}$ | $|\beta_{3}|$ | ${\rm arg}\\{\beta_{3}\\}$ | $|\beta_{4}|$ | ${\rm arg}\\{\beta_{4}\\}$ | $|\beta_{5}|$ | ${\rm arg}\\{\beta_{5}\\}$ | $|\beta_{6}|$ | ${\rm arg}\\{\beta_{6}\\}$ | $|\beta_{7}|$ | ${\rm arg}\\{\beta_{7}\\}$ | $|\beta_{8}|$ | ${\rm arg}\\{\beta_{8}\\}$ | $|\beta_{9}|$ | ${\rm arg}\\{\beta_{9}\\}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 0.460 | 0.0∘ | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | -
2 | 0.947 | 0.0∘ | 0.947 | 128.0∘ | - | - | - | - | - | - | - | - | - | - | - | - | - | -
3 | 1.204 | 0.0∘ | 0.484 | 72.4∘ | 1.204 | 144.8∘ | - | - | - | - | - | - | - | - | - | - | - | -
4 | 0.540 | 0.0∘ | 2.039 | 37.2∘ | 2.039 | 239.7∘ | 0.540 | 276.9∘ | - | - | - | - | - | - | - | - | - | -
5 | 2.248 | 0.0∘ | 2.248 | 161.1∘ | 0.975 | 140.4∘ | 0.975 | 20.8∘ | 0.273 | 80.6∘ | - | - | - | - | - | - | - | -
6 | 1.114 | 0.0∘ | 1.114 | 139.4∘ | 1.171 | 139.2∘ | 3.162 | 346.9∘ | 3.162 | 152.5∘ | 1.171 | 0.2∘ | - | - | - | - | - | -
7 | 3.330 | 0.0∘ | 1.703 | 157.3∘ | 1.108 | 8.1∘ | 1.108 | 158.6∘ | 0.496 | 83.4∘ | 3.330 | 166.7∘ | 1.703 | 9.5∘ | - | - | - | -
8 | 3.620 | 0.0∘ | 0.423 | 84.0∘ | 3.620 | 168.0∘ | 0.423 | 84.0∘ | 2.030 | 159.5∘ | 1.575 | 4.0∘ | 1.575 | 164.0∘ | 2.030 | 8.5∘ | - | -
9 | 2.440 | 0.0∘ | 0.404 | 80.7∘ | 0.753 | 139.4∘ | 4.520 | 355.7∘ | 0.753 | 21.9∘ | 2.440 | 161.3∘ | 2.440 | 0.0∘ | 4.520 | 165.6∘ | 2.440 | 161.3∘
Table A2: Optimum values of the $\beta_{i}$ parameters corresponding to the
solutions presented in Fig. 2 of the main text for $N$ concentric circular
zones of equal area with $N=1-9$. The optimum dispersive phase is $\phi=2\pi
z/z_{T}=\pi/4$ in all cases.

###  III.4 Optimum light-electron coupling parameters for concentric circular
zones of equal area

We use the steepest-gradient method to optimize the degree of coherence for
$m=1$ at the focal point ${\bf r}=0$ [i.e., ${\rm DOC}_{1}(0)$] as an approach
to obtain temporally compressed electron wave function profiles. In
particular, we consider configurations consisting of a number $N$ of
concentric zones with the same area, such that the coefficients $a_{i}$ cancel
out in the evaluation of Eq. (A14) with $M_{m}(0)$ calculated from Eq. (A15).
More precisely, we find

$\displaystyle{\rm DOC}_{1}(0)=\left|\frac{\sum_{ii^{\prime}}{\rm e}^{{\rm
i}(\phi-\varphi_{i^{\prime}}+\chi_{ii^{\prime}})}J_{1}(2b_{ii^{\prime}})}{\sum_{ii^{\prime}}J_{0}(2b_{ii^{\prime}})}\right|^{2}.$

In all cases, we obtain an optimum dispersive phase $\phi=2\pi z/z_{T}=\pi/4$.
The so-obtained optimum light-electron coupling parameters are listed in Table
A2 for $N=1-9$.

Figure A5: Light-electron coupling in a planar multilayer structure. (a)
Parameters defining the multilayer structure and the propagation of light
throughout it. (b) Orientation of the incidence electron and light directions
relative to the multilayer.

###  III.5 Light-electron coupling coefficient in planar multilayers under
plane-wave illumination

A practical realization of $i=1,\cdots,N$ zones featuring different light-
electron coupling coefficients $\beta_{i}$ could be based on illumination by a
uniform light plane wave of amplitude $\mathcal{E}_{0}$, such that each zone
$i$ consists of a planar multilayer with thicknesses adjusted to obtain the
desired coefficient $\beta_{i}$ up to a global factor $\mathcal{E}_{0}$. In
this section, we sketch a calculation of the light-electron coupling
coefficient mediated by a planar multilayer under oblique light and electron
incidence conditions. We set the surface normal along $z$ and, for simplicity,
take the light and electron incidence directions in the $x$-$z$ plane, forming
angles $\theta_{l}$ and $\theta_{e}$ relative to the $z$ axis, respectively,
with the light incident from the $z<0$ region [see Fig. A5(b)].

We consider $n$ layers ($j=1,\cdots,n$) of thicknesses $d_{j}$ and
permittivities $\epsilon_{j}$, as shown in Fig. A5(a). The optical electric
field in each layer $j$ delimited by $z_{j-1}<z<z_{j}$ is expressed in terms
of plane wave coefficients $A_{j}^{\pm}$ by writing ${\bf E}({\bf r},t)=2{\rm
Re}\big{\\{}{\bf E}({\bf r})\,{\rm e}^{-{\rm i}\omega t}\big{\\}}$ with

$\displaystyle{\bf E}({\bf r})=\mathcal{E}_{0}\,\left[A_{j}^{+}\,\hat{\bf
e}^{+}_{j}\,{\rm e}^{{\rm i}k_{jz}(z-z_{j-1})}+A_{j}^{-}\,\hat{\bf
e}^{-}_{j}\,{\rm e}^{-{\rm i}k_{jz}(z-z_{j})}\right]\;{\rm e}^{{\rm
i}k_{x}x},\quad\quad\quad z_{j-1}<z<z_{j}$ (A16a) where the $x$ component of
the wave vector $k_{x}>0$ and the frequency $\omega$ are conserved during
light propagation, $k_{jz}=\sqrt{k_{j}^{2}-k_{x}^{2}+{\rm i}0^{+}}$ with
$k_{j}=(\omega/c)\sqrt{k_{j}}$ and ${\rm Im}\\{k_{jz}\\}>0$ is the out-of-
plane wave vector, and $\hat{\bf e}^{\pm}_{j}=(\pm k_{jz}\hat{\bf
x}-k_{x}\hat{\bf z})/k_{j}$ are upward ($+$ sign) and downward ($-$ sign)
vectors for p polarization inside medium $j$. We dismiss s-polarized fields
because they produce a vanishing light-electron coupling under the geometry of
Fig. A5(b). In the near-side region ($z<0$, $j=0$, $\epsilon_{0}=1$), we write
the field as $\displaystyle{\bf E}({\bf r})=\mathcal{E}_{0}\,\left(\hat{\bf
e}^{+}_{0}\,{\rm e}^{{\rm i}k_{z}z}+R\;\hat{\bf e}^{-}_{0}\,{\rm e}^{-{\rm
i}k_{z}z}\right)\;{\rm e}^{{\rm i}k_{x}x},\quad\quad\quad z<0$ (A16b) where
$R$ is the reflection coefficient of the entire multilayer and $k_{z}\equiv
k_{0z}$. Likewise, the field in the far-side region ($z>z_{n}$, $j=n+1$,
$\epsilon_{n+1}=1$) reads $\displaystyle{\bf E}({\bf
r})=\mathcal{E}_{0}\,T\;\hat{\bf e}^{+}_{0}\,{\rm e}^{{\rm
i}[k_{z}(z-z_{n})+k_{x}x]},\quad\quad\quad z>z_{n}$ (A16c)

where $k_{z}$ and $\hat{\bf e}^{+}_{0}=\cos\theta_{l}\,\hat{\bf
x}-\sin\theta_{l}\,\hat{\bf z}$ are the same as in the near side and $T$ is
the transmission coefficient.

The coefficients $A_{j}^{\pm}$, $R$, and $T$ in Eqs. (A16) are determined by
the boundary conditions at the $z=z_{j}$ interfaces. Here, we consider the
equivalent Fabry-Perot-like expressions

$\displaystyle A_{j}^{+}=r_{j,j-1}\,{\rm e}^{{\rm
i}k_{jz}d_{j}}\,A_{j}^{-}+t_{j-1,j}\,{\rm e}^{{\rm
i}k_{{j-1},z}d_{j-1}}\,A_{j-1}^{+},$ $\displaystyle A_{j}^{-}=r_{j,j+1}\,{\rm
e}^{{\rm i}k_{j,z}d_{j}}\,A_{j}^{+}+t_{j+1,j}\,{\rm e}^{{\rm
i}k_{{j+1},z}d_{j+1}}\,A_{j+1}^{-}$

for $j=1,\cdots,n$, written in terms of the p-polarization Fresnel reflection
and transmission coefficients
$r_{j,j^{\prime}}=(\epsilon_{j^{\prime}}k_{j,z}-\epsilon_{j}k_{j^{\prime},z})/(\epsilon_{j^{\prime}}k_{j,z}+\epsilon_{j}k_{j^{\prime},z})$
and
$t_{j,j^{\prime}}=2\sqrt{\epsilon_{j}\epsilon_{j^{\prime}}}\,k_{j,z}/(\epsilon_{j^{\prime}}k_{j,z}+\epsilon_{j}k_{j^{\prime},z})$,
respectively, at each $j|j^{\prime}$ interface for incidence from the $j$
side. We are left with a linear system of $2n$ equations and variables (the
coefficients $A_{j}^{\pm}$ with $j=1,\cdots,n$), supplemented by $A_{0}^{+}=1$
(corresponding to an incident field amplitude $\mathcal{E}_{0}$) and
$A_{n+1}^{-}=0$ (no incident wave from the far side), as well as the
parameters $d_{0}=d_{n+1}=0$, which are defined such that the above equations
take a compact form. After finding $A_{j}^{\pm}$ by using standard linear
algebra techniques, the reflection and transmission coefficients are obtained
from $R=r_{0,1}+t_{1,0}\,{\rm e}^{{\rm i}k_{1,z}d_{1}}\,A_{1}^{-}$ and
$T=t_{n,n+1}\,{\rm e}^{{\rm i}k_{nz}d_{n}}\,A_{n}^{+}$ [see Fig. A5(a)].

The light-electron coupling coefficient is given by

$\displaystyle\beta=\frac{ev}{\hbar\omega}\int_{-\infty}^{\infty}dt\;\hat{\bf
v}\cdot{\bf E}({\bf r}={\bf v}t)\,{\rm e}^{-{\rm i}\omega t},$ (A17)

where ${\bf v}=v\hat{\bf v}=v_{x}\,\hat{\bf x}+v_{z}\,\hat{\bf z}$ with
$v_{x}=v\sin\theta_{e}$ and $v_{z}=v\cos\theta_{e}$ is the velocity vector
oriented along the direction $\hat{\bf v}=\sin\theta_{e}\,\hat{\bf
x}+\cos\theta_{e}\,\hat{\bf z}$, and we take the electron to cross the $z=0$
plane at $t=0$. This expression reduces to Eq. (3) in the main text when $z$
is chosen along the e-beam direction. Finally, inserting Eqs. (A16) into Eq.
(A17), we find

$\displaystyle\beta=\pm\frac{{\rm
i}ev\mathcal{E}_{0}}{\hbar\omega^{2}}\bigg{\\{}$
$\displaystyle\sum_{j=1}^{n}\bigg{[}A_{j}^{+}\,\hat{\bf v}\cdot\hat{\bf
e}^{+}_{j}\;\frac{{\rm e}^{{\rm i}(k_{x}v_{x}-\omega)z_{j}/v_{z}}{\rm e}^{{\rm
i}k_{jz}d_{j}}-{\rm e}^{{\rm
i}(k_{x}v_{x}-\omega)z_{j-1}/v_{z}}}{1-(k_{x}v_{x}+k_{jz}v_{z})/\omega}$
$\displaystyle\;\;\;\;\,+A_{j}^{-}\,\hat{\bf v}\cdot\hat{\bf
e}^{-}_{j}\;\frac{{\rm e}^{{\rm i}(k_{x}v_{x}-\omega)z_{j}/v_{z}}-{\rm
e}^{{\rm i}(k_{x}v_{x}-\omega)z_{j-1}/v_{z}}{\rm e}^{{\rm
i}k_{jz}d_{j}}}{1-(k_{x}v_{x}-k_{jz}v_{z})/\omega}\bigg{]}$ $\displaystyle-$
$\displaystyle\sin(\theta_{l}-\theta_{e})\;\frac{1-T\,{\rm e}^{{\rm
i}(k_{x}v_{x}-\omega)z_{n}/v_{z}}}{1-\cos(\theta_{l}-\theta_{e})\,v/c}$
$\displaystyle-$
$\displaystyle\sin(\theta_{l}+\theta_{e})\;\frac{R}{1+\cos(\theta_{l}+\theta_{e})\,v/c}\bigg{\\}},$
(A18)

where the overall $+$ and $-$ signs apply to electrons moving along upward
($-\pi/2<\theta_{e}<\pi/2$) or downward ($\pi/2<\theta_{e}<3\pi/2$)
directions, respectively, while the light is propagating upwardly
($0<\theta_{l}<\pi/2$) in all cases.

Figure A6: Two-zone plate design. (a) We consider a structure consisting of a
20 nm Si3N4 film partially coated with Au layers on both sides: 5 nm on top
and a layer of thickness $d$ below. The coated and uncoated regions are taken
to have the same area. The plate is optimized for 200 keV electrons, light
incidence at an angle $\theta_{l}=30^{\circ}$ relative to the surface normal,
and electrons incident with an angle
$\theta_{e}=\arcsin\big{[}(v/c)\sin\theta_{l}\big{]}\approx 20.3^{\circ}$ such
that the light-imprinted and electron-dispersion-corrected phase has a
vanishing overall geometric factor
$\propto(\omega/c)\sin\theta_{l}-(\omega/v)\sin\theta_{e}=0$, independent of
lateral position along the plate. (b) We vary the photon energy $\hbar\omega$
and the lower Au layer thickness $d$ to explore the resulting degree of
coherence DOC${}_{1}(0)$ at the focal point under the conditions of Fig. 2 in
the main text. The obtained values are close to the absolute limit for two-
zone plates over a wide range of thicknesses. (c) The actual value of
DOC${}_{1}(0)$ depends on the light electric field amplitude, which we adjust
for each combination of $\omega$ and $d$ to optimize temporal compression. The
so-obtained values of the field amplitude are attainable with synchronized
light-electron pulses in ultrafast electron microscopes. The plate-focus
distance is taken to be $z_{T}/4=\pi m_{\rm e}v^{3}\gamma^{3}/\hbar\omega^{2}$
in all cases (e.g., $z_{T}/4\approx 23\,$mm at $\hbar\omega=2\,$eV).

###  III.6 Two-zone design

We now use the light-coupling coefficient given by Eq. (A18) for multilayer
structures to optimize a specific plate yielding a reasonable level of
temporal compression. For simplicity, we target a two-zone plate with the
structure presented in Fig. A6(a). Plane-wave illumination requires oblique
incidence to guarantee a nonvanishing light-electron coupling. In addition,
electrons have to impinge obliquely as well to cancel the geometric optical
phase. The phase-cancellation condition is $v\sin\theta_{l}=c\sin\theta_{e}$.
We explore the performance of the plate with different thicknesses of one of
the coating layers for a $1-3\,$eV photon energy range, yielding a maximum
DOC${}_{1}(0)$ close to the absolute limit for two-zone plates over a wide
range of thicknesses and photon energies around 2 eV. Similar results are
obtained for different e-beam energies and geometrical parameters, which can
be varied to move the spectral region showing an optimum degree of coherence.

## References

  * Paul  _et al._ (2001) P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, Science 292, 1689 (2001).
  * Corkum and Krausz (2007) P. B. Corkum and F. Krausz, Nat. Phys. 3, 381 (2007).
  * Krausz and Ivanov (2009) F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 (2009).
  * Li  _et al._ (2020) J. Li, J. Lu, A. Chew, S. Han, J. Li, Y. Wu, H. Wang, S. Ghimire, and Z. Chang, Nat. Commun. 11, 2748 (2020).
  * Sears  _et al._ (2008) C. M. S. Sears, E. Colby, R. Ischebeck, C. McGuinness, J. Nelson, and R. Noble, Phys. Rev. Accel. Beams 11, 061301 (2008).
  * Feist  _et al._ (2015) A. Feist, K. E. Echternkamp, J. Schauss, S. V. Yalunin, S. Schäfer, and C. Ropers, Nature 521, 200 (2015).
  * Priebe  _et al._ (2017) K. E. Priebe, C. Rathje, S. V. Yalunin, T. Hohage, A. Feist, S. Schäfer, and C. Ropers, Nat. Photon. 11, 793 (2017).
  * Morimoto and Baum (2018) Y. Morimoto and P. Baum, Nat. Phys. 14, 252 (2018).
  * Kozák  _et al._ (2018) M. Kozák, N. Schönenberger, and P. Hommelhoff, Phys. Rev. Lett. 120, 103203 (2018).
  * Sears  _et al._ (2005) C. M. S. Sears, E. R. Colby, B. M. Cowan, R. H. Siemann, J. E. Spencer, R. L. Byer, and T. Plettner, Phys. Rev. Lett. 95, 194801 (2005).
  * Gilmour (2011) A. S. Gilmour,  _Klystrons, Traveling Wave Tubes, Magnetrons, Cross-field Amplifiers, and Gyrotrons_ (Artech House, Boston/London, 2011).
  * Andrews and Brau (2004) H. L. Andrews and C. A. Brau, Phys. Rev. Spec. Top.-AC. 7, 070701 (2004).
  * Emma  _et al._ (2010) P. Emma, R. Akre, J. Arthur, R. Bionta, C. Bostedt, J. Bozek, A. Brachmann, P. Bucksbaum, R. Coffee, F.-J. Decker, Y. Ding, D. Dowell, S. Edstrom, A. Fisher, J. Frisch, S. Gilevich, J. Hastings, G. Hays, P. Hering, Z. Huang, R. Iverson, H. Loos, M. Messerschmidt, A. Miahnahri, S. Moeller, H.-D. Nuhn, G. Pile, D. Ratner, J. Rzepiela, D. Schultz, T. Smith, P. Stefan, H. Tompkins, J. Turner, J. Welch, W. White, J. Wu, G. Yocky, and J. Galayda, Nat. Photon. 4, 641 (2010).
  * Hemsing  _et al._ (2014) E. Hemsing, G. Stupakov, and D. Xiang, Rev. Mod. Phys. 86, 897 (2014).
  * Pellegrini  _et al._ (2016) C. Pellegrini, A. Marinelli, and S. Reiche, Rev. Mod. Phys. 88, 015006 (2016).
  * Gover  _et al._ (2019) A. Gover, R. Ianconescu, A. Friedman, C. Emma, N. Sudar, P. Musumeci, and C. Pellegrini, Rev. Mod. Phys. 91, 035003 (2019).
  * Ryabov  _et al._ (2020) A. Ryabov, J. W. Thurner, D. Nabben, M. V. Tsarev, and P. Baum, Sci. Adv. 6, eabb1393 (2020).
  * Smith and Purcell (1953) S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953).
  * Urata  _et al._ (1998) J. Urata, M. Goldstein, M. F. Kimmitt, A. Naumov, C. Platt, and J. E. Walsh, Phys. Rev. Lett. 80, 516 (1998).
  * Weingartshofer  _et al._ (1977) A. Weingartshofer, J. K. Holmes, G. Caudle, E. M. Clarke, and H. Krüger, Phys. Rev. Lett. 39, 269 (1977).
  * Weingartshofer  _et al._ (1983) A. Weingartshofer, J. K. Holmes, J. Sabbagh, and S. L. Chin, J. Phys. B 16, 1805 (1983).
  * Barwick  _et al._ (2009) B. Barwick, D. J. Flannigan, and A. H. Zewail, Nature 462, 902 (2009).
  * García de Abajo  _et al._ (2010) F. J. García de Abajo, A. Asenjo-Garcia, and M. Kociak, Nano Lett. 10, 1859 (2010).
  * Piazza  _et al._ (2015) L. Piazza, T. T. A. Lummen, E. Quiñonez, Y. Murooka, B. Reed, B. Barwick, and F. Carbone, Nat. Commun. 6, 6407 (2015).
  * Kfir  _et al._ (2020) O. Kfir, H. Lourenço-Martins, G. Storeck, M. Sivis, T. R. Harvey, T. J. Kippenberg, A. Feist, and C. Ropers, Nature 582, 46 (2020).
  * Wang  _et al._ (2020) K. Wang, R. Dahan, M. Shentcis, Y. Kauffmann, A. Ben Hayun, O. Reinhardt, S. Tsesses, and I. Kaminer, Nature 582, 50 (2020).
  * Talebi (2020) N. Talebi, Phys. Rev. Lett. 125, 080401 (2020).
  * Di Giulio and García de Abajo (2020) V. Di Giulio and F. J. García de Abajo, Optica 7, 1820 (2020).
  * Tsarev  _et al._ (2021) M. V. Tsarev, A. Ryabov, and P. Baum, Phys. Rev. Research 3, 043033 (2021).
  * Zhao  _et al._ (2021) Z. Zhao, X.-Q. Sun, and S. Fan, “Quantum entanglement and modulation enhancement of free-electron–bound-electron interaction,” (2021).
  * Kfir  _et al._ (2021) O. Kfir, V. Di Giulio, F. J. García de Abajo, and C. Ropers, Sci. Adv. 7, eabf6380 (2021).
  * Di Giulio  _et al._ (2021) V. Di Giulio, O. Kfir, C. Ropers, and F. J. García de Abajo, ACS Nano 15, 7290 (2021).
  * Yalunin  _et al._ (2021) S. V. Yalunin, A. Feist, and C. Ropers, Phys. Rev. Research 3, L032036 (2021).
  * Schwartz  _et al._ (2019) O. Schwartz, J. J. Axelrod, S. L. Campbell, C. Turnbaugh, R. M. Glaeser, and H. Müller, Nat. Methods 16, 1016 (2019).
  * Konečná and García de Abajo (2020) A. Konečná and F. J. García de Abajo, Phys. Rev. Lett. 125, 030801 (2020).
  * García de Abajo and Konečná (2021) F. J. García de Abajo and A. Konečná, Phys. Rev. Lett. 126, 123901 (2021).
  * Madan  _et al._ (2022) I. Madan, V. Leccese, A. Mazur, F. Barantani, T. LaGrange, A. Sapozhnik, P. M. Tengdin, S. Gargiulo, E. Rotunno, J.-C. Olaya, I. Kaminer, V. Grillo, F. J. García de Abajo, F. Carbone, and G. M. Vanacore, ACS Photonics 9, 3215 (2022).
  * Mihaila  _et al._ (2022) M. C. C. Mihaila, P. Weber, M. Schneller, L. Grandits, S. Nimmrichter, and T. Juffmann, Phys. Rev. X 12, 031043 (2022).
  * Park  _et al._ (2010) S. T. Park, M. Lin, and A. H. Zewail, New J. Phys. 12, 123028 (2010).
  * García de Abajo and Di Giulio (2021) F. J. García de Abajo and V. Di Giulio, ACS Photonics 8, 945 (2021).
  * (41) The maximum possible value of ${\rm DOC}_{m}=1$ for any $m$ automatically guarantees ${\rm DOC}_{m}=1$ for all $m$’s. We therefore maximize ${\rm DOC}_{1}$ with $m=1$ as a practical procedure to optimize temporal compression. With this goal in mind, we define ${\rm DOC}_{m}({\bf r})$ in Eq. (A14) as a quantity normalized to the time-averaged electron probability density at a chosen position ${\bf r}$.
  * Vanacore  _et al._ (2018) G. M. Vanacore, I. Madan, G. Berruto, K. Wang, E. Pomarico, R. J. Lamb, D. McGrouther, I. Kaminer, B. Barwick, F. J. García de Abajo, and F. Carbone, Nat. Commun. 9, 2694 (2018).
  * Abramowitz and Stegun (1972) M. Abramowitz and I. A. Stegun,  _Handbook of Mathematical Functions_ (Dover, New York, 1972).