Introduction

Electron microscopes are an indispensable analytic tool, bringing analytic imaging down to an atomic resolution. The rise of the laser-triggered electron (e⁻-)microscope [1–4] added ultrafast dynamics capabilities to the sub-nm spatial sensitivity [5–8] and opened a viable path towards quantum entanglement between free electrons and photons [9–11]. In such instruments, the temporal features of the electron pulses determine the accessible physics. Shrinking the duration of e⁻-pulses can be typically done using RF cavities [12–14] and laser-driven THz fields [15–18]. Alternatively, optical fields can structure the intra-pulse features dramatically by PINEM (Photon-induced nearfield electron microscopy) [19–22]. Under intense laser illumination, the electron forms periodic micro-pulses as short as a femtosecond or even reaching the attosecond scale, separated by the optical-phase cycle [23–26]. Investigation of quantum aspects of PINEM revealed it constitutes a coherent modulation of the e⁻-wavefunction, which energetically extended to exchanges of hundreds of photons [27–29], can shape the e⁻-wavefunction spatially [30–32], and enable subwavelength field holography [33,34]. Such sub-optical-cycle tailoring of a nano-focused e⁻-beam is a novel probe that is phase-locked to the dynamics driven by the same laser [35,36].

A recent, potentially transformative, theoretical prediction suggests exerting quantum-optical control at the atomic scale if electron pulses are bunched both globally and internally. The FEBERI scheme (Free-electron bound-electron resonant interaction) [37–39] claims that if multiple electrons shaped to trains of attosecond pulses pass by a quantum system, they can induce coherent excitation nonlinearly at a frequency defined by the micro-pulse separation, that is, the cycle of the PINEM-driving laser. For a two-level system, the transition amplitude is predicted to be proportional to the number of FEBERI-structured electrons, N. Thus, the transition probability scales as N² or, more generally, as the $\sin ^2\left(g_{Q u} N\right)$ of a Rabi-oscillation cycle, where ${g_{Qu}}$ is the quantum coupling. Doing so within an electron microscope could allow the manipulation of individual quantum systems with high spatial selectivity, in free space and without any physical probe. The bunch duration matters. The N FEBERI electrons should arrive within the dephasing time of the quantum system for their contribution to build up coherently. Temporal compression can enable access to drive short-lived excitations and compensate for the e⁻-pulse Coulomb broadening [40,41]. Hence, FEBERI has a particular set of constraints: (i) high-quality beam for nanoscopic focusing (ii) laser-electron interaction for PINEM (iii) electron compression.

Using few- or single-cycle laser-pumped THz pulses for compressing the electrons is appealing for integrating within an ultrafast electron microscope since it is compact, inherently timed with laser-triggered e⁻-pulses, and a THz cycle fits the duration of short electron pulses (∼200-700 fs [40–42]). Intense terahertz waves are generated from the optical rectification of short pulses in lithium niobate (LiNbO₃) [46]. The radiation forms off-axis beams which are collected and re-focused with high-numerical-aperture (high-NA) optics onto the target. The geometry of the pumping laser pulse, the crystal, and the THz collection play an intricate role in optimizing the THz throughput [45,46]. For a given pump energy and duration, the chosen geometry is dictated by the limit on the peak intensity due to multi-photon absorption. At 1 µm pump wavelength, the limiting intensity is 20-100 GW/cm² [43], above which the THz efficiency diminishes [44]. By focusing a pulse with a tilted front into a LiNbO₃ [39,40] prism the electric fields reach above MV/cm in the few-THz regime [47,48]. But since the efficiency of tilted-front pumping drops for pulse energies below the millijoule range [49], it operates at low repetition rates of one or a few kHz. More recent schemes propose THz generation from pulses propagating in a LiNbO₃ slab, befitting pulses with up to 200 µJ approximately [50,51]. A slab geometry enables either a compensation of the THz-phase jitter [52,53] or an efficient heat dissipation through its surface [50]. Thus, allowing the THz to be pumped by higher average power, that is, with a higher repetition rate.

While the term THz broadly refers to 0.1-100 THz, only sub-THz is relevant for compressing e⁻-pulses with an initial duration of a few hundred femtoseconds [40]. However, delivering radiation in the sub-THz regime is particularly challenging. A detailed quantitative analysis by Tsarev et al. [50], shows that the radiated power efficiency scales cubically with the THz frequency, and the focused power density scales as the fifth power(!) due to the diffraction limit. For e⁻-beam manipulation, the problem is further exacerbated if high-NA optics cannot be used to reach a diffraction-limited focal spot. Since light can be trivially focused to a spot size of a few micrometers, laser beams can provide dramatically higher energy densities. The beat note of such a tightly focused bi-chromatic laser was suggested as a means to compress a portion of the electrons in a bunch [54]. The few-mm region of addressable e⁻-beam in electron microscopes poses a standing issue as a barrier for compression of e⁻-pulses using THz-fields, especially for e⁻-pulses longer than a picosecond.

This work presents a conceptual change for laser-driven sub-THz compression of ps e⁻-pulses: instead of radiation, having a direct interaction between the electrons and the laser-induced dipolar nearfields. Avoiding an intermediary energy conversion to propagating waves omits the unfavorable frequency scaling of generating and transporting sub-THz radiation. We address this topic analytically and numerically. First, the compressive strength of nearfields from µJ-level pulses in LiNbO₃ is compared against optimal radiation and refocusing of THz. The analytic comparison is conducted for quasi-static nonlinear polarization induced in LiNbO₃ by the infrared driving pulse. The approximation holds for electron energies below 5 keV within the chosen parameter regime but provides a rough estimation up to tens of keV. The calculation is benchmarked for optical pulses with 1 µJ energy and a frequency of 0.1 THz (100 GHz). For lower frequencies and small NAs, the nearfield-based e⁻-compression is better by an order of magnitude due to the favorable frequency scaling. We describe an optical pumping scheme that maintains the process efficiency for more energetic infrared pumping. Second, we show numerical calculations that quantify the e⁻-compression by THz nearfields. As an example, we find that fields in this approach can reach 2.4 kV/cm at the challenging regime of 0.1 THz, pumped above the intensity that would saturate THz-emitting crystals. However, we emphasize that our motivation is not to reach the highest THz field, but rather to find a laser-locked approach with a favorable scaling for experiments with tight constraints. We believe that this small-scale scheme opens a path towards in-situ focusing of e⁻-pulses which is imperative for the coherent interaction of multi-electrons with nanoscopic quantum systems.

The outline of this paper is as follows: first, we present the proposed geometry and the analytic derivation for the compressive force using the nonlinear dipolar nearfields in LiNbO₃ and compare the analytic results to the full numerical calculation. Then we compare nearfields to radiation-based electron compression and find the regimes for which the nearfields are superior. We finish by suggesting extensions at higher pumping energies which are unique to the nearfields approach.

Results and discussion

Analytic derivation

THz generation with µJ infrared laser pulses in LiNbO₃ is optimal when implemented with a slab geometry, with a silicon output coupler, where the radiation is ideally collected and refocused by high-NA optics. The temporal profile of the e⁻ energy gain and the resulting e⁻-pulse compression is compared between a direct interaction with the nearfield (Fig. 1(a)) and an optimal scenario of radiation from such a slab (Fig. 1(b)), which interacts with the electron pulse on a distant membrane. To eliminate higher-frequency components we consider a 10-ps-long laser pulse focused near the surface of a Y-cut LiNbO₃ crystal, where the c-axis parallel to the surface. The slab geometry allows for efficient cooling of the LiNbO₃, which handles the thermal load of a high-repetition-rate laser operation and suppresses absorption by thermal phonons [50,55].

 

Fig. 1. Illustration of the proposed paradigm for electron compression. (a) The quasi-static nearfields of an optical rectification polarization (${P_{OR}}$) in LiNbO₃ interact with a traversing electron. The inset shows the calculated electric field (arrows) and potential (colormap) near the induced dipole. The temporally varying field induces a linear velocity-time correlation that compresses the electrons to a short pulse downstream. (b) A typical scheme for e⁻-pulse compression by a THz-irradiated membrane. We consider a slab LiNbO₃ crystal pumped by an elliptical optical beam (FWHM semiaxis marked ${W_x}$, ${W_z}\; $), from which a THz field is emitted at an angle γ and transferred with high-NA optics.

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The analytic derivation of the e⁻-pulse compression by nearfields vs. THz radiation required several assumptions. (i) The electron propagates along the z-axis of the semi-infinite LiNbO₃ crystal. The crystal spans the region $z,y \in ({ - \infty ,\infty } )$ and $x \in ({ - \infty ,0} ]$. (ii) The fields surrounding the laser-induced polarization are quasi-static. We define the regimes for which this assumption holds at a later point. (iii) The problem is two-dimensional. The confocal parameter of the laser and the distance to its entrance to the crystal are much larger than the spacing between the e⁻-trajectory and the crystal. (iv) The spatial profile of the optical pulse is assumed to be a perfect cylinder, creating a flat-top polarization region with a radius ${R_0}$ throughout the crystal depth. Thus, we account only for the two-dimensional dipolar density, without considering the laser beam diffraction or possible interference that a Gaussian pulse would have in proximity to the crystal edge. (v) The temporal profile of the optical pulse is a Gaussian with a full-width-at-half-maximum (FWHM) of ${T_{FHWM}} = 10\; ps$.

Importantly, when approximations are required to keep the calculation analytic, we consistently underestimate the effect of the nearfields while overestimating the radiation. Thus, our quantitative comparative conclusions here are conservative and can be considered the lowest bound.

For generating the nonlinear sub-THz polarization, we consider a laser with a peak intensity of ${I_0} = 20\frac{{GW}}{{c{m^2}}}$ in the LiNbO₃ with a spatial profile of a 25-µm-diameter cylinder and a temporal FWHM span of 10 ps, which has an energy of 1 µJ. Thus, these parameters are a convenient reference. The polarization induced by the optical rectification along the crystal c-axis is ${P_{OR}}(t )= 2{d_{eff}}\frac{{2I(t )}}{{c{n_p}}}\; $, where ${n_p}$ is the laser’s refractive index, ${d_{eff}}$ is the nonlinear coefficient of LiNbO₃ [56,57], c is the speed of light, and $I(t )= {I_0}\exp ({ - 4\ln 2{t^2}/T_{FWHM}^2} )$ is the instantaneous optical intensity. The term $\frac{{2I(t )}}{{c{n_p}}}$ is the square of the electric field that creates the difference-frequency polarization (Eq. (1) and Eq. (23), Ref. [58]) times the vacuum permittivity, ${\varepsilon _0}$. As illustrated in Fig. 1(a), the electron propagates purely in the vacuum subspace, $x > 0$. The quasi-static potential is calculated using the image dipole contribution [59], $\mathrm{\Phi }({x > 0,z,t} )={-} \frac{1}{{2\pi {\varepsilon _0}{\rho ^2}}}\frac{2}{{1 + {\varepsilon _r}}}({\; {{\vec{p}}_{2d}}(t )\cdot \vec{\rho }} )$. $\vec{\rho }$ is the two-dimensional radius vector from a dipole to the point of observation, ${\rho ^2} = {x^2} + {z^2}$ is the square of the radius vector length, ${\varepsilon _r} = n_{THz}^2$ is the relative dielectric constant of LiNbO₃, ${\vec{p}_{2d}}(t )= \pi R_0^2{P_{OR}}(t )\hat{z}$ is the macroscopic nonlinear polarization per unit depth, measured in Coulombs, and $\hat{z}$ is the unit vector pointing parallel to the crystal’s c-axis. The field component relevant for e⁻-pulse compression is the spatial derivative of the quasi-static potential along the propagation axis,

The electron on-axis acceleration depends on the energy it accumulates throughout its path,

Here, q is the electron charge and ${z_{(t )}}$ marks the electron trajectory, simplified as one-dimensional. The energy gain varies with the electron timing $\tau $, and its derivative, $d{U_e}(\tau )/d\tau $ is the figure of merit for e⁻-pulse compression. We also refer to this figure of merit as the compressive strength since a higher value shortens the resulting e⁻-pulse duration and the necessary propagation for reaching a full compression. $\tau $ is the relative delay between the electron passage and the optical pulse, such that $\tau = 0$ represents an electron at $z = 0$ when the nearfield is maximal. For an electron traveling with velocity ${v_e}$ along the z axis, the trajectory is ${z_{(t )}} = {v_e}({t - \tau } )$. Temporally, the field exists for approximately ${T_{FHWM}}$, during which the electron passes a finite distance, ${v_e}{T_{FWHM}}$. Thus, we simplify the integral by assuming a constant field within the period of ${v_e}{T_{FWHM}}$, ${E_z}({{L_d},z,\tau } )\to \frac{1}{2}{E_z}({{L_d},0,\tau } )$, resulting in an underestimated energy gain of ${U_e}(\tau )= ({ - q} ){v_e}{T_{FWHM}}\frac{1}{2}{E_z}({{L_d},0,\tau } )$. The maximal value of $d{U_e}(\tau )/d\tau $ for a temporal Gaussian envelope is ${\left. {\frac{{d{U_e}}}{{d\tau }}} \right|_{max}} = {e^{ - \frac{1}{2}}}\sqrt {8\ln 2} \frac{{{U_{e,max}}}}{{{T_{FWHM}}}} = {e^{ - \frac{1}{2}}}\sqrt {8\ln 2} q{v_e}\frac{1}{2}{E_z}({{L_d},0,0} )$. Figure 2(a) shows the ${E_z}$ component (log scale) along the electron path for energies up to 40 keV for the analytic approximation. Figure 2(b) shows the numerically calculated fields. Substituting the optical rectification dipole we find

 

Fig. 2. Compression dynamics of electrons in the nearfield. The graphs describe the reference example, pumped by a peak optical power of 20 GW/cm², pulse duration of 10 ps, focal diameter of 25 µm located in LiNbO₃, 50 µm from the electron trajectory. (a) Quasistatic and (b) full dynamic calculation of the parallel component of the electric field ${E_z}$ (absolute value, logarithmic color scale) vs. the distance from the dipole and the electron kinetic energy. The dashed white line marks 1 keV on the zoomed insets. The field distribution in (b) is asymmetric due to the radiation field. (c) The cumulative energy gain for electrons traveling through the nearfield vs. their propagation coordinate. Each curve marks different electron timing, ${\tau _e}$, with respect of the peak energy gain (or loss). (d) The final e⁻-energy gain ${U_e}$ vs ${\tau _e}$. The region with $d{U_e}/d{\tau _e}$ > 90% of the maximum defines the threshold duration ${\tau _{th}}$=2.6 ps. A dashed black line tangent to the maximal slope is added for clarity. Both (a)-(b) describe electrons at a kinetic energy of 1 keV. (e) Temporal focus length (${L_{focus}}$, logarithmic scale) for the compression of e⁻-pulse vs. their kinetic energy. The focal length extracted from the time-domain calculation of the electric fields in COMSOL (red curve) converges to the quasi-static approximation (blue curve) at low electron energies. The propagation length is as short as a few cm for e⁻-energies <1 keV at 1 µJ optical pump energy. We note in the text that 1 µJ is not a limit and, for example, electrons at 5 keV for which ${L_{focus}} = 3.3\; m\; @\; 1\mu J$ have a focusing distance $= 33\; mm\; @\; 100\; \mu J$. At 11 keV the electron velocity matches the THz wave velocity, 0.204 c, corresponding to a refractive index of 4.9 in the LiNbO₃.

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Figure 2(c) shows the energy accumulated as the electron traverses the laser-driven nearfield region, numerically. It is calculated for an electron with a kinetic energy of 1 keV. Each curve represents a different timing, $\tau $. The final energy, $U_e^{nearfield}(\tau )$, is approximately a Gaussian (see Fig. 2(d)), matching the laser pulse envelope. This stems from the e⁻-energy being well within the regime that matches the quasi-static approximation (see curve coalescence in Fig. 2(e)). The inflection point of the energy-gain shifts temporally from that of the optical pump (See Fig. 2(d)), i.e., the quasistatic approximation, however, we ignore this constant timing shift since it is trivially compensated for by delaying the optical pump. Figure 2(e) compares the compression figure of merit between the quasistatic approximation and a full dynamical calculation of the fields near the surface of the crystal surface. The parameters are optical pumping with 1 µJ energy and a FWHM duration of 10 ps. We express the results in terms of the optimal (shortest) temporal focal length, ${L_{focus}}$, which is given by ${L_{focus}} = {\left( {{{\left. {\frac{{d{U_e}}}{{d\tau }}} \right|}_{max}}} \right)^{ - 1}}\gamma _r^3{\beta ^3}{c^3}\frac{{{m_e}}}{q}$. This focal length is the spatial propagation at which the electron-pulse duration is compressed to a minimum if it is initially dispersionless [60]. Thus, it is a useful parameter in designing experimental layouts. Here, $\beta $ is the unitless relativistic parameter for the velocity, ${\gamma _r}$ is the relativistic Lorentz factor, and ${m_e}$ is the electron mass. Figure 2(e) shows that for kinetic energies below 5 keV the focal length reduces dramatically (note the logarithmic scale) and the exact calculation of the fields in COMSOL converges to the quasi-static calculation. The calculation details are in Supplement 1, section S.C. In the example we discuss at a later point, the electrons are accelerated to 1 keV (${v_e}$=0.0625c), well within the approximation’s validity range. The exact calculation (red) exhibits a resonant-like deviation from the quasi-static effect (blue) at 11 keV, originating from the match between the electron velocity at these energies and the sub-THz wave velocity in LiNbO₃ (0.204c). Overall, the quasi-static approach provides a good estimation for the compression. For electrons up to 30 keV, the quasi-static calculation deviates by a factor of 3 at its worst.

Although the nearfield compressive scheme is non-harmonic, a comparison to radiation fields necessitates a definition of an effective frequency. An ideal compression of an electron pulse requires a uniform compressive strength, that is, a uniform energy-gain gradient. Thus, we define the frequency by the duration for which the compressive strength is higher than 90% of the maximum value. For harmonic fields, 1/7 of the cycle complies with such a requirement. Thus, we calculate the region for which the nearfield’s energy gain gradient is ${d_\tau }{U_e} > 0.9{({{d_\tau }{U_e}} )_{max}}\; \; $, and define that duration as 1/7 the effective period. For a Gaussian temporal profile with a FWHM duration ${T_{FWHM}}$, the criterion is met for $0.26\; {T_{FWHM}}$, so the effective THz frequency is $f_{THz}^{eff} = 0.55/{T_{FWHM}}$. Thus, nearfields pumped by 10 ps pulses can compress e⁻-pulses as long as 2.6 ps, which is a threshold duration, ${\tau _{th}},$ (see marked in Fig. 2(d)).

Nearfields vs. radiation for e⁻-pulse compression

To define the relative improvement of the nearfields we find an explicit closed-form expression for the figure of merit for e⁻-compression from THz radiation, based on a slab source at optimal conditions,

The detailed derivation is in Supplement 1 Section S.B. ${W_x},{W_y},{W_z}$ are the spatial width of the pump beam and $\gamma $ is the off-axis angle of the THz emission, all of which are marked in Fig. 1(b). ${T_{rad}}$ is the idealized transmission coefficient of the THz power from the LiNbO₃ crystal to free space, ${({NA} )_{THz}}$ is the numerical aperture of the THz focusing optics, and ${\lambda _{THz}}$ is the THz wavelength in a vacuum. Thus, the ratio between the approaches for e⁻-pulse compression is

We can now consider a specific scenario and acquire the added value of the nearfield approach, quantitatively. For LiNbO₃ at a frequency below 0.1 THz ${n_{THz}} = 4.9,\; {\varepsilon _r} = n_{THz}^2$, and $\cos \gamma = 2.3/4.9$. The direct-incident power outcoupling is ${T_{rad}} = 0.43$, thus, the above ratio is $\frac{{{d_t}U_e^{rad}}}{{{d_t}U_e^{nearf}}} \approx 238\frac{{{{({NA} )}_{THz}}{W_x}\sqrt {{W_z}{W_y}} }}{{\lambda _{THz}^2}}{\left( {\frac{{{L_d}}}{{{R_0}}}} \right)^2}$ (see Supplement 1 Section S.B). The radiative contribution is governed by the focusing conditions and the source size, while the nearfield effect is governed by one parameter, the distance of the e⁻-path from the crystal surface with respect to the radius of the nearfield dipole, ${L_d}/{R_0}$. Note that the electron velocity is implicitly reflected in ${L_d}$, which is the distance for which the nearfield of a static dipole is approximately constant over a length ${v_e}{T_{FWHM}}$. Since a dipole field flips its sign 45° from its maximum, the compressive force scales quadratically for a reduced ${L_d}$ only as long as ${L_d} \ge {v_e}{T_{FWHM}}$. For the evaluation of the radiation’s figure of merit, ${d_\tau }U_{e,ref}^{rad}$ per µJ at 0.1 THz, we estimate the optical pump dimensions. The optical intensity is assumed to peak at 20 GW/cm² since at that level the THz conversion efficiency is quadratic in LiNbO₃ (See Fig. 5 in Ref. [43]). The 4-photon absorption length for the laser is ${L_{4ph}} = {({{\delta_{4p\textrm{h}}}{I^3}} )^{ - 1}}$= 416 mm, far longer than a typical crystal. The crystal length is assumed to be 10 mm. We estimated the 4-photon absorption coefficient conservatively, using Ref. [50], ${\mathrm{\delta }_{4\textrm{ph}}}\textrm{} = \textrm{}30 \cdot {10^{ - 7}}\frac{{\textrm{c}{\textrm{m}^5}}}{{\textrm{G}{\textrm{W}^3}}}$, rather than the ${10^{ - 7}}\frac{{\textrm{c}{\textrm{m}^5}}}{{\textrm{G}{\textrm{W}^3}}}$ of Ref. [43]. Transversely, the laser spot should extend to ${W_z} > 2{\lambda _{THz}}$ = 6 mm such that the radiation from the resulting 3-mm-wide THz source can be collected with a NA < 0.5 optics (60° collection angle). The LiNbO₃ should be thin with respect to the THz absorption length ${\alpha _{THz}}$ [56], such that it is weakly affected by propagating at angle $\gamma $ through the slab, hence, ${W_x} < \frac{{\sin \gamma }}{{{\alpha _{THz}}}}\sim \; 0.5\; mm$. For these parameters the optical pulse energy is ${E_p} = 7200\; \mu J$ (see Supplement 1 Section S.B). Thus, per $1\; \mu J$, the ratio of these cases for a given focusing numerical aperture is

The correction factor introduced here, ${C_r}$, accounts for the optical pulse duration that results in the target frequency of 0.1 THz. A short pulse generating THz excitation in a slab produces a central frequency of $\sqrt {2\ln 2} /\pi {T_{FWHM}}$ [50,61]. For 0.1 THz it requires a pulse duration of 3.75 ps. We mentioned above that the nearfield equivalent frequency is $f_{THz}^{eff} = 0.55/{T_{FWHM}}$. Thus, the same final effective frequency requires an energy ratio equal to the pulse-duration ratio, ${C_r} = \frac{{T_{FWHM}^{nearfield}}}{{T_{FWHM}^{rad}}} = \frac{{0.55\pi }}{{\sqrt {2\ln 2} }} = 1.47$.

Even for the ultimate high-NA focusing optics the electron compressive strength from nearfields is 3-fold that of radiation. Importantly, the calculation was systematically biased in favor of the radiative approach, so the actual enhancement would be greater. The focusability of the astigmatic THz-beam shape and aberrations in high-NA optics can add an order of magnitude. The calculations are also conservative for the nearfields. For example, the nearfield effect can be increased by bringing both the optical laser and the e⁻-beam closer to the vacuum-crystal surface, an effect we leave out of the scope of the numerical examples we brought here.

Since our nearfield approach is beneficial for replacing low radiation frequencies, we turn to find the watershed frequency, for which the effect of the two approaches balances. We will refer to the field’s effective frequency or wavelength freely, using their free-space dispersion relation ${f_{THz}} = c\lambda _{THz}^{ - 1}$. The relative efficiency scales as $\lambda _{THz}^{ - 5/2}$ since optimally, ${W_z} \propto {\lambda _{THz}}$, and Eq. (5) divided by the pump energy is proportional to ${\left( {\lambda_{THz}^{ - 2}\sqrt {{W_z}} } \right)^{ - 1}}$. Using the reference case calculated for 0.1 THz per µJ in Eq. (6), the ratio between the radiative and nearfield methods is ${\left( {\frac{{{f_{THz}}}}{{0.1\; THz}}} \right)^{5/2}}0.33{({NA} )_{THz}}$. Thus, they balance for

Considering few focusing geometries, for ${({NA} )_{THz}} = $ 0.5, 0.1, and 0.009 the nearfield approach surpasses the radiative one for frequencies below 0.2 THz, 0.39 THz, and 1 THz, respectively. In terms of the e⁻-pulse duration for compression, these effective frequencies support >90% of the maximal gradient for 1/7 of their cycle, therefore, the nearfield approach would be preferable for compressing electron pulses that span 750 fs, 400 fs, and 140 fs, respectively.

As a final point of the analytical comparison, we claim that the nearfield approach for compressing e⁻-pulses can scale linearly with the pump energy by two approaches. The first one is to simply pump harder. Although seemingly trivial, radiation sources rely on the macroscopic dipole induced throughout the optical pulse propagation and hence their efficiency suffers from 4-photon absorption for intensities above 20 GW/cm² [43]. However, the nearfield acts on the electron directly and locally, over mere tens of microns, thus, the optical penetration depth is irrelevant as long as it is sufficiently long to approximate an infinite dipolar cylinder. Thus, characteristic decay lengths, ${L_{4ph}}$, for intensities 100, 200, and 300 GW/cm² comply with the long-source condition, being 3.3 mm, 416 µm, and 123 µm, respectively. These intensities are far from the conservative parameters we use in this paper, however, they can bring the effective nearfields to a few kV/cm at the challenging sub-THz regime. Importantly, they are experimentally realistic based on the literature on recorded saturation and damage intensities, 400 GW/cm² and 1 TW/cm², respectively [44]. Extrapolating from Fig. 2(b) (that is calculated for 20 GW/cm²) the sub-THz field reaches 2.4 kV/cm for an intensity of 300 GW/cm². We comment that the locally generated heat should be extracted to avoid thermal damage, drift, or expansion due to the average power of a high repetition rate laser.

Alternatively, at a given peak intensity, the optical pumping energy can be increased if the beam is expanded parallel to the crystal surface, forming an ellipse with ${W_z} > {W_x}$. However, since a 1 keV electron passes only 18.7 µm per picosecond, the downstream parts of the elongated beam should be properly delayed to coincide with the e⁻-passage. Thus, the extended elliptical pump should be sheared spatiotemporally according to the electron velocity, such that the nearfields are effectively phase-matched with it. Let us take the reference calculation (induced polarization cylinder, diameter 25 µm, ${T_{FWHM}}$=10 ps, energy of 1 µJ). Stretching the optical mode 10-fold to 0.25 mm allows the pumping of nearfields by 10 µJ laser-pulse energy at the same efficiency. Electrons at 1 keV with a velocity of 18.7 µm/ps pass 0.25 mm in 133 ps. Thus, the temporal shear over ${W_z}$ of the ellipse should be 133 ps. Such a shearing can be achieved by a grating-based stretcher, that is detuned to have a residual spatial dispersion. Electron pulses in this example would fully compress within 16.5 mm. The geometric constraints for the e⁻-compression arise from the maximum extent of such a stretch since we assume the e⁻-beam experiences a uniform energy-gain gradient. For an e⁻-beam semi-convergence angle of 10 milliradians, the beam’s maximal diameter over a distance of 0.25 millimeters is 2.5 µm. The numerical calculation in Fig. 3 shows that ${d_\tau }{U_e}$ decays sub-exponentially away from the surface, approximated by a characteristic e^-1 decay length of 63 µm (red circles). An exponential line is added as a reference. Thus, a uniform interaction can be extended to a few millimeters, allowing the energy efficiency of the nearfield scheme to be maintained up to hundreds of micro joules. The blue crosses in Fig. 3 show that the e⁻-pulse duration can be longer if the e⁻-beam passes further away from the LiNbO₃. Thus, the compressive force can be traded off for an effective lower frequency, and as mentioned above, for accommodating faster electrons. This spatiotemporal spread of the optical pump and intensities above the 4-photon threshold can be combined, for example, by using a smaller beam closer to the LiNbO₃ surface and stretched to improve heat dissipation. New methodologies for ultrafast THz-field mapping by optical microscopy, such as QFIM [62], could quantify experimentally the local sub-THz fields that are presented in Fig. 2. Since our calculation in this work is conservative, we expect that such a comparison would reveal that nearfields are better than the above predictions.

 

Fig. 3. e⁻-beam compression dependence of the distance from the LiNbO₃ surface. When an e⁻-pulse moves parallel to the surface the compressive power decays sub-exponentially with the spacing from the crystal, approximated by an exponent with an e^-1 length of 63 µm, thus the temporal focal length increases.

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To provide several concrete examples of the final e⁻-pulse properties, Fig. 4 presents particle simulations of the electrons at the temporal focus. The plots represent the phase space in which the timing and the relative velocity of every electron is one dot. An ideally compressed e⁻-pulse is a vertical line. The calculation for Fig. 4(a) and Fig. 4(c) assumes that the electron pulse with a FWHM duration, ${\tau _{FHWM}},$ enters the compressing field region at 1 keV, with a stochastic energy uncertainty of $\mathrm{\Delta }{U_s}$=0.1 eV, 0.5 eV. The field corresponds to the marked distance in Fig. 3, albeit with a 10 µJ pump, optimized in one of the approaches mentioned above. From the relativistic velocity, we calculate the classical fly time to the temporal focus and the new timing relative to the mean. We do the calculation for pulses with ${\tau _{FHWM}} = {\tau _{th}}$ (black), shorter (blue, ${\tau _{FHWM}} = \frac{1}{2}{\tau _{th}},\frac{1}{{\sqrt 2 }}{\tau _{th}}$), and longer (red, ${\tau _{FHWM}} = 2{\tau _{th}},\sqrt 2 {\tau _{th}}$). The scatter plots are shifted transversely for clarity. The legend notes the final compressed duration as FWHM, which is calculated here statistically, as $\sqrt {8\ln 2} $ of the standard deviation. Clearly, pulses longer than ${\tau _{th}}$ (red) are poorly compressed. Figure 4(b) and 4(d) consider additional factors. In Fig. 4(b) the electrons were assumed to have undergone Coulomb-induced dispersion, which added time-correlated energy broadening (chirp) of 1 eV FWHM, which requires longer propagation distances, $L > {L_{focus}}$ to achieve temporal focusing. In Fig. 4(d) the compressive field is weaker, precisely as marked in Fig. 3, with a 1 µJ optical pump. These simulations show that it is realistic to use nearfields to compress multi-ps e⁻-pulses to well below one picosecond, in particular, if their linewidth broadening is time-energy correlated.

 

Fig. 4. Phase space particle simulations of the e⁻-pulse properties, velocity vs. timing at the temporal focus. (a) An initial uncorrelated energy spread $\mathrm{\Delta }{U_s} = 0.5\; eV$ FWHM, focused with 10 µJ (${L_{focus}} = 16\; mm$). Black-colored scatter corresponds to an initial e⁻-pulse with a FWHM duration ${\tau _{th}} = 2.6\; ps$, equal to the threshold mentioned in the text, $\frac{{dU}}{{d\tau }}$ > 0.9 $\max \left( {\frac{{dU}}{{d\tau }}} \right)$. Blue and red scatter plots are shorter and longer than the ${\tau _{th}}$ by a factor of $\frac{1}{2},\frac{1}{{\sqrt 2 }}$ and $\sqrt 2 ,2$, respectively. The scatter plots are shifted for clarity. (b) is as (a), with an additional convoluted e⁻-linewidth of 1 eV originating from correlated energy and timing (chirp). The compressed chirped pulses are not significantly longer, however, the propagation distance $L > {L_{focus}}$. (c) is as (a), but with a smaller stochastic energy spread $\mathrm{\Delta }{U_s} = 0.1\; eV$, resulting in shorter final pulses. (d) is as (c), albeit with a weaker compressing field, corresponding to ${L_{focus}} = 160\; mm$, resulting in pulse duration of 450 fs at a minimum. ${L_{focus}}$ of 160 mm and 16 mm, can originate from a laser pulse energy of 1 µJ or 10 µJ, respectively, with a FHWM duration of 10 ps. The calculation is based on an e⁻-path 30 µm from the crystal, marked by a dashed line (for 1 µJ) in Fig. 3. The 10 µJ is assumed to have the same geometrical properties as the 1 µJ, neglecting 4-photon absorption.

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Conclusions

We propose a novel approach for compressing electron pulses using laser-driven fields, exploiting the nearfields emanating from the optically driven crystal directly instead of relying only on re-focused radiated power. Our study shows that analytical quasi-static approximation can be applied for electrons accelerated to below 5 keV (14% the speed of light), assuming an instantaneous dipolar field induced by laser polarization near the surface of a LiNbO₃ crystal. The analytical comparison demonstrates that at few-µJ pulse energy, nearfields are especially advantageous for sub-THz frequencies and small numerical apertures. We also present a tilted-pulse method to match the velocity of the electron, which keeps the effectiveness of the nearfields for laser-pump energies of hundreds of µJ. This approach addresses challenges in producing sub-THz fields in confined regions, with inherent laser-locking and elevated saturation intensities. We believe that these effectively intense sub-THz fields would be bridging a gap in controlling electrons, such as compression, deflection, and acceleration. The e⁻-wavefunction manipulation it enables could be the necessary path for exerting nonlinear optics operation with electrons in free space on nano-confined quantum systems.