Accepted Manuscript

Stimulated electron energy loss and gain in an electron microscopewithout a pulsed electron gun

P. Das, J.D. Blazit, M. Tence, L.F. Zagonel, Y. Auad, Y.H. Lee,X.Y. Ling, A. Losquin, C. Colliex, O. Stephan, F.J. Garcıa de Abajo,M. Kociak

PII: S0304-3991(18)30317-6DOI: https://doi.org/10.1016/j.ultramic.2018.12.011Reference: ULTRAM 12704

To appear in: Ultramicroscopy

Received date: 13 September 2018Revised date: 6 December 2018Accepted date: 16 December 2018

Please cite this article as: P. Das, J.D. Blazit, M. Tence, L.F. Zagonel, Y. Auad, Y.H. Lee, X.Y. Ling,A. Losquin, C. Colliex, O. Stephan, F.J. Garcıa de Abajo, M. Kociak, Stimulated electron energy lossand gain in an electron microscope without a pulsed electron gun, Ultramicroscopy (2018), doi:https://doi.org/10.1016/j.ultramic.2018.12.011

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Stimulated electron energy loss and gain in an electron microscope without apulsed electron gun

P. Das, J. D. Blazit, M. Tencé

Laboratoire de Physique des Solides, Université Paris-Sud, CNRS-UMR 8502, Orsay 91405, France

L. F. Zagonel, Y. Auad

"Gleb Wataghin" Institute of Physics, University of Campinas – UNICAMP 13083-859, Campinas, São Paulo, Brazil

Y. H. Lee, X. Y. Ling

Division of Chemistry and Biological Chemistry, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore

637371, Singapore

A. Losquin, C. Colliex, O. Stéphan

Laboratoire de Physique des Solides, Université Paris-Sud, CNRS-UMR 8502, Orsay 91405, France

F. J. García de Abajo

ICFO-Institut de Ciences Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona) and ICREA-Institució

Catalana de Recerca i Estudis Avancats, Passeig Lluis Companys, 23, 08010 Barcelona, Spain

M. Kociak

Laboratoire de Physique des Solides, Université Paris-Sud, CNRS-UMR 8502, Orsay 91405, France

Abstract

We report on a novel way of performing stimulated electron energy-loss and energy-gain spectroscopy(sEELS/sEEGS) experiments that does not require a pulsed gun. In this scheme, a regular scanning transmissionelectron microscope (STEM) equipped with a conventional continuous electron gun is fitted with a modified EELSdetector and a light injector in the object chamber. The modification of the EELS detector allows one to expose theEELS camera during tunable time intervals that can be synchronized with nanosecond laser pulses hitting the sample,therefore allowing us to collect only those electrons that have interacted with the sample under light illumination.Using ∼ 5 ns laser pulses of ∼ 2 eV photon energy on various silver samples, we obtain evidence of sEELS/sEEGSthrough the emergence of up to two loss and gain peaks in the spectra at ±2 and ±4 eV. Because it does not involveany modification of the gun, our method retains the original performances of the microscope in terms of energy res-olution and spectral imaging with and without light injection. Compared to pulsed-gun techniques, our method ismainly limited to a perturbative regime (typically no more that one gain event per incident electron), which allows usto observe resonant effects, in particular when the plasmon energy of a silver nanostructure matches the laser photonenergy. In this situation, both EELS and EEGS signals are enhanced in proportion to n + 1 and n, respectively, wheren is the average plasmon population due to the external illumination. The n term is associated with stimulated lossand gain processes, and the term of 1 corresponds to conventional (spontaneous) loss. The EELS part of the spectrumis therefore an incoherent superposition of spontaneous and stimulated EEL events. This is confirmed by a properquantum-mechanical description of the electron/light/plasmon incorporating light-plasmon and plasmon-electron in-teractions, as well as inelastic plasmon decay.

Keywords: electron energy gain

Preprint submitted to Ultramicroscopy December 17, 2018

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1. Introduction

The three scientists honored in this special issue, oneof them being an author of this article, have shared manycommon interests in electron microscopy and contributedwith pioneering advances. They encompass in particu-lar inelastic scattering, coherence, various sorts of fieldsmapping, the search for new electron-based techniquesand the development of innovative instrumentation. Itis therefore no surprise that we decided to present anew instrumental development in a relatively new field ofelectron microscopy mixing inelastic scattering of elec-trons and the influence of electromagnetic optical fieldsin an electron microscope, namely stimulated electronenergy-loss spectroscopy (sEELS) and stimulated elec-tron energy-gain spectroscopy (sEEGS).

The idea of shining a light beam onto a sample of in-terest and observing sEELS/sEEGS was prophesied byArchie Howie in a short proceeding for a conference(1). In that paper, he was screening some potential tech-niques able to help analyzing defects in semi-conductors,including photoelectron emission microscopy (PEEM),cathodoluminescence (CL) and emission of light by scan-ning tunneling microscopy. Interestingly, among all thesetechniques, only cathodoluminescence proved to be reli-able for single-defect detection. In particular, point de-fects behaving as single photon emitters can now be de-tected (2; 3). Nevertheless, the other techniques foundimpressive applications in nanophotonics. In particular,Howie pointed out how PEEM could be utilized for plas-mon mapping, and later it was brought to produce highresolution plasmon maps (4; 5) and ultrafast plasmon dy-namics (6; 5). Archie Howie also discussed tip-enhancedRaman scattering (TERS), which can now image sub-molecular features with a Raman signal (7).

However, Howie’s most striking prophecy was aboutelectron energy gain. At the time, electron energy gainshad been already detected several decades earlier in differ-ent configurations. Those include thermal EEGS, whereelectrons are accelerated by thermal phonons, detected al-most 50 years ago with no spatial resolution (8; 9) - ther-mal EEGS can now be acquired with nanometer resolu-tion (10). Acceleration of electrons with non-thermallyexcited quasi-particles (i. e., not phonons, but ratherhigher energy quasi-particles such as for example plas-mons) has also been proved a long time ago: Schilling

and Raether (11) managed to observe EEGS from high-energy plasmons using an high-intensity electron beam;with such a beam, the incoming electrons pump the sys-tem, so that trailing electrons in the same beam could pickup the energy deposited by leading electrons and be accel-erated. Also, acceleration of electron beams with a lightbeam was discussed in the context of the inverse Ceren-chov effect (12). But Howie predicted a brand new phys-ical mechanism of electron energy gain, namely, the ac-celeration of electrons by the field scattered by nanoscaleobjects illuminated by a laser beam (1).

Almost ten years later, two of us calculated the prob-ability for an electron to be accelerated close to a plas-monic nanoparticle (13), showing that it could be possibleto see a peak in the gain part of the EELS spectrum usingreasonably high-power continuous-wave (CW) laser light.

The electron energy-gain spectroscopy (EEGS) pro-posed in Ref. (13) was concerned with resonant structuresexcited by light and probed at the nanometer scale. Thiswork focused on the increase in spectral resolution, whichcould be only limited by the laser spectral width, and notby the zero-loss-peak (ZLP) energy width of the electronmicroscope, which is orders of magnitude higher. Theprinciple of electron energy gain in laser-excited sampleswas experimentally demonstrated one year later by Bar-wick and collaborators (14). In contrast to the perturbativeregime that we studied (13), the effect demonstrated in theexperiment conducted by the group of the late Ahmed Ze-wail was not a small perturbation, but led to a spectrumoverwhelmed by replicas, with multiple peaks spaced bythe exciting laser energy, on the gain and loss parts ofthe spectrum in a nearly symmetric distribution relativeto the ZLP. Indeed, the experiment was performed usinga femtosecond pulsed laser able to both trigger electronemission and sample light irradiation. With proper syn-chronization, the overlap between the electron and laserpulses could be made sufficiently large to make the inten-sity of the replica peaks similar to that of the ZLP. Usingthis high signal-to-noise ratio spectra, images of the elec-tromagnetic field scattered by the nano-object could beresolved. Such microscopy technique was therefore bap-tized as photon-induced near-field electron microscopy(PINEM). The occurrence of the multiple replicas oc-curring in PINEM was soon described theoretically (15).PINEM was further used to map electromagnetic fields ofvarious structures, including surface plasmons polaritons

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(16). It was also recently extended to tunable femtosec-ond lasers, allowing spectrally resolved plasmon imagingto be performed (17).

In Ref. (15), the special case in which the photon pulseis much longer than the electron pulse was also analyzed,showing even stronger effects, including a vanishing ofthe ZLP and quantum-billiard transitions between spectralpeaks associated with different net numbers of exchangedphotons. These effects were brilliantly demonstrated in2015 by Feist and collaborators (18), who interpreted theresults in terms of Rabi oscillations of the fast electronbetween the free propagation eigenstates shifted by mul-tiples of the laser excitation. We note that Feist et al. usedan ultrafast high-brightness/high-coherence gun (a Schot-tky field emission gun in this case) in order to be able toobserve this effect with high spatial resolution (note thatsimilar effects can be retrieved without spatial resolutionwith a low brightness gun (19)). More generally, the useof a high coherence gun should be a pre-requisite for bothtime-resolved holography and spatially resolved analyticmicroscopy (20). For spectroscopic works at high spatialresolution, as the one of the present paper, high brightnessindeed allows to get high current densities for incident an-gles small enough such as not inducing spatial resolutiondegradation due to geometrical aberrations.

Now, the development of a high-brightness pulsed elec-tron gun is a demanding task (21). Switching betweencontinuous and pulsed configurations through the electrongun is usually not easy. Additionally, changing the gunin an existing microscope involves a complex and costlyeffort. Therefore, one may wonder if a more versatileand cheaper solution is possible. On a more fundamen-tal level, one may also want to explore a simpler phys-ical configuration, in which the interaction between theelectron and the photon, as mediated by a nano-object,is essentially linear, and where the spectral resolution inthe sEELS/sEEGS spectra is closer to that used in reg-ular EELS for nano-optics, which is significantly higherthan the one currently reported in pulsed-gun-technologystudies. In this way, gain spectroscopy, which has beenelusive until now, could be made possible.

In this paper, we present a technique in which, insteadof exciting a sample with synchronized sub-picosecondpulsed electron and laser beams (pulsed excitation tech-nique), we are irradiating the sample with a stochasticelectron beam and a pulsed laser, the later synchronized

with the detection (pulsed detection technique). In thepulsed excitation technique, all electrons have the possi-bility to experience stimulated gain or loss and then con-tribute to the collected spectrum. In the pulsed detectiontechnique that we introduce here, most of the electronsdo not experience sEEL and sEEG, and therefore only in-elastic events arising when the sample is exposed to thelaser have to be detected through synchronized detection.We show that indeed sEELS/sEEGS replicas can be ob-served in this way. We report a spectral resolution bet-ter than 500 meV, as measured by the full width at max-imum (FWHM) of the ZLP. Compared to the pulsed ex-citation technique, the pulsed detection approach showssEELS/sEEGS peak intensities that are approximatelylinear with the laser intensity. We then demonstrate thepossibility to perform spectral imaging interchangeablywith and without sEELS/sEEGS signals. By workingwith a nanostructure exhibiting a mode in resonance withthe incident laser, we could further unveil basic proper-ties of resonant sEELS/sEEGS. First, compared to theoff-resonance situation probed in the same experiment,sEELS/sEEGS peaks are largely enhanced due to the res-onant matching of the laser and the structure mode en-ergy. Additionally, the EELS peak intensity is higher thanits EEGS counterpart. As confirmed by a detailed theo-retical treatment presented below, this discrepancy withthe current literature can be simply explained by the factthat in our case loss and gain events should be propor-tional to n + 1 and n, respectively, where n is the averagenumber of plasmons created by the incident laser. Thisdependence describes the probability of creating and de-stroying a bosonic plasmon mode, with the term n beingassociated to stimulated loss and gain processes, respec-tively, and the term of 1 representing spontaneous losses(i.e., conventional EELS). The EELS events are thereforeincoherent superpositions of spontaneous and stimulatedEEL ones.

2. Experimental details

The principle of a pulsed-detection sEELS/sEEGS set-up is given in Figure 1, together with a timeline in Fig-ure 2. A conventional (S)TEM (a dedicated VG HB501in this paper) is fitted with a light injection device (basedon the Attolight Mönch 4107 in the present case) throughwhich a laser beam can be focused on the sample. The

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microscope is also fitted with a modified EELS spectrom-eter and detector. The main modification of the EELSdetection scheme is the addition of a fast deflector afterthe last quadrupole of a modified 666 Gatan spectrom-eter. Combined with a slit in front of the scintillator,the deflector allows us to periodically obtain a stationarybeam hitting the scintillator during a customizable periodas short as τON = 18 ns (ON time). In the present case,the deflection period is chosen to match the repetition rateR = 1/τR = 10 kHz of a nanosecond-pulse high-powerSYRAH Credo laser. This laser emits 582 nm (2.13 eV)pulses of δ = 5 ns duration. We usually set τON to itsminimum (i.e, as close as possible to the value of δ) andadjust the delay ∆ between the laser and deflector pulsesso that all the electrons that have been emitted during theperiod in which the sample was irradiated by the laser arefinally detected (see Figure 2). The setup is prepared tosend an average power tunable between 0 and more than400 mW after coupling into an optical fiber. Using an op-tical fiber simplifies alignments at the price of reducingthe maximum power density achievable. If needed, how-ever, nothing prevents the experiment from being madefree-space with the light injection system used (21). Weused a field programmable gate array (FPGA) for syn-chronization electronics to synchronize the triggering ofthe laser emission and the beam deflection. The delay ∆

between the two of them can be fine-tuned with a pro-grammable 5 ns precision. EELS spectra, or more gener-ally EELS, sEELS and sEEGS spectra, are acquired on aPrinceton proEM electron multiplying charge coupled de-vice (EMCCD) camera, and simultaneously, high-angle-annular-dark-field (HAADF) data can also be detected.The EMCCD camera was used in its low-noise mode(non-EM mode, similar to a regular CCD camera). Forthe remaining of this article, we used 15 mrd incidenceand acceptance angles, and a typical current of 50 pA inthe probe.

A very important point in the present setup is that theoperation of the microscope is exactly the same as a regu-lar one, even if of course the sample can be affected phys-ically by the laser illumination (see below). For example,for electron dose sensitive sample, a regular pre-specimenblanker (with µs resolution in our setup) can be synchro-nized with the end of the ON time. The probe stability isnot affected by the illumination, and the HAADF imagingis only slightly perturbed by stray light coming from the

laser. The stray light is due to the design of the VG micro-scope, where no electron optics exists between the sampleand the HAADF detector to protect the later. However, itwould be naturally screened from light in a more mod-ern microscope. Additionally, it is important to note thatthe EELS detection system is not affected by our pulseddetection approach. Indeed, pulsed detection does not in-duce extra aberrations in the spectrometer, as it stands af-ter all the electron optics (see a demonstration below).EELS spectra are acquired with typically millisecond toseconds dwell time, during which an arbitrary number Nof light pulses and ON deflector times take place (see Fig-ure 2). This means that typically N = 10 to N = 104

pulses have interacted during the dwell time. Apart fromthe fact that the effective current reaching the EELS cam-era is reduced by a factor 104 × 5 × 10−9 = 5 × 10−5

(the accumulated time during which the signal reachesthe detector in one second is the repetition rate multipliedby the pulse width), the EELS detector works the sameway as without pulsed operation. Therefore, spectral andHAADF imaging can transparently be performed in thepulsed or non-pulsed configurations with no modificationof the setup, which is a practical major advantage over theregular, pulsed-gun configuration.

For light injection alignments, we usually first pre-aligned the light injector to maximize the cathodolumi-nescence signal. Then, we brought the laser power to apoint where holes could be drilled in some sacrificial partof the sample and realign this way, if necessary.

3. Results and discussion

Samples were prepared by Joule-evaporating a 250 nmthick silver film on top of a regular TEM carbon holeygrid. Silver nanocubes in solution were further deposited.In many places the cubes could stick out in a hole, yetbeing in thermal contact with the film. This allows usto collect a substantial sEELS/sEEGS signature withoutmelting the sample.

Figure 3 shows a proof-of-principle of sEELS/sEEGSacquired with the beam positioned in the vacuum nextto a folded piece of silver film. A clear sEELS/sEEGSsignature is observed, revealing the presence of peaksat multiples of the photon energy. In this particularinstance, two sEELS and two sEEGS peaks (replicas)could be resolved. However, in most cases, the power

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Figure 1: Schematics of the pulsed-detection sEELS/sEEGS setup. Thescanning coils and spectral-imaging electronics are omitted for clarity.a. Side view including the dispersive axis (E). b. bottom view includingthe non-dispersive axis (y).

needed to observe the replicas (here around 432 mW or6.35 × 1014 W/m2, as determined by the modelization ofthe light injection configuration) led to the destruction ofthe sample. Even in the present extreme case, we note thatonly two replicas could be observed, with a maximum in-tensity (area under the peak) of ≈ 20% of the ZLP. Thisis much smaller than what is usually observed in pulsed-gun sEELS/sEEGS experiments (14; 16; 18), where theinteraction can be so strong that the ZLP eventually dropsto zero (18). In our pulsed-detection measurements, thesEELS/sEEGS signal is generally of the order of the reg-ular EELS signal from plasmons in large nanoparticles(i.e., a fraction ∼ 10−3 − 10−2 of the ZLP).

We present in Figure 4 the effect of changing the delaybetween the laser pulse and the fast deflector pulse. Inthis case, we set τON = 28 ns. We see that the signaldrops to zero within ±10 ns. This is slightly smaller thanexpected from the overlap of the ON time and laser pulseduration, possibly because of an effective ON time shorterthan expected. We are currently investigating this effect.

Knowing that the observed signal corresponds indeedto sEELS/sEEGS, we turn our attention to a few differ-ences with respect to the pulsed-gun technology. Fig-ure 5 presents a sEELS/sEEGS spectrum acquired closeto a silver nanocube. The FWHM of the ZLP and thereplicas is slightly smaller than 0.5 eV. This is very closeto the nominal value of the FWHM of the ZLP on thismicroscope, as seen from the overlapped ZLP obtainedunder regular conditions in Figure 5. The main limitsin this particular microscope stem from the lack of sta-bility of the high acceleration voltage and spectrometerpower supplies, the quality of the scintillator and the rel-atively high incidence and acceptance angles (and relatedaberrations) used here. A modern microscope and spec-trometer combination typically reach better than 300 meVon a routine basis, see e. g. (22). This is in contrastto the pulsed-gun approach, which is currently limitedabove 600 meV (20; 23), owing in part to energy broad-ening due to the electrons photoemission physics and inpart to the stability of the electron microscope and spec-trometer. We understand that there is no reason to ex-pect that our type of setup could not be implemented in amonochromated electron microscope and reach state-of-the-art spectral resolution (currently in the sub-10 meVrange (24)). A second aspect that appears as a strikingdifference with respect to the pulsed-gun sEELS/sEEGS

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Figure 2: Synchronization scheme of the pulsed detection. From top to bottom, we indicate the relevant elements in the electron column of Figure 1.Note that the photon pulses and electron deflection shapes have been schematized as rectangular gate functions.

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Figure 3: Proof-of-principle pulsed-detection sEELS/sEEGS with tworeplicas. The data have been collected with the beam position fixed closeto a thick silver layer (see inset) with a laser power density of 6.35 ×1014 W/m2 and an acquisition time of 1 s.

Figure 4: Effect of the delay between laser excitation and fast deflec-tion. The electron beam is placed close to a melted part of a silver filmLeft: series of spectra taken at various delays, as indicated in the legend.The dwell time is 1000 ms, the detection ON time τON is 28 ns. A totalof 20 spectra have been acquired in this configuration, realigned withrespect to their ZLP position and averaged. Right: Integrated intensityof the first replica (EEGS and EELS) as a function of delay. The in-tegrated intensity has been calculated by removing a linear backgroundcontribution, then summing the intensity under each of the peaks.

technologies is the linearity of the phenomenon in ourpulsed-detection scheme. More precisely, as exemplifiedin Figure 6, the intensity of a replica scales linearly withthe injected power. Although pulsed-gun setups can ofcourse be operated in this regime, they are commonlyused in a strong nonlinear regime in which the summedsEEGS is larger than the ZLP, leading to an excellent sig-nal to noise ratio for application in e.g. PINEM. In ourcase, trying to reach a nonlinear regime leads to destruc-tion of our samples, due to the necessary much higherpower. The quality of the data in the pulsed-detectionscheme, where the sEELS/sEEGS signal is of the orderof magnitude of the regular EELS signal, i.e several orderof magnitudes smaller than a typical PINEM signal, reliestherefore on the same optimization of the detection setupas in conventional EELS (25).

Figure 5: Typical spectrum extracted from spectral imaging performedon a silver nanocube (see Figure 7). In the pulsed-detection mode (ma-genta spectrum), the ZLP as well as the replicas have a FWHM slightlysmaller than 0.5 eV. A spectrum obtained in regular mode (blue) is alsoshown for comparison.

We next turn our attention to the study of resonantsEELS/sEEGS signals. In the preceding examples, pointspectra were taken at positions relevant to unveil themain properties of the pulsed-detection sEELS/sEEGS

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Figure 6: sEELS/sEEGS experiments in the linear regime. Left: spec-tra measured with an electron beam close to coalescent silver nanocubes(see inset) for various laser powers. Right: area under the sEEGS andsEELS first replica peaks as a function of laser power, showing an ap-proximately linear dependence.

approach. We selected points where the laser energy wasnot matching any resonance of the system. In order tostudy the effect of resonantly exciting a plasmon mode,we performed regular EELS spectral-imaging on a silvernanocube lying attached to a silver film edge with part ofit on the vacuum side, as shown in Figure 7 (top inset). Inthis spectral image, the laser was left on, but the ON stateof the deflector was set to 9000 ns, much larger than the10 ms acquisition dwell time of individual spectra. Thisway, the obtained EELS spectra were directly comparablewith the pulsed-detection mode (i.e., the sample was irra-diated the same way). The spectra of this object, whichhave been normalized to the total number of counts inorder to quantitatively compare the signals in both con-figurations, present two major features, one at ≈2 eV andlocalized on the suspended part of the cube, and anotherone at ≈3.2 eV on the film side, as shown in the filteredmaps of the inset to Figure 7 (top). Although the samplewas irradiated by a laser beam, no sEELS/sEEGS signalwas detected, because it was overwhelmed by the ZLPand the regular EELS signal originating from the time pe-riods in which the sample was not illuminated (i.e., mostof the time due to the large ON state of the deflector).As already mentioned above, triggering the pulse detec-tion operation does not affect the overall performance ofthe microscope, which makes it easy to compare regu-lar EELS and sEELS/sEEGS signals in spectral-imagingmode. We therefore performed a second spectral image ina pulsed-detection mode where the ON time (18 ns) wasslightly larger than the laser pulse duration. The results

are displayed in Figure 7 (bottom). A spectrum takenclose to the film exhibits the ≈3.2 eV mode, in additionto a small sEELS/sEEGS signal at the laser energy. Thefiltered map at this energy (not shown) is quite compara-ble to that of Figure 7 (top). The only noticeable differ-ence lies in the signal-to-noise ratio, which is obviouslymuch worse in the sEELS/sEEGS mode, an effect thatwe attribute to the ≈ 5 × 10−4 drop in the signal reach-ing the EELS detector (everything else being equal) in thepulsed detection mode (i.e., the small fraction of time inwhich the detector is ON). The situation is quite differ-ent for the ≈2 eV plasmon peak, which is nearly in reso-nance with the laser injection (i.e., the energy differencebetween plasmon and incident photon energies is 0.13 eV,which is small compared with the FWHM of the plas-mon). First, the sEELS/sEEGS signal is now largely en-hanced to the point that it is much larger than the ≈3.2 eVsignal or the EELS signal of the ≈2 eV plasmon peak.Second, the sEELS and sEEGS maps, despite the noise,closely resemble each other. However, they slightly differfrom their EELS counterpart. The signal is now more lo-calized on the two tips of the cube, which is a characteris-tic difference between EELS and sEELS/sEEGS (see the-oretical description below). Finally, the spectral weight ofthe sEELS and sEEGS spectra are clearly different. Thisis in stark contrast with the available experimental liter-ature, in which sEELS/sEEGS replicas are totally sym-metric.

4. Theoretical description

We can obtain insight into the electron/light/plasmoninteraction processes through a proper quantum-mechanical description incorporating light-plasmon andplasmon-electron interactions, as well as inelastic plas-mon decay, taking into account that light and electronscannot directly interact due to their energy-momentummismatch. We thus write the Hamiltonian of the systemas

H =~ωp a†a + ~∑

k

εkb†kbk (1)

+ g(t)(a† + a

)+

∑

kk′

(gkk′b

†kbk′a + g∗kk′b

†k′bka†

),

where a† and a represent creation and annihilation oper-ators of the plasmon, respectively, b†k and bk create and

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TFigure 7: sEELS/sEEGS spectral-imaging experiments in the resonantregime. The laser power is 73 mW (≈ 1014 W/m2). Top: EELS spectral-imaging with 900 ms ON time. In this configuration, the sample isilluminated, but most of the detected electrons do not undergo anysEELS/sEEGS event. Inset: HAADF image of the structure (coloredsquares indicate the position of the beam where the spectra of corre-sponding colors in the main panel have been acquired) and filtered im-ages for two main modes. Bottom: sEELS/sEEGS spectral-imagingwith shorter ON time. Specifically, the ON time is close to the laserpulse duration, leading to a clear observation of sEELS/sEEGS features.Inset: HAADF image of the structure (colored squares indicate the po-sition of the beam where the spectra of corresponding colors in the mainpanel have been acquired) and filtered images around the energies of thesEELS/sEEGS first peak.

annihilate an electron of wave vector k and kinetic energy~εk along the beam direction, and ωp is the plasmon fre-quency. Terms in the first line of eq. (1) describe the non-interacting plasmon and electron, respectively, whereasthe second line incorporates interaction terms. In particu-lar, the externally applied light of frequency ω0, approxi-mated to be continuous wave, interacts with the plasmonthrough a time-varying rate g(t) = 2Re

{g0e−iω0t

}, where

g0 = −E0 · p is written in terms of the optical electricfield amplitude E0 and an effective plasmon dipole p. Forsimplicity, we assume in what follows a single plasmonmode.

Before we discuss the effect of inelastic decay of theplasmon, we can examine the Hamiltonian (1) to extractsome general characteristics of the electron spectrum. As-suming the initial state to contain n plasmons (i.e., a plas-mon Fock state |n〉) and neglecting light-plasmon interac-tion during the passage of the electron, we can use theFermi golden rule to first order in the electron-plasmoninteraction gkk′ to find that the electron spectrum con-tains two peaks separated by energies ±~ωp from the ZLP,whose magnitudes are proportional to n + 1 and n in theloss and gain peaks associated with matrix elements fortransitions to plasmon states |n + 1〉 and |n − 1〉, respec-tively. The loss peak is therefore more intense than thegain peak (in agreement with experiment), as both of themcontain a term proportional to the plasmon population n(stimulated process), but the loss peak has an additionalterm (+1) representing spontaneous (i.e., conventional)loss events. We thus conclude that direct comparison be-tween loss and gain peaks in the measured spectra allowsus to determine the number of plasmons n continuouslyexcited by the external illumination before arrival of theelectron, as the peak ratio is simply given by (n + 1)/n. Inthe case of Figure 7, for example, the comparison of thearea under the gain and loss peaks in the illuminated andnon-illuminated case gives an average number of plasmonof n ≈ 1.2, in agreement with the linear regime reached inthe present experiments. We note that this value is slightlyunderestimated, as the ON time is larger than the δ time,therefore there are more spontaneous events counted aris-ing during the time where the electrons are detected butthe laser is not illuminating the sample.

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4.1. Inelastic plasmon losses

In a realistic scenario, plasmons have a finite lifetimethat affects the profile of the resulting electron spectrum.We introduce this effect through a finite plasmon decayrate γ by resorting to the density matrix of the system ρ,which is subject to the equation of motion

dρdt

=i~

[ρ,H]

+γ

2

(2aρa† − a†aρ − ρa†a

). (2)

It is then convenient to express the density matrix in theinteraction picture as

ρ =∑

ll′

∑

kk′αll′,kk′ |l〉〈l′| ⊗ |k〉〈k′| ei(l′−l)ωpt+i(εk′−εk)t,

where |l〉 and |k〉 denote a Fock state of the plasmon withoccupancy l and a momentum state of the electron withwave vector k, respectively. Plugging this expression intoeq. (2), we find the explicit differential equations

dαll′,kk′

dt=

ig(t)~

[αll′+1,kk′

√l′ + 1eiωpt + αll′−1,kk′

√l′e−iωpt

− αl+1l′,kk′√

l + 1e−iωpt − αl−1l′,kk′√

leiωpt]

+γ

2

[2√

(l + 1)(l′ + 1)αl+1l′+1,kk′ − (l + l′)αll′,kk′]

+i~

∑

k′′

[g∗k′k′′ αll′+1,kk′′

√l′ + 1 ei(εk′′−εk′+ωp)t

+ gk′′k′ αll′−1,kk′′√

l′ ei(εk′′−εk′−ωp)t

− gkk′′ αl+1l′,k′′k′√

l + 1 ei(εk−εk′′−ωp)t

− g∗k′′k αl−1l′,k′′k′√

l ei(εk−εk′′+ωp)t] (3)

for the evolution of the density matrix coefficients,which readily leads to the hermiticity condition αll′,kk′ =

(αl′l,k′k)∗.We solve eq. (3) perturbatively in the electron-plasmon

interaction, but to all orders in the light-plasmon interac-tion, so that we decompose α = α(0) +α(1) + · · · , where theterms α(m) have an m-order dependence on the coefficientsgkk′ . In this scheme, as explained for example in Ref. (26),the density matrix to zeroth order in the electron-plasmoninteraction (i.e., in the infinite past) reads

ρ(0) = |k0〉〈k0| ⊗ |ξ〉〈ξ|, (4)

where k0 is the wave vector of the incident electron, while|ξ〉 = e−|ξ|

2/2 ∑∞l=0(ξl/

√l!)e−ilωpt |l〉 is a coherent plasmon

state characterized by the time-dependent amplitude

ξ =−i~

∫ t

−∞dt′g(t′)eiωpt′−γ(t−t′)/2

=−1~

g0ei(ωp−ω0)t

ωp − ω0 − iγ/2+

g∗0ei(ωp+ω0)t

ωp + ω0 − iγ/2

. (5)

We note that this state corresponds to a number ofoptically-excited plasmons given by n = |ξ|2. Underthe approximations |ωp + ω0| � γ (narrow width) and|ωp−ω0| � γ (resonant excitation), we find n ≈ |2g0/~γ|2.

Using eqs. (4) and (5), we can readily write the unper-turbed (from the electron-plasmon-interaction viewpoint)coefficients of the density matrix as

α(0)ll′,kk′ = δkk0δk′k0 e−|ξ|

2 ξl(ξl′ )∗√l!l′!

.

We assume this value of the coefficients at an initial timet = t0 and integrate eq. (3) by retaining terms of the sameorder on both sides of the equation, so that α(m) for eachorder m is obtained from the coefficients for orders < m.Repeating this procedure twice (to first obtain α(1) fromα(0), and then α(2) from α(1)), and integrating up to a timet such that γ(t − t0) � 1, we calculate the electron transi-tion rate to the lowest nonvanishing order in the electron-plasmon interaction as

Pk = limt−t0�1/γ

∑

l

α(2)ll,kk

t − t0.

This expression gives the population (i.e., the diagonal el-ements of the density matrix) of states in which the elec-tron has evolved to a new wave vector k, with the plas-mons traced out.

For a dipole-like plasmon (e.g., the dipolar mode of ametallic sphere, or also a dipolar mode in one of the silvercubes under consideration), the electron-plasmon interac-tion coefficient can be obtained from the electric field pro-duced by the electron acting on the plasmon dipole p. Wefind (27)

gkk′ =2eγ2

r Lp · [|q|γrK1(|q|R/γr)R − iqK0)(|q|R/γr)z

],

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where q = k − k′, v is the electron velocity, γr =

1/√

1 − v2/c2 accounts for relativistic corrections, R is thedistance from the dipole to the electron trajectory, and L isa quantization length along the beam direction. This ex-pression assumes the non-recoil approximation, in whichthe electron velocity is taken to remain nearly unchangedduring interaction with the plasmon under the assumption|k−k0| � k0, which directly impliesω ≈ (k−k0)v from en-ergy and momentum conservation. Multiplying the tran-sition rate Pk by the interaction time L/v and using theconversion

∑k → (L/2π)

∫dω, we finally obtain the elec-

tron spectrum

Γ(ω) =L2

2πvPk,

where the L2 factor correctly cancels the 1/L dependenceof the gkk′ coupling coefficients to m = 2 order in thecalculation of Pk. Nevertheless, for the present discus-sion, the magnitude of the coupling coefficient gkk′ is es-sentially absorbed in an overall constant in the resultingelectron spectral intensity.

We present characteristic examples of calculation inFigure 8 for different values of the light frequency ω0

(vertical axis) relative to the plasmon frequency ωp in aregime of weak interaction between the electron and theplasmon (i.e., a regime in which the ZLP is not substan-tially depleted by loss and gain events). In these calcula-tions, the illumination intensity is expressed in terms ofthe parameter n = |2g0/~γ|2 (i.e., the population of theplasmon mode), which we take to be independent of ω0

for the sake of this analysis, although in practice this pa-rameter should follow a Lorentzian profile as the lightfrequency is swept across the plasmon resonance. Weobserve an illumination-independent conventional EELSfeature in the loss part of the spectrum corresponding to aplasmon with a broadening determined by the parameterγ (= 0.3ωp in the present case), accompanied by symmet-ric narrow peaks originating in stimulated loss and gain atfrequenciesω = ±ω0 relative to the ZLP. In contrast to thebroad spontaneous EELS feature, the width of the stimu-lated peaks is roughly inversely proportional to 1/(t − t0),with the simulation time t taken here to be t − t0 = 100/γ.We thus observe a spectrum consisting of a regular EELSspectrum (independent of illumination), on which narrowsEELS and sEEGS peaks are superimposed. This result isin good agreement with the experiment, keeping in mind

that stimulated peaks are broadened by the ZLP widthand, to a smaller extend, by the laser linewidth. It can beunderstood in terms of energy conservation because stim-ulated processes involve discrete energy transfers betweenelectrons and photons mediated by the plasmon. Also, thelack of interference between regular EELS and sEELS isconsistent with the fact that they are associated with mu-tually incoherent processes. Finally, the integral of thesignal in the plots of Figure 8 for ω > 0 (loss) and ω < 0(gain) is approximately given, within the numerical un-certainty, by (n + 1)/n, so the conclusion extracted abovein the absence of losses is still maintained, although thespectral distribution of the n and +1 terms follow differ-ent characteristics.

5. Conclusion

We have presented an alternative solution for perform-ing sEELS/sEEGS experiments. In our approach, we usea nanosecond laser instead of a femtosecond laser, andthe electron detection rather than the electron excitationis pulsed. sEELS/sEEGS signals could be unambigu-ously detected. Although they resemble those obtainedwith a pulsed excitation, the coupling efficiency is muchsmaller, and most of the phenomena can be understoodin a perturbative manner. This configuration allows us togenuinely switch from a normal EELS operation modeto sEELS/sEEGS operation mode with comparable en-ergy resolution. It also makes it easier to perform spec-tral imaging. Indeed, we report two-dimensional spectralimages acquired with a sEELS/sEEGS signal. It allowsus to accurately study scenarios in which the excitationis in resonance with a particular plasmon of the sample.Under such circumstances, both EELS and EEGS signalsare amplified. However, the EELS signal is larger and itsratio to the EEGS signal given by sEELS/sEEGS = n+1

n ,where n is the average number of light-induced plasmons.This is because the EELS signal is an incoherent super-position of the spontaneous and stimulated EEL events.In order to describe these phenomena, the plasmon decaywhich is usually not included in EEGS theory, has beenintroduced in a new theoretical approach.

The versatility of the setup should allow its directimplementation in any modern microscope, thus givesus some hope that sEELS/sEEGS physics will be morewidely studied in the coming years. In particular, the

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Nor

mal

ized

pho

ton

ener

gy w

0/wp

Normalized energy loss w/wp Normalized energy loss w/wp Normalized energy loss w/wp

(a) n=0.05 (b) n=0.5 (c) n=1.5

Figure 8: Calculated spectra for interaction of an electron beam with an illuminated plasmon mode, as described in the formalism presented inthe main text. We present the evolution of spectra for different values of the light frequency ω0. The loss and light frequencies (horizontal andvertical axes, respectively) are both normalized to the plasmon frequency ωp. The plasmon width is assumed to be γ = 0.3ωp. The plasmon-lightinteraction strength is quantified in terms of the parameter mean number of plasmons n = |2g0/~γ|2 (see labels).

study of resonant effects, first demonstrated here at oneparticular excitation wavelength, should be straightfor-wardly extended using tunable lasers. Spectroscopy withspectral resolution only limited by the energy width of thelaser, which is orders of magnitude better than the ZLPwidth, would then be possible by studying the laser wave-length dependence of the sEELS/sEEGS peaks (13; 27).

6. Acknowledgments

MK wants to thank F. Houdellier and A. Arbouet forfruitful discussions. This work has received support fromthe National Agency for Research under the programof future investment TEMPOS-CHROMATEM with theReference No. ANR-10-EQPX-50 and FEMTOTEMANR-14-CE26-0013. FJGA acknowledges support fromERC (Advanced Grant 789104-eNANO) and the Span-ish MINECO (MAT2017-88492-R and SEV2015-0522).LFZ and YA acknowledges support from FAPESP (grants2014/23399-9 and 2017/00259-5).

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