#  $\mu$eV electron spectromicroscopy using free-space light

Yves Auad  Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides,
91405, Orsay, France  Eduardo J. C. Dias  ICFO-Institut de Ciencies
Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain  Marcel Tencé  Université Paris-Saclay, CNRS,
Laboratoire de Physique des Solides, 91405, Orsay, France  Jean-Denis Blazit
Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405,
Orsay, France  Xiaoyan Li  Université Paris-Saclay, CNRS, Laboratoire de
Physique des Solides, 91405, Orsay, France  Luiz Fernando Zagonel  Applied
Physics Department, “Gleb Wataghin” Institute of Physics, University of
Campinas – UNICAMP, 13083-859 Campinas, SP, Brazil  Odile Stéphan  Université
Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France
Luiz H. G. Tizei  Université Paris-Saclay, CNRS, Laboratoire de Physique des
Solides, 91405, Orsay, France  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  Mathieu Kociak
<EMAIL_ADDRESS>Université Paris-Saclay, CNRS,
Laboratoire de Physique des Solides, 91405, Orsay, France

###### Abstract

The synergy between free electrons and light has recently been leveraged to
reach an impressive degree of simultaneous spatial and spectral resolution,
enabling applications in microscopy and quantum optics. However, the required
combination of electron optics and light injection into the spectrally narrow
modes of arbitrary specimens remains a challenge. Here, we demonstrate
microelectronvolt spectral resolution in the nanoscale mapping of photonic
modes with quality factors as high as $10^{4}$. We rely on mode-matching of a
tightly focused laser beam to whispering gallery modes to achieve a
$10^{8}$-fold increase in light-electron coupling efficiency. By adapting the
shape and size of free-space optical beams to address specific physical
questions, our approach allows us to interrogate any type of photonic
structure with unprecedented spectral and spatial detail.

##  I Introduction

Thanks to a sustained series of impressive advances in instrumentation Batson
_et al._ (2002); Krivanek  _et al._ (2014); Lagos  _et al._ (2017); Hage  _et
al._ (2020); Li  _et al._ (2021), electron microscopes can currently focus
60-300 keV electrons down to sub-Ångström focal spots with an energy spread of
just a few meV. As impressive as this spectral resolution might seem, optical
modes of high quality factor $Q$, which are of utmost importance for
applications including quantum optics and optical metrology, possess
substantially smaller linewidths and, therefore, are unresolvable by state-of-
the-art electron spectroscopies such as energy-loss spectroscopy (EELS),
cathodoluminescence (CL) Righini  _et al._ (2011); Gómez  _et al._ (2013);
Losquin  _et al._ (2015), and photon-induced near-field electron microscopy
(PINEM) Barwick  _et al._ (2009); García de Abajo  _et al._ (2010); Park  _et
al._ (2010); Feist  _et al._ (2015); Kfir  _et al._ (2020).

The so-called electron energy-gain spectroscopy (EEGS) was proposed García de
Abajo and Kociak (2008) as a technique that can dramatically enhance electron-
based spectroscopies by inheriting the spectral resolution of laser sources
while retaining the spatial resolution of electron beams (e-beams). EEGS data
thus consist of a series of conventional EELS spectra that are acquired as one
scans the wavelength of an external laser irradiating the specimen. Electron-
light coupling is mediated by near-field optical components, whose strength is
dependent on the optical response of the sample. The latter is consequently
retrieved from the intensity associated with light-induced electron energy-
gain events as a function of laser wavelength García de Abajo and Kociak
(2008); Asenjo-Garcia and García de Abajo (2013), with an energy resolution
that is only limited by the energy-time photon uncertainty ($\sim\hbar\sim
1\,$eV fs). Early attempts to demonstrate EEGS measurements were performed
with 100s-fs laser pulses Pomarico  _et al._ (2018); Wang  _et al._ (2020),
therefore reaching a spectral resolution of tens of meV, yet not overtaking
the few-meV resolution of modern electron monochromators Krivanek  _et al._
(2014). In parallel, efforts were undertaken to develop EEGS using nanosecond
Das  _et al._ (2019) or continuous-wave Liu  _et al._ (2019) visible laser
sources combined with continuous e-beams, although these attempts did not
introduce any wavelength tunability. Recently, a spectral resolution of a few
$\mu$eV has been demonstrated using on-purpose designed photonic waveguides
operating in the near-infrared Henke  _et al._ (2021). By injecting continous
laser light, this work elegantly circumvented the fundamental problem of
optical coupling to a high-$Q$ cavity. Nevertheless, a sample holder equipped
with a dedicated optical fiber was needed to achieve efficient coupling to a
photonic device of 10s $\mu$m in size, thus limiting its applicability to a
limited range of specimens.

Here, we demonstrate high-spectral-resolution EEGS enhanced by mode matching
between a free-space laser beam and the sample, which renders the technique
generally applicable to any kind of specimen. Specifically, we excite
whispering-gallery modes (WGMs) of well-defined angular momenta in spherical
resonators by means of a focused off-axis laser Gaussian beam using a high-
numerical-aperture mirror. We first concentrate on $\sim 4~{}\mu$m silica
spheres with quality factors $Q\sim~{}100-300$, in which EELS and CL
characterization reveals sharp resonances Auad  _et al._ (2022a) that are
corroborated by EEGS with a higher spectral resolution of $\sim 2~{}$meV. This
test system shows that optimum laser-mode coupling is achieved via
conservation of angular momentum, resulting in a $10^{8}$-fold enhancement of
the coupling efficiency relative to irradiation by an unfocused light plane
wave. The improved sensitivity of EEGS is also explained in terms of the sharp
laser linewidth ($7~{}\mu$eV) and high numerical aperture of the light
injection system. We then demonstrate the full potential of this approach by
controlling the laser beam position with sub-$\mu$m accuracy and resolving
narrow optical modes ($Q\sim 10^{4}$) in EEGS spectra of $\sim 8~{}\mu$m
polystyrene spheres, which are unobservable by EELS or CL. Our technique is
readily applicable to study arbitrary structures and represents an increase by
more than two orders of magnitude in spectral resolution relative to state-of-
the-art EELS with the same spatial resolution.

Figure 1: Versatile EEGS experiments using a continuous electron gun
microscope. (A) Schematics of the setup. A laser beam is focused down to a
$\sim 1~{}\mu$m spot and positioned with sub-$\mu$m precision at ${\bf R}_{f}$
on the sample of interest by using a parabolic mirror. A monochromatized 200
keV electron is also focused on the sample and scanned to acquire EELS, EEGS,
and CL signals. EEGS measurements are taken by synchronizing light (through a
laser trigger) and electrons (through an e-beam blanker) at the detector. (B)
A series of EELS spectra is acquired for a given electron probe position as
the laser wavelength is scanned over the spectral region of interest. The
strength of the energy-gain signal is boosted when the light wavelength is
close to an optical resonance of the specimen. (C) Using a wavelength-
independent light power, the energy-gain signal yields the EEGS intensity as a
function of light wavelength. Figure 2: EEGS measurements in WGM resonators
with $Q\sim$ 100–300. (A) Measured series of EELS gain spectra as a function
of light wavelength (vertical axis), showing two distinct resonances at 586.27
nm (2.1296 eV) and 601.02 nm (2.0631 eV) with quality factors of 244 and 194,
respectively. (B) Cascade of one in every four spectra taken from (A), where
both the stimulated loss and gain sidebands are observable. (C) EEGS, EELS,
and CL measured spectra (solid curves) compared to theory (dashed curves) for
the same e-beam probe position (blue circle at ${\bf R}_{e}$ in the inset) and
mirror focal position.  Figure 3: Free-space optical mode matching. Dependence
of the EEGS probability on the illumination conditions for a SiO2 sphere of
4122 nm diameter. (A) EEGS probability around a 601.5 nm resonance for plane-
wave and mirror-focused illumination, normalized to the light intensity
incident on the sphere and mirror, respectively. (B) Dependence of the EEGS
probability on light wavelength and focal spot position. The vertical axis
corresponds to the distance from the focal spot to the sphere center as it
moves along the $x$ axis (see inset). (C and D) EEGS probability map at the
peak wavelength of (A) with ${\bf R}_{f}=(x_{f},y_{f})$ scanned for incident
light polarized along $y$ and $z$, respectively. The sphere contour is shown
as a white circle. The color scale is shared by panels (B) to (D). (E) Optimum
optical-focus position $R_{f}$ as a function of orbital momentum number $l$
for resonances in PS (solid circles) and SiO2 (open circles) spheres of
diameters 8000 nm and 4122 nm, respectively. Symbols show all modes of high-
quality factor within the $\lambda=570-620~{}$nm spectral region (see color-
coordinated scale). The EEGS probability is indicated by the symbol size (see
legend). The dashed straight lines correspond to $2\pi R_{f}/\lambda$ equal to
$l$ and $l+3$. In (A), (B), and (E), the e-beam passes at a fixed position
$\sim 80~{}$nm away from the sphere surface on the $x$ axis, as indicated in
the insets. The intensity $I_{0}$ and power $P_{0}$ of the light incident on
the mirror are related through $P_{0}=A\,I_{0}$, where $A=18.74~{}$mm2 is the
mirror area.  Figure 4: High quality factor in large polystyrene spheres. (A)
Two laser wavelength series of EELS spectra, taken at the same electron probe
position over an increasingly small wavelength range with step sizes of 50 pm
and 2 pm, respectively. (B) EEGS spectrum constructed from the rightmost EELS
series in (A), along with Lorentzian fittings revealing three resonances (1–3)
of quality factors 7430, 10541, and 9178, respectively. The uncertainties for
the measured quality factor are $<0.1$ for the three resonances.

##  II RESULTS AND DISCUSSION

A sketch of the experimental setup is displayed in Figure 1. Experiments were
carried out in a modified Nion Hermes 200 transmission electron microscope
(Chromatem) working at 200 keV with a subnanometer e-beam. A nanosecond-laser
beam was focused on the sample and synchronized with the EELS detection in
order to record PINEM spectra, as detailed in the Appendix. Critical
improvements were implemented relative to our previous work Das  _et al._
(2019). Specifically, the microscope was fitted with a high-numerical-aperture
(NA$\sim 0.5$) Attolight Mönch light detection/injection system able to focus
down to a 1 $\mu$m spot size with sub-$\mu$m accuracy. The time-averaged laser
input power of $1-5~{}$mW used in experiments resulted in a typical
$10^{9-10}~{}$W/m2 optical spot intensity. We used the mirror to position the
laser spot at the edge of the WGM resonators. In addition, the laser
wavelength was tuned to spectrally map the resonances, with a wavelength
resolution of 2 pm ($\sim 7~{}\mu$eV at 585 nm), limited by the laser
specifications. Because the EEGS signal was weak ($\sim 10^{-4}$ of the
measured ZLP), a large increase of signal-to-noise ratio was needed, which we
achieved through a slight monochromation of $30-50~{}$meV over the 350 meV
initial e-beam energy spread. This led to a strong suppression of the ZLP
tails, which would otherwise produce a substantial background. We recorded
spectra with a direct electron detector (MerlinEM, from Quantum Detectors)
using an effective current of $\sim 0.2~{}$fA (see Appendix), comparable to
PINEM experiments. The WGM resonators were drop-cast on a lacey-carbon sample
grid, which was coated with 60 nm of silver to improve charge and thermal
dissipation. Finally, alignment of the laser spot to the microscope optical
axis was achieved with $\sim 1~{}\mu$m precision by maximizing the electron
EEGS signal from a featureless silver film. A more detailed description of the
setup is offered in the Appendix.

The EEGS electron-light coupling is described by a single parameter García de
Abajo and Di Giulio (2021):

$\displaystyle\beta({\bf
R}_{e},\omega)=\frac{e}{\hbar\omega}\int_{-\infty}^{\infty}dz\,E_{z}({\bf
R}_{e},z,\omega)\,e^{-i\omega z/v},$ (1)

where ${\bf R}_{e}$ is the transverse electron probe position, $v$ is the
electron velocity, $\omega$ is the angular frequency of the external light,
and $E_{z}$ is the optical electric field component along the e-beam direction
$z$, corresponding to a time-varying field $E_{z}({{\bf R}}_{e},t)=2{\rm
Re}\\{E_{z}({\bf R}_{e},z,\omega)e^{-i\omega t}\\}$, which is dependent on the
focal beam profile and position relative to the specimen (see Appendix). In
our experiment, we use a low peak-intensity illumination, such that
$|\beta|^{2}\ll 1$ is the probability for the electron to gain one photon
quantum (i.e., the EEGS signal is essentially a perturbation).

To illustrate and validate the principle of EEGS and its relation to other
spectroscopies, we start by studying $\sim 4~{}\mu$m silica spheres (Figure
2), which are known to exhibit quality factors $Q\sim 10^{2}$ under similar
experimental conditions and good electron-WGM coupling for the employed 200
keV electrons Auad  _et al._ (2022a). In Figure 2A, we plot a measured series
of spectra acquired for varying light wavelengths (vertical axis, in steps of
250 pm, corresponding to a 0.92 meV photon-energy interval at 580 nm) with a
constant laser power of $\sim 1.5~{}$mW and the e-beam probe positioned as
indicated by the blue circle ${\bf R}_{e}$ in Figure 2C. We observe two
distinct WGMs with $Q=244$ and $Q=194$ separated by a spectral distance of
66.4 meV. Due to the high monochromaticity of the e-beam, it is possible to
resolve the energy-gain resonance shifting in energy as we raster the laser
wavelength (dashed white line in Figure 2A). One of every four of these
spectra is shown in the cascade in Figure 2B, where we note the presence of
both stimulated electron energy gain and loss features. The gain signal
accounts for a fraction $\sim 5\times 10^{-4}$ of the integrated measured
spectrum (see color bar in Figure 2A), implying a gain probability of $\sim
5\times 10^{-3}$ after correcting for the e-beam blanker time window ($\sim
10$ times the laser pulse duration). Furthermore, there are no visible higher-
order replicas at multiples of the photon energy $\pm n\hbar\omega$, as
expected in the low-intensity regime García de Abajo  _et al._ (2010); Park
_et al._ (2010). In Figure 2C, we display the measured EEGS intensity (solid
red curve) obtained by integrating the EELS spectra series between 1.9 eV and
2.3 eV for each light wavelength. A similar result is obtained by integrating
the stimulated loss peak, although the gain side is free from loss features
and thus has a better signal-to-noise ratio. In this particular sample, the
$\sim 30~{}$meV spectral resolution of our setup is already enough to unveil
the same gallery-modes in EELS, and additionally, modes can be identified by
collecting light leakage from the resonator to the far field to record a CL
spectrum. The three spectroscopies (EEGS, EELS, and CL) are performed for the
same e-beam probe position ${\bf R}_{e}$ and mirror focal spot ${\bf R}_{f}$.
As expected from the reciprocity of Maxwell’s equations, the CL and EEGS
spectral variations look similar García de Abajo and Di Giulio (2021) (see
Appendix). Also, in non-dissipative systems, the EELS and CL probabilities
should be identical because radiation losses are the only source of energy
losses García de Abajo (2010), although deviations between the two of them can
arise because we are collecting only a fraction of the emission solid angles
Schmidt  _et al._ (2021). Nevertheless, no spectral shifts can be discerned
between EELS (the equivalent of optical extinction Losquin  _et al._ (2015)),
CL (scattering), and EEGS within the single-pixel uncertainty of the EELS
channels. Theoretical modelling of the EEGS intensity (Figure 2C, dashed
curves; see details in Appendix) matches the WGM positions for a fitted sphere
diameter of 4122 nm (consistent with the experimentally determined diameter),
although the predicted quality factors are $\sim 1500$, almost one order of
magnitude higher than the experimental results, presumably because of the
effect of losses produced at the supporting carbon structure. The reported
EEGS features with a full width at half maximum (FWHM) of $\sim 2~{}$meV
separated by 66.4 meV already demonstrate a spectral resolution one order of
magnitude better than the used EELS resolution at 200 keV electron energy, but
also better than the ultimate spectral resolution of the machine ($\sim
5~{}$meV at 60 keV).

It should be noted that CL yields a faint signal, which we accumulate for
$\sim 30~{}$s without electron monochromation (i.e., using 10–20 times more
current than in the monochromated experiments). Although this is faster than
EEGS acquisition, the high electron currents used can produce larger sample
damage. In addition, the EEGS signal can be enhanced by increasing the
incident laser power, which is still well below the sample damage threshold.

We next interrogate the potential of EEGS for the investigation of high-Q
photonic modes (i.e., those in which a high spectral resolution is actually
required). These modes are weakly coupled to the far field, and therefore, a
clear understanding of how to maximize light-electron coupling is needed. We
start by presenting theoretical calculations that illustrate the benefits of
using a position-controlled high-numerical-aperture focusing system.

The EEGS simulations presented in Figure 3A for the SiO2 sphere studied in
Figure 2 show an enhancement in probability by eight orders of magnitude when
moving from plane wave illumination to focused illumination (assuming the same
power and initial laser beam extension over the mirror area in both
scenarios). Besides a clear improvement due to the focusing effect of the
mirror, the principle behind such huge increase in coupling efficiency can be
found in mode-matching between a free-space optical beam and WGMs when the
former is focused near the edge of a dielectric sphere Zambrana-Puyalto  _et
al._ (2021). Such a mode-matching can be well-understood in terms of a
preferential orbital angular number $l\sim 2\pi R_{f}/\lambda$ produced when
the focal spot is at a distance $R_{f}$ from the sphere center and $\lambda$
is the light wavelength. Near the edge of the silica sphere discussed in
Figure 2, we have $R_{f}=2060\pm 500$ nm, leading to preferential coupling to
modes of angular momentum number $l=22\pm 5$, in agreement with the angular
order $l=26$ deduced from theory (see Appendix). The beneficial effects of
mode-matching can be further evidenced in several ways. The EEGS signal for an
electron passing near the edge of a dielectric sphere is represented in Figure
3B as a function of the optical beam position for a fixed wavelength,
revealing that the signal is strikingly peaked at the position of optimum
mode-matching for the value of $l$ corresponding to the mode that is resonant
at a wavelength $\lambda$. In Figure 3(C&D), the mirror position is scanned
with a fixed electron probe position ${\bf R}_{e}$ for the two possible light
polarizations, again evidencing the mode-matching condition through an optimal
mirror position ${\bf R}_{f}$. These results are further corroborated by
examining multiple resonances comprised in the 570–620 nm range both for the
already discussed $4.122~{}\mu$m SiO2 sphere and for a bigger $8~{}\mu$m
polystyrene (PS) sphere. The relation between the focal point and the angular
momentum number is linear as expected Zambrana-Puyalto  _et al._ (2021),
although shifted by 3, presumably as a result of the finite beam size. More
generally, mode matching between focused light and WGM resonators is known to
be similar to light coupling to a waveguide Zou  _et al._ (2013) and reach
coupling efficiencies up to 20$\%$. We thus anticipate that high-$Q$ cavities
could be studied with nanometer-scale resolution in a way similar to
waveguides Henke  _et al._ (2021), but with higher flexibility regarding the
type of specimen and arrangement.

To validate this hypothesis, we examined a larger PS sphere of $8~{}\mu$m in
diameter in search for high-$Q$ resonances. In the one considered in Figure 4,
EEGS measurements with a laser power of $\sim 1~{}$mW revealed quality factors
as high as $10^{4}$ (FWHM of $194~{}\mu$eV). In the leftmost wavelength series
of Figure 4A, the laser step was fixed at 50 pm, scanning a relatively broad
energy range $\sim 88.34~{}$meV from 580 nm to 605 nm. After identifying a
sharp resonance close to 592.6 nm, the energy range and the laser step were
reduced to the limit of our laser, rastering a wavelength range of 0.7 nm in
steps of 2 pm (i.e., a spectral step of approximately $7~{}\mu$eV). The yellow
dashed rectangle illustrates the energy range used for the sequential
acquisitions. This series of measurements demonstrates the potential of EEGS
to map a significant range of resonance linewidths by adapting the laser
wavelength scan range and the spectral step. Interestingly, the acquisition of
each wavelength series took $\sim 8~{}$min, showing remarkable repeatability
of the experiments, with minor changes between them. The rightmost series of
Figure 4A is integrated along the electron energy axis for each laser
wavelength to produce the EEGS intensity curve shown in Figure 4B, as well as
the Lorentzian fittings for the three sharpest resonances, yielding quality
factors of 7430, 10541, and 9178. The peak separation is $\sim 300~{}\mu$eV
apart, and the highest-$Q$ mode has a FWHM of $194~{}\mu$eV.

We remark that EELS cannot resolve such fine features due to its limited
spectral resolution. Also, although CL could in principle be performed with a
sufficiently accurate light spectrometer, it cannot resolve high-$Q$ features
in practice because of its smaller signal count rate. Indeed, the ratio of
integrated CL and EEGS probabilities scales as $\Gamma_{\rm CL}/\Gamma_{\rm
EEGS}\sim 1/Q$ when measuring a mode of quality factor $Q$ (see detailed
derivation in the Appendix). In this work, the illumination intensity is $\sim
10^{8}~{}$W/m2 and the resonance energy $\sim 2~{}$eV, and hence, we have
$\Gamma_{\rm CL}/\Gamma_{\rm EEGS}=2.5$ for $Q=200$, thus explaining why CL
can resolve the WGMs in the smaller silica spheres discussed in Figure 2. In
contrast, $\Gamma_{\rm CL}/\Gamma_{\rm EEGS}=0.05$ for $Q=10^{4}$, therefore
yielding an undetectable CL signal in the larger spheres. This constitutes a
compelling argument supporting the superior signal-to-noise ratio of EEGS
relative to CL. It is worth mentioning that the same arguments limit the
sensitivity of EEGS when it is performed with a laser of small spectral
resolution compared to the WGM linewidth (e.g., when using femtosecond light
pulses, in which most of the injected photons lie outside the resonance, thus
resulting in a dramatic loss of coupling efficiency).

##  III CONCLUSIONS

The present work demonstrates EEGS with nanometer spatial resolution and down
to $200~{}\mu$eV spectral resolution on arbitrary optical dielectric cavities,
therefore leveraging the spatial resolution of free electrons, the versatility
of electron microscopy, and the spectral resolution of laser light sources.
Our results are made possible by using a small laser bandwidth compared with
the widths of the probed modes, as well as by adapting the symmetry, size, and
shape of the laser beam to that of the excitations in those cavities, all in a
free-space configuration. More general strategies for laser-to-cavity mode
matching could rely on light beams sculpted in amplitude and phase by slide
projection (e.g., through spatial light modulators). This versatility holds
potential for imaging at the ultimate limits of resolution, as required for
the characterization of vanguard photonic structures. It should be noted that
EEGS is not limited to the visible spectrum, but it should be useful to probe
mid- Kurman  _et al._ (2021) and far-infrared modes, going well beyond the
spectral resolution reached by electron monochromation, which is now reaching
the limits prophesied by their creators Krivanek  _et al._ (2009).
Applications in vibration mapping at such high resolution should directly
impact biological applications Rez  _et al._ (2016). Fast beam blanking
technologies are also becoming increasingly available in distinct operation
frequencies and duty cycles Verhoeven  _et al._ (2018); Zhang  _et al._
(2019), enabling this experiment to be implemented in different microscope
configurations. In addition, beyond the current design, energy-gain
experiments using pulsed laser sources could be performed without e-beam
blankers by relying on time-resolved electron detectors Auad  _et al._
(2022b). The use of nanosecond-based time-resolved detectors could further
increase the spectroscopic portfolio in electron microscopy by means of
temporal correlation between photons and electrons, as recently demonstrated
in experiment Varkentina  _et al._ (2022); Feist  _et al._ (2022). The
integration of such new techniques holds promise for a thriving future in
free-electron-based nanooptics.

##  Appendix A Experimental setup for continuous-wave EEGS spectroscopy

###  A.1 Light detection

For the CL experiments, light detection was performed with an Attolight Mönch
system fitted with a $\approx$ 0.5 numerical aperture. The light collected
through the mirror was focused into a single-step-index multimode fiber of 100
$\mu$m in diameter (model FG105LVA Thorlabs Inc.) coupled to an optical
spectrometer of similar numerical aperture, roughly 0.1. The fiber
transmission efficiency was more than 95% within the entire spectral range
used throughout this work.

###  A.2 Laser source and laser injection parameters

For light injection, a dye laser beam (Pyromethane 597 organic solution, with
peak efficiency at 585 nm) was focused on the sample by means of a high-
numerical-aperture parabolic reflector after being spatially filtered by a
single-mode optical fiber. The focused light beam, of approximately $\sim$ 1
$\mu$m in diameter (see Sec. B.2 for a numerical analysis), could be precisely
positioned at the edge of a whispering-gallery mode resonator (WGMR; a large
dielectric sphere) thanks to a Mönch three-dimensional stage with an ultimate
50 nm precision. The pulsed dye laser operated at a repetition rate of 10 kHz
and delivered pulses with a duration in the 20–30 ns range. In contrast to
conventional PINEM experiments in ultrafast electron microscopes, the electron
and photon temporal synchronization was carried out by means of a fast blanker
placed immediately before the detector, as shown in Fig. 1 in the main text
and discussed elsewhere in more detail Das  _et al._ (2019). The nanosecond-
resolved beam blanker selected electrons that crossed the sample during the
laser pulse duration, with the beam blanker time window fixed at 250 ns in
order to accommodate a slight variation of the laser pulse delay on the pump
laser current. Notice that only a tenth of the transmitted electrons were
exposed to the laser action. Typical time-averaged input powers were in the
range of 1–5 mW, thus resulting in 108-9 W/m2 light intensities. The
acquisition time per EELS spectrum in the wavelength series was typically 100
ms. Over this time interval, a set of 5–25 measurements were performed (and
subsequently averaged) per wavelength. A direct electron detector (Merlin,
from Quantum Detectors) was used, which improved the quantum efficiency and
the signal-to-noise ratio of the acquired data. The laser pulse duty cycle
(i.e., the product of its pulse length and its repetition rate) was $2\times
10^{-4}$, implying that the average current available for generating an EEGS
signal was typically $2\times 10^{-16}$ A.

Figure 5: Scheme of the optical components used in our EEGS experiment. The
laser output is focused on a gradient wheel density filter. The telescopic
lenses L1 and L2 produce an output with unity magnification. The beam splitter
BS1 is used to collect 10% of the incident power in the power meter PM1.
Finally, the laser beam is focused on the optical fiber. The other side of the
output fiber is coupled to L4. M2 and BS2 are used to send the light beam to
the parabolic mirror, focusing the beam on the sample plane (dashed line). The
boundary between the atmosphere and vacuum (inside the microscope) is
represented by the microscope edge ME. L1, L2: achromatic lenses; L3, L4:
achromatic aspheric lenses; BS1, BS2: 90:10 beamsplitter; M1, M2: silver
mirror; PM1, PM2: power meters.

###  A.3 Details on the optical setup used to perform EEGS experiments

The elements of the EEGS experimental setup are shown in Fig. 5. The optical
setup consists of two main parts: the coupling of the dye-laser light beam
into the optical fiber (Fig. 5a) and the other side of the optical fiber
attached to the microscope light injection (or collection) system (Fig. 5b).
The achromatic lenses L1 and L2 in Fig. 5a, which have the same focal point
and, thus, form a telescopic imaging system with unity magnification, are used
to produce an optical cross-over in which a continuous neutral density
gradient wheel is placed. The latter is linked to a servo motor by a
mechanical belt and two homemade 3D-printed rotating shafts. Additionally, an
Arduino UNO is used to control the servo motor through the microcontroller
standard library. The beamsplitter cube BS1 reflects 90% of the incident
light. The remaining 10% is sent to the power meter PM1 (Thorlabs S120C sensor
with Thorlabs PM100USB interface). The PM1 element is mostly used to determine
the transmission percentage across the optical fiber, which is helpful during
alignment and allow us to estimate the reminiscent chromatic aberration of the
optical system. The silver mirror M1 and the achromatic aspheric lens L3 are
used to focus the laser beam onto the fiber. In M1, there are two angular
degrees of freedom controlled by piezoelectric actuators (Thorlabs PIAK10),
while L3 is placed on a lens mount with 5 degrees of freedom (3 spatial and 2
angular). The fiber ends at a standard fiber SMA connector and is mounted on a
3-axis Thorlabs NanoMaxTM flexure stage using manual differential micrometers.
Alignment is mainly carried out by playing with M1 and the fiber 3-axis stage.
For a single-mode fiber, typical transmission values are $\sim$ 4-10% across
the Pyrromethene dye range (566–611 nm wavelength), although these values are
properly corrected by the gradient-wheel-based control system. To estimate the
focusing power of the parabolic mirror, the field of view of the single-mode
optical fiber is directly measured by rastering a $<100$ nm diamond crystal
containing multiple nitrogen-vacancy (NV) centers, which are known to
efficiently emit light in the wavelength range of the dye laser used, thus
reducing the effect of chromatic aberration. As shown in Fig. 5c, when moving
an NG center by approximately 0.8 $\mu$m away from the optical axis of the
electron microscope, the integrated collected signal is reduced by a factor of
two.

##  Appendix B Analytical theory for the EEGS probability

The interaction of a swift electron with a spherical object has previously
been studied using analytical theory methods to calculate the EELS and
cathodoluminescence (CL) emission probabilities García de Abajo (2010), which
should actually coincide when absorption by the probed materials is negligible
(e.g., for the spheres considered in the present work). We derive below an
analytical expression for the EEGS probability due to the interaction with an
illuminated sphere by relating it to the CL far-field amplitude via the
reciprocity theorem.

Considering a specimen subject to external monochromatic illumination of
frequency $\omega$, the effect of the optical field on an electron moving with
constant velocity $v$ along the $z$ direction is encapsulated in the coupling
parameter García de Abajo and Di Giulio (2021)

$\displaystyle\beta_{\rm
EEGS}(\omega)=\frac{e}{\hbar\omega}\int_{-\infty}^{\infty}dz\,E_{z}(x_{e},y_{e},z)\,{\rm
e}^{-{\rm i}\omega z/v},$ (2)

where $(x_{e},y_{e})$ defines the electron beam (e-beam) position in the
transverse plane, while the amplitude ${\bf E}({\bf r})$ is taken such that
the time-dependent electric field reads $2{\rm Re}\big{\\{}{\bf E}({\bf
r}){\rm e}^{-{\rm i}\omega t}\big{\\}}$. We explicitly indicate the frequency
dependence of $\beta_{\rm EEGS}(\omega)$ because this is a key ingredient in
EEGS. After interaction, the incident zero-loss peak ($\ell=0$) splits into a
series of peaks separated by multiples of the photon energy $\ell\hbar\omega$,
with associated probabilities $P_{\ell}(\omega)=J_{\ell}^{2}(2|\beta_{\rm
EEGS}(\omega)|)$ satisfying $\sum_{\ell=-\infty}^{\infty}P_{\ell}(\omega)=1$.
In the present study, the coupling is relatively weak ($|\beta_{\rm
EEGS}(\omega)|\ll 1$), and therefore, only the first gain ($\ell=1$) and loss
($\ell=-1$) peaks are experimentally studied in the transmitted electron
spectra, both of them having the same probability
$J_{1}^{2}\big{[}2|\beta_{\rm EEGS}(\omega)|\big{]}\approx|\beta_{\rm
EEGS}(\omega)|^{2}$.

Figure 6: Sketch of the geometry involved in the application of the
reciprocity theorem. We relate the EEGS coupling coefficient $\beta_{\rm
EEGS}(\omega)$ (red) to the CL far-field amplitude (blue) by means of Eq. (3).
The propagation directions of electron and light are both reversed in CL
relative to EEGS.

Rather than calculating $\beta_{\rm EEGS}(\omega)$ by first obtaining the
total (incident+scattered) field ${\bf E}$ and then integrating Eq. (2), we
exploit the reciprocity theorem to relate $\beta_{\rm EEGS}(\omega)$ to the CL
far-field amplitude ${\bf f}^{\rm CL}(\omega)$ as García de Abajo and Di
Giulio (2021)

$\displaystyle\beta_{\rm EEGS}(\omega)=\frac{{\rm
i}c^{2}}{\hbar\omega^{2}}{\bf f}^{\rm CL}_{\hat{\bf r}}(\omega)\cdot{\bf
E}^{\rm ext},$ (3)

where ${\bf E}^{\rm ext}$ is the EEGS incident electric field amplitude in the
absence of the specimen. In this expression, ${\bf f}^{\rm CL}(\omega)$ needs
to be computed after reversing the electron velocity vector relative to EEGS,
and with the CL emission direction pointing to the EEGS light source, as
indicated in Fig. 6. In particular, the CL emission direction is $\hat{\bf
r}$, but the propagation direction of the EEGS incident light is $-\hat{\bf
r}$ (see Fig. 6).

###  B.1 Spherical specimen under plane wave illumination

We now take a previously derived analytical expression for the CL far-field
amplitude of a sphere in vacuum García de Abajo (1999):

$\displaystyle{\bf f}^{\rm CL}_{\hat{\bf
r}}(\omega)=\frac{c}{\omega}\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\bigg{[}t_{l}^{M}\psi_{lm}^{M,{\rm
ext}}\;\vec{\zeta}_{lm}(\hat{\bf r})\,+t_{l}^{E}\psi_{lm}^{E,{\rm
ext}}\;\hat{\bf r}\times\vec{\zeta}_{lm}(\hat{\bf r})\,\bigg{]},$ (4)

where $l$ and $m$ run over multipolar components, $t_{l}^{\nu}$ are the
$m$-independent Mie scattering coefficients for electric ($\nu=$E) and
magnetic ($\nu=$M) polarization, and $\psi_{lm}^{\nu,{\rm ext}}$ are expansion
coefficients that express the external evanescent field carried by the
electron as a sum over multipoles around the sphere center. In Eq. (4), we use
the vector spherical harmonics

$\displaystyle\vec{\zeta}_{lm}(\hat{\bf
r})=\frac{1}{2}\big{[}C_{lm}^{-}Y_{lm-1}(\hat{\bf
r})+C_{lm}^{+}Y_{lm+1}(\hat{\bf r})\big{]}\,\hat{\bf x}\;+\frac{{\rm
i}}{2}\big{[}C_{lm}^{-}Y_{lm-1}(\hat{\bf r})-C_{lm}^{+}Y_{lm+1}(\hat{\bf
r})\big{]}\,\hat{\bf y}\;+mY_{lm}(\hat{\bf r})\,\hat{\bf z},$

where $C_{lm}^{\pm}=\sqrt{(l\pm m+1)(l\mp m)}$. Also, we consider the Mie
scattering coefficients for a self-standing homogenous sphere of radius $a$
and permittivity $\epsilon_{d}$ centered at the origin, which are given by

$\displaystyle t_{l}^{M}$
$\displaystyle=\frac{-j_{l}(\rho_{0})\rho_{1}j_{l}^{\prime}(\rho_{1})+\rho_{0}j_{l}^{\prime}(\rho_{0})j_{l}(\rho_{1})}{h_{l}^{(+)}(\rho_{0})\rho_{1}j_{l}^{\prime}(\rho_{1})-\rho_{0}[h_{l}^{(+)}(\rho_{0})]^{\prime}j_{l}(\rho_{1})},$
$\displaystyle t_{l}^{E}$
$\displaystyle=\frac{-j_{l}(\rho_{0})\tilde{j}_{l}(\rho_{1})^{\prime}+\epsilon_{d}\tilde{j}_{l}(\rho_{0})^{\prime}j_{l}(\rho_{1})}{h_{l}^{(+)}(\rho_{0})\tilde{j}_{l}(\rho_{1})^{\prime}-\epsilon_{d}\tilde{h}_{l}^{(+)}(\rho_{0})^{\prime}j_{l}(\rho_{1})},$

where $\rho_{0}=\omega a/c$, $\rho_{1}=\sqrt{\epsilon_{d}}\omega a/c$, $j_{l}$
and $h_{l}^{(+)}$ are spherical Bessel and Hankel functions Messiah (1966), we
have defined the functions $\tilde{j}_{l}(\rho)=\rho\,j_{l}(\rho)$ and
$\tilde{h}_{l}^{(+)}(\rho)=\rho\,h_{l}^{(+)}(\rho)$, and the prime denotes
differentiation with respect to the argument. Finally, the incident field
supplied by the electron is represented through the coefficients García de
Abajo (2000)

$\displaystyle\left[\begin{array}[]{c}\psi_{lm}^{M,{\rm ext}}\\\ \\\
\psi_{lm}^{E,{\rm ext}}\end{array}\right]=\frac{-2\pi\,e\,{\rm
i}^{1-l}\omega}{l(l+1)\,c^{2}}\,K_{m}\bigg{[}\frac{\omega
R_{e}}{v\gamma}\bigg{]}\;{\rm e}^{-{\rm
i}m\varphi_{e}}\,\times\left[\begin{array}[]{c}2mA_{lm}v/c\\\ \\\
B_{lm}/\gamma\end{array}\right],$ (11)

where $(R_{e},\varphi_{e})$ are the polar coordinates of the electron
transverse vector $(x_{e},y_{e})$, $K_{m}$ is the modified Bessel function of
order $m$, $\gamma=1/\sqrt{1-(v/c)^{2}}$ is the Lorentz factor, and the
coefficients $A_{lm}$ and $B_{lm}$ only depend on the normalized electron
velocity $v/c$. More precisely Winther and Alder (1979),

$\displaystyle A_{lm}$ $\displaystyle={\rm
i}^{l+m}(2m-1)!!\,\sqrt{\dfrac{(2l+1)}{\pi}\dfrac{(l-m)!}{(l+m)!}}\,\dfrac{(c/v)^{m+1}}{\gamma^{m}}\,C_{l-m}^{m+1/2}(c/v),\;\;\;\;\;m\geq
0,$ $\displaystyle B_{lm}$
$\displaystyle=A_{l,m+1}\sqrt{(l+m+1)(l-m)}\,-\,A_{l,m-1}\sqrt{(l-m+1)(l+m)},$

and $C^{\mu}_{\nu}$ are Gegenbauer polynomials Abramowitz and Stegun (1972).
We use the property $A_{l,-m}=(-1)^{m}A_{lm}$ to calculate the $A_{lm}$
coefficients for negative $m$.

These results allow us to calculate the EEGS probability $|\beta_{\rm
EEGS}(\omega)|^{2}$ for an electron moving downwards and passing close to a
homogeneous sphere under plane-wave light irradiation from Eq. (3) by using
the CL amplitude ${\bf f}_{\hat{\bf r}}^{\rm CL}(\omega)$ given by Eq. (4) for
an electron moving along the positive $z$ axis (see Fig. 6).

###  B.2 Illumination by means of a parabolic mirror

Illumination of the specimen in the microscope is introduced through a
parabolic aluminum mirror, producing a focal spot that can be regarded as a
superposition of plane waves resulting from reflection of the incident laser
on different regions of the metal surface. We model the system by describing
the reflection from each mirror surface position ${\bf s}$ through the Fresnel
coefficients of a tangent planar mirror at that position.

Figure 7: Illumination configuration used in the present work. We assume light
incident from the left and propagating towards the positive $x$ direction, the
e-beam is injected through a hole and moves downwards along the negative $z$
direction, and the specimen is placed near the focal spot at ${\bf r}=0$. The
mirror surface is defined by the equation $x=p/2-(y^{2}+z^{2})/2p$. In the
present experiment, we have $p=2\,$mm, and the mirror surface is limited by
the conditions $x>x_{0}=-5\,$mm and $z_{0}=0.3\,$mm $<z<z_{1}=2.3\,$mm. The
hole is ignored in our calculations. In this work, we consider a laser field
amplitude ${\bf E}_{0}$ oriented along either the $y$ or $z$ directions.

In what follows, we consider the e-beam to move along the negative $z$
direction and the parabolic mirror to be illuminated by a laser plane wave
propagating along the positive $x$ direction, as illustrated in Fig. 7. To
obtain the field reaching the specimen region after reflection by the mirror,
we assimilate the response of each surface element to effective ${\bf
s}$-dependent electric and magnetic dipole densities ${\bf p}_{\bf s}$ and
${\bf m}_{\bf s}$, the magnitudes of which are obtained in the limit of a
planar mirror, as we mentioned above.

We now analyze the equivalent dipole sources produced by reflection on a
planar mirror. More precisely, for an incident light plane wave ${\bf
E}^{i}{\rm e}^{{\rm i}{\bf k}^{i}\cdot{\bf r}}$ of wave vector ${\bf k}^{i}$
and electric field amplitude ${\bf E}^{i}$, the field reflected by a planar
mirror that contains the origin ${\bf r}=0$ and has an outer normal unit
vector $\hat{\bf n}$ is given by ${\bf E}^{r}{\rm e}^{{\rm i}{\bf
k}^{r}\cdot{\bf r}}$, where ${\bf k}^{r}={\bf k}^{i}-2({\bf
k}^{i}\cdot\hat{\bf n})\hat{\bf n}$ is the reflected wave vector, and ${\bf
E}^{r}=r_{\rm s}({\bf E}^{i}\cdot\hat{\bf e}_{s})\,\hat{\bf e}_{s}+r_{\rm
p}({\bf E}^{i}\cdot\hat{\bf e}_{p}^{i})\,\hat{\bf e}_{p}^{r}$ is the reflected
field amplitude written in terms of the Fresnel coefficients $r_{\rm s}$ and
$r_{\rm p}$ for $s$ and $p$ polarization. Here, $\hat{\bf e}_{s}$ and
$\hat{\bf e}_{p}^{\nu}$ are the corresponding unit polarization vectors. We
note that the $p$-polarization vector $\hat{\bf e}_{p}^{\nu}$ is different for
incidence ($\nu=i$) and reflection ($\nu=r$) directions (see below).
Translational invariance of the planar mirror allows us to incorporate the
${\bf s}$ dependence of the effective dipoles through ${\bf p}_{\bf s}={\bf
p}\,{\rm e}^{{\rm i}{\bf k}_{\parallel}\cdot{\bf s}}$ and ${\bf m}_{\bf
s}={\bf m}\,{\rm e}^{{\rm i}{\bf k}_{\parallel}\cdot{\bf s}}$, where ${\bf
k}_{\parallel}={\bf k}^{i}-({\bf k}^{i}\cdot\hat{\bf n})\hat{\bf n}$ is the
in-plane component of the incident wave vector. In terms of these effective
sources, and assuming a permittivity $\epsilon_{0}$ ($=1$ in the present work)
in the medium outside the mirror, the reflected field becomes

$\displaystyle\int d^{2}{\bf s}\;\Big{[}k^{2}\,{\bf p}_{\bf
s}+\frac{1}{\epsilon_{0}}({\bf p}_{\bf s}\cdot\nabla_{\bf r})\nabla_{\bf
r}-{\rm i}k\,{\bf m}_{\bf s}\times\nabla_{\bf r}\Big{]}\,\dfrac{{\rm e}^{{\rm
i}k\sqrt{\epsilon_{0}}|{\bf r}-{\bf s}|}}{|{\bf r}-{\bf s}|}=\frac{2\pi{\rm
i}k^{2}}{k_{\perp}}\,{\rm e}^{{\rm i}{\bf k}^{r}\cdot{\bf r}}\,\Big{[}{\bf
p}-({\bf p}\cdot\hat{{\bf k}}^{r})\hat{{\bf k}}^{r}+\sqrt{\epsilon_{0}}\;{\bf
m}\times\hat{{\bf k}}^{r}\Big{]},$ (12)

where $k=\omega/c$, $k_{\perp}=-{\bf k}^{i}\cdot\hat{\bf n}$, and the ${\bf
s}$ integral is extended over the plane normal to $\hat{\bf n}$. Comparing
this result with the reflected field ${\bf E}^{r}{\rm e}^{{\rm i}{\bf
k}^{r}\cdot{\bf r}}$ and noticing that ${\bf E}^{r}\cdot{\bf k}^{r}=0$, we
find that we can take ${\bf m}=0$ and ${\bf p}=(-{\rm i}k_{\perp}/2\pi
k^{2})\,{\bf E}^{r}$. Finally, inserting the expression given above for ${\bf
E}^{r}$ in terms of the Fresnel coefficients, we have ${\bf p}=(-{\rm
i}k_{\perp}/2\pi k^{2})\,\big{[}r_{\rm s}({\bf E}^{i}\cdot\hat{\bf
e}_{s})\,\hat{\bf e}_{s}+r_{\rm p}({\bf E}^{i}\cdot\hat{\bf
e}_{p}^{i})\,\hat{\bf e}_{p}^{r}\big{]}$.

For a parabolic mirror defined by $x=p/2-(y^{2}+z^{2})/2p$ (focal distance
$p/2$), as shown in Fig. 7, we define the coordinates ${\bf r}_{\perp}=(y,z)$
in the transverse plane relative to the rotation axis $x$ and write the
elements involved in the calculation of the reflected field at each position
${\bf s}$ as

$\displaystyle\hat{\bf n}=-({\bf r}_{\perp}+p\hat{\bf
x})/\sqrt{r_{\perp}^{2}+p^{2}},$ (13a) $\displaystyle\hat{{\bf
k}}^{i}=\hat{\bf x},$ (13b) $\displaystyle\hat{{\bf
k}}^{r}=[(r_{\perp}^{2}-p^{2})\hat{\bf x}-2p{\bf
r}_{\perp})]/(r_{\perp}^{2}+p^{2}),$ (13c) $\displaystyle\hat{\bf
e}_{s}=(-z\hat{\bf y}+y\hat{\bf z})/r_{\perp},$ (13d) $\displaystyle\hat{\bf
e}_{p}^{i}=\hat{{\bf r}}_{\perp},$ (13e) $\displaystyle\hat{\bf
e}_{p}^{r}=[(r_{\perp}^{2}-p^{2}){\bf r}_{\perp}+2pr_{\perp}\hat{\bf
x})]/(r_{\perp}^{2}+p^{2}),$ (13f)

and ${\bf k}^{\nu}=k\sqrt{\epsilon_{0}}\,\hat{{\bf k}}^{\nu}$. In addition,
the normal wave vector becomes
$k_{\perp}=k\sqrt{\epsilon_{0}}\,p/\sqrt{r_{\perp}^{2}+p^{2}}$. Using these
elements, we can calculate the electric field ${\bf E}({\bf r})$ produced by
the mirror at a position ${\bf r}$ near the focal point (${\bf r}=0$) from an
expression similar to the left-hand side of Eq. (12), but now with ${\bf s}$
integrated over the surface of the curved mirror. More precisely,

$\displaystyle{\bf E}({\bf r}_{f})$ $\displaystyle=\int_{\rm
mirror}\\!\\!\\!\\!\\!\\!\\!\\!d^{2}{\bf s}\;\Big{[}k^{2}\,{\bf p}_{\bf
s}+\frac{1}{\epsilon_{0}}({\bf p}_{\bf s}\cdot\nabla_{{\bf
r}_{f}})\nabla_{{\bf r}_{f}}\Big{]}\,\dfrac{{\rm e}^{{\rm
i}k\sqrt{\epsilon_{0}}|{\bf r}_{f}-{\bf s}|}}{|{\bf r}_{f}-{\bf s}|}$ (14a)
$\displaystyle\approx{\rm e}^{{\rm i}kp\sqrt{\epsilon_{0}}}\int_{\rm
mirror}\\!\\!\\!\\!\\!\\!\\!\\!d^{2}{\bf s}\;\Big{[}k^{2}\,{\bf p}_{\bf
s}-\frac{1}{\epsilon_{0}}({\bf p}_{\bf s}\cdot{\bf k}^{r}){\bf
k}^{r}\Big{]}\,\dfrac{{\rm e}^{{\rm i}{\bf k}^{r}\cdot{\bf r}_{f}}}{s}$ (14b)
$\displaystyle=-{\rm e}^{{\rm i}kp\sqrt{\epsilon_{0}}}\,\frac{{\rm
i}}{2\pi}\int_{\rm mirror}\\!\\!\\!\\!\\!\\!\\!\\!d^{2}{\bf
s}\;\frac{k_{\perp}}{s}\,{\rm e}^{{\rm i}{\bf k}^{r}\cdot{\bf
r}_{f}}\,\big{[}r_{\rm s}({\bf E}_{0}\cdot\hat{\bf e}_{s})\,\hat{\bf
e}_{s}+r_{\rm p}({\bf E}_{0}\cdot\hat{\bf e}_{p}^{i})\,\hat{\bf
e}_{p}^{r}\big{]}$ (14c) $\displaystyle=-{\rm e}^{{\rm
i}kp\sqrt{\epsilon_{0}}}\,\frac{{\rm i}k\sqrt{\epsilon_{0}}}{2\pi}\int_{\rm
mirror}\\!\\!\\!\\!\\!d{\bf r}_{\perp}\;\frac{1}{s}\,{\rm e}^{{\rm i}{\bf
k}^{r}\cdot{\bf r}_{f}}\,\big{[}r_{\rm s}({\bf E}_{0}\cdot\hat{\bf
e}_{s})\,\hat{\bf e}_{s}+r_{\rm p}({\bf E}_{0}\cdot\hat{\bf
e}_{p}^{i})\,\hat{\bf e}_{p}^{r}\big{]},$ (14d)

where (14b) is obtained from (14a) by adopting the $kp\gg 1$ limit (i.e.,
considering that the focal distance $p/2$ is large compared with the light
wavelength) and using the relation $\hat{{\bf k}}^{r}=-\hat{{\bf s}}$ to
connect the reflected wave vector to the direction of the surface position
${\bf s}$ (as seen from the focus) in the integral. In doing so, we are
assuming an incident plane wave ${\bf E}_{0}\,{\rm e}^{{\rm i}kx}$
illuminating the mirror, such that the sum of light propagation distances from
the $x=0$ plane to the mirror surface and from this to the focal point amounts
to a path length given by $p$ and emerging as an irrelevant global phase
factor in Eq. (14b). Then, Eq. (14c) is obtained by noticing that ${\bf
p}_{\bf s}\cdot{\bf k}^{r}=0$ because ${\bf p}_{\bf s}\propto{\bf E}^{r}$ and
the reflected field must be transverse. In addition, we insert the explicit
expression for ${\bf p}$ derived above for the planar mirror in terms of the
Fresnel coefficients, with ${\bf E}^{i}$ substituted by ${\bf E}_{0}$. We note
that the ${\bf s}$-dependent phase in ${\bf E}^{i}$ is already included in the
global phase factor ${\rm e}^{{\rm i}kp\sqrt{\epsilon_{0}}}$. Finally, Eq.
(14d) results from a change of integration variables ${\bf s}\to{\bf
r}_{\perp}$ after noticing that the corresponding Jacobian times $k_{\perp}$
reduces to $k\sqrt{\epsilon_{0}}$. The electric field in the focal region is
then given by Eq. (14d) using the quantities defined in Eqs. (13) in terms of
${\bf r}_{\perp}=(y,z)$ as well as $s=(p^{2}+r_{\perp}^{2})/2p$. Also, we use
the Fresnel coefficients

$\displaystyle r_{\rm
s}=(k_{\perp}-k^{\prime}_{\perp})/(k_{\perp}+k^{\prime}_{\perp}),$ (15a)
$\displaystyle r_{\rm
p}=(\epsilon_{m}k_{\perp}-\epsilon_{0}k^{\prime}_{\perp})/(\epsilon_{m}k_{\perp}+\epsilon_{0}k^{\prime}_{\perp}),$
(15b)

where
$k^{\prime}_{\perp}=\sqrt{k^{2}(\epsilon_{m}-\epsilon_{0})+k_{\perp}^{2}}$ and
$\epsilon_{m}$ is the metal permittivity.

Figure 8: Focal spot produced in the $z=0$ plane by the mirror in Fig. 7. We
consider incident light polarization along either the $y$ (left) or the $z$
(right) directions. The electric field intensity is normalized to the maximum
$|E_{\rm max}|^{2}$, with values of the latter relative to the incident field
intensity $|E_{0}|^{2}$ also indicated. The calculation is made for an
aluminum mirror at a photon energy of 2 eV (i.e., a wavelength
$\lambda=620\,$nm). Similar results are obtained when assimilating the metal
to a perfect conductor, so these results describe well the focal spot at
frequencies up to the visible range. The mirror has a projected area on the
$yz$ plane of 18.74 mm${}^{2}=4.87\times 10^{7}\,\lambda^{2}$. The effective
projected area including absorption in aluminum is 16.51 mm2 and 17.35 mm2 for
${\bf E}_{0}\parallel\hat{\bf y}$ and ${\bf E}_{0}\parallel\hat{\bf z}$,
respectively.

In our experimental setup, we have an aluminum mirror ($\epsilon_{m}(\omega)$
taken from optical data Palik (1985)) in vacuum ($\epsilon_{0}=1$) with a
focal distance of 1 mm (i.e., $p=2\,$mm) and the metal surface limited by the
conditions stated in Fig. 7. We adopt these parameters throughout the present
paper. Then, the focal spot size is about 1$\,\mu$m, with the actual shape
depending on the polarization of the incident light, as shown in Fig. 8.

###  B.3 EEGS under illumination by a parabolic mirror

Like the focal spot, the electron–light coupling coefficient $\beta_{\rm
EEGS}(\omega)$ can also be regarded as a superposition of contributions due to
different reflected plane waves. Using Eq. (3) for the contribution of each of
these waves, integrating over the mirror surface just like in Eq. (14d), and
ignoring any global phase, we find

$\displaystyle\beta_{\rm EEGS}(\omega)=\frac{c}{\pi\hbar\omega}\int_{\rm
mirror}\\!\\!\\!\\!\\!d{\bf r}_{\perp}\;$
$\displaystyle\frac{p}{p^{2}+r_{\perp}^{2}}\;{\bf f}^{\rm CL}_{\hat{{\bf
s}}}(\omega)\cdot\big{[}r_{\rm s}({\bf E}_{0}\cdot\hat{\bf e}_{s})\,\hat{\bf
e}_{s}+r_{\rm p}({\bf E}_{0}\cdot\hat{\bf e}_{p}^{i})\,\hat{\bf
e}_{p}^{r}\big{]}$ (16a)
$\displaystyle=\frac{c}{\pi\hbar\omega}\int_{z_{0}}^{z_{1}}dz\int_{-y_{0}(z)}^{y_{0}(z)}dy\;$
$\displaystyle\frac{p}{p^{2}+r_{\perp}^{2}}\;{\bf f}^{\rm CL}_{\hat{{\bf
s}}}(\omega)\cdot\big{[}r_{\rm s}({\bf E}_{0}\cdot\hat{\bf e}_{s})\,\hat{\bf
e}_{s}+r_{\rm p}({\bf E}_{0}\cdot\hat{\bf e}_{p}^{i})\,\hat{\bf
e}_{p}^{r}\big{]},$ (16b)

involving $\hat{{\bf s}}=-\hat{{\bf k}}^{r}$ as well as different quantities
defined in Eqs. (13) and (15). We remark that Eq. (16a) is general for any
parabolic mirror geometry, while Eq. (16b) is specialized to our mirror
dimensions, as described in Fig. 7, such that the limits of integration are
$z_{0}=0.3\,$mm, $z_{1}=2.3\,$mm, and $y_{0}(z)=\sqrt{p^{2}-2px_{0}-z^{2}}$
with $x_{0}=-5\,$mm and $p=2\,$mm. Also, Eqs. (16) can be applied to any
specimen by plugging the corresponding CL amplitude ${\bf f}^{\rm
CL}_{\hat{{\bf s}}}(\omega)$. In the present study, we calculate the latter
for a sphere by means of Eq. (4).

Figure 9: Extinction cross-section spectrum of a large SiO2 sphere. (a) Sketch
of the sphere with a diameter $2a=4122\,$nm and a permittivity
$\epsilon_{d}\approx 2.13$. (b) Total optical extinction cross section
$\sigma^{\rm ext}$ (black curve) normalized to the geometrical cross section
$\pi a^{2}$, along with the corresponding partial contributions due to
electric (red) and magnetic (blue) modes, as a function of light wavelength.
The vertical dashed lines indicate the two resonance wavelengths observed in
the experimental results of Fig. 2 in the main text. (c) Partial extinction
cross sections within the shaded region of panel (b), allowing us to identify
two electric modes with $l=26$ and $27$ (red), as well as a magnetic mode with
$l=27$ (blue). The vertical dashed green lines are the same as in panel (b).
(d) Total extinction cross section due to electric modes (top curve) and
partial contributions from different multipolar orders $l$ (see labels on the
right axis). A vertical offset between different curves is introduced for
clarity. (e) Same as panel (d), but for magnetic modes. Figure 10: Extinction
cross-section spectrum of a large polystyrene (PS) sphere. (a) Sketch of the
sphere with a diameter $2a=8000\,$nm and a permittivity $\epsilon_{d}\approx
2.53\times(1+{\rm i}\eta)$, where losses are phenomenologically introduced
through a small imaginary part $\eta=10^{-4}$ in the multiplying factor. (b)
Total optical extinction cross section $\sigma^{\rm ext}$ (black curve)
normalized to the geometrical cross section $\pi a^{2}$, along with the
corresponding partial contributions due to electric (red) and magnetic (blue)
modes, as a function of light wavelength. The vertical dashed lines indicate
the position of the resonances observed in the experimental measurements (see
Fig. 3 in the main text). (c) Partial cross sections contributed by magnetic
modes with $l=46,47$ (blue) and electric modes with $l=45$-47 (red), giving
rise to the resonances observed in panel (b). The vertical dashed green lines
are the same as in panel (b). (d) Quality factors of electric (red) and
magnetic (blue) Mie resonances as a function of their multipolar order $l$.
Solid and open circles are calculated with $\eta=10^{-4}$ (partially lossy PS)
and $\eta=0$ (nonlossy PS), respectively. (e) Total extinction cross section
due to electric modes (top curve) and partial contributions from different
multipolar orders $l$ (see labels on the right axis). A vertical offset
between different curves is introduced for clarity. (f) Same as panel (e), but
for magnetic modes.

##  Appendix C Mie modes in large dielectric spheres

To characterize the large dielectric spheres studied in this work, we first
calculate their optical extinction cross sections as a function of light
wavelength $\lambda=2\pi c/\omega$ from Mie theory Mie (1908) using the
expression

$\displaystyle\sigma^{\rm
ext}(\lambda)=\frac{\lambda}{2\pi\sqrt{\epsilon_{0}}}\sum_{l=1}^{\infty}(2l+1)\big{[}{\rm
Im}\\{t_{l}^{M}\\}+{\rm Im}\\{t_{l}^{E}\\}\big{]},$

where we set $\epsilon_{0}=1$ (particle in vacuum).

For the $4122\,$nm silica sphere, we fine-tune the diameter to match the two
prominent resonances observed in the EEGS/EELS/CL experiments. The results
presented in Fig. 9 show that such resonances have electric polarization with
multipolar orders $l=27$ and $26$ at wavelengths of $\sim 582\,$nm and $\sim
602\,$nm, respectively. There is an additional magnetic resonance at $\sim
593\,$nm with $l=27$, which is missed in the experiments for symmetry reasons,
as we argue below. The quality factors of these modes are $\sim 1500$,
substantially larger than those observed in experiment $\sim 200$, in which
losses at the support or due to surface contamination can affect the actual
resonances.

A larger polysterene (PS) sphere is also studied, for which we predict modes
with quality factors reaching $10^{6}$ when the particle is self-standing
(Fig. 10d). This is in contrast to the values $\sim 10^{4}$ observed in
experiment, a disagreement that could be attributed to losses originating in
the effect of the supporting film. In addition, we cannot find a good match
between the calculated Mie resonances and the three peaks reported in Fig. 4
of the main text even when fine-tuning the sphere diameter. A possible reason
for this discrepancy is that the particle (including the effect of the
support) deviates from the spherical geometry assumed in Mie theory, such that
the three observed peaks originate in the splitting of a single high-order
multipolar mode relative of the idealized sphere. We therefore search for a
possible Mie resonance in the sphere that has large optical strength in the
measured spectral range. To compare with experiment, we introduce a factor
$1+10^{-4}\,{\rm i}$ in the material permittivity, so that the maximum quality
factors of the Mie resonances are brought down to $\sim 10^{4}$ (see Fig.
10d). The resulting multipolar decomposition of this slightly lossy sphere
reveals a magnetic resonance of multipolar order $l=51$ and an electric one of
order $l=50$ near the observed features (Fig. 10c), so these are possible
candidates to explain the measurements if a small shift and splitting is
produced by the lack of sphericity or the effect of the support. Such effects
would be less relevant in the smaller silica sphere studied above because of
the larger width of its resonances.

Figure 11: EELS and EEGS in the silica sphere of Fig. 9. (a) EELS probability calculated from Eq. (26) as a function of light wavelength $2\pi c/\omega$. (b) EEGS probability $|\beta_{\rm EGGS}(\omega)|^{2}$ obtained from Eq. (16b) and normalized either to the laser intensity $I_{0}$ that is incident on the mirror described in Fig. 7 ($|\beta_{\rm EGGS}(\omega)|^{2}/I_{0}$, left vertical scale) or to the laser power $P_{0}$ ($|\beta_{\rm EGGS}(\omega)|^{2}/P_{0}$, right vertical scale, with $P_{0}=I_{0}A$, where $A=18.74\,$mm2 is the area of the mirror projection on the $yz$ plane), with the laser focal spot aimed at positions that are 57 nm inside the projected sphere contour; the EEGS probability is calculated for two different azimuthal locations of the focal spot and the e-beam (A and B in the inset, with the focal spot and the e-beam sharing the same azimuthal position in each case), it vanishes in A for incident light polarized along $y$ due to symmetry considerations, and it is multiplied by a factor of 3 in B. The orientation of the Cartesian axes is the same as in Fig. 7. (c) EEGS probability computed from Eq. (3) and normalized to the incident light intensity for plane-wave illumination with two different directions of light incidence (see color-coordinated thick arrows in the inset). Electrons have an energy of 200 keV and pass at a distance of 39 nm from the sphere surface in all cases. Figure 12: Spatial maps of the EEGS probability for the silica sphere of Fig. 9 (4122 nm diameter). We plot $|\beta_{\rm EGGS}(\omega)|^{2}$ as a function of e-beam position $(x_{e},y_{e})$ (see upper-left frame, coordinated with Fig. 7) for illumination on resonance with the three Mie modes of Fig. 9c (see the corresponding wavelengths $\lambda=2\pi c/\omega$ on the left) and 200 keV electron energy. We consider two different orientations of the incident field ${\bf E}_{0}$ referred to the mirror in Fig. 7 ($\parallel\hat{\bf y}$ or $\parallel\hat{\bf z}$, see labels on the left). Results are presented for four different positions of the laser focal spot (blue dots) relative to the sphere (white contours). The EEGS probability $|\beta_{\rm EGGS}(\omega)|^{2}$ (color log scale) is normalized to the laser intensity $I_{0}=c|E_{0}|^{2}/2\pi$ incident on the mirror (i.e., $|\beta_{\rm EGGS}(\omega)|^{2}/I_{0}$, lower scale) and the laser power $P_{0}=I_{0}A$ incident on the mirror (i.e., $|\beta_{\rm EGGS}(\omega)|^{2}/P_{0}$, upper scale), where $A=18.74\,$mm2 is the area of the mirror projection on the $yz$ plane. We only plot $|\beta_{\rm EGGS}(\omega)|^{2}$ for e-beam positions that do not intersect the particle. The focal-spot intensity at $z=0$ (taken from Fig. 8) is superimposed for each ${\bf E}_{0}$ orientation (color scale from maximum intensity (blue) down to a fraction $1/e$ of it (white contour)). Figure 13: Same as Fig. 12, but with a smaller mirror defined by $x_{0}=0.5\,$mm (see Fig. 7). The smaller mirror area ($A=2.30\,$mm2 when projected on the $yz$ plane) produces larger laser focal spots. The color scale for $|\beta_{\rm EGGS}(\omega)|^{2}/P_{0}$ runs over similar values as in Fig. 12 (i.e., we have a similar coupling to the electron for the same power incident on the mirror in each case), while the scale for $|\beta_{\rm EGGS}(\omega)|^{2}/P_{0}$ shows numbers that are roughly one order of magnitude smaller than in Fig. 12, in agreement with the ratio of projected mirror areas between both mirrors. Figure 14: Dependence of the EEGS spectra on the position of the optical focal spot. We plot the EEGS probability for the silica sphere of Fig. 9 (4122 nm diameter) as a function of laser wavelength (horizontal axes) and radial position of the laser focal spot $R_{f}$. The latter is placed along the positive $x$ axis. The sphere edge is indicated by dashed horizontal lines. The e-beam passes 50 nm outside the sphere surface, crossing either the $x$ axis in (a) or a line that forms $45^{\circ}$ with the $x$ axis in (b,c) (see upper-left insets). The laser is polarized along $z$ in (a,c) and along $y$ in (b) (see Fig. 7). The silica permittivity is supplemented with an additional imaginary part of $10^{-3}\,{\rm i}$ to slightly broaden the resonances. $\begin{matrix}\text{mode}\\\ \text{wavelength}\end{matrix}$ |  $\begin{matrix}\text{4122\,nm}\\\ \text{silica sphere}\\\ \text{grazing}\end{matrix}$ |  $\begin{matrix}\text{4122\,nm}\\\ \text{silica sphere}\\\ \text{50\,nm distance}\end{matrix}$ |  $\begin{matrix}\text{8636\,nm}\\\ \text{PS sphere}\\\ \text{grazing}\end{matrix}$ |  $\begin{matrix}\text{8636\,nm}\\\ \text{PS sphere}\\\ \text{50\,nm distance}\end{matrix}$
---|---|---|---|---
$582\,$nm |  0.0100 |  0.0031 |  - |  -
$593\,$nm |  0.0032 |  0.0096 |  - |  -
$602\,$nm |  0.0219 |  0.0070 |  - |  -
$592\,$nm |  - |  - |  $8.5\times 10^{-5}$ |  $2.8\times 10^{-5}$
$594\,$nm |  - |  - |  0.0003 |  0.0001
Table 1: CL probabilities for 200 keV electrons passing either grazingly or at
a distance of 50 nm from the surface of the 4122 nm sphere of Fig. 9 or the
8636 nm PS sphere of Fig. 10. The probabilities are integrated over the
spectral peaks at the wavelengths indicated in the left column. Incidentally,
the CL probability for the PS sphere is nearly independent on the parameter
$\eta$ used to introduce losses (see Fig. 10) when comparing $\eta=0$ and
$\eta=10^{-4}$, although the EELS probability is substantially increased with
the latter choice.

##  Appendix D EEGS in a large silica sphere

We start by calculating the EELS probability for the 4122 nm silica sphere
considered in Sec. C (Fig. 11a), which coincides with the CL emission
probability because the material is lossless. Using the elements introduced in
Sec. B.1, these probabilities are given by García de Abajo (2010)

$\displaystyle\left[\begin{array}[]{c}\Gamma_{\rm EELS}(\omega)\\\ \\\
\Gamma_{\rm
CL}(\omega)\end{array}\right]=\frac{\alpha}{\omega}\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\frac{1}{l(l+1)}K_{m}^{2}\bigg{(}\frac{\omega
R_{e}}{v\gamma}\bigg{)}\left(\frac{4m^{2}v^{2}}{c^{2}}|A_{lm}|^{2}\left[\begin{array}[]{c}{\rm
Im}\\{t_{l}^{M}\\}\\\ \\\
|t_{l}^{M}|^{2}\end{array}\right]+\frac{1}{\gamma^{2}}|B_{lm}|^{2}\left[\begin{array}[]{c}{\rm
Im}\\{t_{l}^{E}\\}\\\ \\\ |t_{l}^{E}|^{2}\end{array}\right]\right),$ (26)

such that the total frequency-integrated probability per electron is
$\int_{0}^{\infty}d\omega\,\Gamma_{\rm EELS,CL}(\omega)$. Incidentally, these
expressions are made unit-independent by using the fine structure constant
$\alpha\approx 1/137$. The calculated spectrum is dominated by the two
electric Mie modes discussed in Fig. 9c, with an additional small contribution
from the intermediate magnetic mode.

Likewise, the EEGS probability $|\beta_{\rm EGGS}(\omega)|^{2}$, obtained from
Eqs. (4) and (16b), is dominated by the same two electric modes (Fig. 11b). In
particular, for the configuration A (see inset), the magnetic mode only
produces a residual feature in the EEGS spectral profile. Incidentally, the
effect of light concentration by the mirror gives rise to an enhancement by
five orders of magnitude in the EEGS probability relative to direct light
plane-wave illumination for a given value of the light intensity $I_{0}$ (cf.
Fig. 11b and Fig. 11c).

In Fig. 12, we present spatial maps of the calculated EEGS probability
$|\beta_{\rm EGGS}(\omega)|^{2}$ for the 4122 nm silica sphere. Each pixel in
the map corresponds to a different e-beam position for a fixed light focal
spot (blue dots). We only consider e-beam positions that do not intersect the
sphere. Results are presented for the three Mie modes discussed in Fig. 9c and
for four different positions of the focal spot. These plots demonstrate that
substantial electron-light coupling takes place even when the e-beam passes at
a large distant (but still near the sphere surface) relative to the focal
spot. This is a manifestation of the delocalized nature of the Mie modes under
consideration.

Incidentally, when reducing the size of the mirror, the EEGS probability
$|\beta_{\rm EGGS}(\omega)|^{2}$ decreases roughly in proportion to the mirror
area, as shown in Fig. 13. However, when normalized to the total laser power
$P_{0}$ collected by the mirror (see upper color scale), we find a similar
order of magnitude of $|\beta_{\rm EGGS}(\omega)|^{2}/P_{0}$ as for the larger
mirror, indicating that the electron couples to Mie modes that are relatively
delocalized, and thus, although a larger mirror is beneficial for delivering
more power in practice, the size of the focal spot is not too relevant for the
present specimen.

Another signature of optical field delocalization is found by examining the
EEGS spectra produced at a constant e-beam position as we move the optical
spot. The results shown in Fig. 14 reveal two interesting trends. (1) When the
e-beam and laser spot cross the same azimuthal direction (the positive $x$
axis in Fig. 14a), the EEGS probability takes comparatively large values,
which tend to increase as the optical spot is placed further apart from the
sphere center, such that it is able to excite Mie resonances of higher angular
momentum following the rule $l\sim\omega R_{f}/c$, which yields an orbital
number $l\sim 26$ at $R_{l}=2.5\,\mu$m, matching well the two prominent modes
at $l=26$ and 27 observed in Fig. 14 (see Fig. 11). (2) In addition, when the
azimuthal positions of the e-beam and optical spots are different (Fig.
14b,c), such radial dependence becomes much weaker and the EEGS probability is
comparatively reduced, indicating that the e-beam misses the path followed by
ballistic-like whispering gallery modes launched along a sphere meridian when
the laser beam is grazingly incident.

Finally, we compare the CL emission probability integrated over frequency
within the Mie resonances under consideration for the silica and PS spheres
considered in Figs. 9 and 10. The results, presented in Table 1, show that the
probability is much higher in the silica particle, thus providing an
additional piece of information to explain why only this one can be resolved
in our CL experiments.

##  Appendix E On the question of spectral resolution and signal strength in
EELS, CL, and EEGS

The spectral resolution in EELS is generally limited by the width of the zero-
loss peak and the energy analyzer. In contrast, the resolution in CL depends
on the precision of the light spectrometer. Finally, the EEGS resolution is
controlled by the width of the external laser. One could argue that CL and
EEGS should be ultimately comparable in precision, as they rely on optical
tools. However, a problem arises regarding the ability of these techniques to
detect narrow resonances: indeed, our experimental CL system is unable to
observe the narrow resonances of the PS particle; in contrast, EEGS can neatly
resolve them with observed quality factors $Q\sim 10^{4}$. We also note that
for the silica sphere one observes $Q\sim 100$, and both techniques allow us
to resolve the resonances, although EEGS provides a superior spectral
precision.

To address this puzzle, we argue that EEGS has the advantage that, under
continuous-wave illumination, the spectral width of the laser can be made much
narrower than the resonance, so that the electrons interact with a strongly
enhanced field. For large quality factors $Q$ such as those of the PS sphere,
the CL spectrometer is eventually unable to resolve the spectral shape. Even
if an ultraprecise spectrometer is used, capable of resolving resonances with
large $Q$, we find it instructive to compare in what follows the total CL
signal integrated over the resonance as a way to quantify the number of
electron counts in CL relative to EEGS.

For simplicity, we consider a narrow resonance of frequency $\omega_{0}$ and
quality factor $Q$ in the specimen, for which the spectral dependence of the
CL far-field amplitude ${\bf f}^{\rm CL}_{\hat{\bf r}}(\omega)$ in Eq. (3) can
generally be approximated as a Lorentzian, such that

$\displaystyle\big{|}{\bf f}^{\rm CL}_{\hat{\bf
r}}(\omega)\big{|}^{2}\approx\frac{\big{|}{\bf f}^{\rm CL}_{\hat{\bf
r}}(\omega_{0})\big{|}^{2}}{4Q^{2}(\omega/\omega_{0}-1)^{2}+1}.$

Now, the angle- and frequency-dependent photon emission probability in CL is
given by García de Abajo (2010) $\Gamma(\hat{\bf
r},\omega)=(c/4\pi\hbar\omega)\,\big{|}{\bf f}^{\rm CL}_{\hat{\bf
r}}(\omega)\big{|}^{2}$, so that the total probability (i.e., integrated over
emission angles and frequencies within the resonance peak) reduces to

$\displaystyle\Gamma_{\rm CL}=\int d^{2}\hat{\bf r}\int_{\rm
peak}\\!\\!\\!\\!\\!d\omega\,\big{|}{\bf f}^{\rm CL}_{\hat{\bf
r}}(\omega)\big{|}^{2}\approx\frac{c}{8Q\hbar}\int d^{2}\hat{\bf
r}\,\big{|}{\bf f}^{\rm CL}_{\hat{\bf r}}(\omega_{0})\big{|}^{2}.$

Likewise, from the EEGS probability $\Gamma_{\rm EEGS}(\omega)=|\beta_{\rm
EGGS}(\omega)|^{2}$, using Eq. (3) for plane-wave illumination, we find

$\displaystyle\Gamma_{\rm
EEGS}=\frac{c^{4}}{8\pi\hbar^{2}\omega_{0}^{4}}\,\big{|}E^{\rm
ext}\big{|}^{2}\int d^{2}\hat{\bf r}\,\big{|}{\bf f}^{\rm CL}_{\hat{\bf
r}}(\omega_{0})\big{|}^{2},$

where, for the sake of our argument, we assume resonant illumination
($\omega=\omega_{0}$) and introduce an average over orientations of the light
field ${\bf E}^{\rm ext}$ and the directions $\hat{\bf r}$ from which light is
coming (i.e., we integrate over directions $\hat{\bf r}$, divide by the full
$4\pi$ solid angle of the sphere, and also divide by a factor of 2 to account
for the average over light polarizations). We have now quantities that can
directly be compared to find the dimensionless quantity

$\displaystyle\mathcal{R}^{\rm EEGS}_{\rm CL}=\frac{Qc^{3}\big{|}E^{\rm
ext}\big{|}^{2}}{\pi\hbar\omega_{0}^{4}}=\frac{2Qc^{2}I^{\rm
ext}}{\hbar\omega_{0}^{4}},$

which is the ratio of EEGS-to-CL counts normalized per incident electron.
Here, we use the relation $I^{\rm ext}=c\big{|}E^{\rm ext}\big{|}^{2}/2\pi$
between the incident light electric field amplitude and the light intensity.
For $\hbar\omega_{0}=2\,$eV, this expression can be written as

$\displaystyle\mathcal{R}^{\rm EEGS}_{\rm CL}=Q\,\frac{I^{\rm ext}}{5\times
10^{10}\,{\rm W}/{\rm m}^{2}}.$

Then, for a plausible light intensity of $10^{8}\,$W/m2 in our experiment, the
ratio with $Q=200$ (silica sphere) is $\mathcal{R}^{\rm EEGS}_{\rm CL}=0.4$,
and therefore, both CL and EEGS render comparable signals. In contrast, with
$Q=10^{4}$ (PS sphere), the ratio becomes $\mathcal{R}^{\rm EEGS}_{\rm
CL}=20$, so that EEGS is much stronger and capable of resolving the
resonances, whereas the CL signal is too weak. Finally, we note that the
actual ratio $\mathcal{R}^{\rm EEGS}_{\rm CL}=0.4$ should depend on the
specific illumination directions, and also on the effect of the mirror.

###### Acknowledgements.

We thank T. Lovejoy for discussions on the functioning of the fast blanker on
the Chromatem machine. We are also thankful to H. Lourenço-Martins, S. Meuret,
and A. Arbouet for fruitful discussions. This project has been funded in part
by the European Union through the Horizon 2020 Research and Innovation Program
(grant agreement No. 101017720 (EBEAM)), the French National Agency for
Research under the program of future investment TEMPOS-CHROMATEM (reference
No. ANR-10-EQPX-50) and QUENOT (ANR-20-CE30-0033), the Spanish MICINN
(PID2020–112625GB-I00 and SEV2015-0522), the ERC (Advanced Grants 789104-eNANO
and 787510-4DBIOSERS), the Catalan CERCA Program, and Fundació Privada Cellex.
Author contributions: M.K., Y.A., O.S., M.T., L.H.G.T, and F.J.G.d.A designed
the experiment. Y.A., M.T., and J.-D.B. developed the light injection and
synchronization setup with support from L.F.Z. and L.T. Y.A. performed
experiments with support from L.H.G.T, X.L., and M.K. Y.A. analyzed the data
with support from L.H.G.T. F.J.G.d.A. developed the theory and carried out
numerical simulations with support from E.D. All authors participated in the
results analysis and contributed to the preparation of the manuscript.

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