High-gradient modulation of microbunchings using a minimized system driven by a vortex laser

Shufa Hao; Zhengxing Lv; Hao Dong; Jianzhi He; Nanshun Huang; Fengyu Sun; Zhiyong Shi; Hao Sun; Wenpeng Wang; Yuxin Leng; Ruxin Li; Zhizhan Xu

doi:10.3788/COL202321.093801

Abstract

This study entailed the development of a high-gradient modulation of microbunching for traditional radiation frequency accelerators using a minimized system driven by a relativistic Laguerre–Gaussian (LG) laser in three-dimensional particle-in-cell (PIC) simulations. It was observed that the LG laser could compress the transverse dimension of the beam to within a 0.7 µm radius (

divergence \approx 4.3 mrad

), which is considerably lower than the case tuned by a Gaussian laser. In addition, the electron beam could be efficiently modulated to a high degree of bunching effect (

> 0.5

) within

\sim 21 fs

(

\sim 7 μm

) in the longitudinal direction. Such a high-gradient density modulation driven by an LG laser for pre-bunched, low-divergence, and stable electron beams provides a potential technology for the system minimization of X-ray free-electron lasers (XFELs) and ultrashort-scale (attosecond) electron diffraction research.

Keywords

electron bunch modulation relativistic Laguerre–Gaussian laser X-ray free-electron lasers

1. Introduction

Owing to the requirement for high-quality electron beams or electromagnetic radiation in basic research fields^[1], an increasing number of methods for improving their properties have been proposed and implemented. One of the major challenges in science is to observe atoms in motion with femtosecond time resolution^[2,3] and angstrom spatial resolution^[4,5]. Electron diffraction and synchrotron radiation have become indispensable detection methods for the study of ultrafast dynamics and ultrahigh spatial resolution. However, they have weak coherence and pulses with long relaxation times, which is a big challenge for high-resolution spectroscopy and imaging experiments in the physical and chemical fields.

To solve these issues, advanced accelerator-based lights^[6-9] and electronic compression mechanisms^[10-12] have been developed to generate ultra-intense ultrafast full-coherence electron beams or radiation. For example, external nonrelativistic femtosecond Gaussian lasers^[8,13,14] are usually combined with flexible magnetic devices^[12,15,16] such as doglegs and magnetic chicanes to manipulate the relativistic electron distributions and precisely tailor the properties of the radiation pulse. First, a femtosecond Gaussian laser was used as an external seed to modulate the electron beam energy at the scale of the wavelength of the optical laser in the first undulator. Subsequently, a magnetic chicane (dispersion section) converts the formed energy modulation into an associated density modulation (microbunching)^[17-21]. Finally, these electron microbunches are further input into the undulator to generate coherent harmonic emissions in the high-gain harmonic generation (HGHG)^[22-24] or echo-enabled harmonic generation (EEHG)^[25-29] schemes. During this process, beam control is an important factor for X-ray free-electron laser (XFEL) equipment, which usually depends on the complex insertion assembly and magnetic focusing devices. This equipment is considerably large and expensive; the development of a compact and economical mechanism for convenient XFEL applications is therefore needed.

One solution that has been implemented is to use the laser wake-field scheme driven by the relativistic Gaussian laser to accelerate electrons with extremely high acceleration gradient. In the past decades, the relativistic vortex laser has drawn increasing attention in the laser acceleration. It has been theoretically verified that the Laguerre–Gaussian (LG) laser could generate collimated electron beams through the interaction with an ultrathin target^[30,31]. Attosecond electron bunches generated from a nanofiber driven by LG lasers were also investigated^[32-35].

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2. Results and Discussion

In this study, strong coherent electron microbunching was realized by simply using an intense LG laser interacting with the beam source from a storage ring or traditional radio frequency gun in three-dimensional (3D) particle-in-cell (PIC) simulations. It can compact the energy and density modulation processes, from the large traditional magnetic chicanes and undulators, to tens of micrometers and femtoseconds. Studies have shown that electron beam density can be compressed from the initial density of $\sim 0.07 n_{c}$ to $\sim 0.7 n_{c}$ and the beam-bunching effect is sharply enhanced ( $> 0.5$ ) within the electron beam. In addition, the electron beam is accelerated from 1 MeV to hundreds of MeV with low divergence, which is expected to further compact the traditional accelerator section in future mimic XFEL facilities for various applications.

In 3D PIC simulations, a circularly polarized (CP) intense LG ( $l = 1$ , $p = 0$ , $σ_{z} = - 1$ ) laser was incident from the left boundary along the positive $x$ -axis direction. The incident laser amplitude can be expressed as $E_{⊥} = E_{0} \sqrt{\frac{2 p!}{π (p + l)!}} \frac{1}{w (x)} {[\frac{r \sqrt{2}}{w (x)}]}^{l} \exp [\frac{r^{2}}{w^{2} (x)}] L_{p}^{l} [\frac{2 r^{2}}{w^{2} (x)}] \exp (i l Φ) \exp [\frac{i k_{0} r^{2} x}{2 (x^{2} + x_{R}^{2})}] [e_{y} + \exp (\frac{i π σ_{z}}{2}) e_{z}] \exp [- i (2 p + l + 1) \arctan (\frac{x}{x_{R}}) + φ],$ (1)where $E_{0} = a_{0} m_{e} ω c / e$ is the dimensionless amplitude of the laser electric field; $a_{0} = 40$ is the normalized laser intensity, which corresponds to a laser intensity of $\sim 7 \times 10^{21} W / {cm}^{2}$ ; $e$ is the electron charge; $m_{e}$ is the electron mass; $ω$ is the laser frequency; and $c$ is the speed of light in vacuum. $w (x) = w_{0} {(1 + x^{2} / x_{R}^{2})}^{1 / 2}$ is the beam waist width of the laser; $w_{0} = 4 μm$ is the full width at half-maximum of the laser focal spot; $x_{R} = π w_{0}^{2} / λ$ is the Rayleigh length of the laser pulse; and $σ_{z} = - 1$ represents the right CP mode. The laser pulse had a central wavelength $λ_{0} = 0.8 μm$ . The pulse width was approximately 30 fs. The duration of the incident electron beam duration was $7 T$ ( $T = λ_{0} / c$ ). The density of the electron beam was $n_{e} = 0.07 n_{c}$ , and it was uniformly distributed in the range $- 1 μm < y < 1 μm$ and $- 1 μm < z < 1 μm$ . The electrons within the beam have an initial energy of 1 MeV ( $0.94 c$ ), and the electron temperature was $8 \times 10^{4} eV$ . For the conventional accelerator electron beam divergence angle of $10^{- 6} mrad$ , the initial divergence angle we set is 0 mrad, which is an ideal state. An incident laser pulse was injected at $t = 16 T$ . The grid scale of the simulation space was $1200 \times 600 \times 600$ , and the corresponding coordinate length was $30 μm (x) \times 30 μm (y) \times 30 μm (z)$ . The simulation used a moving window at time $t = 20 T$ , which moved along the laser direction at the speed of the light.

To show the detailed process of beam bunching, the 3D PIC code EPOCH^[36] was used to simulate the relativistic LG laser propagation and its interaction with the input electron beam in Figs. 1(a) and 2. Previously, relativistic LG light was proposed using plasma holograms^[37] or a large reflected phase plate for various applications in the laboratory^[30,38,39]. In this study, the reflected scheme was devised to generate an intense LG laser ( $\sim 7 \times 10^{21} W / {cm}^{2}$ ) to investigate the compact beam modulation system (see Fig. 1). It was found that the incident electron beam was cut into slices by the periodic concentrating and emanating electric fields in the transverse direction [see Fig. 1(d)] and oscillating acceleration fields in the longitudinal direction at $t = 20 T$ [see Fig. 2(c)]. In contrast, the incident electron beam is sinusoidally distributed by a Gaussian sinusoidal electric field in the transverse direction [see Figs. 1(c) and 1(e)]. At $t = 100 T$ (267 fs), a string of coherent microbunches was generated in the coherent duration of the laser wavelength $λ_{0}$ [see Fig. 2(b)] in the LG laser case, whereas the electron beam was transversely dispersed to a broadened region ( $x = \pm 10 μm$ ) for the Gaussian laser [see Fig. 2(f)]. The main reason for this is that the hollow intensity distribution of the LG laser provides a concentrating ponderomotive force for the electron beam in the central singularity, which finally bunches most of the electrons along the $y$ axis ( $- 0.25 μm < y < 0.25 μm$ ) [see Fig. 2(b)]. Here, the divergence angle of each electron bunch is $\sim 4.3 mrad$ , which is remarkably smaller than that in the case driven by the Gaussian laser [see Fig. 2(h)]. Notably, this beam collimation process is similar to the quadrupole magnet effect, which is used as an insertion device to focus the electron beam in conventional accelerators^[40]. However, the quadrupole magnetic field length is on the submeter scale and is even larger in the superconducting quadrupole magnet^[41]. The relativistic LG laser can generate an evident focusing effect at tens of micrometers, which can be considered to work as an “electron focusing lens,” which may enable the miniaturization of conventional focusing systems in future experiments. Although this scheme can effectively compress the electron bunching, it cannot be directly applied in free-electron lasers (FELs) and requires further beam shaping.

Figure 1.(a) Schematic layout of the microbunching scheme and its potential application. (b), (c) Electric field (E_y) of LG and Gaussian laser. The corresponding vector plot of the electric fields for (d1)–(d4) ${LG}_{0}^{1}$ (σ = −1) and (e1)–(e4) Gaussian laser.

Figure 2.(a), (b) Electron density maps generated by the LG laser at 20T and 100T. (e), (f) Electron density maps generated by the Gaussian laser under the same initial conditions as the LG laser. (c), (g) Longitudinal phase space distributions of electrons in (b) and (f). (d), (h) Comparison of the energy spectrum distribution and electron divergence generated by the LG and Gaussian lasers.

In addition, the electron beam is modulated by the oscillating acceleration field $E_{x}$ in the longitudinal direction, as shown in Fig. 2(c). This shows that $E_{x} \approx 10^{13} V / m$ can be obtained on the beam axis ( $x$ axis), which can reach the order of the accelerating field in the target normal sheath field acceleration mechanism^[42], indicating that it can be used to efficiently accelerate electrons there. It is observed that most electrons are locked in the accelerating phase of $E_{x}$ [Fig. 2(c)] and can be accelerated for hundreds of femtoseconds and hundreds of micrometers. At $t = 100 T$ , the central electron beam energy can reach up to 400 MeV, and its cutoff energy is slightly higher than the case driven by the Gaussian laser [see Fig. 2(b)]. However, the electron beam has been expanded to tens of micrometers in the transverse direction in the later Gaussian laser case, which requires further modulation and maintenance of the focusing state using quadrupole magnets. It is interesting that the hollow intensity distribution of the LG laser successfully resolves this issue, where the peak current intensity of the microbunching exceeds ten times the initial current at $100 T$ [Fig. 3(a)]. Meanwhile, we could find that the charge and electron density distribution of adjacent microbunchings are similar [Fig. 3(a)], showing a Gaussian-like distribution in the transverse space. Therefore, the focusing effect of microbunchings located in the nearest phase is identical. The most important observation is that such dense electron slices can be continuously compressed, in the central oscillating electric field, to 0.12 µm (415 as), which is of great significance in the study of attosecond dynamics. The spot size of the laser also has an impact on the longitudinal morphology of the electron microbunching. An excessively large or small spot size can cause the broadening of the electron slice, which is related to the symmetry of the electric field along the optical axis. We select a relatively optimal solution with spot size of 4 µm here.

Figure 3.(a) Current distribution modulated by the LG laser. Here, λ₀ = 0.8 µm. (b) Transverse electron distribution characteristics at 100T. (c) Trajectories of electrons in the multi-particle model. The black and red solid lines represent the electron trajectory in decelerating phase and accelerating phase, respectively. (d) Phase space distribution of electrons and longitudinal electric field (red line) distribution on the x axis at 5T (the hollow black circle represents the tested electrons with a uniform phase gap of 0.25π). Here, areas outlined in red and blue represent accelerating phase and decelerating phase, respectively.

To understand the density modulation mechanism for the attosecond electron slice in the PIC simulations, we used a multi-particle model to calculate the one-dimensional electron motion on the beam axis ( $x$ axis). The initial longitudinal energies of the tested electrons were set to 1 MeV, which was similar to the simulation in Fig. 2. The one-dimensional longitudinal electric field along the $x$ axis can be expressed as $E_{x} (x, t) = E_{peak} \cos (ω t - k x + π / 2), - 2.5 λ_{0} + c t < x < 2.5 λ_{0} + c t,$ (2)where $E_{peak} \approx 1.5 \times 10^{13} V / m$ is the peak amplitude of the laser electric field [see Fig. 3(c)]. Then, the particle acceleration and modulation can be expressed as follows: $\frac{d}{d t} (\frac{β}{\sqrt{1 - β^{2}}}) = \frac{e}{m_{e}} E_{x} (x, t),$ (3) $\frac{d}{d t} x (t) = β (t) c,$ (4)where $E_{x}$ is the laser electric field, $m_{e}$ is the electron mass, and $β = v / c$ is the dimensionless velocity. Nine typical electrons were uniformly distributed in the decelerating ( $- 2 λ_{0} < x < - 1.5 λ_{0}$ ) and accelerating phases ( $- 2.5 λ_{0} < x < - 2 λ_{0}$ ) in the $E_{x}$ field [Fig. 3(c)]. It can be observed that electrons in the deceleration phase are dragged into the acceleration phase within $1 T$ , and then stably accelerated by the positive electric field [Fig. 3(c)]. Meanwhile, the electrons initially in the acceleration phase are continuously accelerated and finally remain together with the electrons from the deceleration phase, resulting in a compressed electron layer, as shown in Fig. 3(d). All these results indicate that the electrons can be accelerated without dephasing for at least hundreds of femtoseconds, which can be proven by the dephasing rate $R$ experienced by the accelerated electron in this study. For example, as indicated by Fig. 3(d), $R = (c - v_{x}) / c \sim 10^{- 6}$ can be obtained at $t = 5 T$ , where $v_{x}$ is the electron velocity, and the electric field $E_{x}$ transports at light speed $c$ , indicating that the electrons can be stably locked in the acceleration phase [Fig. 3(c)]. This is consistent with the simulation results shown in Fig. 2(c), which proves that the longitudinal electric field on the beam axis plays an important role in the acceleration and compression of the tuned electron beam.

In synchrotron radiation applications, a higher bunching effect for electron beams is usually required to enhance the intensity of the output radiation using the coherent harmonic generation (CHG) technique^[22,43]. A phase-merging enhanced harmonic generation (PEHG) mechanism has also been proposed to significantly increase the bunching factor [ $〈 e^{i θ} 〉 = 〈 e^{i (k \bar{x} - ω t + φ)} 〉 = \frac{1}{N} \sum_{i = 1}^{N} e^{i (k \bar{x} - ω t + φ)}$ ] by using a transverse-gradient undulator^[13]. However, these methods require meter-scale devices. In our study, the LG laser could solve the miniaturization issue to a certain extent, where the electrons were directly tuned to a thin slice with a sharp bunching effect ( $\sim 0.6$ ) within $8 T$ ( $\sim 7 μm$ ) (see Fig. 4). The electron-bunching effect can be further increased up to $\sim 0.8$ at $t = 100 T$ . Compared with the microbunching treatment in traditional undulators, the scheme driven by the LG laser has an evident microsize advantage. However, the energetic spectra in this study are slightly broader, which can be optimized by choosing an appropriate incident electron beam and LG laser parameters, as well as depending on the coordination of the complex achromatic system to further improve the quality of the electron beam.

Figure 4.Comparison of the degree of bunching effects produced using a conventional undulator (CHG)^[22], PEHG scheme^[13], and the method proposed in this study. The red solid line represents the fitting line of the bunching effect. The x axis represents the modulation time of the electron beam and laser field.

3. Conclusions

In conclusion, a compact and low-cost scheme was proposed to generate ultrashort electron microbunching with a high bunching effect and low divergence using a relativistic LG laser without depending on large magnets and sophisticated connection schemes of numerous insertion devices used in the traditional electronic modulation process. It was found that the incident electron beam density could be compressed to a near-critical density within a short time. The beam-bunching effect is sharply enhanced ( $> 0.5$ ), which is comparable to the compression mechanism in traditional accelerator systems. Simultaneously, the incident electron beam could be accelerated from 1 MeV to hundreds of MeV in tens of micrometers, which is expected to compact the traditional accelerator section. Due to the extreme sensitivity to energy dispersion in the application of FEL, the electron beam generated here is clearly unable to directly generate FEL radiation, which can be possibly improved by the mature beam compression schemes. In addition, the beam divergence could also remain at $\sim 5 mrad$ in this study, similar to a quadrupole commonly used in synchrotron radiation. This may benefit the possible realization and applications of mimic XFEL facilities in the future.

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