1. Introduction

Compact electron sources underlie a wide range of scientific research including free-electron lasers [1], ultrafast electron diffraction [2], and advanced cancer therapies [3]. Historically, radio frequency (RF) has been the workhorse for fueling charge particles to high brightness and energy. But they are also associated with large, expensive facilities limited by a few tens of MV/m breakdown threshold [4]. Terahertz is a specific frequency band located between radio frequencies and optics. Since the first pioneering work which theoretically laid the basic framework of terahertz-driven acceleration using a dielectric-lined waveguide (DLW) a decade ago [5], increasing interest has been attracted by its merits of large acceleration gradient and pC-level charge load. As a result, footsteps toward THz-driven particle accelerator and even electron gun have achieved milestones so far [6–8]: from the earliest 7 keV electron energy gain utilizing a 10 µJ driving THz pulse [9] to the most recent 170 keV result in a single-stage segmented structure [10]. Multiple functionalities, including focusing, compression [11], deflection, and streaking [12], have been demonstrated by altering the relative phase of the electron and the THz. So far, THz-driven electron pulse compression from over 1 picosecond to 100 femtoseconds [13], and from 105 to 39 femtoseconds have been reported, bringing its compression capability to femtosecond level while decreasing timing jitter from 76 femtoseconds to 31 femtoseconds [11]. In addition, a few methods have demonstrated the optical generation [14] and compression [15] of attosecond electron pulse trains, which were, nevertheless, prone to energy divergence and other quality problems and must be enhanced for practical applications.

Achieving even better acceleration performance necessitates THz pulse energies greater than one millijoule to improve the acceleration gradient and account for the non-negligible energy loss during energy collection, transit, and coupling to the dielectric-loaded waveguide (DLW). Very recently, the generation of 13.9 mJ THz radiation in cryogenically cooled lithium niobate was reported, bringing the energy of crystal-based THz sources from sub-millijoules to over 10 mJ [16]. Such sources hold the promise of tremendously upscaling the electron energy gain once their radiation energies can be fully used. However, intensifying the available field inside a DLW remains challenging due to the energy losses during transport, mode conversion and coupling, etc.

New capabilities are necessary for obviating the difficulty with alternative radiation mode and millijoule-scale pulse energy. In this regard, the recent discovery of THz surface plasmon polariton (SPP) amplification via free electron pumping [17] sheds light on a completely new method of driving the electrons. In comparison with free-space radiations, THz SPPs are tightly confined to the material surface and can propagate with minor attenuation and dispersion [18]. These advantages are especially welcome when the THz SPP is bound to a metal wire surface, because of the radial polarization (or TM01 mode [19,20]) property. While conventional THz SPP suffers from the low energy produced by free-space coupling, the free-electron amplification process has demonstrated coherent THz SPP energy gain from laser-driven electrons, whose picosecond-period helical trajectories around the wire determines their radiation in the THz spectral band [21]. Recently, the mechanism has already demonstrated stable far-field THz pulses with kHz repetition [22] and high radiation energy of 3 mJ when irradiated by an intense femtosecond laser [23]. In electron acceleration, the surface wave nature of the THz SPPs has also been shown to accelerate the laser-induced electrons with MeV scale energy gain through the simple combination of a wire and circular DLW [24].

Here, we propose a scheme to achieve flexible manipulation of high-quality high-energy electron beams by replacing the wire with a hollow metal tube. The approach retains the waveguide characteristics of the metal cylinder, and exempts the need for bulky optics of THz generation, collection, transportation, and focusing. By extending the electron source from laser-produced electron pulse to available electron sources, we expect the proposed scheme to serve as a possible means for beam control capable of acceleration, and compression within the DLW. Specifically, by assuming a 2.9 mJ THz pulse in simulation, the tunable electron energy reaches up to ${\pm} $ 860 keV using a 5 MeV injection electron pulse. Alternatively, by changing the delay to the temporal compression mode, the pulse width can be compressed from 100fs to 14.6 fs for the same electron pulse. Using lower injection energies of 50 keV and 200 keV typical of the direct current (DC) electron gun, enhanced compression from 400 fs to less than 70 fs is demonstrated, which could facilitate the applications of ultrafast electron diffractions.

2. Results

2.1 Femtosecond laser-driven terahertz source and relative phase control

The basic idea of our proposed scheme is based on the millijoule-level, radially-polarized, and low-loss waveguide features of the particular THz source. When a femtosecond laser pulse is focused on a metallic cylinder, the abrupt charge separation gives rise to a weak THz SPP, and forces the electrons to move in a helical manner over the cylinder surface [21]. Importantly, in this process, the electrons remain in the deceleration phase of the THz SPP, resulting in a nearly net transfer of energy from the electrons to the THz SPP. Our recent study on the characterization of the SPP amplification during the process has demonstrated coherent energy growth of the THz SPP, which may facilitate access to ultra-intense surface wave light sources. Meanwhile, the THz SPP can also be excited via coupling with a free-space terahertz source [19,20] and use structure-enhanced coupling efficiency [25]. Particularly, the employment of an axis-symmetric cylinder allows for the simultaneous combination of efficient energy transmission, coupling, a high source energy, and the preferred mode for acceleration purposes. In this mechanism, the frequency range of the THz SPP is dictated by electron trajectories whose picosecond circling period predescribe the typical center frequency of approximately 0.25 THz.

Figure 1(a) schematically represents the design of our proposed scheme. The setup comprises a circular dielectric-lined waveguide (DLW), a horn coupler linked to the in-coupling of the DLW, and an additional thin hollow metal tube whose front-end aperture is slightly positioned into the DLW. Such an arrangement prevents the more than half energy loss of the THz in a free-space coupling situation. Higher electric field strengths inside the DLW take benefits at least from two aspects: first, the THz energy is strongly spatially confined due to the subwavelength characteristic of the THz Sommerfeld wave; second, the similar modes prevent energy loss caused by imperfect mode conversion and the undesired reflection loss by embedding the tube in the DLW. Electrons are supposed to be injected from external mature electron sources, such as a DC or radio frequency (RF) electron gun. Due to the metal shielding of the cylinder shell, interaction between the electron and THz field only takes place within the DLW, similar to conventional THz-driven electron accelerators.

 

Fig. 1. (a) Schematic of the setup. Together, the radially polarized surface THz wave and electron propagate in the same direction on the surface and inner channel of the hollow tube, respectively, and are steered into the interior of the waveguide. The electrons can be compressed or accelerated within the waveguide by appropriate delay changes. (b)-(e) and (h)-(k) respectively illustrates the phase-space evolutions of acceleration and compression scenarios of a 1.3 MeV electron bunch. The zero position is defined at the entrance of the hollow tube, which has 6-mm length outside the DLW and 3-mm inside. The colored plots depict the axial field components along the electron propagation path.

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In the current design, the phase delay between THz and electrons can be tuned by adjusting the exit time of the electrons as they enter the DLW. Figure 1(b)-(e) intuitively depicts the phase position of electrons within a THz field in the acceleration scenario. When the electron beam encounters the THz pulse, the electron beam catches up to the THz pulse from the pulse's tail and is injected into the acceleration phase. By tuning the wall thickness of the dielectric, a phase-matched interaction can be realized, that is, the THz phase velocity is approximately commensurate with the electron velocity, whereas its group velocity generally falls behind. Consequently, the electron continues to acquire energy (maintained in the acceleration phase) before “walking off”. The corresponding compression scenarios are presented in Fig. 1(h)-(k). Here, the electron beam is injected into the zero-crossing of the THz field, where its beam head experiences deceleration and its beam tail undergoes acceleration. Subsequently, the negative chirped electron beam will become squeezed in time until the high-energy electrons catch up with the low-energy electrons, and reach a minimum pulse width.

To demonstrate the proposed scheme, the copropagating THz surface wave and electrons as well as their interaction inside the DLW were simulated using the particle-in-cell solver of CST [26]. The THz settings were adapted from our experimentally reported results. Specifically, a quasi-single-cycle waveform with a center frequency of 0.25 THz, and spectral width spanning 0.1 to 0.4 THz was assumed in the simulation. The THz surface wave propagates along the outer surface of the 300-µm-diameter metal tube. Efficient THz-electron interaction can be achieved through adjusting the DLW with different dielectric thickness and their materials. In our simulation, a respective 0.13 mm and 0.2 mm-thick quartz, and a 0.15 mm-thick alumina with the vacuum channel radius of 0.47 mm, 0.6 mm and 0.65 mm were adopted, resulting in the transmitted TM₀₁ mode with the THz phase velocity v_p of 0.99c, 0.70c, and 0.41c (where c is the vacuum speed of light) inside a 12-mm-long circular DLW.

2.2 Electron energy modulation

Assume that the electron beam from the RF gun with a central energy of 5 MeV has a charge of the order of pC and a pulse width of 100 fs. When altering the delay between the electron beam and the THz, the electrons and their energies would undergo a periodic phase change within the DLW. Using the quasi-single-cycle waveform described above, this produces the quasi-sinusoidal electron energy modulation (root mean square, rms) depicted in Fig. 2(a). When the terahertz surface wave carries 2.9 mJ energy, continuous energy modulation from 4.1 MeV to 5.9 MeV can be achieved, allowing for applications such as electron diffraction. Note that, these results show improved results than our recent experimental report, which relied on approximately ten millijoules THz surface wave energy for achieving MeV scale electron accelerations [24]. A major cause responsible for the difference is that the electron propagating inside the hollow tube leaves out the requirement for the excessive room for electron injection from outer the wire. As a result, the 0.47 mm vacuum channel diameter of the DLW concentrates the field much more strongly than the 1.5 mm case in the experiment, resulting in 910 MV/m peak field intensity and elevated acceleration gradient which are beneficial for accelerations.

 

Fig. 2. Modulation of the electron energy. (a) Root-mean-square energy of an electron pulse as a function of the relative delay between the electron pulse and the THz wave. Within the DLW, the electron energy may be altered continuously from 4.1 MeV to 5.9 MeV with a discernible periodicity of 4 ps, which corresponds to the THz frequency employed. (b) The maximum amount of energy gain as a function of the THz energy. (c) Energy evolution of a 5 MeV, 1% dispersive electron bunch at maximal energy gain delay. The inset depicts the pulse width(left) and energy dispersion (right) of the electrons as a function of propagation distance.

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To take a deeper look at the electron bunch evolution at the maximum acceleration, Fig. 2(c) depicts the terahertz-driven energy gain for an electron bunch with 1% energy full-width-half maximum (FWHM) spread as a function of propagation distance. Electrons enter the DLW from the inside of the hollow tube at z = 3 mm and encounter the emitted THz at z = 5.9 mm during its highest acceleration phase. The entire process can be divided into three stages: in the first stage (z < 5.9 mm), the electron beam trails behind the terahertz envelope; in the second stage (5.9 mm < z < 8.6 mm), the electron beam catches up with the terahertz envelope and remains almost synchronized during the acceleration phase, where the electron beam experiences a large energy boost inside the DLW; and in the third stage (z > 8.6 mm), the electron walks away with the terahertz envelope before leaving the waveguide. The complete interaction process happened across a travel distance of 2.7 mm in the second stage, indicating an average acceleration gradient of up to 318 MeV/m. The inset in Fig. 2(c) displays the electron beam energy dispersion as a function of distance.

The modulation depth of our suggested apparatus can be further scaled using more powerful terahertz energy. The electron energy gain as a function of the terahertz pulse energy is plotted in Fig. 2(b). The electron energy gain rises with increasing terahertz energy, from 0.2 MeV at 0.18 mJ to 1.7 MeV at 11.5 mJ. From the previous experiment which measured 3 mJ far-field THz radiation by using a 600 mJ pump laser pulse, the far-field THz energy is estimated to be 1 mJ using 200 mJ femtosecond laser energy [24]. This result corresponds to 3 mJ of the THz surface wave based on simulation. Thus, the scheme not only can effectively reduce the THz energy loss, but also presents an efficient way of generating THz waves owing to the mechanism of coherent electron amplification [17].

2.3 Pulse compression

The longitudinal electron momentum modulation provides a means of overcoming the coulombic expansion during propagation. This is illustrated in Fig. 1(j)-(k), in which the faster electrons ahead of the zero crossing of the THz field experience a decelerating field while the slower electrons are accelerated, allowing the longitudinal distribution to be narrowed without a net energy gain. Clearly, higher electric field gradient at the zero crossing allows for tighter electron beam compressions. To demonstrate compression in our proposed scheme, we further evaluated its ability to compress electron beams with relatively low energy dispersion. As shown in Fig. 3(a), (b), and (c), when an appropriate delay is provided, the electron beam will be compressed from 0.4 ps to 63 fs or 66 fs for 48.5 keV and 200 keV electrons, and from 0.1 ps to 14.6 fs for relativistic electrons of 1.3 MeV energy, exceeding the current minimum RF compression of 28fs [27]. In addition, the THz energy can also influence the pulse width of the electron beam at the exit of the DLW. With increasing THz energy, the compression effect improves as well, resulting in a reduction in the width of the electron pulse at the exit. However, with a further rise in field intensity, the electron pulse width at the exit will increase due to over-compression, resulting in the waveguide's minimum pulse width.

 

Fig. 3. Electron pulse compression. (a), (b), (c) Electron beam pulse width at the zero-crossing phase point as a function of beam drift distance at 48.5 keV, 400fs, 0.2 MeV, 400fs, 1.3 MeV, and 100fs, respectively. (d)-(f) are snapshots of the appropriate outgoing electron beam pulse widths.

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Figure 3(d)–(i) shows several snapshots of the beam width after it passes through the DLW. At the minimum pulse width, the phase-space distributions of the compressed electrons (Fig. 3(d)-(i)) are spatially aligned, suggesting that the electrons experience spatial compression in the terahertz field as compared to the uncompressed electrons (Fig. 3(d)-(i)).

To further upscale the electron energy and comply with the application of future developments, a cascade acceleration scheme is proposed as shown in Fig. 4. In this arrangement, the electrons from the RF source are transmitted into the DLW through the channel of the hollow tube in the same direction as the THz pulse. The relative phase between the electron beam and the THz pulse can be appropriately controlled via their temporal delays. Since the phase velocity is approximately equal to v_p∼c, with this scheme, in theory, the electrons can experience continuous acceleration via cascading. Figure 4(b) presents an example of this scenario. Since the velocity of the relativistic electron experiences minor changes at energies greater than 5 MeV, a stable energy gain of 0.86 MeV is expected for each acceleration stage.

 

Fig. 4. THz electron acceleration cascade. (a) Schematic diagram of single-stage acceleration. (b) Illustration of cascaded electron acceleration, the first DLW provides a 5.9 MeV electron beam, which is accelerated to 6.7 MeV in the second DLW, and the subsequent cascade will obtain a stable energy gain of 0.8 MeV.

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To envision such prospects, we estimate the maximal energy gain of an electron bunch using laser pulses up to 10 J that are currently available. The sum of far-field THz pulses from a 10 J laser is about 100 mJ using the 1% energy conversion efficiency [21], whereas the surface waves could carry up to 330 mJ energy considering the near-to-far field transfer. Thus, by dividing the total energy into multiple stages with 3-mJ energy, an upper limit of the electron energy gain should be the addition of multiple 0.86 MeV energy gain per stage, which in aggregate yields an energy gain of 100 MeV. It should be pointed out that, however, the implementation of such a configuration should take into consideration beam deflection and expansion, which can degrade the acceleration result especially at sub-relativistic energies. To address this challenge, alternating phase focusing as proposed in radio-frequency accelerators offers a possible solution and has been recently demonstrated in dielectric laser accelerations [28–30]. Meanwhile, stability and repetition rate of the terahertz source should be improved, such as, by coupling of stable terahertz sources [16] with structure-enhanced coupling efficiency [25]. Both these methods should allow for the prospect of cascaded acceleration with careful arrangement of the acceleration and spatiotemporal reshaping with the aid of swift phase shift designs [31].

3. Conclusion

In summary, we have demonstrated efficient electron manipulation using terahertz surface waves and hollow tube waveguides. The concept exploits the intense terahertz source based on coherent amplification of the surface wave by free electrons, including its radial polarization, high conversion efficiency (about 1%) and high terahertz energy availability. The controllable modulation of electron energy and pulse width is realized by using an externally injected electron source and through delay control. Simulation results show that for an electron of 5 MeV, a final energy range between 4.1 MeV and 5.9 MeV is obtained. Furthermore, the electron energy spread and pulse width at the maximum energy are well preserved in future applications. Meanwhile, the initial electron beam with a pulse width of 0.1 ps can be compressed from 0.1 ps to 14.6 fs, indicating that its original time distribution is almost completely compressed. Thanks to the external injection mode, we simulate very good whole-beam acceleration and compression results for electron beam pulses exceeding pC, which should be easily applicable to various ultrafast applications.

Moreover, the extended application of terahertz electron acceleration is outlooked via the cascaded acceleration. In comparison with our previous work which made use of laser-generated electrons, the feasibility of such an arrangement is only possible with the current design where low emittance electron traverses inside the tube. Transporting the electrons from the center of the tube not only suppresses the spatial asymmetries experienced by the electron, but also allows for narrowing down the vacuum channel size and hence the field strength inside the DLW. Meanwhile, the independent control of THz and the electrons adds more flexibility to the delay adjustment, which is essential for the implementation of the cascaded beam control. Hence, Under the existing conditions, the electron beam output up to 100 MeV can be achieved by cascading, and its laboratory size can be conveniently used in medical and other applications. Going forward, our proposed all-optical integrated terahertz electron accelerator concept may pave the way for ultra-compact, high-energy tunable electron sources.