Introduction

Upon irradiation of a material with sufficiently intense monochromatic light, a polarization is induced that contains harmonics of the incident frequency1. In addition, when the material is illuminated with polychromatic light, wave-mixed components also appear at the sum and difference of the external optical frequencies. These nonlinear phenomena can be leveraged to map microscopic spatial variations in the response of different material structures, enabling suppression of the background associated with linear scattering through spectral filtering2,3,4. However, the spatial resolution of these methods is fundamentally limited by diffraction to approximately half the light wavelength. One approach to overcoming this problem involves the use of localized optical fields, such as those scattered from sharp tips5,6, which, in addition, can enhance the near-field strength and, therefore, dramatically increase the nonlinearly scattered signal7,8,9. Nevertheless, the introduction of external nanostructures generally perturbs the intrinsic response of the specimen.

A less conventional way of generating nonlinear signals consists in mixing external light with the broadband evanescent field accompanying a swift electron. In this work, we explore this possibility as a means to perform spectromicroscopy with a combination of high spectral and spatial resolutions. Specifically, we envision the illumination and detection of visible and near-infrared (vis-NIR) light to probe an idle free-electron field in the mid- and far-infrared spectral ranges. This phenomenon could be investigated in electron microscopes equipped with external illumination capabilities, which are becoming increasingly popular in the context of ultrafast electron microscopy10,11,12,13,14,15,16,17.

Low-frequency excitations are commonly accessed by far-field FTIR and THz time-domain spectroscopy, which offer broadband coverage down to the mid/far-IR but are diffraction-limited to a micrometer-scale resolution18,19. Scattering-type scanning near-field optical microscopy (s-SNOM) and related spectroscopies (including tip-enhanced and photothermal implementations) overcome the diffraction limit via localized fields, routinely reaching a spatial resolution of tens of nanometers at the cost of placing a tip near the sample20,21,22. Surface second-order nonlinear optics has also been shown to achieve monolayer sensitivity23,24, although it is limited by diffraction, weak signals, and the presence of a nonresonant background, requiring mid/far-IR sources and careful filtering. In contrast, electron beams (e-beams) offer a less invasive approach to achieving a high spatial resolution in electron microscopes, as they can be focused down to sub-ångström regions25,26. These probes permit electron energy-loss spectroscopy (EELS) to be performed27, rendering information on the excitation modes in a specimen with a state-of-the-art spectral resolution of a few meV28,29. However, EELS requires thin samples that are transparent to electrons, and it cannot resolve far-infrared excitations because they are overshadowed by the tail of the so-called zero-loss peak. The cathodoluminescence (CL) emission originating from linear scattering of the evanescent electron field provides an alternative spectromicroscopy method that can be applied to thicker samples, combined with a high spectral resolution through the determination of the emitted light wavelength30. Nevertheless, CL spectroscopy suffers from a low signal-to-noise ratio due to the relatively low photon emission probability compared to energy-loss events. In addition, the detection of low-frequency photons remains challenging because of the poor efficiency of current detectors.

In photon-induced near-field electron microscopy10,11,12,13,14,15,16 (PINEM), electron spectra are recorded to reveal the absorption or emission of multiple photons as the electrons traverse the near field of an illuminated specimen. A similar effect has been used in electron energy-gain spectroscopy (EEGS) to achieve a combination of high spectral and spatial resolutions by scanning the frequency of the incident light31,32,33,34,35,36,37,38. Additionally, spectral asymmetries in the observed electron energy sidebands associated with different numbers of exchanged photons have been claimed to provide information on the nanoscale nonlinear optical response of the specimen39. These methods rely on the ability to identify spectral features in the transmitted electrons, which lie beyond the capabilities of current electron spectroscopy for far-infrared excitations. Alternatively, stimulated electron–light schemes with optical readout have been proposed, such as electron- and light-induced stimulated Raman (ELISR), where an electron-excited plasmonic near field is intended to produce a Stokes feature in the presence of an optical pump40. While this technique avoids the need for low-frequency detectors or high-resolution electron spectrometers, it requires a large Raman response as well as optical background suppression, which have so far escaped experimental demonstration.

Here, we introduce wave-mixing cathodoluminescence (WMCL) as a disruptive approach for performing spectromicroscopy of low-frequency excitations, based on the nonlinear interaction between external light and the evanescent field of free electrons. Upon illumination with monochromatic light, WMCL arises from the nonlinear response of a specimen, producing outgoing light components with frequency shifts corresponding to energy losses and gains experienced by the electrons. When these frequency shifts coincide with resonances in the specimen, the electron field is enhanced and imprints distinct spectral features onto the nonlinearly scattered light. Considering a silver nanorod coated with retinal, we theoretically demonstrate that this method can reveal spectral information on the far-infrared vibrational fingerprints of this molecule through optical illumination and detection in the visible range. We further present a tutorial description of this unconventional wave mixing process for small, non-centrosymmetric particles exhibiting a nonlinear bulk response. Beyond its fundamental interest as a previously unexplored phenomenon, WMCL provides a unique opportunity to map low-frequency modes by avoiding the low efficiency of optical detectors in such a spectral range and only relying on visible-light illumination and detection.

Results

Theoretical framework

We consider a nanostructure simultaneously exposed to external monochromatic light of frequency ωpump and a focused e-beam passing through or near the material, as depicted in Fig. 1. Wave mixing between light and the evanescent electron field can take place, assisted by the nonlinear response of the specimen and giving rise to WMCL. This phenomenon manifests in the emission of photons with an output frequency ωWM resulting from a linear combination of ωpump and the different frequency components ω of the electron field. WMCL at the lowest (second) order can be anticipated to comprise three different possible channels, as summarized in Fig. 1:

  1. (i)

    sum-frequency generation (SFG): ωWM = ωpump + ω (Fig. 1, top);

  2. (ii)

    difference-frequency generation at low electron frequencies (DFG1): ωWM = ωpump − ω for ω < ωpump (Fig. 1, middle);

  3. (iii)

    difference-frequency generation at high electron frequencies (DFG2): ωWM = ω − ωpump for ω > ωpump (Fig. 1, bottom).

Fig. 1: Wave mixing cathodoluminescence (WMCL).
Fig. 1: Wave mixing cathodoluminescence (WMCL).
Full size image

a We consider a specimen simultaneously exposed to light and free-electron fields. The schemes illustrate different WMCL processes involving incident photons of frequency ωpump and the electromagnetic field component of frequency ω produced by a passing electron. The interaction is mediated by the nonlinear second-order susceptibility χ(2) of a specimen, giving rise to inelastically scattered photons with an output frequency ωWM > 0. We identify three different WMCL channels: (top) sum-frequency generation (SFG), with ωWM = ωpump + ω; and (middle and bottom) difference-frequency generation, with either (b) ωWM = ωpump − ω (DFG1) or c ωWM = ω − ωpump (DFG2) for ωpump > ω or ωpump < ω, respectively.

To map far-infrared modes (i.e., low ω’s) using visible light, only channels (i) and (ii) need to be considered, but we retain channel (iii) in our analysis for completeness.

We introduce the external light as a plane wave of wave vector kpump, with kpump = ωpump/c, and an electric field distribution \({{{\bf{E}}}}_{{{\rm{pump}}}}^{{{\rm{ext}}}}\,{{{\rm{e}}}}^{{{\rm{i}}}({{{\bf{k}}}}_{{{\rm{pump}}}}\cdot {{\bf{r}}}-{\omega }_{{{\rm{pump}}}}t)}+{{\rm{c.c.}}}\), where \({{{\bf{E}}}}_{{{\rm{pump}}}}^{{{\rm{ext}}}}\) is the amplitude of the incident field. Upon interaction with the structure, the linear optical field can be written as \({{{\bf{E}}}}_{{{\rm{pump}}}}({{\bf{r}}})\,{{{\rm{e}}}}^{-{{\rm{i}}}{\omega }_{{{\rm{pump}}}}t}+{{\rm{c.c.}}}\), exhibiting a more involved spatial dependence of the total field amplitude Epump(r).

Without loss of generality, electrons are considered to move with velocity \({{\bf{v}}}=v\,\hat{{{\bf{z}}}}\) within an e-beam focused at a transverse position R0 in the x − y plane. The external electric field associated with an electron has a broadband distribution that can be decomposed as \({{{\bf{E}}}}_{{{\rm{electron}}}}^{{{\rm{ext}}}}({{\bf{r}}},t)={(2\pi )}^{-1}\int \,d\omega \,{{{\bf{E}}}}_{{{\rm{electron}}}}^{{{\rm{ext}}}}({{\bf{r}}},\omega )\,{{{\rm{e}}}}^{-{{\rm{i}}}\omega t}\) in terms of the frequency-resolved components27

$${{{\bf{E}}}}_{{{\rm{electron}}}}^{{{\rm{ext}}}}({{\bf{r}}},\omega )=\frac{2{{\rm{i}}}e\omega }{{v}^{2}{\gamma }^{2}}\,{{{\rm{e}}}}^{{{\rm{i}}}\omega z/v}\,\left[{K}_{0}(\zeta )\hat{{{\bf{z}}}}+{{\rm{i}}}\gamma {K}_{1}(\zeta )\hat{{{\bf{b}}}}\right],$$

where \(\gamma=1/\sqrt{1-{v}^{2}/{c}^{2}}\) is the Lorentz factor, K0 and K1 are modified Bessel functions evaluated at ζ = ωb/vγ with b = R − R0, and we introduce the notation r = (R, z) with R = (x, y). We disregard electron recoil (i.e., v is assumed to be constant) as a safe approximation for the energetic e-beams under consideration. The self-consistent linear electron field Eelectron(r) generally has a more complex spatial dependence once the interaction with the specimen is accounted for.

The material responds with a nonlinear polarization density PWMCL(r) at different output frequencies ωWM according to1

$${{{\bf{P}}}}_{{{\rm{WMCL}}}}^{\nu }({{\bf{r}}},{\omega }_{{{\rm{WM}}}})=\left\{\begin{array}{ll}{\chi }^{(2)}({{\bf{r}}},{\omega }_{{{\rm{pump}}}},\omega,{\omega }_{{{\rm{WM}}}}):{{{\bf{E}}}}_{{{\rm{pump}}}}({{\bf{r}}})\,{{{\bf{E}}}}_{{{\rm{electron}}}}({{\bf{r}}},\omega ),\hfill & \,\,\nu=\,{{\rm{SFG}}}\\ {\chi }^{(2)}({{\bf{r}}},{\omega }_{{{\rm{pump}}}},-\omega,{\omega }_{{{\rm{WM}}}}):{{{\bf{E}}}}_{{{\rm{pump}}}}({{\bf{r}}})\,{{{\bf{E}}}}_{{{\rm{electron}}}}^{*}({{\bf{r}}},\omega )\,\Theta ({\omega }_{{{\rm{pump}}}}-\omega ),& \,\,\nu={{{\rm{DFG}}}}_{1}\\ {\chi }^{(2)}({{\bf{r}}},-{\omega }_{{{\rm{pump}}}},\omega,{\omega }_{{{\rm{WM}}}}):{{{\bf{E}}}}_{{{\rm{pump}}}}^{*}({{\bf{r}}})\,{{{\bf{E}}}}_{{{\rm{electron}}}}({{\bf{r}}},\omega )\,\Theta (\omega -{\omega }_{{{\rm{pump}}}}),& \,\,\nu={{{\rm{DFG}}}}_{2}\end{array}\right.$$
(1)

where χ(2) is the position- and frequency-dependent second-order susceptibility tensor corresponding to the three different WMCL channels under consideration, indexed by ν, and the step function Θ(x) is defined as 1 for x≥0 and 0 otherwise. The WMCL field amplitude can be written as \({{{\bf{E}}}}_{{{\rm{WM}}}}^{\nu }({{\bf{r}}},t)={(2\pi )}^{-1}\int \,d{\omega }_{{{\rm{WM}}}}\,{{{\bf{E}}}}_{{{\rm{WM}}}}^{\nu }({{\bf{r}}},{\omega }_{{{\rm{WM}}}})\,{{{\rm{e}}}}^{-{{\rm{i}}}{\omega }_{{{\rm{WM}}}}t}\), where the integral over ωWM indicates that it inherits a broadband character from the field of the electron (i.e., from the integral over ω). The frequency-resolved emission amplitude must then incorporate the linear response of the structure at each output frequency ωWM, which is captured by the electromagnetic Green tensor \({{\mathcal{G}}}({{\bf{r}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})\), defined by ref. 41

$$\nabla \times \nabla \times {{\mathcal{G}}}({{\bf{r}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})-{k}_{{{\rm{W}}}M}^{2}{{\mathcal{G}}}({{\bf{r}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})=4\pi {k}_{{{\rm{W}}}M}^{2}\delta ({{\bf{r}}}-{{{\bf{r}}}}^{{\prime} }),$$

where kWM = ωWM/c. More precisely,

$${{{\bf{E}}}}_{{{\rm{WM}}}}^{\nu }({{\bf{r}}},{\omega }_{{{\rm{WM}}}})={\int _{V}}{d}^{3}{{{\bf{r}}}}^{{\prime} }\,{{\mathcal{G}}}({{\bf{r}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})\cdot {{{\bf{P}}}}_{{{\rm{WM}}}}^{\nu }({{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}}),$$

where we introduce the nonlinear polarization density defined in Eq. (1). In the far field, the output electric field becomes \({{{\bf{E}}}}_{{{\rm{WM}}}}^{\nu }({{\bf{r}}},{\omega }_{{{\rm{WM}}}})\approx {{{\bf{f}}}}_{{{\rm{WM}}}}^{\nu }({\Omega }_{\hat{{{\bf{r}}}}})\,{{{\rm{e}}}}^{{{\rm{i}}}{k}_{{{\rm{WM}}}}r}/r\), where \({\Omega }_{\hat{{{\bf{r}}}}}\) denotes the emission direction \(\hat{{{\bf{r}}}}\) and

$${{{\bf{f}}}}_{{{\rm{WM}}}}^{\nu }({\Omega }_{\hat{{{\bf{r}}}}})={\int _{V}}{d}^{3}{{{\bf{r}}}}^{{\prime} }\,g({\Omega }_{\hat{{{\bf{r}}}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})\cdot {{{\bf{P}}}}_{{{\rm{WM}}}}^{\nu }({{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})$$
(2)

is the far-field WMCL amplitude. In these expressions, the \({{{\bf{r}}}}^{{\prime} }\) integral extends over the volume V occupied by the nonlinear material, and the kernel in Eq. (2) is implicitly defined by the far-field limit (kWMr ≫ 1) of the Green tensor \({{\mathcal{G}}}({{\bf{r}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})\approx g({\Omega }_{\hat{{{\bf{r}}}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})\,{{{\rm{e}}}}^{{{\rm{i}}}{k}_{{{\rm{W}}}M}r}/r\). For a given emission direction, we use the reciprocity principle to conveniently obtain \(g({\Omega }_{\hat{{{\bf{r}}}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})\) from the near field produced by an incident light plane wave (see Methods). We note that Eq. (2) includes phase matching implicitly through the Green tensor, which rigorously accounts for wave propagation.

We calculate the nonlinearly emitted energy by integrating the far-field Poynting vector over time and emission directions \({\Omega }_{\hat{{{\bf{r}}}}}\). Plancherel’s theorem transforms the time integral into a frequency integral, which allows us to separate the emitted energy into frequency components. After dividing each of those components by the photon energy, we finally obtain the total emission probability as27\({\Gamma }_{{{\rm{WM}}}}^{\nu }={\int }_{0}^{\infty }\,d{\omega }_{{{\rm{WM}}}}\,{\Gamma }_{{{\rm{WM}}}}^{\nu }({\omega }_{{{\rm{WM}}}})\), where

$${\Gamma }_{{{\rm{WM}}}}^{\nu }({\omega }_{{{\rm{WM}}}})=\frac{c}{4{\pi }^{2}\hslash {\omega }_{{{\rm{WM}}}}}\int \,{{\rm{d}}}{\Omega }_{\hat{{{\bf{r}}}}}\,| {{{\bf{f}}}}_{{{\rm{WM}}}}^{\nu }({\Omega }_{\hat{{{\bf{r}}}}}){| }^{2}$$
(3)

is the probability distribution for emitting WMCL photons normalized per electron and unit of photon frequency ωWM for each of the three channels ν = SFG, DFG1, and DFG2 under consideration [see Eq. (1) and Fig. 1].

Light–electron wave mixing in a small particle

As a tutorial configuration, we study a specimen consisting of a spherical particle of small radius a made of a homogeneous, non-centrosymmetric material, such that it exhibits a bulk second-order nonlinearity. For simplicity, we only consider the dipolar response for a field orientation and geometry as depicted in Fig. 2a, with a nonlinear polarization density \({{{\bf{P}}}}_{{{\rm{WM}}}}={P}_{{{\rm{WM}}}}\hat{{{\bf{x}}}}\) associated with the xxx component of the second-order susceptibility tensor χ(2).

Fig. 2: WMCL and WM by a small particle.
Fig. 2: WMCL and WM by a small particle.
Full size image

a Sketch of the geometry under consideration, including a spherical particle of radius a that features a dipolar resonance at frequency ω0. The electron moves with velocity v and passes at a distance b from the particle center. The directions of electron and light propagation are indicated by arrows relative to the lower-left frame. b Probability of nonlinear WM photon emission \({\Gamma }_{{{\rm{WM}}}}^{\nu }\) normalized to ΓCL as a function of incident and output frequencies ωpump and ωWM normalized to ω0. The solid gray line divides regions of ν = SFG (ωWM = ωpump + ω) and ν = DFG1 (ωWM = ωpump − ω) emission (see labels). Pink and blue lines mark the main emission features. c Same as (b), but for the probability of nonlinear WM photon emission \({\Gamma }_{{{\rm{WM}}}}^{\nu }\) without normalizing to CL. d Dependence of \({\Gamma }_{{{\rm{WMCL}}}}^{{{\rm{SFG}}}}\) on damping η for ωpump and ωWM at the green dot in (b) (solid curve), compared with a ∝ η−4 dependence (dashed line). e, f Cuts of panels (b, c) along the color-coordinated lines (see legends). The gray solid curves represent the CL photon emission probability ΓCL. We set χ(2) = 10−10 m/V, Epump = 108 V/m, a = 10 nm, b = 12 nm, v = c/10 (2.6 keV), f = 1, ω0 = 0.1 eV, and η/ω0 = 0.01 in all panels [see Eqs. (4)-(6)]. In panels (b, c), the minimum and maximum values on the color maps are saturated to the values indicated in the corresponding legends.

We adopt the electrostatic limit (assuming that a is small compared with the relevant optical wavelengths) and describe the material through a generic Drude-Lorentz permittivity

$$\epsilon (\omega )=1+f{\omega }_{{{\rm{r}}}}^{2}/[{\omega }_{{{\rm{r}}}}^{2}-\omega (\omega+{{\rm{i}}}\eta )],$$
(4)

where f is a dimensionless factor, ωr is an intrinsic resonance frequency, and η is an inelastic damping rate. The optical and electron fields inside the particle are then uniform and approximately given by the external fields at the sphere center times a factor 3/(ϵ + 2) evaluated at the frequencies ωpump and ω, respectively. The near field is thus enhanced under the condition ϵ = − 2, or equivalently, when the frequency approaches the sphere resonance \({\omega }_{0}={\omega }_{{{\rm{r}}}}\sqrt{1+f/3}\), as obtained by directly applying Eq. (4). Likewise, the kernel of Eq. (2) reduces to \(g({\Omega }_{\hat{{{\bf{r}}}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})=\left(3{k}_{{{\rm{W}}}M}^{2}/[\epsilon ({\omega }_{{{\rm{WM}}}})+2]\right)\,(1-\hat{{{\bf{r}}}}\otimes \hat{{{\bf{r}}}})\) in the electrostatic limit (see Methods). Inserting these elements in Eqs. (2) and (3), the WMCL emission probability directly becomes

$${\Gamma }_{{{\rm{WM}}}}^{\nu }({\omega }_{{{\rm{WM}}}})= 3888\,\alpha \,{V}^{2}\,{I}^{{{\rm{ext}}}}\,| {\chi }^{(2)}{| }^{2}\,\frac{{\omega }_{{{\rm{WM}}}}^{3}{\omega }^{2}}{{c}^{3}{v}^{4}{\gamma }^{2}}\,{K}_{1}^{2}\left(\frac{\omega b}{v\gamma }\right)\\ \times \frac{1}{| [\epsilon ({\omega }_{{{\rm{pump}}}})+2][\epsilon ({\omega }_{{{\rm{WM}}}})+2][\epsilon (\omega )+2]{| }^{2}},$$
(5)

where Iext = (c/2π)∣Epump∣2 is the incident light intensity, V = 4πa3/3 is the particle volume, α = e2/ℏc ≈ 1/137 is the fine structure constant, and b is the e-beam distance to the particle center (see Fig. 2a). We consider only aloof electron trajectories that do not penetrate the sampled material. A nonlinear emission channel ν is selected through the choice of ωWM and the frequency parameters of χ(2) [see Eq. (1)]. Incidentally, from this analysis, it is clear that the WMCL probability scales as ∝ ∣χ(2)∣2Iext.

Since both regular CL and WMCL can give rise to the emission of photons with the same frequency, the ratio of their respective probabilities becomes a relevant quantity that we explore in Fig. 2. The CL probability is also given by Eqs. (2) and (3), but taking ωWM = ω and writing the polarization density as \({{{\bf{P}}}}_{{{\rm{WM}}}}^{{{\rm{ext}}}}({{{\bf{r}}}}^{{\prime} },\omega )={{{\bf{E}}}}_{{{\rm{electron}}}}^{{{\rm{ext}}}}(-b\,\hat{{{\bf{x}}}},\omega )\,[\epsilon (\omega )-1]/4\pi\) from the linear response to the electron field. This leads to

$${\Gamma }_{{{\rm{CL}}}}(\omega )=\frac{3\alpha \,{V}^{2}\,{\omega }^{5}}{2{\pi }^{3}{c}^{2}{v}^{4}{\gamma }^{2}}{\left| \frac{\epsilon (\omega )-1}{\epsilon (\omega )+2}\right| }^{2} \left[{K}_{1}^{2}\left(\frac{\omega b}{v\gamma }\right)+\frac{1}{{\gamma }^{2}}{K}_{0}^{2}\left(\frac{\omega b}{v\gamma }\right)\right],$$
(6)

which agrees with previous analyses42.

Figure 2b shows the \({\Gamma }_{{{\rm{WM}}}}^{\nu }({\omega }_{{{\rm{WM}}}})/{\Gamma }_{{{\rm{CL}}}}({\omega }_{{{\rm{WM}}}})\) ratio as a function of input and output light frequencies (ωin and ωWM) for the ν = SFG (ωWM > ωpump) and ν = DFG1 (ωpump > ωWM) channels, assuming a particle resonance ℏω0 = 0.1 eV, such that the e-beam can produce strong frequency shifts with a large probability (see also Supp. Fig. S1). Figure 2c shows the same data without normalization to the CL intensity, assuming typical orders of magnitude for the values of χ(2) ∼ 10−10 m/V43 (see Table S1 in the Supplementary Information) and Epump ∼ 108 V/m. In particular, this value for Epump is routinely reached in nonlinear experiments using short pulses (100s fs) without causing sample damage44,45,46. In both panels, we can identify some lines where the WMCL emission is boosted (marked in pink/red for SFG/DFG1 channels; see also cuts along these lines in Fig. 2e). These features correspond to resonances marked by the condition ϵ + 2 ≈ 0 in the denominator of Eq. (5) when either ωpump = ω0 (horizontal line) or ω = ω0 (oblique lines). Note that the ωWM = ω0 resonance is removed in the \({\Gamma }_{{{\rm{WM}}}}^{\nu }/{\Gamma }_{{{\rm{CL}}}}\) ratio because ΓCL is also peaked at that frequency [see Eq. (6)]. For completeness, Supp. Fig. S1 shows \({\Gamma }_{{{{\rm{DFG}}}}_{2}}({\omega }_{{{\rm{WM}}}})/{\Gamma }_{{{\rm{CL}}}}({\omega }_{{{\rm{WM}}}})\) and \({\Gamma }_{{{{\rm{DFG}}}}_{2}}({\omega }_{{{\rm{WM}}}})\) for the alternative channel where ωWM = ω − ωpump, which also exhibits resonances at ωpump = ω0 and ωpump + ωWM = ω0 (dashed and solid lines). Cuts along these lines in Supp. Figs. S1c, d reveal a behavior similar to that of DFG1 in Fig. 2e,f, respectively. Finally, we note that the apparent increase of ΓWM/ΓCL with ωWM in Figs. 2b,e (and ΓWM in Fig. 2c,f) originates from a broad high-energy resonance associated with \({\omega }_{{{\rm{WM}}}}^{3}{\omega }^{2}{K}_{1}^{2}(\omega b/v\gamma )\) in Eq. (5).

The boosts of WMCL induced by material resonances are strongly dependent on optical damping [η in Eq. (4)]. For the specific form of the permittivity in Eq. (4), each resonant denominator (∣ϵ + 2∣2 ≈ 0) in the WMCL probability [Eq. (1)] contributes a factor \(\propto {({\omega }_{0}/\eta )}^{2}\). In particular, at the double-resonance crossing signaled by the green dot in Fig. 2b (DFG1 with ωpump = ω = ω0), we expect a \({\Gamma }_{{{\rm{WM}}}}^{{{{\rm{DFG}}}}_{1}}\propto {\eta }^{-4}\), as corroborated by Fig. 2d.

We present above WMCL probabilities normalized to CL because photons generated by these two processes cannot be directly distinguished in an experiment. However, unlike CL, the WMCL contribution is proportional to the intensity of the external pump Iext, and thus, it can be resolved by switching the external illumination on and off (e.g., in a lock-in amplifier). In Supplementary Fig. S2, we show both ΓWM (summed over all nonlinear processes), ΓCL, and their sum as a function of the output light frequency ωWM, for ωpump = ω0 and different values of Epump. Additional channels not present in this figure correspond to the direct pump field scattered by the sample and its harmonics, which can be experimentally removed by filtering (analogously to the suppression of Rayleigh scattering in Raman scattering). In addition, the WMCL and pump-scattered signals generally exhibit different far-field angular and polarization patterns, suggesting that they could be angularly separated (e.g., using directional nanoantennas). Finally, harmonics of the electron field are also allowed, but we expect them to be too weak to be measurable.

While in Fig. 2 we maintain the analysis general for tutorial purposes, the results therein presented are readily applicable to nonlinear materials that are non-centrosymmetric (and therefore exhibit a nonzero bulk χ(2)) and whose permittivity is well described by the Drude-Lorentz model in Eq. (4). For example, we show in Supp. Table S1 a representative list of materials fitting these criteria, for which the results in this section readily apply. The table contains the material-dependent orders of magnitude of χ(2) within different spectral ranges47,48,49,50,51,52,53,54,55, along with references for fitting parameters of the Drude-Lorentz permittivity56,57,58,59,60,61,62,63,64. We note that a straightforward extension of the present theory can deal with permittivities constructed as a sum of multiple Lorentz oscillators, which can be more accurate for some of the materials in Supp. Table S1.

Detection of far-infrared molecular fingerprints in a plasmonic nanoparticle

We are interested in extending the WMCL analysis to a specimen hosting multiple resonances, and in particular, one at high energy supported by a metallic nanoparticle (enhancing the overall WMCL signal for input and output light in the visible domain), and another one at low frequency (down to the far infrared), driven by molecular fingerprints of an analyte. As a practical implementation of this idea, we consider a metallic particle coated with a molecular layer of retinal. The particle is aimed at producing a strong near-field enhancement of incident light at a visible resonant frequency. In contrast, the broadband electron field extends down to low frequencies and can be enhanced by the response of the molecules. The output WMCL signal should then be boosted for ωpump and ωWM close to a nanorod resonance, while its fine structure should reveal features at frequency shifts ωWM − ωpump corresponding to the molecular fingerprints.

We take a silver nanorod (length L = 100 nm, radius a = 10 nm, hemispherical caps) covered by a layer of retinal (thickness t = 5 nm), as schematically sketched in Fig. 3a. The system is illuminated by a light plane wave of frequency ωpump. An electron with velocity v passes near the structure as indicated in Fig. 3a. We focus on the SFG and DFG1 processes by which the electron gains or loses an amount of energy ℏω upon interaction with the sample, which results in the emission of a photon of frequency ωWM = ωpump ± ω. Since silver is a centrosymmetric material, its second-order nonlinear response is only coming from the nanorod surface, and is quantified by a nonlinear surface susceptibility χ(2) for which we only retain the dominant ⊥⊥⊥ component. Symmetry breaking in the applied electric field is produced by the position of the electron, near one of the ends of the structure. In our analysis, we use the boundary-element method65 (BEM) to calculate the self-consistent linear fields produced by the light and the electron, from which we obtain the WMCL probability through Eqs. (2) and (3) (see Methods).

Fig. 3: Probing low-energy excitations through WMCL.
Fig. 3: Probing low-energy excitations through WMCL.
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a Scheme of the structure under consideration, consisting of a silver nanorod (length L = 100 nm, radius a = 10 nm) coated by a retinal layer (thickness t). The structure is irradiated by a light plane wave of frequency ωpump and polarization along the rod. The electron moves with a velocity v = 0.45 c and passes at a distance b = 3 nm from one end of the coated rod. SFG and DFG produce output photon energies ωWM = ωpump ± ω through the second-order susceptibility \({\chi }_{\perp \perp \perp }^{(2)}({\omega }_{{{\rm{pump}}}},\pm \omega,{\omega }_{{{\rm{WM}}}})\) at the silver surface. b CL spectra for bare (t = 0) and coated (t = 5 nm) silver nanorods under the geometry depicted in (a). c Normalized probability of SFG and DFG photon emission as a function of electron energy change. We plot the quantity \({{\mathcal{R}}}(\omega )/{\xi }^{2}\) for bare and retinal-coated nanorods, with \({{\mathcal{R}}}(\omega )={\Gamma }_{{{\rm{SFG/DFG}}}}({\omega }_{{{\rm{WM}}}})/{\Gamma }_{{{\rm{CL}}}}({\omega }_{{{\rm{WM}}}})\), ω = ± (ωWM − ωpump), and \({\xi }^{2}=| {\chi }_{\perp \perp \perp }^{(2)}{E}_{{{\rm{pump}}}}{| }^{2}/A\) combining the surface area of the nanorod A = 2πaL and the incident light-field amplitude Epump. The upper horizontal scale indicates the output photon energy for the retinal-coated nanorod (blue curve), with the bare-nanorod counterpart (green curve) shifted by  + 184.5 meV. d Comparison of \({\partial }_{\omega }{{\mathcal{R}}}(\omega )\) to the imaginary part of the permittivity of retinal as a function of absolute electron frequency change ω = ∣ωWM − ωpump∣.

In Fig. 3b, we present the CL spectrum of the structure in Fig. 3a, exhibiting a strong dipolar resonance that shifts due to the presence of the retinal layer. For WMCL, we take the incident light frequency ωpump to match the maximum of the resonance in this plot for a given layer thickness. In Fig. 3c, we show the probability of emitting SFG and DFG photons as a function of electron energy change, normalized to both the CL emission probability and the dimensionless parameter \({\xi }^{2}=| {\chi }_{\perp \perp \perp }^{(2)}{E}_{{{\rm{pump}}}}{| }^{2}/A\), where A is the surface area of the nanorod. In Supplementary Fig. S3a, we present the same results without normalization, while in Supplementary Fig. S3b, we show analogous calculations, but for a silica coating instead of retinal. In all cases, we use an attainable pump field amplitude Epump = 108V/m (see discussion above) and \({\chi }_{\perp \perp \perp }^{(2)} \sim 4\times 1{0}^{-16}\) m2/V. The latter results from extrapolating the measured visible-range second-harmonic susceptibility for silver66,67 to the lower-frequency range here considered (see Methods). We concentrate on the electron energy loss and gain region in the  − 8 to 8 meV range [that is, in terms of the electron frequency ω = ± (ωWM − ωpump)], where retinal exhibits strong vibrational resonances. Such resonances are imprinted onto the WMCL spectrum of the structure with retinal, emerging as weak modulations in the normalized emission probability, which are instead absent from the spectrum for the bare nanorod. Note that, in contrast, the CL spectrum (without external illumination) to which we are normalizing the WMCL signal in this energy range is dominated by the silver nanorod, so it is featureless even in the presence of a retinal layer, which only produces a nearly rigid redshift of the plasmon resonance (Fig. 3b). The modulations in probability produced by the retinal vibrational modes on the WMCL spectrum (Supplementary Fig. S3a) have a similar order of magnitude as the CL peak (Fig. 3b), and although they are clearly visible, to make such features more prominent, we take the derivatives of the WMCL spectra with respect to ω = ∣ωWM − ωpump∣, and compare the result to the absorption spectrum of retinal (Fig. 3d), finding excellent agreement.

The WMCL signal can be improved by adequately choosing the electron trajectory to traverse the near-field hotspots produced upon illumination of the sample, as shown by the spatial map in Supplementary Fig. S4. This method can also be applied to other types of analytes. For example, for a silver nanorod coated with a thin silica layer, we find analogous features associated with vibrational resonances in the mid-infrared spectral range (see Supplementary Figs. S3b and S5). We thus conclude that WMCL is capable of identifying the chemical nature of analytes characterized by low-frequency spectral fingerprints by resorting to external illumination and light detection in the visible regime.

Discussion

In conclusion, we show that free electrons can be combined with light to produce WMCL and map low-frequency optical excitations by triggering a nonlinear response associated with sum- and difference-frequency generation. Unlike other low-frequency spectroscopy techniques, WMCL benefits from the broadband character of the field carried by fast electrons while up-converting far-infrared fingerprints to the visible, eliminating the need for low-frequency light sources and detectors, and thus, enabling spectroscopy with standard visible-range optical detectors in nanoscale specimens. This method can thus resolve far-infrared features with nanometer resolution, for which no other technique is available. Although we have considered uniform molecular coatings for simplicity, in a practical situation, WMCL could be applied to examining inhomogeneous molecular distributions, benefiting from the tightly confined electron field. In addition, WMCL is not limited to molecular vibrations, as the same wave-mixing principle applies to other low-frequency excitations, such as phonon-polaritons in the Reststrahlen bands of polaritonic materials (as shown in Supp. Fig. S5), free-carrier plasmons in doped semiconductors or graphene, and spin excitations (THz magnons) supported, for example, by some antiferromagnets. In all cases, low-frequency modes can be up-converted to visible sidebands, similarly to our demonstration, enabling nanoscale readout with standard detectors.

The successful design of experiments based on electron–photon wave mixing requires that the output signal originating from this process is larger than (and thus distinguishable from) the regular CL emission produced in the absence of external illumination. This condition depends on the second-order nonlinear susceptibility of the specimen or a neighboring structure (e.g., retinal and a silver nanorod in our illustrative example), while in addition, the WMCL intensity is proportional to the employed light intensity. The latter can be boosted by using synchronized ultrafast electron and laser pulses similar to PINEM and EEGS. Other strategies to increase the WMCL signal could rely on near-field enhancement (e.g., via pump field focusing or engineering of sample geometry and choice of materials), improving the second-order nonlinear response of the sample (i.e., higher χ(2) in optimized materials and geometries), and adjusting the electron trajectory to enhance interaction (e.g., using small impact parameters and aligning the trajectory to increase field overlap and phase matching).

Realistic candidate materials for WMCL experiments include non-centrosymmetric dielectrics with large bulk χ(2) and low loss at visible pump wavelengths, such as thin-film LiNbO3, GaP, and III-nitrides (AlN/GaN), which are standard platforms for efficient second-harmonic/SFG and, therefore, natural targets for WMCL68,69,70. In phonon-polaritonic dielectrics such as hBN, SiC, and α-MoO3, which provide strong mid-IR near-field enhancements, symmetry forbids bulk second-order processes, thus leading to the vanishing of WMCL in the electrostatic limit (e.g., for the small particle studied in Fig. 2). However, a sizeable wave mixing could be obtained in larger particles, like in the ones explored in Fig. 3, based on their second-order surface response71,72. In that direction, noble metals such as Au or Ag could be useful because they exhibit strong, well-characterized surface second-order responses at vis-NIR frequencies67. Heavily doped semiconductors (e.g., n-GaAs) are also plasmonic mid-IR candidates, although their field enhancement at low frequencies is limited by inelastic damping73,74,75,76.

Besides rendering spectral information on far-infrared excitations, WMCL should also provide quantitative measurements of the material-dependent second-order susceptibility for any of the materials listed above. Although we have restricted our analysis to second-order nonlinear processes, WMCL could also be extended to higher orders, which should become dominant in small centrosymmetric samples.

Methods

Calculation of the far-field electromagnetic Green tensor using the reciprocity principle

To calculate \(g({\Omega }_{\hat{{{\bf{r}}}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})\) as a function of far-field emission direction \({\Omega }_{\hat{{{\bf{r}}}}}\) and position \({{{\bf{r}}}}^{{\prime} }\) within a nanostructure, we consider a dipole \({{\bf{p}}}=p{\hat{{{\bf{u}}}}}_{1}\) oscillating with frequency ωWM, placed at \({{{\bf{r}}}}^{{\prime} }\), and oriented along the unit vector \({\hat{{{\bf{u}}}}}_{1}\). The electric field component along \({\hat{{{\bf{u}}}}}_{2}\) at a far-field position r (with \(r\gg {r}^{{\prime} }\)) is given by

$${\hat{{{\bf{u}}}}}_{2}\cdot {{\mathcal{G}}}({{\bf{r}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})\cdot {{\bf{p}}}\approx \left[{\hat{{{\bf{u}}}}}_{2}\cdot g({\Omega }_{\hat{{{\bf{r}}}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})\cdot {\hat{{{\bf{u}}}}}_{1}\right]\,p\,\frac{{{{\rm{e}}}}^{{{\rm{i}}}{k}_{{{\rm{W}}}M}r}}{r}.$$

By invoking the principle of reciprocity [i.e., \({\hat{{{\bf{u}}}}}_{1}\cdot {{\mathcal{G}}}({{\bf{r}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})\cdot {\hat{{{\bf{u}}}}}_{2}={\hat{{{\bf{u}}}}}_{2}\cdot {{\mathcal{G}}}({{{\bf{r}}}}^{{\prime} },{{\bf{r}}},{\omega }_{{{\rm{WM}}}})\cdot {\hat{{{\bf{u}}}}}_{1}\)], this quantity must coincide with the \({\hat{{{\bf{u}}}}}_{1}\) component of the field produced at \({{{\bf{r}}}}^{{\prime} }\) by a distant dipole \(p{\hat{{{\bf{u}}}}}_{2}\) placed at r. We only need to consider \({\hat{{{\bf{u}}}}}_{2}\) directions perpendicular to r because the far field is transverse. The external field produced by such a dipole near the structure is \({k}_{{{\rm{W}}}M}^{2}p{\hat{{{\bf{u}}}}}_{2}\,{{{\rm{e}}}}^{-{{\rm{i}}}{{{\bf{k}}}}_{{{\rm{W}}}M}\cdot {{{\bf{r}}}}^{{\prime} }}\,{{{\rm{e}}}}^{{{\rm{i}}}{k}_{{{\rm{W}}}M}r}/r\), where \({{{\bf{k}}}}_{{{\rm{W}}}M}={k}_{{{\rm{W}}}M}\hat{{{\bf{r}}}}\). From these considerations, we formulate the following prescription to calculate the components of the tensor \(g({\Omega }_{\hat{{{\bf{r}}}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})\): (1) we consider a plane wave illuminating the structure with wave vector  − kWM and unit electric field amplitude \({\hat{{{\bf{u}}}}}_{2}\); (2) we calculate the resulting self-consistent near field \({{\bf{E}}}({{{\bf{r}}}}^{{\prime} })\) as a function of \({{{\bf{r}}}}^{{\prime} }\); (3) we then write

$${\hat{{{\bf{u}}}}}_{2}\cdot g({\Omega }_{\hat{{{\bf{r}}}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})\cdot {\hat{{{\bf{u}}}}}_{1}={k}_{{{\rm{W}}}M}^{2}\,{\hat{{{\bf{u}}}}}_{1}\cdot {{\bf{E}}}({{{\bf{r}}}}^{{\prime} }).$$

To obtain the full tensor, this procedure needs to be repeated for \({\hat{{{\bf{u}}}}}_{1}=\hat{{{\bf{x}}}}\), \(\hat{{{\bf{y}}}}\), and \(\hat{{{\bf{z}}}}\) for all values of the emission direction, while \({\hat{{{\bf{u}}}}}_{2}\) can be set to the p and s unit polarization vectors corresponding to the emission direction \({\Omega }_{\hat{{{\bf{r}}}}}\).

For a sphere in the electrostatic limit, a unit incident field \({\hat{{{\bf{u}}}}}_{2}\) produces a uniform field \(3{\hat{{{\bf{u}}}}}_{2}/(\epsilon+2)\) inside the material, where ϵ is the permittivity at the frequency ωWM under consideration. From this identity, applying the procedure formulated above, we directly write

$$g({\Omega }_{\hat{{{\bf{r}}}}},{{{\bf{r}}}}^{{\prime} },{\omega }_{{{\rm{WM}}}})=\frac{3{k}_{{{\rm{W}}}M}^{2}}{\epsilon+2}\,(1-\hat{{{\bf{r}}}}\otimes \hat{{{\bf{r}}}}),$$

where we have inserted \((1-\hat{{{\bf{r}}}}\otimes \hat{{{\bf{r}}}})\) to account for the fact that the far field is transverse.

For the (coated) silver nanorod, we also follow the procedure above and calculate the near field upon plane-wave illumination using an implementation of the boundary-element method65 specialized for axially symmetric structures.

Material permittivities

We use the tabulated permittivity of silver taken from experimental measurements77. The permittivity of retinal is modeled as a sum of six Lorentzians78:

$$\begin{array}{r}{\epsilon }_{{{\rm{retinal}}}}(\omega )={\epsilon }_{\infty }+{\sum }_{j=1}^{6}\frac{{S}_{j}{\omega }_{j}^{2}}{{\omega }_{j}^{2}-\omega (\omega+{{\rm{i}}}{\eta }_{j})},\end{array}$$
(7)

with parameters ϵ∞ = 2.157, S1 = 0.018, ω1 = 46.6 cm−1, η1 = 5.2 cm−1, S2 = 0.011, ω2 = 54.4 cm−1, η2 = 4.7 cm−1, S3 = 0.001, ω3 = 61.0 cm−1, η3 = 2.8 cm−1, S4 = 0.002, ω4 = 66.2 cm−1, η4 = 3.4 cm−1, S5 = 0.001, ω5 = 69.3 cm−1, η5 = 2.4 cm−1, S6 = 0.008, ω6 = 90.6 cm−1, and η6 = 4.9 cm−1.

Nonlinear susceptibility for wave mixing of far-infrared and visible light

While literature values of \({\chi }_{\perp \perp \perp }^{(2)}\) for noble metals at vis-NIR frequencies are available66,67, the mixing of visible and far-infrared fields remains poorly studied79. To address this gap for the present proposal, we extrapolate vis-NIR measurements by considering the scaling of this type of process as a function of the mixed frequencies ω1 and ω2 (e.g., \({\chi }_{\perp \perp \perp }^{(2)}\propto {[{\omega }_{1}({\omega }_{1}+{\omega }_{2})]}^{-1}\) for SFG in a metal1). Considering typical values of \({\chi }_{\perp \perp \perp }^{(2)} \sim 1{0}^{-18}\) m2/V for second-harmonic generation with ℏω1 = 1.55 eV light in silver66,67, we expect the nonlinear susceptibility to be enhanced by approximately two orders of magnitude when one of the frequencies is shifted to the far infrared. More precisely, assuming this frequency scaling and taking ℏω2 < 8 meV while keeping ω1 at a similar value as indicated above (i.e., under the same conditions as in Fig. 3), \({\chi }_{\perp \perp \perp }^{(2)}\) is expected to scale by a factor ∼ 2ω1/ω2 ∼ 400. Accordingly, we use a value \({\chi }_{\perp \perp \perp }^{(2)}=4\times 1{0}^{-16}\) m2/V for wave mixing of far-infrared ( < 8 meV) and vis-NIR ( ∼ 1.6 eV) light to plot the WMCL probability in Supp. Fig. S3 under the conditions of Fig. 3.