Electron matter waves with internal torque

MainInvestigating the structure and dynamics of materials on atomic dimensions requires access to the motion of atoms and electrons in space and time1,2. If rotation is involved, torque must be applied3. Shaped laser pulses with a chiral electromagnetic field can exert angular momentum onto a material via the photon spin4, phase vortex beams5 or, in the most general case, optical beams with self-torque6,7,8. However, coupling the photon torque with an individual atom inside a material is challenging because the wavelength of laser light lies in the nanometre range, far above the size of atoms or molecules. Even if higher harmonics are applied6, their wavelength is still too long to probe or control the rotation of a material on the level of the single atoms or unit cells. In contrast to the photons of laser light, electrons carry a rest mass and can, therefore, easily have a wavelength in the picometre range, ten times smaller than an atomic size. Structured electrons in which the wavefunction is shaped in space and time9,10,11,12,13 are, therefore, useful for studying and manipulating materials on atomic scales14,15,16,17. In terms of angular momentum, researchers have so far demonstrated electron vortex beams18,19,20,21,22, femtosecond vortex pulses23,24, twisted currents25,26, and coils of charge and mass27. A matter wave with a time-dependent chirality remains to be observed, although it would be a useful and general concept for free-electron quantum optics and for the investigation and control of the ultrafast dynamics of angular momentum on an atomic scale.Here we report the creation and observation of electrons with an ultrafast internal torque. Figure 1 depicts our concept and experiment. We use an ultrafast transmission electron microscope with a Schottky field emitter source28,29 at an electron energy of E0 = 70 keV. Femtosecond laser pulses (green) hit the emitter tip (grey) and create a femtosecond electron beam. We use only ~0.01 electrons per pulse (~5 fA) to increase spatial coherence and reduce temporal space charge effects29. The formation and measurement of self-torqued electrons are realized by three steps (Fig. 1a). We first devise a chiral coherent quantum walk that creates correlated energy vortex states (light red area). We then let the wave packet disperse (light blue area) by propagation in free space. Finally, we characterize the resulting matter wave in energy and time.Fig. 1: Concept of generating an electron beam with internal torque.The alternative text for this image may have been generated using AI.Full size imagea, Experimental scenario. Ultrashort electron pulses (blue) are generated by photoemission with an ultrashort laser beam (green). Next, we apply a chiral quantum walk (light red area) to modulate the electron matter wave (blue) by an optical vortex beam (red). A SiN membrane (brown) provides the necessary interaction for discrete energy gain or loss. The blue rings indicate the resulting charge density and the spiral profiles indicate the phase of the de Broglie wave. Angular momentum becomes a function of electron energy E. Then, we apply dispersive propagation in free space (light blue area). The velocity difference Δv of the different electron partial waves with respect to the central velocity v0 separates them in time. In this way, we form an electron pulse with internal torque. b, Chiral quantum walk. In the interaction of a chiral laser photon with a non-chiral electron, energy gain or loss becomes proportional to gain or loss of OAM. E0 is the central electron energy and ℏω is the photon energy. The red arrows show several possible transition paths. c, Dispersion of the resulting electron partial waves after propagation. I, intensity; t, time. Electrons with different angular momentum (colours) are separated in the time domain.In our chiral quantum walk, we intersect the electron beam with a laser vortex beam at an ultrathin membrane as a modulation element to lock the angular momentum to energy and time. The resulting coherent laser–electron interaction30 imprints the matter wave of the electrons with a series of coherent sidebands in the energy–momentum domain30,31,32. To create energy sidebands with orbital angular momentum (OAM) of integer amount, we apply a first-order, left-handed optical vortex beam (red) that carries an optical OAM of ℓphℏ per photon, where ℓph = 1. The laser wavelength is λph = 1,030 nm, the frequency is ω = 2πc/λph and the photon energy is Eph = ℏω ≈ 1.2 eV, where c is the speed of light. The free-standing silicon nitride membrane (brown) changes the optical phase of the laser beam on transmission and, therefore, imprints an instantaneous momentum change onto the electron de Broglie wave33. If the temporal coherence of the electrons exceeds the optical cycle period34, spectral interference creates a series of discrete and coherent electron energy sidebands at integers of the photon energy30,31. In our chiral photon–electron interaction, angular momentum must be conserved, and the absorption of one chiral laser photon implies the creation of one quantum of OAM in the electron’s matter wave. Multiple photon absorption and photon emission pathways are known to create a complex shape of intensities31, but the correlation of the final energy with the final angular momentum should always be maintained. We, therefore, obtain a lock between OAM and energy; the nth electron partial wave at an energy of E0 + nℏω has in sum absorbed n photons from the optical vortex beam and should, therefore, carry an OAM of nℏ per electron (Fig. 1b). A negative n corresponds to an effective emission of chiral photons into the optical beam. After the quantum walk, we, therefore, expect that each sideband acquires a left-handed or right-handed helical phase of its matter wave (Fig. 1a, blue spirals) in proportion to the number of absorbed or emitted laser photons. The necessary transverse momentum changes are provided by the optical near field at the modulation membrane. Due to the optical singularity in the middle, the electron beam profile should acquire a doughnut mode18,19 of increasing clarity but of identical ring radius (Fig. 1a, blue rings).Next, the electron wavefunction acquires dispersion by propagation in free space. Higher-energetic parts travel faster in a vacuum than lower-energetic parts35. For small energy changes ΔE = nℏω with ℏω = 1.2 eV at around E0 = 70 keV, the temporal separation is Δt(n) ≈ ‒nLEphmel1/2[E0(γ + 1)]‒3/2, where L = 18 cm is the propagation distance, mel is the rest mass of the electron and γ = 1.137 is the Lorentz factor. This approximation produces an error of less than 0.04 fs for |n| 300 fs, the characterizing laser beam does not interact with any of the chiral sidebands, and we see the original spectrum from the generation stage (compare Fig. 2b). Around Δt = 0, the incoming chiral sideband spectrum gets additional broadening that emerges for different initial sidebands with distinct time delays (blue dots). The slight chirp that our initial femtosecond electron pulses have obtained from acceleration in the electron source28 is not relevant for this metrology (Extended Data Fig. 6, right column) because it only tilts the sidebands themselves, not the distribution of arrival times. A coherent cycling between sidebands has a minor influence on our arrival time measurements (Extended Data Fig. 7). Figure 3e,f shows the details of the measured arrival time data for final n = ±3 and n = ±4. A peak shift of 30 fs for the ±third energy sidebands (Fig. 3e) implies a 30-fs delay between the +second and the ‒second partial waves of the incoming shaped electron. The delay between the ‒third and the third partial waves is 45 fs (Fig. 3f). Figure 3g shows a summary of all time-domain results. We see a quantized linear delay of the sideband arrival time with respect to n that matches well with the analytical dispersion theory (dashed line). The slope of the fit (red) is \(n/\Delta t\) = (−0.11 ± 0.01) fs−1. Due to the proportionality between nℏω and nℏ (Fig. 2), angular momentum also arrives linearly with time.Therefore, the combined results show that the electrons in our experiment are converted into a matter wave with internal torque (Fig. 4a). The local chirality within our electrons changes from left handed at the beginning into achiral in the middle towards right handed in the end. Figure 4b,c shows the OAM and the internal torque as a function of arrival time of the electron wave at a material (Methods). The observed time dependencies and absolute magnitudes (blue) agree with the simulation results (black). The OAM changes from ‒5ℏ to 5ℏ within 400 fs at a peak torque of ~65ℏ/ps. In the spatial domain, this electron self-torque has a ring shape (Extended Data Fig. 8). Although the eigenstates of OAM are, in principle, integer multiples of ℏ, different sidebands partially overlap in time. Considering the complete electron beam, the relative weights of their superposition (Fig. 1c) make the instantaneous angular momentum continuous, and we obtain a smooth gradient with a fractional value of the effective, local angular momentum in time and space. This self-torque is an intrinsic property of each electron in the beam and will transfer to materials, photons or other electrons during each single-electron scattering event.Fig. 4: Electron matter waves with internal torque.The alternative text for this image may have been generated using AI.Full size imagea, Schematic of the phase of our single electrons, showing a left-handed chirality (L) at the beginning of the pulse and a right-handed chirality (R) later on. The instantaneous OAM is depicted with the colour scale. We assume here an incoming electron at the peak of the laser intensity in time. b, Reconstructed OAM (blue line) as a function of arrival time, compared with theory (dashed line). c, Reconstructed internal torque (blue solid line) as a function of t, compared with theory (dashed line). We assume a 20-fs coherence time, which is evaluated from a subcycle coherence measurement (Extended Data Fig. 9). Incoherent jitter of electron injection will shift the arrival time. The light blue areas show the statistical error of the experimental results (standard deviations).Our concept and experiment provide several ways to control the amount, range and timing of the angular momentum and, therefore, the properties of the internal torque. For example, if we adjust the intensity of the optical vortex beam or its phase ℓph, we can control the bandwidth of the generated sideband spectrum and, therefore, the range of angular momentum in the beam. If we apply an intermediate energy filter after beam formation, or control our electrons with multicolour laser fields40, we can control the internal shape of the sideband spectrum to customize the time dependence of the resulting self-torque wave. If we alter the central energy E0 of the electron beam or the propagation distance L, we can control how the shaped energy spectrum converts into an internal torque in time. If we adjust the coherence length of the electrons34, we can tune between a smooth gradient (Fig. 4b,c) and a temporally localized torque that changes step by step (Extended Data Fig. 10). Incoherent jitter of electron injection from the source into the laser beam will produce almost the same single-electron self-torque at shifted time. If we generate self-torque electrons with a continuous electron source and a continuous laser wave43, every single electron will still obtain internal torque. Our electrons can, therefore, be produced at almost the full beam brightness of modern electron microscopes. If we assume an optical vortex beam with ℓph ≈ 20 and fifth-order sidebands, we expect an OAM of ~100ћ per electron and a self-torque of ~1,300ℏ ps−1.Free-electron quantum optics31,41,44 and attosecond electron microscopy42 aim at studying and utilizing the quantized properties of electrons for understanding basic quantum mechanics and for investigating or controlling materials or photon states with shaped electrons. Our results contribute here by providing an electron matter wave with self-torque on femtosecond scales. In contrast to laser beams and photons with related properties6,7,8, our electrons have a substantially smaller wavelength below atomic diameters. Also, they carry a rest mass and an electric charge. Their interaction with chiral materials is, therefore, complementary to optical or X-ray experiments. Generally, any inelastic interaction of a shaped electron wave packet with a material will interlink the free-electron state with the quantum-mechanical properties of the material45. For example, if our self-torque electrons are absorbed or scattered by a single atom or molecule in scanning transmission electron microscopy from condensed-matter crystals46, we expect that each scattering process will exert a positive and then a negative torque onto an electronic state or a nuclear core in a very short time, inducing a finite impulsive rotational displacement that can serve as a trigger or probe of the subsequent evolution of the material. Post-selection of electron energy loss45 can reveal if a valence electron, an inner electron or a nuclear core has been modified or probed. Excitation of cold atoms with self-torque electrons and the collection of spontaneous emission could reveal the scattering paths of energy transfer47. Specially designed magnetic lenses can convert our beam into an ultrafast angular momentum streak in space48 that might be useful for manipulating the left and right parts of a nanostructure or molecule at different times49. A collision in time29 between differently shaped electrons will transfer time-dependent angular momentum through twisted Coulomb effects27. Sidebands with integer values of angular momentum under the control of laser light may serve as sensors or carriers of information in free-electron quantum technology41,50. If we make the laser–electron interaction much more efficient, for example, by whispering gallery modes51, we might become capable of seeing the imprint of the electron beam’s self-torque and OAM onto the emitted photons52. In general, the ability to shape an electrically charged and mass-bearing elementary particle like the electron into an almost arbitrary rotational form will enable to transfer almost any chiral-optical or dichroic principle of laser spectroscopy to the domain of electron microscopy, for studying chirality with atomic resolution in space and time.MethodsExperimentWe operate the experiments with an ultrafast transmission electron microscope with a Schottky field emitter source (JEM F200, JEOL) at an electron energy of 70 keV. At this energy, electron dispersion is substantial for centimetre-long propagation lengths. The femtosecond laser (CARBIDE, LIGHT CONVERSION) has a 1,030-nm wavelength, 2-MHz repetition rate and the pulses are 250 fs long. The laser beam is split into three paths. In the first path, the beam is frequency doubled with a beta barium borate crystal to trigger photoemission from the Schottky field emitter29. The electron pulse duration is ~270 fs (ref. 29) and the energy spread is about ~0.6 eV (full-width at half-maximum). In the second path, the laser beam is converted into an optical vortex beam by a spiral phase plate with ℓp = 1 (Vortex Photonics) and focused into the electron microscope with an f = 350 mm lens, similar to non-chiral beam-shaping experiments53. The focal spot size is ~200 µm (diameter of 1/e2 intensity). The SiN membrane is 50 nm thick. The polarization of the optical vortex beam points along the direction of the electrons, and the angle between the electron beam and the membrane is ~56°. This angle deviates from the velocity-matching angle θ = atan(c/vel) = 25.4° (ref. 33), where vel is the velocity of the electrons, and the electron direction is orthogonal to the laser direction. Therefore, no two-stage attosecond interferences are produced42. The slight velocity mismatch introduces a linear spatial phase delay but does not change the OAM of the electrons. The membrane is transparent to the laser beam. Its absorption is <10−5 and the temperature increase per laser pulse is <1 K. We did not see any damage throughout many months of experiments. Flat membranes directly imprint the laser’s spatiotemporal phase onto the electrons42, but more complex modulation elements are also useable. In the third laser path, an off-axis parabolic mirror focuses the beam onto another 50-nm-thick silicon nitride membrane at a distance of L = 18 cm. The probe laser is approximately a plane wave with a diameter of ~8 µm (full-width at half-maximum), and the average power is ~1 mW. Finally, the electrons are guided into a magnetic post-column electron energy analyser (CEFID, CEOS) and measured with an event-based direct electron detector (Timepix3, Amsterdam Scientific Instruments) that can measure every single electron54. We use a one-stage interaction (Fig. 2) and a two-stage interaction (Fig. 3). The electron beam from the source is filtered for less than a single electron per pulse with a 300-µm condenser-lens aperture. After laser interaction, the shaped electron beam is focused to the sample position, 18 cm after the modulation stage, and then magnified onto the screen. The electron beam diameter is ~300 μm at the modulation stage (image plane) and ~3 μm at the second stage (far-field plane), mediated by an effective focal distance of 17 m in a diffraction geometry. The convergence angle is roughly 0° at the first stage (image plane) and less than 0.01° at the second stage (far-field plane). The energy-filtered images shown in Fig. 2 are acquired with a 1-eV energy window in the energy plane of the spectrometer. In Fig. 3 and Extended Data Fig. 6, we scan the time delay of the probe laser in 5-fs steps, expose for 10 s and repeat the scan five times for each dataset. The spectrometer does not introduce an additional time delay. We use only ~0.01 electrons per pulse or ~104 electrons per second, corresponding to a beam current of ~5 fA, to achieve a high degree of spatial coherence in the beam. The number of electrons are measured on the detector. The integration time is 5 min for the spectrum shown in Fig. 2b, 30 min for each pattern shown in Fig. 2a,d, 10 s for the spectrum shown in Fig. 3b and ~1.5 h for the scanning of the data shown in Fig. 3d. Although we argue that our experiments reveal all necessary properties of self-torque electrons, a holographic reconstruction or attosecond electron microscopy could expose additional details. The creation of self-torque electrons requires a different velocity-matching condition, different interaction strength, different electron energy and different characterization methodology compared with the creation of electrons with chiral mass and charge27. Filtering one particular sideband yields a conventional, static electron vortex beam at a selectable angular momentum.Data processing and fitIn Fig. 3, we minimize the influences of laser and electron beam instabilities by normalizing the spectrum at each Δt by its total intensity. The probe laser only redistributes the population among sidebands but does not change the total number of electrons within each acquisition. Therefore, this normalization does not artificially change the evaluation of delays in any other than a random way. Then, we use a nonlinear least squares algorithm to fit the raw data in the time domain with a Gaussian-like function, yn = a1 × exp{‒[(Δt‒a2)/a3]2} + a4, where yn is integrated over an energy interval of 1 eV to select the nth-order sideband. The four fitting parameters, namely, a1, a2, a3 and a4, correspond to the amplitude, peak moment, standard deviation and intensity offset, respectively.Optical bandwidth and potential spatial dispersionThe bandwidth of our laser pulses is ~10 nm (full-width at half-maximum), which corresponds to a 12-meV energy difference when absorbed by the electrons. In the spectral domain, such an energy difference is trivial compared with our electron spectrum (Fig. 2b). In the time domain, the time-of-flight difference over 18-cm propagation between 70 keV and 70 keV + 12 meV is ~0.1 fs. Our results shown in Fig. 3g demonstrate that the measured time delay is two orders of magnitude larger than any artificial time delay that could be introduced by the spatial dispersion effects. Thus, a potential spatial dispersion of the optical vortex beam is not relevant to our results.Simulations of self-torque electrons in timeWe simulate the chiral quantum walk shown in Fig. 1 by solving the relativistically corrected Schrödinger equation, iћ∂tψ(z, t) = Ĥψ(z, t) (ref. 34). The Hamiltonian is given by Ĥ = Ĥ0 + ĤI, in which Ĥ0 = E0 + (p̂ ‒ p0)v0 + (p̂ ‒ p0)2/(2γ3mel) is the non-perturbed Hamiltonian and ĤI ≈ ‒eAzp̂/(γmel) is the interaction part. There is no substantial nonlinear, ponderomotive coupling at our optical field strength. Here ψ(z, t) is the electron wavefunction; z′ is a local coordinate around the centre of the beam, E0 is the central electron energy, \(\hat{p}\) is the momentum operator, p0 is the central electron momentum, v0 is the central velocity of the electrons, γ is the Lorentz factor, mel is the rest mass of electron, e is the charge of the electron and Az is the vector potential of the local laser field. After solving the quantum-walk interaction, we obtain a phase-modulated electron wavefunction ψ(z′). In the modulation, we account for multiple reflections between the front and back surfaces33. For propagation, we use a Fourier transform to obtain the electron wavefunction in momentum space, ѱ̃(k) = ∫∞–∞ѱ(z′)exp(‒ikz′)dz′. Replacing k with the electron energy, we obtain the spectrum of the electrons (Fig. 2b). Then, we solve the Schrödinger equation for free propagation, iћ∂tψ(k) = Ĥ0ψ(k) to obtain the electron wavefunction ψL(k) at a distance L. The electron wavefunction in temporal or longitudinal coordinates is calculated by applying the inverse Fourier transform, ψ(z′) = ∫k2–k1ψ(k)exp(ikz′)dk. By selecting the upper limit k1 and lower limit k2 of the integration, we can inspect the temporal properties at selected energy sidebands.Simulation of vortex ringsTo account for the spatial shape of the optical vortex beam, we first use a Laguerre–Gaussian mode for the incident optical vortex and then explicitly calculate the optical field distribution in the vicinity of the membrane by taking into account the refraction and reflection of the optical vortex at the membrane interface. The optical field amplitude and phase are given by the Laguerre–Gaussian modes of the laser beam and a sine2-shaped temporal envelope. We then solve the relativistically corrected Schrödinger equation for each point in the plane of laser beam. The resulting field, expressed as a function of time and spatial coordinates, is then used in the interaction Hamiltonian. After using the Jacobi–Anger expansion for the resolved time-domain wavefunction, we have an explicit mapping correlation between the sideband order and the angular phase gradient of electron:$${\psi }_{0}(x,y,z=0)=\mathop{\sum }\limits_{n}{J}_{n}(|g|){{\rm{e}}}^{\mathrm{in}\cdot {\text{arg}}(g)}\exp [-(E{\prime} -n\hslash \omega )/4{\sigma }_{E}^{2}],$$where g is the coupling constant, whose phase is arg(g) ≈ ℓphφ, inherited from the spiral phase of optical vortex, exp(iℓphφ) (φ, azimuthal angle). Jn is the Bessel function of the first kind and σE is the energy spread of the electron pulse. The above expression anchors the selection rule that the OAM of nth electron sideband is nℓph.We then simulate the spatial intensity of the electrons after free-space propagation by approximating the magnetic lens system in our electron microscope between the generation and characterization stage as a single focusing lens. Utilizing the Fraunhofer diffraction theory55, the electron wavefunction at the focal plane, r = (xf, yf, zf), is given by$${\psi }_{f}\,({x}_{f},{y}_{f},{z}_{f})=\frac{{{\rm{e}}}^{{\rm{i}}\alpha }}{{\rm{i}}{\lambda }_{\mathrm{el}}\,f}\iint {\psi }_{0}(x,y,z=0)\exp [-2\uppi{\mathrm{ i}}(x{q}_{x}+y{q}_{y})]{\rm{d}}x{\rm{d}}y,$$ (1) where f is the focal length of this effective lens; λel is the de Broglie wavelength of the electron; ψ0(x, y, z = 0) is the wavefunction in the incident plane of the effective lens, obtained by solving the relativistically corrected Schrödinger equation at each point in the image plane; qx = xf/(fλel) and qy = yf/(fλel) are the components of the spatial frequency; and α = [πh′p(z)/λelhp(z)](xf2 + yf2) in which hp(z) is a particular solution of paraxial equation of the electron trajectory in the vicinity of the optical axis. We optimize f to fit the ring size of the simulation result with our measurement. The results are plotted in Fig. 2e. Rings of increasing diameter can also be obtained in a beam with spherical aberrations and defocus, but the needed aberration coefficients would be unrealistically large.Simulation of electron profiles between the image and far-field planeWe use the Huygens–Fresnel principle to simulate the electron beam profiles:$${\psi }_{s}({x}_{s},{y}_{s},{z}_{s})=\frac{1}{{\rm{i}}{\lambda }_{\mathrm{el}}}\iint \psi (x,y,z=0)\frac{{{\rm{e}}}^{{\rm{i}}{k}_{\mathrm{el}}R}}{R}K(\,\chi ){\rm{d}}x{\rm{d}}y,$$ (2) where R = [(xs − x)2 + (ys − y)2+zs2]1/2 describes the distance between a source point (x, y, 0) and an observation point (xs, ys, zs), ψ(x, y, 0) = ψ0(x, y, 0)exp(−ikelRf) is the incident wavefunction of the electron matter wave, exp(−ikelRf) denotes a spherical phase accounting for the focusing of the electrons by the effective magnetic lens, Rf = (x2 + y2 + f2)1/2 is the distance between a source point (x, y, 0) and the lens focus (0, 0, f), and K(χ) = 1 + zs/R is the obliquity factor. To calculate the rapidly oscillating exponential term exp(−ikelRf) and exp(−ikelR), we expand them into Taylor series and keep the first four terms. As a result, equation (2) is written as$$\begin{array}{l}{\varPsi }_{s}({x}_{s},{y}_{s},{z}_{s})\\ =\displaystyle\frac{{{\rm{e}}}^{{\rm{i}}{{k}_{\mathrm{el}}}^{({Z}_{s}-f\,)}}}{{\rm{i}}{\lambda }_{\mathrm{el}}}\displaystyle\iint {\varPsi }_{0}(x,y,z=0)\displaystyle\frac{1}{R}\left(1+\displaystyle\frac{{z}_{s}}{R}\right)\\\quad\,\,{{\rm{e}}}^{-{\rm{i}}{{k}_{\mathrm{el}}}{\left(\displaystyle\frac{{\rho }_{f}^{2}}{2f}-\displaystyle\frac{{\rho }_{f}^{4}}{8{f}^{3}}+\displaystyle\frac{{\rho }_{f}^{6}}{16{f}^{5}}\right)}}{{\rm{e}}}^{{\rm{i}}{{k}_{\mathrm{el}}}{\left(\displaystyle\frac{{\rho }^{2}}{2{z}_{s}}-\displaystyle\frac{{\rho }^{4}}{8{z}_{s}^{3}}+\displaystyle\frac{{\rho }^{6}}{16{z}_{s}^{5}}\right)}}{\rm{d}}x{\rm{d}}y,\end{array}$$ (3) in which ρf and ρ are given by ρf = (x2 + y2)1/2 and ρ = [(xs − x)2 + (ys − y)2]1/2. The simulation results are displayed in Extended Data Fig. 2.Simulation of the effect of spatial coherenceWe estimate the effect of a limited spatial coherence by simulating electron beam profiles If(xf, yf, zf) with different finite coherence lengths according to an incoherent superposition$$\begin{array}{l}{I}_{f}\;({x}_{f},{y}_{f},{z}_{f})=\displaystyle \iint \left| \displaystyle\frac{{\rm{e}}^{{\rm{i}}\alpha}}{{\rm{i}}{\lambda}_{\mathrm{el}}\;f}\displaystyle\iint {\varPsi}_{0}(x,y,z=0)g(x{\prime},y{\prime}) \right.\\ \left. \qquad\qquad\qquad\quad {\rm{exp}}[-2{{\pi}i}(x{q}_{x}+y{q}_{y})]{\rm{d}}x{\rm{d}}y\right|^{2}{\rm{d}}x{\prime}{\rm{d}}y{\prime},\end{array}$$ (4) where g(x′, yʹ) is the coherent spatial envelope and (xʹ, yʹ) is the peak position in incident plane. Here g(xʹ, yʹ) is given by a Gaussian function$$g(x{\prime} ,y{\prime} )=\exp \left\{-\frac{4\,\mathrm{ln}2}{{\sigma }^{2}}\left[{(x-x{\prime} )}^{2}-{(\,y-y{\prime} )}^{2}\right]\right\},$$ (5) where σ denotes the coherence length. Results are plotted in Extended Data Fig. 5. A match to the experiment is found for \(\sigma \approx\) 70 µm. An independent estimation of the degree of coherence from the known brightness of Schottky field emitter sources (~107 A m−2; ref. 28) and the number of electrons per pulse in our experiment (~0.01) yield comparable results.Reconstruction of OAM and self-torqueIn Fig. 4, we extract the intensity of all electron partial waves from Fig. 2b. The timing of the partial waves is read from Fig. 3g, where the missing experimental data for the ±first and zeroth partial waves are obtained by interpolation and theory (dashed line). This intensity and timing together with an estimated coherence time34 are then used to reconstruct the OAM and the internal torque.Confusion matrix and purity analysisWe first fit our data (Fig. 2b) with an 11-peak Gaussian formula, f(ΔE) = ∑iAiexp{−4ln(2)[(ΔE − μi)/σi]2}, where i is the sideband order, A is the amplitude, μ is the peak position and σ is the full-width at half-maximum. The fits and fitting results are displayed in Extended Data Fig. 4e–g. For each order, we apply a 1-eV window centred at its peak (Extended Data Fig. 4e, grey area), and then calculate the confusion matrix by integrating over the window for each sideband order. Finally, the confusion matrix is normalized along each column.

Extended Data Fig. 4f shows the resulting 11 × 11 confusion matrix. It is strongly diagonal; typical purities exceed 0.97 for almost all orders.