Sub-Terahertz Nearfields for Electron-Pulse Compression

The advent of ultrafast science with pulsed electron beams raised the need in controlling the temporal features of the electron pulses. One promising suggestion is the nano-selective quantum optics with multi-electrons, which scales quadratically with the number of electrons within the coherence time of the quantum system. Terahertz (THz) radiation from optical nonlinear crystals is an attractive methodology to generate the rapidly varying electric fields necessary for electron compression, with an advantage of an inherent temporal locking to laser-triggered electrons, such as in untrafast electron microscopes. Longer (picosecond-) pulses require sub-THz field for their compression, however, the generation of such low frequencies require pumping with energetic optical pulses and their focusability is fundamentally limited by their mm-wavelength. This work proposes electron-pulse compression with sub-THz fields directly in the vicinity of their dipolar origin, thereby avoiding mediation through radiation. We analyze the merits of nearfields for compression of slow electrons particularly in challenging regimes for THz radiation, such as small numerical apertures, micro-joule-level optical pump pulses, and low frequencies. This sheme can be implemented within the tight constraints of electron microscopes and reach fiels of a few kV/cm below 0.1 THz at high repetition rates. Our paradigm offers a realistic approach for controlling electron pulses spatially and temporally in many experiments, opening the path of flexible multi-electron manipulation for analytic and quantum sciences.

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][38][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 ( ) of a Rabi-oscillation cycle, where 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 coherence 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 eˉ-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 eˉ-pulses, and a THz cycle fits the duration of short electron pulses (~200-700 fs [40][41][42]). Intense terahertz waves are generated from the optical rectification of short pulses in lithium niobate (LiNbO3). 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.
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 1µm 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 eˉpulses 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 eˉ-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 the one or 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 eˉ-beam in electron microscopes poses a standing issue as a barrier for compression of eˉ-pulses using THz-fields, especially for eˉ-pulses longer than a picosecond.
This work presents a conceptual change for laser-driven sub-THz compression of ps eˉ-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 eˉ-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 eˉ-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 radiative THz. 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 eˉ-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 radiationbased 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.

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 eˉ energy gain and the resulting eˉ-pulse compression is compared between a direct interaction with the nearfield (Figure 1a) and an optimal scenario of radiation from such a slab (Figure 1b), 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].
The electron on-axis acceleration depends on the energy it accumulates throughout its path, Here, is the electron charge and ( ) marks the electron trajectory, simplified as one-dimensional.
The energy gain varies with the electron timing , and its derivative,  Figure 2b shows the numerically calculated fields. Substituting the optical rectification dipole we find (3) Figure 2c 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, . The final energy, ( ), is approximately a Gaussian (see Figure 2d), matching the laser pulse envelope. This stems from the eˉ-energy being well within the regime that matches the quasi-static approximation (see curve coalescence in Figure 2e). The inflection point of the energy-gain shifts temporally from that of the optical pump (See Figure 2d), i.e., the quasistatic approximation, however, we ignore this constant timing shift since it is trivially compensated for by delaying the optical pump. Figure  . This focal length is the spatial propagation at which the electron-pulse duration is compressed to a minimum if it was initially dispersionless [60]. Thus, it is a useful parameter in designing experimental layouts. Here, is the unitless relativistic parameter for the velocity, is the relativistic Lorentz factor, and is the electron mass. Figure 2e 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. In the following, we use an example of electrons

Nearfields vs. radiation for eˉ-pulse compression
To define the relative improvement of the nearfields we find an explicit closed-form expression for the figure of merit for eˉ-compression from THz radiation, based on a slab source at optimal conditions, The detailed derivation is in Appendix B. , , are the spatial width of the pump beam and is the off-axis angle of the THz emission, all of which are marked in Figure 1b. is the idealized transmission coefficient of the THz power from the LiNbO3 crystal to free space, ( ) is the numerical aperture of the THz focusing optics, and is the THz wavelength in a vacuum. Thus, the ratio between the approaches for eˉ-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 Ref. [43]). The 4-photon absorption length for the laser is 4 ℎ = ( 4 h 3 ) −1 = 416 mm, far longer than a typical crystal 1 . The crystal length is assumed to be 10 mm. Transversely, the laser spot should extend to > 2 = 6 mm collect the radiation from the resulting 3-mm-wide THz source with a NA<0.5 optics (60° collection angle). The LiNbO3 should be thin with respect to the THz absorption length [56], such that it is weakly affected by propagating at angle through the slab, hence, < sin~ 0.5 . For these parameters the optical pulse energy is = 7200 (see Appendix B). Thus, per 1 , the ratio of these cases for a given focusing numerical aperture is 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 1 We used the conservative 4-photon absorption coefficient from ref. [50], δ 4ph = 30 ⋅ 10 −7 cm 5 GW 3 , rather than the 10 −7 cm 5 GW 3 of Ref. [43]. field's effective frequency or wavelength freely, using their free-space dispersion relation = −1 . The relative efficiency scales as −5/2 since optimally, ∝ , and eq. (5) divided by the pump energy is proportional to ( −2 √ ) −1 . Using the reference case calculated for 0.1 THz per µJ in eq. (6), the ratio between the radiative and nearfield methods is 0.33( ) . Thus, they balance for ℎ = 0.1 (0.33( ) ) 2/5 .
As an example of a few focusing geometries, for ( ) = 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 eˉ-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 eˉ-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, 4 ℎ , 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 Figure 2b (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. The extended elliptical pump should be sheared spatiotemporally according to the electron velocity, such that the nearfields are  Figure 3 shows that 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 Figure 3 show that the eˉ-pulse duration can be longer if the eˉ-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 Figure 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.

Conclusion
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 refocused 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 eˉ-wavefunction manipulation it enables could be the necessary path for exerting nonlinear optics operation with electrons in free space on nanoconfined quantum systems.

Appendix A -Evaluating compression using THz radiation from a LiNbO 3 slab
This section derives this figure of merit for a radiative THz. We do so by calculating the radiated power and the resulting field at a tight focus, followed by integrating over the electron trajectory.
We consider a source based on LiNbO3 slab, having a silicon prism on one of its facets for efficient coupling to free-space radiation (see Figure 1b and refs. [50,61]). The beam shape of the driving laser is a transverse ellipse and constant over its propagation. We mark its width, height, and propagation length in the crystal as , and , respectively, as in ref. [50]. We use the axes notations as the standard in the literature, where is along the laser pulse's propagation direction, whereas in the nearfield calculation the laser propagates along the y-axis. To simplify the quantitative calculation, we assume a rectangular intensity profile, flat within these dimensions.
We further assume a collection of the radiation emitted towards the positive x-axis. Note that the emission is tilted towards the z-axis by an angle , determined by phase matching between the THz radiation and the laser-pulse group velocity in the LiNbO3, cos = , [45]. The angle is marked on the inset in Figure 1b. For a wavelength of 1030 in LiNbO3 the group index is , = 2.3 [63] and we use the measured sub-THz refractive index in LiNbO3, = 4.9 [64], hence = 62 . We assume optimal handling of the radiation, that is, a perfect out-coupling (e.g., with a silicon prism), aberration-less collimation at high-NA, and an ideal focusing of the radiated THz power into a Gaussian focal spot. The temporal peak of the radiated power can be calculated as a difference-frequency generation (DFG) source (see Table 39 for the DFG efficiency in Ref. [58]), cos .
Since the radiation propagates in a tilted angle, the effective medium depth we used for the THz is / cos , which contributes quadratically. We wrote the above expression in terms of power rather than efficiency, by accounting for the spatial extent of the radiated beam, ⋅ cos (see inset of Figure 1b). is the THz wavelength in free-space. Our calculation assumes implicitly that the source is effectively two-dimensional and that the THz radiates as a wide beam. In practice sizes larger than a wavelength suffice, as long as they can be collected by optics with a sufficient numerical aperture, that is, cos , > / . Although we assume flat-top pump, in practice, the transverse size of the THz source is /2 due to the quadratic nonlinearity. in the above expression is bounded from below by the wavelength, regardless of the NA. For < the source width smaller than /2, and hence the radiation geometry is cylindrical and the radiated power scales differently, as −5/2 .For the out-coupling, we assume an ideal transmission to free space at a direct incidence, yielding a transmission of = ( As illustrated in Figure 1b, we assume that the perfect THz collection is followed by a perfect refocusing onto a tilted membrane. For simplicity, we account for a 45° tilt, although for electron beams with a finite emittance, there is an optimal angle for transverse velocity matching [17,65]. A traversing electron experiences a sinusoidal acceleration field ( , ) = ( ) = sin (2 − ). Note that the reflected field is polarized perpendicular to the electron trajectory and does not contribute to an on-axis acceleration. Here = 0 represents the electron trajectory that passes the thin surface when the field is nullified. The maximal energy gain is = ∫ sin ( 2 ) /2 0 = 2 2 , and the maximal gradient is | | = 2 . Hence we get eq. (4) , where we consider the FHWM width, which is given by = √2 ln 2 ω . We can set the intensity to be just below the appearance of 4-photon absorption, 0 = 4 = 20 2 , hence 0 = . We now substitute = 0.5 , = 6 , = 10 and get = 0.0072 = 7200 .
To combine with eq. (5), and receive a numerical value, we rewrite it, Thus, the ratio per µJ is 0.228, which when multiplied by = 1.47 results in 0.335, as in eq. (6).